hypothesis space example

What’s a Hypothesis Space?

Last updated: March 18, 2024

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1. Introduction

Machine-learning algorithms come with implicit or explicit assumptions about the actual patterns in the data. Mathematically, this means that each algorithm can learn a specific family of models, and that family goes by the name of the hypothesis space.

In this tutorial, we’ll talk about hypothesis spaces and how to choose the right one for the data at hand.

2. Hypothesis Spaces

Let’s say that we have a binary classification task and that the data are two-dimensional. Our goal is to find a model that classifies objects as positive or negative. Applying Logistic Regression , we can get the models of the form:

which estimate the probability that the object at hand is positive.

2.1. Hypotheses and Assumptions

The underlying assumption of hypotheses ( 1 ) is that the boundary separating the positive from negative objects is a straight line. So, every hypothesis from this space corresponds to a straight line in a 2D plane. For instance:

2.2. Regression

3. expressivity of a hypothesis space.

We could informally say that one hypothesis space is more expressive than another if its hypotheses are more diverse and complex.

We may underfit the data if our algorithm’s hypothesis space isn’t expressive enough. For instance, linear hypotheses aren’t particularly good options if the actual data are extremely non-linear:

So, training an algorithm that has a very expressive space increases the chance of completely capturing the patterns in the data. However, it also increases the risk of overfitting. For instance, a space containing the hypotheses of the form:

would start modelling the noise, which we see from its decision boundary:

Such models would generalize poorly to unseen data.

3.1. Expressivity vs. Interpretability

Additionally, even if a complex hypothesis has a good generalization capability, it may be unusable in practice because it’s too complicated to understand or compute. What’s more, intricated hypotheses offer limited insight into the real-world process that generated the data. For example, a quadratic model:

4. How to Choose the Hypothesis Space?

We need to find the right balance between expressivity and simplicity. Unfortunately, that’s easier said than done. Most of the time, we need to rely on our intuition about the data.

So, we should start by exploring the dataset, using visualizations as much as possible. For instance, we can conclude that a straight line isn’t likely to be an adequate boundary for the above classification data. However, a high-order curve would probably be too complex even though it might split the dataset into two classes without an error.

A second-degree curve might be the compromise we seek, but we aren’t sure. So, we start with the space of quadratic hypotheses:

We get a model whose decision boundary appears to be a good fit even though it misclassifies some objects:

Since we’re satisfied with the model, we can stop here. If that hadn’t been the case, we could have tried a space of cubic models. The idea would be to iteratively try incrementally complex families until finding a model that both performs well and is easy to understand.

4. Conclusion

In this article, we talked about hypotheses spaces in machine learning. An algorithm’s hypothesis space contains all the models it can learn from any dataset.

The algorithms with too expressive spaces can generalize poorly to unseen data and be too complex to understand, whereas those with overly simple hypotheses may underfit the data. So, when applying machine-learning algorithms in practice, we need to find the right balance between expressivity and simplicity.

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Hypothesis in Machine Learning

The concept of a hypothesis is fundamental in Machine Learning and data science endeavours. In the realm of machine learning, a hypothesis serves as an initial assumption made by data scientists and ML professionals when attempting to address a problem. Machine learning involves conducting experiments based on past experiences, and these hypotheses are crucial in formulating potential solutions.

It’s important to note that in machine learning discussions, the terms “hypothesis” and “model” are sometimes used interchangeably. However, a hypothesis represents an assumption, while a model is a mathematical representation employed to test that hypothesis. This section on “Hypothesis in Machine Learning” explores key aspects related to hypotheses in machine learning and their significance.

Table of Content

How does a Hypothesis work?

Hypothesis space and representation in machine learning, hypothesis in statistics, faqs on hypothesis in machine learning.

A hypothesis in machine learning is the model’s presumption regarding the connection between the input features and the result. It is an illustration of the mapping function that the algorithm is attempting to discover using the training set. To minimize the discrepancy between the expected and actual outputs, the learning process involves modifying the weights that parameterize the hypothesis. The objective is to optimize the model’s parameters to achieve the best predictive performance on new, unseen data, and a cost function is used to assess the hypothesis’ accuracy.

In most supervised machine learning algorithms, our main goal is to find a possible hypothesis from the hypothesis space that could map out the inputs to the proper outputs. The following figure shows the common method to find out the possible hypothesis from the Hypothesis space:

Hypothesis Space (H)

Hypothesis space is the set of all the possible legal hypothesis. This is the set from which the machine learning algorithm would determine the best possible (only one) which would best describe the target function or the outputs.

Hypothesis (h)

A hypothesis is a function that best describes the target in supervised machine learning. The hypothesis that an algorithm would come up depends upon the data and also depends upon the restrictions and bias that we have imposed on the data.

The Hypothesis can be calculated as:

[Tex]y = mx + b [/Tex]

• m = slope of the lines
• b = intercept

To better understand the Hypothesis Space and Hypothesis consider the following coordinate that shows the distribution of some data:

Say suppose we have test data for which we have to determine the outputs or results. The test data is as shown below:

We can predict the outcomes by dividing the coordinate as shown below:

So the test data would yield the following result:

But note here that we could have divided the coordinate plane as:

The way in which the coordinate would be divided depends on the data, algorithm and constraints.

• All these legal possible ways in which we can divide the coordinate plane to predict the outcome of the test data composes of the Hypothesis Space.
• Each individual possible way is known as the hypothesis.

Hence, in this example the hypothesis space would be like:

The hypothesis space comprises all possible legal hypotheses that a machine learning algorithm can consider. Hypotheses are formulated based on various algorithms and techniques, including linear regression, decision trees, and neural networks. These hypotheses capture the mapping function transforming input data into predictions.

Hypothesis Formulation and Representation in Machine Learning

Hypotheses in machine learning are formulated based on various algorithms and techniques, each with its representation. For example:

• Linear Regression : [Tex] h(X) = \theta_0 + \theta_1 X_1 + \theta_2 X_2 + … + \theta_n X_n[/Tex]
• Decision Trees : [Tex]h(X) = \text{Tree}(X)[/Tex]
• Neural Networks : [Tex]h(X) = \text{NN}(X)[/Tex]

In the case of complex models like neural networks, the hypothesis may involve multiple layers of interconnected nodes, each performing a specific computation.

Hypothesis Evaluation:

The process of machine learning involves not only formulating hypotheses but also evaluating their performance. This evaluation is typically done using a loss function or an evaluation metric that quantifies the disparity between predicted outputs and ground truth labels. Common evaluation metrics include mean squared error (MSE), accuracy, precision, recall, F1-score, and others. By comparing the predictions of the hypothesis with the actual outcomes on a validation or test dataset, one can assess the effectiveness of the model.

Hypothesis Testing and Generalization:

Once a hypothesis is formulated and evaluated, the next step is to test its generalization capabilities. Generalization refers to the ability of a model to make accurate predictions on unseen data. A hypothesis that performs well on the training dataset but fails to generalize to new instances is said to suffer from overfitting. Conversely, a hypothesis that generalizes well to unseen data is deemed robust and reliable.

The process of hypothesis formulation, evaluation, testing, and generalization is often iterative in nature. It involves refining the hypothesis based on insights gained from model performance, feature importance, and domain knowledge. Techniques such as hyperparameter tuning, feature engineering, and model selection play a crucial role in this iterative refinement process.

In statistics , a hypothesis refers to a statement or assumption about a population parameter. It is a proposition or educated guess that helps guide statistical analyses. There are two types of hypotheses: the null hypothesis (H0) and the alternative hypothesis (H1 or Ha).

• Null Hypothesis(H 0 ): This hypothesis suggests that there is no significant difference or effect, and any observed results are due to chance. It often represents the status quo or a baseline assumption.
• Aternative Hypothesis(H 1 or H a ): This hypothesis contradicts the null hypothesis, proposing that there is a significant difference or effect in the population. It is what researchers aim to support with evidence.

Q. How does the training process use the hypothesis?

The learning algorithm uses the hypothesis as a guide to minimise the discrepancy between expected and actual outputs by adjusting its parameters during training.

Q. How is the hypothesis’s accuracy assessed?

Usually, a cost function that calculates the difference between expected and actual values is used to assess accuracy. Optimising the model to reduce this expense is the aim.

Q. What is Hypothesis testing?

Hypothesis testing is a statistical method for determining whether or not a hypothesis is correct. The hypothesis can be about two variables in a dataset, about an association between two groups, or about a situation.

Q. What distinguishes the null hypothesis from the alternative hypothesis in machine learning experiments?

The null hypothesis (H0) assumes no significant effect, while the alternative hypothesis (H1 or Ha) contradicts H0, suggesting a meaningful impact. Statistical testing is employed to decide between these hypotheses.

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Introduction to the hypothesis space and the bias-variance tradeoff in machine learning.

In this post, we introduce the hypothesis space and discuss how machine learning models function as hypotheses. Furthermore, we discuss the challenges encountered when choosing an appropriate machine learning hypothesis and building a model, such as overfitting, underfitting, and the bias-variance tradeoff.

The hypothesis space in machine learning is a set of all possible models that can be used to explain a data distribution given the limitations of that space. A linear hypothesis space is limited to the set of all linear models. If the data distribution follows a non-linear distribution, the linear hypothesis space might not contain a model that is appropriate for our needs.

To understand the concept of a hypothesis space, we need to learn to think of machine learning models as hypotheses.

The Machine Learning Model as Hypothesis

Generally speaking, a hypothesis is a potential explanation for an outcome or a phenomenon. In scientific inquiry, we test hypotheses to figure out how well and if at all they explain an outcome. In supervised machine learning, we are concerned with finding a function that maps from inputs to outputs.

But machine learning is inherently probabilistic. It is the art and science of deriving useful hypotheses from limited or incomplete data. Our functions are not axioms that explain the data perfectly, and for most real-life problems, we will never have all the data that exists. Accordingly, we will not find the one true function that perfectly describes the data. Instead, we find a function through training a model to map from known training input to known training output. This way, the model gradually approximates the assumed true function that describes the distribution of the data. So we treat our model as a hypothesis that needs to be tested as to how well it explains the output from a given input. We do this using a test or validation data set.

The Hypothesis Space

During the training process, we select a model from a hypothesis space that is subject to our constraints. For example, a linear hypothesis space only provides linear models. We can approximate data that follows a quadratic distribution using a model from the linear hypothesis space.

Of course, a linear model will never have the same predictive performance as a quadratic model, so we can adjust our hypothesis space to also include non-linear models or at least quadratic models.

The Data Generating Process

The data generating process describes a hypothetical process subject to some assumptions that make training a machine learning model possible. We need to assume that the data points are from the same distribution but are independent of each other. When these requirements are met, we say that the data is independent and identically distributed (i.i.d.).

Independent and Identically Distributed Data

How can we assume that a model trained on a training set will perform better than random guessing on new and previously unseen data? First of all, the training data needs to come from the same or at least a similar problem domain. If you want your model to predict stock prices, you need to train the model on stock price data or data that is similarly distributed. It wouldn’t make much sense to train it on whether data. Statistically, this means the data is identically distributed . But if data comes from the same problem, training data and test data might not be completely independent. To account for this, we need to make sure that the test data is not in any way influenced by the training data or vice versa. If you use a subset of the training data as your test set, the test data evidently is not independent of the training data. Statistically, we say the data must be independently distributed .

Overfitting and Underfitting

We want to select a model from the hypothesis space that explains the data sufficiently well. During training, we can make a model so complex that it perfectly fits every data point in the training dataset. But ultimately, the model should be able to predict outputs on previously unseen input data. The ability to do well when predicting outputs on previously unseen data is also known as generalization. There is an inherent conflict between those two requirements.

If we make the model so complex that it fits every point in the training data, it will pick up lots of noise and random variation specific to the training set, which might obscure the larger underlying patterns. As a result, it will be more sensitive to random fluctuations in new data and predict values that are far off. A model with this problem is said to overfit the training data and, as a result, to suffer from high variance .

To avoid the problem of overfitting, we can choose a simpler model or use regularization techniques to prevent the model from fitting the training data too closely. The model should then be less influenced by random fluctuations and instead, focus on the larger underlying patterns in the data. The patterns are expected to be found in any dataset that comes from the same distribution. As a consequence, the model should generalize better on previously unseen data.

But if we go too far, the model might become too simple or too constrained by regularization to accurately capture the patterns in the data. Then the model will neither generalize well nor fit the training data well. A model that exhibits this problem is said to underfit the data and to suffer from high bias . If the model is too simple to accurately capture the patterns in the data (for example, when using a linear model to fit non-linear data), its capacity is insufficient for the task at hand.

When training neural networks, for example, we go through multiple iterations of training in which the model learns to fit an increasingly complex function to the data. Typically, your training error will decrease during learning the more complex your model becomes and the better it learns to fit the data. In the beginning, the training error decreases rapidly. In later training iterations, it typically flattens out as it approaches the minimum possible error. Your test or generalization error should initially decrease as well, albeit likely at a slower pace than the training error. As long as the generalization error is decreasing, your model is underfitting because it doesn’t live up to its full capacity. After a number of training iterations, the generalization error will likely reach a trough and start to increase again. Once it starts to increase, your model is overfitting, and it is time to stop training.

Ideally, you should stop training once your model reaches the lowest point of the generalization error. The gap between the minimum generalization error and no error at all is an irreducible error term known as the Bayes error that we won’t be able to completely get rid of in a probabilistic setting. But if the error term seems too large, you might be able to reduce it further by collecting more data, manipulating your model’s hyperparameters, or altogether picking a different model.

Bias Variance Tradeoff

We’ve talked about bias and variance in the previous section. Now it is time to clarify what we actually mean by these terms.

Understanding Bias and Variance

In a nutshell, bias measures if there is any systematic deviation from the correct value in a specific direction. If we could repeat the same process of constructing a model several times over, and the results predicted by our model always deviate in a certain direction, we would call the result biased.

Variance measures how much the results vary between model predictions. If you repeat the modeling process several times over and the results are scattered all across the board, the model exhibits high variance.

In their book “Noise” Daniel Kahnemann and his co-authors provide an intuitive example that helps understand the concept of bias and variance. Imagine you have four teams at the shooting range.

Team B is biased because the shots of its team members all deviate in a certain direction from the center. Team B also exhibits low variance because the shots of all the team members are relatively concentrated in one location. Team C has the opposite problem. The shots are scattered across the target with no discernible bias in a certain direction. Team D is both biased and has high variance. Team A would be the equivalent of a good model. The shots are in the center with little bias in one direction and little variance between the team members.

Generally speaking, linear models such as linear regression exhibit high bias and low variance. Nonlinear algorithms such as decision trees are more prone to overfitting the training data and thus exhibit high variance and low bias.

A linear model used with non-linear data would exhibit a bias to predict data points along a straight line instead of accomodating the curves. But they are not as susceptible to random fluctuations in the data. A nonlinear algorithm that is trained on noisy data with lots of deviations would be more capable of avoiding bias but more prone to incorporate the noise into its predictions. As a result, a small deviation in the test data might lead to very different predictions.

To get our model to learn the patterns in data, we need to reduce the training error while at the same time reducing the gap between the training and the testing error. In other words, we want to reduce both bias and variance. To a certain extent, we can reduce both by picking an appropriate model, collecting enough training data, selecting appropriate training features and hyperparameter values. At some point, we have to trade-off between minimizing bias and minimizing variance. How you balance this trade-off is up to you.

The Bias Variance Decomposition

Mathematically, the total error can be decomposed into the bias and the variance according to the following formula.

Remember that Bayes’ error is an error that cannot be eliminated.

Our machine learning model represents an estimating function \hat f(X) for the true data generating function f(X) where X represents the predictors and y the output values.

Now the mean squared error of our model is the expected value of the squared difference of the output produced by the estimating function \hat f(X) and the true output Y.

The bias is a systematic deviation from the true value. We can measure it as the squared difference between the expected value produced by the estimating function (the model) and the values produced by the true data-generating function.

Of course, we don’t know the true data generating function, but we do know the observed outputs Y, which correspond to the values generated by f(x) plus an error term.

The variance of the model is the squared difference between the expected value and the actual values of the model.

Now that we have the bias and the variance, we can add them up along with the irreducible error to get the total error.

A machine learning model represents an approximation to the hypothesized function that generated the data. The chosen model is a hypothesis since we hypothesize that this model represents the true data generating function.

We choose the hypothesis from a hypothesis space that may be subject to certain constraints. For example, we can constrain the hypothesis space to the set of linear models.

When choosing a model, we aim to reduce the bias and the variance to prevent our model from either overfitting or underfitting the data. In the real world, we cannot completely eliminate bias and variance, and we have to trade-off between them. The total error produced by a model can be decomposed into the bias, the variance, and irreducible (Bayes) error.

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Best Guesses: Understanding The Hypothesis in Machine Learning

• February 22, 2024
• General , Supervised Learning , Unsupervised Learning

Machine learning is a vast and complex field that has inherited many terms from other places all over the mathematical domain.

It can sometimes be challenging to get your head around all the different terminologies, never mind trying to understand how everything comes together.

In this blog post, we will focus on one particular concept: the hypothesis.

While you may think this is simple, there is a little caveat regarding machine learning.

The statistics side and the learning side.

Don’t worry; we’ll do a full breakdown below.

You’ll learn the following:

What Is a Hypothesis in Machine Learning?

• Is This any different than the hypothesis in statistics?
• What is the difference between the alternative hypothesis and the null?
• Why do we restrict hypothesis space in artificial intelligence?
• Example code performing hypothesis testing in machine learning

In machine learning, the term ‘hypothesis’ can refer to two things.

First, it can refer to the hypothesis space, the set of all possible training examples that could be used to predict or answer a new instance.

Second, it can refer to the traditional null and alternative hypotheses from statistics.

Since machine learning works so closely with statistics, 90% of the time, when someone is referencing the hypothesis, they’re referencing hypothesis tests from statistics.

Is This Any Different Than The Hypothesis In Statistics?

In statistics, the hypothesis is an assumption made about a population parameter.

The statistician’s goal is to prove it true or disprove it.

This will take the form of two different hypotheses, one called the null, and one called the alternative.

Usually, you’ll establish your null hypothesis as an assumption that it equals some value.

For example, in Welch’s T-Test Of Unequal Variance, our null hypothesis is that the two means we are testing (population parameter) are equal.

This means our null hypothesis is that the two population means are the same.

We run our statistical tests, and if our p-value is significant (very low), we reject the null hypothesis.

This would mean that their population means are unequal for the two samples you are testing.

Usually, statisticians will use the significance level of .05 (a 5% risk of being wrong) when deciding what to use as the p-value cut-off.

What Is The Difference Between The Alternative Hypothesis And The Null?

The null hypothesis is our default assumption, which we are trying to prove correct.

The alternate hypothesis is usually the opposite of our null and is much broader in scope.

For most statistical tests, the null and alternative hypotheses are already defined.

You are then just trying to find “significant” evidence we can use to reject our null hypothesis.

These two hypotheses are easy to spot by their specific notation. The null hypothesis is usually denoted by H₀, while H₁ denotes the alternative hypothesis.

Example Code Performing Hypothesis Testing In Machine Learning

Since there are many different hypothesis tests in machine learning and data science, we will focus on one of my favorites.

This test is Welch’s T-Test Of Unequal Variance, where we are trying to determine if the population means of these two samples are different.

There are a couple of assumptions for this test, but we will ignore those for now and show the code.

You can read more about this here in our other post, Welch’s T-Test of Unequal Variance .

We see that our p-value is very low, and we reject the null hypothesis.

What Is The Difference Between The Biased And Unbiased Hypothesis Spaces?

The difference between the Biased and Unbiased hypothesis space is the number of possible training examples your algorithm has to predict.

The unbiased space has all of them, and the biased space only has the training examples you’ve supplied.

Since neither of these is optimal (one is too small, one is much too big), your algorithm creates generalized rules (inductive learning) to be able to handle examples it hasn’t seen before.

Here’s an example of each:

Example of The Biased Hypothesis Space In Machine Learning

The Biased Hypothesis space in machine learning is a biased subspace where your algorithm does not consider all training examples to make predictions.

This is easiest to see with an example.

Let’s say you have the following data:

Happy  and  Sunny  and  Stomach Full  = True

Whenever your algorithm sees those three together in the biased hypothesis space, it’ll automatically default to true.

This means when your algorithm sees:

Sad  and  Sunny  And  Stomach Full  = False

It’ll automatically default to False since it didn’t appear in our subspace.

This is a greedy approach, but it has some practical applications.

Example of the Unbiased Hypothesis Space In Machine Learning

The unbiased hypothesis space is a space where all combinations are stored.

We can use re-use our example above:

This would start to breakdown as

Happy  = True

Happy  and  Sunny  = True

Happy  and  Stomach Full  = True

Let’s say you have four options for each of the three choices.

This would mean our subspace would need 2^12 instances (4096) just for our little three-word problem.

This is practically impossible; the space would become huge.

So while it would be highly accurate, this has no scalability.

More reading on this idea can be found in our post, Inductive Bias In Machine Learning .

Why Do We Restrict Hypothesis Space In Artificial Intelligence?

We have to restrict the hypothesis space in machine learning. Without any restrictions, our domain becomes much too large, and we lose any form of scalability.

This is why our algorithm creates rules to handle examples that are seen in production. 

This gives our algorithms a generalized approach that will be able to handle all new examples that are in the same format.

Other Quick Machine Learning Tutorials

At EML, we have a ton of cool data science tutorials that break things down so anyone can understand them.

Below we’ve listed a few that are similar to this guide:

• Instance-Based Learning in Machine Learning
• Types of Data For Machine Learning
• Verbose in Machine Learning
• Generalization In Machine Learning
• Epoch In Machine Learning
• Inductive Bias in Machine Learning
• Understanding The Hypothesis In Machine Learning
• Zip Codes In Machine Learning
• get_dummies() in Machine Learning
• Bootstrapping In Machine Learning
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Hypothesis Space

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• Hendrik Blockeel  

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Model space

The hypothesis space used by a machine learning system is the set of all hypotheses that might possibly be returned by it. It is typically defined by a Hypothesis Language , possibly in conjunction with a Language Bias .

Motivation and Background

Many machine learning algorithms rely on some kind of search procedure: given a set of observations and a space of all possible hypotheses that might be considered (the “hypothesis space”), they look in this space for those hypotheses that best fit the data (or are optimal with respect to some other quality criterion).

To describe the context of a learning system in more detail, we introduce the following terminology. The key terms have separate entries in this encyclopedia, and we refer to those entries for more detailed definitions.

A learner takes observations as inputs. The Observation Language is the language used to describe these observations.

The hypotheses that a learner may produce, will be formulated in...

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De Raedt, L. (1992). Interactive theory revision: An inductive logic programming approach . London: Academic Press.

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Nédellec, C., Adé, H., Bergadano, F., & Tausend, B. (1996). Declarative bias in ILP. In L. De Raedt (Ed.), Advances in inductive logic programming . Frontiers in artificial intelligence and applications (Vol. 32, pp. 82–103). Amsterdam: IOS Press.

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Claude Sammut

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Geoffrey I. Webb

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Blockeel, H. (2011). Hypothesis Space. In: Sammut, C., Webb, G.I. (eds) Encyclopedia of Machine Learning. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30164-8_373

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VC Dimensions

This is a continuation of my notes on Computational Learning Theory .

One caveat mentioned at the end of that lecture series was that the formula for the lower sample complexity bound breaks down in the case of an infinite hypothesis space. That formula follows. In particular, as $|H| \rightarrow \infty$, $m \rightarrow \infty$.

$$m \geq \frac{1}{\epsilon}(ln|H| + ln \frac{1}{\delta})$$

This is problematic because each of the following hypothesis spaces are infinite: linear separators , artificial neural networks , and decision trees (continuous inputs) . An example of a finite hypothesis space is a decision tree (discrete inputs)

Inputs, $X:\ \lbrace1,2,3,4,5,6,7,8,9,10\rbrace$ Hypotheses, $h(x)=x \geq \theta$, where $\theta \in \mathbb R$, so $|H|=\infty$

Syntactic hypothesis space: all the things you could write. Semantic hypothesis space: the actual different functions you are practically representing. IE, meaningfully different.

So, the hypothesis space for this example is syntactically infinite but semantically finite. This is because we can arrive at the same answer as tracking an infinite number of hypotheses by only tracking non-negative integers ten or below.

Power of a Hypothesis Space

What is the largest set of inputs that the hypothesis class can label in all possible ways?

Consider the set of inputs $S=\lbrace 6 \rbrace$. This hypothesis class can be labeled in all possible ways (either True or False) with different values of $\theta$ assigned. Consider the larger set of inputs $S=\lbrace 5, 6 \rbrace$ There are four ways this hypothesis class could be labeled {(False, False), (False, True), (True, False), (True, True)}. In practice only 3 of these four are accessible because there is not a $\theta$ that is less than 5 and greater than 6.

The technical term for “labeling in all possible ways” is to “ shatter ” the input space.

The size of that input space is called the VC dimension, where VC stands for Vapnik-Chervonenkis. In this example, size of the largest set of inputs this hypothesis class can shatter is 1 element.

Inputs: $X=\mathbb R$ Hypothesis Space: $H=\lbrace h(x)=x\in [a,b]\rbrace$ where the space is parametrized by $a,b \in \mathbb R$

Based on the figure below, the VC dimension is clearly greater than or equal to 2.

Based on the figure below, the VCC dimensions is clearly less than 3, so the VC dimension is 2.

To show the VC dimension is at least a given number, it is only necessary to find one example that can be shattered. However, to show the VC dimension is less than a given number, it is necessary to prove there are no examples with that can be shattered.

Inputs: $X=\mathbb R^2$ Hypothesis Space: $H=\lbrace h(x)=W^T x \geq \theta\rbrace$

Based on the example below, the VC dimension is at least 3, because lines can be drawn to categorize the points in all possible ways.

Once there are four inputs, however, separating the points becomes impossible.

Summarizing the examples outlined above:

The VC dimension is often the number of parameters. For a d-dimensional hyperplane, the parameter count must be d+1.

Sample Complexity and VC Dimensions

If the training data has at least $m$ samples, then that will be sufficient to get $\epsilon$ error with probability $\delta$, where $VC(H)$ is the VC dimension of the hypothesis class.

Infinite case : $$m \geq \frac{1}{\epsilon}(8VC(H) \log_2 \frac{13}{\epsilon}+4\log_2 \frac{2}{\delta})$$

Finite case : $$m \geq \frac{1}{\epsilon}(ln|H| + ln \frac{1}{\delta})$$

What is VC-Dimension of a finite H?

Let $d$ equal the VC dimension of some finite hypothesis $H$: $d = VC(H)$. This implies that there exists $s^d$ distinct concepts, because each gets a different hypothesis. It follows that $2^d \leq |H|$, by manipulation $d \leq \log_2 |H|$.

From similar reasoning we get the following theorem. H is PAC-learnable if and only if the VC dimension is finite.

So, something is PAC-learnable if it has a finite VC dimension. Inversely, something with infinite VC dimension cannot be learned.

The VC dimension captures in one quantity the notion of PAC-Learnability.

For Fall 2019, CS 7461 is instructed by Dr. Charles Isbell . The course content was originally created by Dr. Charles Isbell and Dr. Michael Littman .

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What is the difference between hypothesis space and representational capacity?

I am reading Goodfellow et al Deeplearning Book . I found it difficult to understand the difference between the definition of the hypothesis space and representation capacity of a model.

In Chapter 5 , it is written about hypothesis space:

One way to control the capacity of a learning algorithm is by choosing its hypothesis space, the set of functions that the learning algorithm is allowed to select as being the solution.

And about representational capacity:

The model specifies which family of functions the learning algorithm can choose from when varying the parameters in order to reduce a training objective. This is called the representational capacity of the model.

If we take the linear regression model as an example and allow our output $y$ to takes polynomial inputs, I understand the hypothesis space as the ensemble of quadratic functions taking input $x$ , i.e $y = a_0 + a_1x + a_2x^2$ .

How is it different from the definition of the representational capacity, where parameters are $a_0$ , $a_1$ and $a_2$ ?

• machine-learning
• terminology
• computational-learning-theory
• hypothesis-class

3 Answers 3

Consider a target function $f: x \mapsto f(x)$ .

A hypothesis refers to an approximation of $f$ . A hypothesis space refers to the set of possible approximations that an algorithm can create for $f$ . The hypothesis space consists of the set of functions the model is limited to learn. For instance, linear regression can be limited to linear functions as its hypothesis space, or it can be expanded to learn polynomials.

The representational capacity of a model determines the flexibility of it, its ability to fit a variety of functions (i.e. which functions the model is able to learn), at the same. It specifies the family of functions the learning algorithm can choose from.

• 1 $\begingroup$ Does it mean that the set of functions described by the representational capacity is strictly included in the hypothesis space ? By definition, is it possible to have functions in the hypothesis space NOT described in the representational capacity ? $\endgroup$ –  Qwarzix Commented Aug 23, 2018 at 8:43
• $\begingroup$ It's still pretty confusing to me. Most sources say that a "model" is an instance (after execution/training on data) of a "learning algorithm". How, then, can a model specify the family of functions the learning algorithm can choose from? It doesn't make sense to me. The authors of the book should've explained these concepts in more depth. $\endgroup$ –  Talendar Commented Oct 9, 2020 at 13:09

A hypothesis space is defined as the set of functions $\mathcal H$ that can be chosen by a learning algorithm to minimize loss (in general).

$$\mathcal H = \{h_1, h_2,....h_n\}$$

The hypothesis class can be finite or infinite, for example a discrete set of shapes to encircle certain portion of the input space is a finite hypothesis space, whereas hpyothesis space of parametrized functions like neural nets and linear regressors are infinite.

Although the term representational capacity is not in the vogue a rough definition woukd be: The representational capacity of a model, is the ability of its hypothesis space to approximate a complex function, with 0 error, which can only be approximated by infinitely many hypothesis spaces whose representational capacity is equal to or exceed the representational capacity required to approximate the complex function.

The most popular measure of representational capacity is the $\mathcal V$ $\mathcal C$ Dimension of a model. The upper bound for VC dimension ( $d$ ) of a model is: $$d \leq \log_2| \mathcal H|$$ where $|H|$ is the cardinality of the set of hypothesis space.

A hypothesis space/class is the set of functions that the learning algorithm considers when picking one function to minimize some risk/loss functional.

The capacity of a hypothesis space is a number or bound that quantifies the size (or richness) of the hypothesis space, i.e. the number (and type) of functions that can be represented by the hypothesis space. So a hypothesis space has a capacity. The two most famous measures of capacity are VC dimension and Rademacher complexity.

In other words, the hypothesis class is the object and the capacity is a property (that can be measured or quantified) of this object, but there is not a big difference between hypothesis class and its capacity, in the sense that a hypothesis class naturally defines a capacity, but two (different) hypothesis classes could have the same capacity.

Note that representational capacity (not capacity , which is common!) is not a standard term in computational learning theory, while hypothesis space/class is commonly used. For example, this famous book on machine learning and learning theory uses the term hypothesis class in many places, but it never uses the term representational capacity .

Your book's definition of representational capacity is bad , in my opinion, if representational capacity is supposed to be a synonym for capacity , given that that definition also coincides with the definition of hypothesis class, so your confusion is understandable.

• 1 $\begingroup$ I agree with you. The authors of the book should've explained these concepts in more depth. Most sources say that a "model" is an instance (after execution/training on data) of a "learning algorithm". How, then, can a model specify the family of functions the learning algorithm can choose from? Also, as you pointed out, the definition of the terms "hypothesis space" and "representational capacity" given by the authors are practically the same, although they use the terms as if they represent different concepts. $\endgroup$ –  Talendar Commented Oct 9, 2020 at 13:18

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ID3 Algorithm and Hypothesis space in Decision Tree Learning

The collection of potential decision trees is the hypothesis space searched by ID3. ID3 searches this hypothesis space in a hill-climbing fashion, starting with the empty tree and moving on to increasingly detailed hypotheses in pursuit of a decision tree that properly classifies the training data.

In this blog, we’ll have a look at the Hypothesis space in Decision Trees and the ID3 Algorithm. 

ID3 Algorithm: 

The ID3 algorithm (Iterative Dichotomiser 3) is a classification technique that uses a greedy approach to create a decision tree by picking the optimal attribute that delivers the most Information Gain (IG) or the lowest Entropy (H).

What is Information Gain and Entropy?  

Information gain: .

The assessment of changes in entropy after segmenting a dataset based on a characteristic is known as information gain.

It establishes how much information a feature provides about a class.

We divided the node and built the decision tree based on the value of information gained.

The greatest information gain node/attribute is split first in a decision tree method, which always strives to maximize the value of information gain. 

The formula for Information Gain: 

Entropy is a metric for determining the degree of impurity in a particular property. It denotes the unpredictability of data. The following formula may be used to compute entropy:

S stands for “total number of samples.”

P(yes) denotes the likelihood of a yes answer.

P(no) denotes the likelihood of a negative outcome.

• Calculate the dataset’s entropy.
• For each feature/attribute.

Determine the entropy for each of the category values.

Calculate the feature’s information gain.

• Find the feature that provides the most information.
• Repeat it till we get the tree we want.

Characteristics of ID3: 

• ID3 takes a greedy approach, which means it might become caught in local optimums and hence cannot guarantee an optimal result.
• ID3 has the potential to overfit the training data (to avoid overfitting, smaller decision trees should be preferred over larger ones).
• This method creates tiny trees most of the time, however, it does not always yield the shortest tree feasible.
• On continuous data, ID3 is not easy to use (if the values of any given attribute are continuous, then there are many more places to split the data on this attribute, and searching for the best value to split by takes a lot of time).

Over Fitting:  

Good generalization is the desired property in our decision trees (and, indeed, in all classification problems), as we noted before. 

This implies we want the model fit on the labeled training data to generate predictions that are as accurate as they are on new, unseen observations.

Capabilities and Limitations of ID3:

• In relation to the given characteristics, ID3’s hypothesis space for all decision trees is a full set of finite discrete-valued functions.
• As it searches across the space of decision trees, ID3 keeps just one current hypothesis. This differs from the prior version space candidate Elimination approach, which keeps the set of all hypotheses compatible with the training instances provided.
• ID3 loses the capabilities that come with explicitly describing all consistent hypotheses by identifying only one hypothesis. It is unable to establish how many different decision trees are compatible with the supplied training data.
• One benefit of incorporating all of the instances’ statistical features (e.g., information gain) is that the final search is less vulnerable to faults in individual training examples.
• By altering its termination criterion to allow hypotheses that inadequately match the training data, ID3 may simply be modified to handle noisy training data.
• In its purest form, ID3 does not go backward in its search. It never goes back to evaluate a choice after it has chosen an attribute to test at a specific level in the tree. As a result, it is vulnerable to the standard dangers of hill-climbing search without backtracking, resulting in local optimum but not globally optimal solutions.
• At each stage of the search, ID3 uses all training instances to make statistically based judgments on how to refine its current hypothesis. This is in contrast to approaches that make incremental judgments based on individual training instances (e.g., FIND-S or CANDIDATE-ELIMINATION ).

Hypothesis Space Search by ID3: 

• ID3 climbs the hill of knowledge acquisition by searching the space of feasible decision trees.
• It looks for all finite discrete-valued functions in the whole space. Every function is represented by at least one tree.
• It only holds one theory (unlike Candidate-Elimination). It is unable to inform us how many more feasible options exist.
• It’s possible to get stranded in local optima.
• At each phase, all training examples are used. Errors have a lower impact on the outcome.

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Could anyone explain the terms "Hypothesis space" "sample space" "parameter space" "feature space in machine learning with one concrete example?

I am confused with these machine learning terms, and trying to distinguish them with one concrete example.

for instance, use logistic regression to classify a bunch of cat images.

assume there are 1,000 images with labels indicating the corresponding image is or is not a cat image.

each image has a size of 100*100.

given above, is my following understanding right?

the sample space is the 1,000 images.

the feature space is 100*100 pixels.

the parameter space is a vector that has a length of 100*100+1.

the Hypothesis space is the set of all the possible hyperplanes that have some attribute that I have no idea.

• machine-learning
• classification
• data-mining

2 Answers 2

People are a bit loose with their definitions (meaning different people will use different definitions, depending on the context), but let me put what I would say. I will do so more in the context of modern computer vision.

First, more generally, define $X$ as the space of the input data, and $Y$ as the output label space (some subset of the integers or equivalently one-hot vectors). A dataset is then $D=\{ d=(x,y)\in X\times Y \}$ , where $d\sim P_{X\times Y}$ is sampled from some joint distribution over the input and output space.

Now, let $\mathcal{H}$ be a set of functions such that an element $f \in \mathcal{H}$ is a map $f: X\rightarrow Y$ . This is the space of functions we will consider for our problem. And finally, let $g_\theta \in \mathcal{H}$ be some specific function with parameters $\theta\in\mathbb{R}^n$ , such that we denote $\widehat{y} = g_\theta(x|\theta)$ .

Finally, lets assume that any $f\in\mathcal{H}$ consists of a sequence of mappings $f=f_\ell\circ f_{\ell-1}\circ\ldots\circ f_2\circ f_1$ , where $f_i: F_{i}\rightarrow F_{i+1}$ and $F_1 = X, \, F_{\ell+1}=Y$ .

Ok, now for the definitions:

Hypothesis space (HS): the HS is the abstract function space you consider in solving your problem. Here it is denoted $\mathcal{H}$ . I find that this term does not appear very often in applied ML, rather, it is mostly used in theoretical contexts (e.g., PAC theory ). Sample space (SS): the sample space is simply the input (or instance) space $X$ . This is the same as in probability theory, regarding each training input as a random sample instance 1 . Parameter space (PS): for a fixed classifier $g_\theta$ , the PS is simply the space of possible values of $\theta$ . It defines the space covered by the single architecture that you train 2 . Usually it does not include hyper -parameters when people say it. Feature space (FS): for many models, there are multiple feature spaces. I've denoted them here as $F_2,\ldots, F_\ell$ . They are essentially the intermediate outputs due to the model's layered processing (but see note 1 ). For CNNs, these "feature maps" at different layers are often used for different things, hence distinction is important.

For your example:

The HS is almost the same as the PS once you've chosen logistic regression (except that the HS includes the models arising from different hyper-parameters as well, whereas the PS is fixed for a given set of hyper-parameters). Indeed, here, the HS is the set of all hyperplanes (and the PS could be as well, depending on the presence of e.g. regularization parameters).

The sample space is the set of all possible cat images; i.e., $X$ . It is not usually restricted in meaning to be $D$ , which is usually just called the training set.

The feature space in your case is indeed $F_1 = X$ , assuming that you feed the raw pixels to the logistic regression (so $\ell = 1$ ). 3

1 Some people treat some processed form of the input as the input. E.g., replacing an image $I$ with its HOG or wavelet features $u(I)$ . Then they define the sample space $X_u = \{ u(I_k) \;\forall\; k \}$ , i.e., as the features rather than the images. However, I would argue that you should leave $I\in X$ and simply set $F_1 = X_u$ , i.e., treat it as the first feature space.

2 Note that each $\theta$ defines a different trained model, which is in the HS. However, not all members of $\mathcal{H}$ can be reached by varying the parameter vector. For instance, you might search over the number of layers in a CNN, but the parameter space of a single CNN will not cover that. (Though note again that $\mathcal{H}$ tends to be used more in theoretical contexts). One distinction between HS and PS appears in the context of error decompositions of approximation vs estimation noise .

3 Normally (in "older" computer vision) you would extract features from the image and feed that to e.g. logistic regression. The modern version of this is attaching a fully connected (linear) layer with a softmax at the end of a CNN.

I'll approach this from a more colloquial point of view:

The sample space consists of your sample-level input data, which are instances of specific values in feature space. In your example, your sample space consists of 1000 images.

The feature space consists of the individual components that make up a sample, and potentially intermediate, derived features that express combinations of the raw features. In your example, the feature space is the 10,000 pixels and the color values they can take.

The hypothesis space covers all potential solutions that you could arrive at with your choice of model. A model that draws a linear boundary in feature space, for example, does not have any nonlinear solutions in its hypothesis space. In most cases, you can't enumerate the hypothesis space, but it's useful to know what types of solutions it's even possible for your model to generate.

The parameter space covers the possible values that the model parameters can take, which will vary depending on your model. A logistic regression, for example, will have a weight parameter for every feature that varies between -Inf and +Inf. You could also build a coin flip model that guesses "cat" randomly with probability X, where X is the single parameter that varies from 0 to 100.

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hypothesis space - linear and logistic regression

I am new to machine learning and I came across the term "hypothesis space". I am trying to grasp what is it and especially am interested in dimension of this "space." For example in the context of linear regression, trying to fit a linear polynomial to the data, would the dimension of the hypothesis space be $2$ ? What about in the context of logistic regression?

• machine-learning

• $\begingroup$ How was the term used? $\endgroup$ –  Michael Hardy Commented Apr 29, 2020 at 4:53
• $\begingroup$ One often speaks of a "parameter space". In the simplest logistic regression problems, one has $$ \operatorname{logit} \Pr(Y_i=1) = \alpha + \beta x_i $$ where $$\operatorname{logit} p = \log \frac p {1-p}$$ and $\Pr(Y_i\in\{0,1\}) = 1.$ Then the parameter space is the set of all possible values of the two parameters $\alpha,\beta.$ And one considers hypotheses concerning the values of these two parameters. $\endgroup$ –  Michael Hardy Commented Apr 29, 2020 at 4:56
• $\begingroup$ @MichaelHardy I think hypothesis space has more to do with function space as opposed to parameter space. I am unsure though if both end up have the same dimension. $\endgroup$ –  funmath Commented Apr 29, 2020 at 16:24
• $\begingroup$ As I said: How was the term used? $\endgroup$ –  Michael Hardy Commented Apr 29, 2020 at 16:30
• $\begingroup$ @MichaelHardy A hypothesis space refers to the set of possible approximations that algorithm can create for f. The hypothesis space consists of the set of functions the model is limited to learn. For instance, linear regression can be limited to linear functions as its hypothesis space. $\endgroup$ –  funmath Commented Apr 29, 2020 at 16:40

In the simplest instances of logistic regression one has independent random variables $Y_1,\ldots,Y_n$ for which $$ \begin{cases} \operatorname{logit} \Pr(Y_i=1) = \phantom{+(}\alpha + \beta x_i \\[8pt] \operatorname{logit} \Pr(Y_i=0) = -(\alpha+\beta x_i) \end{cases} $$ where $$ \operatorname{logit} p = \log \frac p {1-p}, $$ and

• $\{(x_i, Y_i) : i=1,\ldots,n\}$ are observed;
• $\alpha,\beta$ are not observed and are to be estimated based on the above observed data;
• As mentioned, $Y_i$ are random variables. On the other had $x_i$ are treated as constant, i.e. non-random, despite the fact that they may change if a new sample of $n$ observations is taken, the justification being that one is really interested in the conditional distribution of $Y$ given $x.$

Least squares is not the method used for estimating $\alpha$ and $\beta;$ maximum likelihood is, and the MLE is found by iteratively re-weighted least squares.

The function of most interest my be $$ p = \operatorname{logit}^{-1} (\alpha + \beta x) = \frac 1 {1 + e^{-(\alpha+\beta x)}}. $$ Every such function is completely determined by the values of $\alpha$ and $\beta.$ And in this case $\alpha$ and $\beta$ can be any real numbers at all.

Therefore the hypothesis space, if that is defined as the set of functions the model is limited to learn, is a $2$ -dimensional manifold homeopmorphic to the plane.

When the mapping from the parameter space to the hypothesis space is one-to-one and continuous, then the dimension of the hypothesis space is the same as the dimension of the parameter space. And "continuous" may be best defined in this context in such a way that it's always continuous, i.e. the mapping itself determines the topology on the hypothesis space.

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Step-by-step guide to hypothesis testing in statistics

Hypothesis testing in statistics helps us use data to make informed decisions. It starts with an assumption or guess about a group or population—something we believe might be true. We then collect sample data to check if there is enough evidence to support or reject that guess. This method is useful in many fields, like science, business, and healthcare, where decisions need to be based on facts.

Learning how to do hypothesis testing in statistics step-by-step can help you better understand data and make smarter choices, even when things are uncertain. This guide will take you through each step, from creating your hypothesis to making sense of the results, so you can see how it works in practical situations.

What is Hypothesis Testing?

Table of Contents

Hypothesis testing is a method for determining whether data supports a certain idea or assumption about a larger group. It starts by making a guess, like an average or a proportion, and then uses a small sample of data to see if that guess seems true or not.

For example, if a company wants to know if its new product is more popular than its old one, it can use hypothesis testing. They start with a statement like “The new product is not more popular than the old one” (this is the null hypothesis) and compare it with “The new product is more popular” (this is the alternative hypothesis). Then, they look at customer feedback to see if there’s enough evidence to reject the first statement and support the second one.

Simply put, hypothesis testing is a way to use data to help make decisions and understand what the data is really telling us, even when we don’t have all the answers.

Importance Of Hypothesis Testing In Decision-Making And Data Analysis

Hypothesis testing is important because it helps us make smart choices and understand data better. Here’s why it’s useful:

• Reduces Guesswork : It helps us see if our guesses or ideas are likely correct, even when we don’t have all the details.
• Uses Real Data : Instead of just guessing, it checks if our ideas match up with real data, which makes our decisions more reliable.
• Avoids Errors : It helps us avoid mistakes by carefully checking if our ideas are right so we don’t make costly errors.
• Shows What to Do Next : It tells us if our ideas work or not, helping us decide whether to keep, change, or drop something. For example, a company might test a new ad and decide what to do based on the results.
• Confirms Research Findings : It makes sure that research results are accurate and not just random chance so that we can trust the findings.

Here’s a simple guide to understanding hypothesis testing, with an example:

1. Set Up Your Hypotheses

Explanation: Start by defining two statements:

• Null Hypothesis (H0): This is the idea that there is no change or effect. It’s what you assume is true.
• Alternative Hypothesis (H1): This is what you want to test. It suggests there is a change or effect.

Example: Suppose a company says their new batteries last an average of 500 hours. To check this:

• Null Hypothesis (H0): The average battery life is 500 hours.
• Alternative Hypothesis (H1): The average battery life is not 500 hours.

2. Choose the Test

Explanation: Pick a statistical test that fits your data and your hypotheses. Different tests are used for various kinds of data.

Example: Since you’re comparing the average battery life, you use a one-sample t-test .

3. Set the Significance Level

Explanation: Decide how much risk you’re willing to take if you make a wrong decision. This is called the significance level, often set at 0.05 or 5%.

Example: You choose a significance level of 0.05, meaning you’re okay with a 5% chance of being wrong.

4. Gather and Analyze Data

Explanation: Collect your data and perform the test. Calculate the test statistic to see how far your sample result is from what you assumed.

Example: You test 30 batteries and find they last an average of 485 hours. You then calculate how this average compares to the claimed 500 hours using the t-test.

5. Find the p-Value

Explanation: The p-value tells you the probability of getting a result as extreme as yours if the null hypothesis is true.

Example: You find a p-value of 0.0001. This means there’s a very small chance (0.01%) of getting an average battery life of 485 hours or less if the true average is 500 hours.

6. Make Your Decision

Explanation: Compare the p-value to your significance level. If the p-value is smaller, you reject the null hypothesis. If it’s larger, you do not reject it.

Example: Since 0.0001 is much less than 0.05, you reject the null hypothesis. This means the data suggests the average battery life is different from 500 hours.

7. Report Your Findings

Explanation: Summarize what the results mean. State whether you rejected the null hypothesis and what that implies.

Example: You conclude that the average battery life is likely different from 500 hours. This suggests the company’s claim might not be accurate.

Hypothesis testing is a way to use data to check if your guesses or assumptions are likely true. By following these steps—setting up your hypotheses, choosing the right test, deciding on a significance level, analyzing your data, finding the p-value, making a decision, and reporting results—you can determine if your data supports or challenges your initial idea.

Understanding Hypothesis Testing: A Simple Explanation

Hypothesis testing is a way to use data to make decisions. Here’s a straightforward guide:

1. What is the Null and Alternative Hypotheses?

• Null Hypothesis (H0): This is your starting assumption. It says that nothing has changed or that there is no effect. It’s what you assume to be true until your data shows otherwise. Example: If a company says their batteries last 500 hours, the null hypothesis is: “The average battery life is 500 hours.” This means you think the claim is correct unless you find evidence to prove otherwise.
• Alternative Hypothesis (H1): This is what you want to find out. It suggests that there is an effect or a difference. It’s what you are testing to see if it might be true. Example: To test the company’s claim, you might say: “The average battery life is not 500 hours.” This means you think the average battery life might be different from what the company says.

2. One-Tailed vs. Two-Tailed Tests

• One-Tailed Test: This test checks for an effect in only one direction. You use it when you’re only interested in finding out if something is either more or less than a specific value. Example: If you think the battery lasts longer than 500 hours, you would use a one-tailed test to see if the battery life is significantly more than 500 hours.
• Two-Tailed Test: This test checks for an effect in both directions. Use this when you want to see if something is different from a specific value, whether it’s more or less. Example: If you want to see if the battery life is different from 500 hours, whether it’s more or less, you would use a two-tailed test. This checks for any significant difference, regardless of the direction.

3. Common Misunderstandings

• Clarification: Hypothesis testing doesn’t prove that the null hypothesis is true. It just helps you decide if you should reject it. If there isn’t enough evidence against it, you don’t reject it, but that doesn’t mean it’s definitely true.
• Clarification: A small p-value shows that your data is unlikely if the null hypothesis is true. It suggests that the alternative hypothesis might be right, but it doesn’t prove the null hypothesis is false.
• Clarification: The significance level (alpha) is a set threshold, like 0.05, that helps you decide how much risk you’re willing to take for making a wrong decision. It should be chosen carefully, not randomly.
• Clarification: Hypothesis testing helps you make decisions based on data, but it doesn’t guarantee your results are correct. The quality of your data and the right choice of test affect how reliable your results are.

Benefits and Limitations of Hypothesis Testing

• Clear Decisions: Hypothesis testing helps you make clear decisions based on data. It shows whether the evidence supports or goes against your initial idea.
• Objective Analysis: It relies on data rather than personal opinions, so your decisions are based on facts rather than feelings.
• Concrete Numbers: You get specific numbers, like p-values, to understand how strong the evidence is against your idea.
• Control Risk: You can set a risk level (alpha level) to manage the chance of making an error, which helps avoid incorrect conclusions.
• Widely Used: It can be used in many areas, from science and business to social studies and engineering, making it a versatile tool.

Limitations

• Sample Size Matters: The results can be affected by the size of the sample. Small samples might give unreliable results, while large samples might find differences that aren’t meaningful in real life.
• Risk of Misinterpretation: A small p-value means the results are unlikely if the null hypothesis is true, but it doesn’t show how important the effect is.
• Needs Assumptions: Hypothesis testing requires certain conditions, like data being normally distributed . If these aren’t met, the results might not be accurate.
• Simple Decisions: It often results in a basic yes or no decision without giving detailed information about the size or impact of the effect.
• Can Be Misused: Sometimes, people misuse hypothesis testing, tweaking data to get a desired result or focusing only on whether the result is statistically significant.
• No Absolute Proof: Hypothesis testing doesn’t prove that your hypothesis is true. It only helps you decide if there’s enough evidence to reject the null hypothesis, so the conclusions are based on likelihood, not certainty.

Final Thoughts 

Hypothesis testing helps you make decisions based on data. It involves setting up your initial idea, picking a significance level, doing the test, and looking at the results. By following these steps, you can make sure your conclusions are based on solid information, not just guesses.

This approach lets you see if the evidence supports or contradicts your initial idea, helping you make better decisions. But remember that hypothesis testing isn’t perfect. Things like sample size and assumptions can affect the results, so it’s important to be aware of these limitations.

In simple terms, using a step-by-step guide for hypothesis testing is a great way to better understand your data. Follow the steps carefully and keep in mind the method’s limits.

What is the difference between one-tailed and two-tailed tests?

 A one-tailed test assesses the probability of the observed data in one direction (either greater than or less than a certain value). In contrast, a two-tailed test looks at both directions (greater than and less than) to detect any significant deviation from the null hypothesis.

How do you choose the appropriate test for hypothesis testing?

The choice of test depends on the type of data you have and the hypotheses you are testing. Common tests include t-tests, chi-square tests, and ANOVA. You get more details about ANOVA, you may read Complete Details on What is ANOVA in Statistics ?  It’s important to match the test to the data characteristics and the research question.

What is the role of sample size in hypothesis testing?  

Sample size affects the reliability of hypothesis testing. Larger samples provide more reliable estimates and can detect smaller effects, while smaller samples may lead to less accurate results and reduced power.

Can hypothesis testing prove that a hypothesis is true?  

Hypothesis testing cannot prove that a hypothesis is true. It can only provide evidence to support or reject the null hypothesis. A result can indicate whether the data is consistent with the null hypothesis or not, but it does not prove the alternative hypothesis with certainty.

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• Neuroscience

Reevaluating the Neural Noise Hypothesis in Dyslexia: Insights from EEG and 7T MRS Biomarkers

Agnieszka glica, katarzyna wasilewska, julia jurkowska, jarosław żygierewicz, bartosz kossowski.

• Katarzyna Jednoróg author has email address
• Laboratory of Language Neurobiology, Nencki Institute of Experimental Biology, Polish Academy of Sciences, Pasteur 3 Street, 02-093 Warsaw, Poland
• Faculty of Physics, University of Warsaw, Pasteur 5 Street, 02-093 Warsaw, Poland
• Laboratory of Brain Imaging, Nencki Institute of Experimental Biology, Polish Academy of Sciences, Pasteur 3 Street, 02-093 Warsaw, Poland
• https://doi.org/ 10.7554/eLife.99920.1
• Open access
• Copyright information

The neural noise hypothesis of dyslexia posits an imbalance between excitatory and inhibitory (E/I) brain activity as an underlying mechanism of reading difficulties. This study provides the first direct test of this hypothesis using both indirect EEG power spectrum measures in 120 Polish adolescents and young adults (60 with dyslexia, 60 controls) and direct glutamate (Glu) and gamma-aminobutyric acid (GABA) concentrations from magnetic resonance spectroscopy (MRS) at 7T MRI scanner in half of the sample. Our results, supported by Bayesian statistics, show no evidence of E/I balance differences between groups, challenging the hypothesis that cortical hyperexcitability underlies dyslexia. These findings suggest alternative mechanisms must be explored and highlight the need for further research into the E/I balance and its role in neurodevelopmental disorders.

eLife assessment

The authors combined neurophysiological (electroencephalography [EEG]) and neurochemical (magnetic resonance spectroscopy [MRS]) measures to empirically evaluate the neural noise hypothesis of developmental dyslexia. Their results are solid , supported by consistent findings from the two complementary methodologies and Bayesian statistics. Additional analyses, particularly on the neurochemical measures, are necessary to further substantiate the results. This study is useful for understanding the neural mechanisms of dyslexia and neural development in general.

• https://doi.org/ 10.7554/eLife.99920.1.sa3
• Read the peer reviews
• About eLife assessments

Introduction

According to the neural noise hypothesis of dyslexia, reading difficulties stem from an imbalance between excitatory and inhibitory (E/I) neural activity ( Hancock et al., 2017 ). The hypothesis predicts increased cortical excitation leading to more variable and less synchronous neural firing. This instability supposedly results in disrupted sensory representations and impedes phonological awareness and multisensory integration skills, crucial for learning to read ( Hancock et al., 2017 ). Yet, studies testing this hypothesis are lacking.

The non-invasive measurement of the E/I balance can be derived through assessment of glutamate (Glu) and gamma-aminobutyric acid (GABA) neurotransmitters concentration via magnetic resonance spectroscopy (MRS) ( Finkelman et al., 2022 ) or through global, indirect estimations from the electroencephalography (EEG) signal ( Ahmad et al., 2022 ).

Direct measurements of Glu and GABA yielded conflicting findings. Higher Glu concentrations in the midline occipital cortex correlated with poorer reading performance in children ( Del Tufo et al., 2018 ; Pugh et al., 2014 ), while elevated Glu levels in the anterior cingulate cortex (ACC) corresponded to greater phonological skills ( Lebel et al., 2016 ). Elevated GABA in the left inferior frontal gyrus was linked to reduced verbal fluency in adults ( Nakai and Okanoya, 2016 ), and increased GABA in the midline occipital cortex in children was associated with slower reaction times in a linguistic task ( Del Tufo et al., 2018 ). However, notable null findings exist regarding dyslexia status and Glu levels in the ACC among children ( Horowitz-Kraus et al., 2018 ) as well as Glu and GABA levels in the visual and temporo-parietal cortices in both children and adults ( Kossowski et al., 2019 ).

Both beta (∼13-28 Hz) and gamma (> 30 Hz) oscillations may serve as E/I balance indicators ( Ahmad et al., 2022 ), as greater GABA-ergic activity has been associated with greater beta power ( Jensen et al., 2005 ; Porjesz et al., 2002 ) and gamma power or peak frequency ( Brunel and Wang, 2003 ; Chen et al., 2017 ). Resting-state analyses often reported nonsignificant beta power associations with dyslexia ( Babiloni et al., 2012 ; Fraga González et al., 2018 ; Xue et al., 2020 ), however, one study indicated lower beta power in dyslexic compared to control boys ( Fein et al., 1986 ). Mixed results were also observed during tasks. One study found decreased beta power in the dyslexic group ( Spironelli et al., 2008 ), while the other increased beta power relative to the control group ( Rippon and Brunswick, 2000 ). Insignificant relationship between resting gamma power and dyslexia was reported ( Babiloni et al., 2012 ; Lasnick et al., 2023 ). When analyzing auditory steady-state responses, the dyslexic group had a lower gamma peak frequency, while no significant differences in gamma power were observed ( Rufener and Zaehle, 2021 ). Essentially, the majority of studies in dyslexia examining gamma frequencies evaluated cortical entrainment to auditory stimuli ( Lehongre et al., 2011 ; Marchesotti et al., 2020 ; Van Hirtum et al., 2019 ). Therefore, the results from these tasks do not provide direct evidence of differences in either gamma power or peak frequency between the dyslexic and control groups.

The EEG signal comprises both oscillatory, periodic activity, and aperiodic activity, characterized by a gradual decrease in power as frequencies rise (1/f signal) ( Donoghue et al., 2020 ). Recently recognized as a biomarker of E/I balance, a lower exponent of signal decay (flatter slope) indicates a greater dominance of excitation over inhibition in the brain, as shown by the simulation models of local field potentials, ratio of AMPA/GABA a synapses in the rat hippocampus ( Gao et al., 2017 ) and recordings under propofol or ketamine in macaques and humans ( Gao et al., 2017 ; Waschke et al., 2021 ). However, there are also pharmacological studies providing mixed results ( Colombo et al., 2019 ; Salvatore et al., 2024 ). Nonetheless, the 1/f signal has shown associations with various conditions putatively characterized by changes in E/I balance, such as early development in infancy ( Schaworonkow and Voytek, 2021 ), healthy aging ( Voytek et al., 2015 ) and neurodevelopmental disorders like ADHD ( Ostlund et al., 2021 ), autism spectrum disorder ( Manyukhina et al., 2022 ) or schizophrenia ( Molina et al., 2020 ). Despite its potential relevance, the evaluation of the 1/f signal in dyslexia remains limited to one study, revealing flatter slopes among dyslexic compared to control participants at rest ( Turri et al., 2023 ), thereby lending support to the notion of neural noise in dyslexia.

Here, we examined both indirect (1/f signal, beta, and gamma oscillations during both rest and a spoken language task) and direct (Glu and GABA) biomarkers of E/I balance in participants with dyslexia and age-matched controls. The neural noise hypothesis predicts flatter slopes of 1/f signal, decreased beta and gamma power, and higher Glu concentrations in the dyslexic group. Furthermore, we tested the relationships between different E/I measures. Flatter slopes of 1/f signal should be related to higher Glu level, while enhanced beta and gamma power to increased GABA level.

No evidence for group differences in the EEG E/I biomarkers

We recruited 120 Polish adolescents and young adults – 60 with dyslexia diagnosis and 60 controls matched in sex, age, and family socio-economic status. The dyslexic group scored lower in all reading and reading-related tasks and higher in the Polish version of the Adult Reading History Questionnaire (ARHQ-PL) ( Bogdanowicz et al., 2015 ),where a higher score indicates a higher risk of dyslexia (see Table S1 in the Supplementary Material). Although all participants were within the intellectual norm, the dyslexic group scored lower on the IQ scale (including nonverbal subscale only) than the control group. However, the Bayesian statistics did not provide evidence for the difference between groups in the nonverbal IQ.

We analyzed the aperiodic (exponent and offset) components of the EEG signal at rest and during a spoken language task, where participants listened to a sentence and had to indicate its veracity. Due to a technical error, the signal from one person (a female from the dyslexic group) was not recorded during most of the language task and was excluded from the analyses. Hence, the results are provided for 119 participants – 59 in the dyslexic and 60 in the control group.

First, aperiodic parameter values were averaged across all electrodes and compared between groups (dyslexic, control) and conditions (resting state, language task) using a 2×2 repeated measures ANOVA. Age negatively correlated both with the exponent ( r = -.27, p = .003, BF 10 = 7.96) and offset ( r = -.40, p < .001, BF 10 = 3174.29) in line with previous investigations ( Cellier et al., 2021 ; McSweeney et al., 2021 ; Schaworonkow and Voytek, 2021 ; Voytek et al., 2015 ), therefore we included age as a covariate. Post-hoc tests are reported with Bonferroni corrected p -values.

For the mean exponent, we found a significant effect of age ( F (1,116) = 8.90, p = .003, η 2 p = .071, BF incl = 10.47), while the effects of condition ( F (1,116) = 2.32, p = .131, η 2 p = .020, BF incl = 0.39) and group ( F (1,116) = 0.08, p = .779, η 2 p = .001, BF incl = 0.40) were not significant and Bayes Factor did not provide evidence for either inclusion or exclusion. Interaction between group and condition ( F (1,116) = 0.16, p = .689, η 2 p = .001, BF incl = 0.21) was not significant and Bayes Factor indicated against including it in the model.

For the mean offset, we found significant effects of age ( F (1,116) = 22.57, p < .001, η 2 p = .163, BF incl = 1762.19) and condition ( F (1,116) = 23.04, p < .001, η 2 p = .166, BF incl > 10000) with post-hoc comparison indicating that the offset was lower in the resting state condition ( M = -10.80, SD = 0.21) than in the language task ( M = -10.67, SD = 0.26, p corr < .001). The effect of group ( F (1,116) = 0.00, p = .964, η 2 p = .000, BF incl = 0.54) was not significant while Bayes Factor did not provide evidence for either inclusion or exclusion. Interaction between group and condition was not significant ( F (1,116) = 0.07, p = .795, η 2 p = .001, BF incl = 0.22) and Bayes Factor indicated against including it in the model.

Next, we restricted analyses to language regions and averaged exponent and offset values from the frontal electrodes corresponding to the left (F7, FT7, FC5) and right inferior frontal gyrus (F8, FT8, FC6), as well as temporal electrodes, corresponding to the left (T7, TP7, TP9) and right superior temporal sulcus, STS (T8, TP8, TP10)( Giacometti et al., 2014 )( Scrivener and Reader, 2022 ). A 2×2×2×2 (group, condition, hemisphere, region) repeated measures ANOVA with age as a covariate was applied. Power spectra from the left STS at rest and during the language task are presented in Figure 1A and C , while the results for the exponent, offset, and beta power are presented in Figure 1B and D .

Overview of the main results obtained in the study. (A) Power spectral densities averaged across 3 electrodes (T7, TP7, TP9) corresponding to the left superior temporal sulcus (STS) separately for dyslexic (DYS) and control (CON) groups at rest and (C) during the language task. (B) Plots illustrating results for the exponent, offset, and the beta power from the left STS electrodes at rest and (D ) during the language task. (E) Group results (CON > DYS) from the fMRI localizer task for words compared to the control stimuli (p < .05 FWE cluster threshold) and overlap of the MRS voxel placement across participants. (F) MRS spectra separately for DYS and CON groups. (G) Plots illustrating results for the Glu, GABA, Glu/GABA ratio and the Glu/GABA imbalance. (H ) Semi-partial correlation between offset at rest (left STS electrodes) and Glu controlling for age and gray matter volume (GMV).

For the exponent, there were significant effects of age ( F (1,116) = 14.00, p < .001, η 2 p = .108, BF incl = 11.46) and condition F (1,116) = 4.06, p = .046, η 2 p = .034, BF incl = 1.88), however, Bayesian statistics did not provide evidence for either including or excluding the condition factor. Furthermore, post-hoc comparisons did not reveal significant differences between the exponent at rest ( M = 1.51, SD = 0.17) and during the language task ( M = 1.51, SD = 0.18, p corr = .546). There was also a significant interaction between region and group, although Bayes Factor indicated against including it in the model ( F (1,116) = 4.44, p = .037, η 2 p = .037, BF incl = 0.25). Post-hoc comparisons indicated that the exponent was higher in the frontal than in the temporal region both in the dyslexic ( M frontal = 1.54, SD frontal = 0.15, M temporal = 1.49, SD temporal = 0.18, p corr < .001) and in the control group ( M frontal = 1.54, SD frontal = 0.17, M temporal = 1.46, SD temporal = 0.20, p corr < .001). The difference between groups was not significant either in the frontal ( p corr = .858) or temporal region ( p corr = .441). The effects of region ( F (1,116) = 1.17, p = .282, η 2 p = .010, BF incl > 10000) and hemisphere ( F (1,116) = 1.17, p = .282, η 2 p = .010, BF incl = 12.48) were not significant, although Bayesian statistics indicated in favor of including them in the model. Furthermore, the interactions between condition and group ( F (1,116) = 0.18, p = .673, η 2 p = .002, BF incl = 3.70), and between region, hemisphere, and condition ( F (1,116) = 0.11, p = .747, η 2 p = .001, BF incl = 7.83) were not significant, however Bayesian statistics indicated in favor of including these interactions in the model. The effect of group ( F (1,116) = 0.12, p = .733, η 2 p = .001, BF incl = 1.19) was not significant, while Bayesian statistics did not provide evidence for either inclusion or exclusion. Any other interactions were not significant and Bayes Factor indicated against including them in the model.

In the case of offset, there were significant effects of condition ( F (1,116) = 20.88, p < .001, η 2 p = .153, BF incl > 10000) and region ( F (1,116) = 6.18, p = .014, η 2 p = .051, BF incl > 10000). For the main effect of condition, post-hoc comparison indicated that the offset was lower in the resting state condition ( M = -10.88, SD = 0.33) than in the language task ( M = -10.76, SD = 0.38, p corr < .001), while for the main effect of region, post-hoc comparison indicated that the offset was lower in the temporal ( M = -10.94, SD = 0.37) as compared to the frontal region ( M = -10.69, SD = 0.34, p corr < .001). There was also a significant effect of age ( F (1,116) = 20.84, p < .001, η 2 p = .152, BF incl = 0.23) and interaction between condition and hemisphere, ( F (1,116) = 4.35, p = .039, η 2 p = .036, BF incl = 0.21), although Bayes Factor indicated against including these factors in the model. Post-hoc comparisons for the condition*hemisphere interaction indicated that the offset was lower in the resting state condition than in the language task both in the left ( M rest = -10.85, SD rest = 0.34, M task = -10.73, SD task = 0.40, p corr < .001) and in the right hemisphere ( M rest = -10.91, SD rest = 0.31, M task = -10.79, SD task = 0.37, p corr < .001) and that the offset was lower in the right as compared to the left hemisphere both at rest ( p corr < .001) and during the language task ( p corr < .001). The interactions between region and condition ( F (1,116) = 1.76, p = .187, η 2 p = .015, BF incl > 10000), hemisphere and group ( F (1,116) = 1.58, p = .211, η 2 p = .013, BF incl = 1595.18), region and group ( F (1,116) = 0.27, p = .605, η 2 p = .002, BF incl = 9.32), as well as between region, condition, and group ( F (1,116) = 0.21, p = .651, η 2 p = .002, BF incl = 2867.18) were not significant, although Bayesian statistics indicated in favor of including them in the model. The effect of group ( F (1,116) = 0.18, p = .673, η 2 p = .002, BF incl < 0.00001) was not significant and Bayesian statistics indicated against including it in the model. Any other interactions were not significant and Bayesian statistics indicated against including them in the model or did not provide evidence for either inclusion or exclusion.

Then, we analyzed the aperiodic-adjusted brain oscillations. Since the algorithm did not find the gamma peak (30-43 Hz) above the aperiodic component in the majority of participants, we report the results only for the beta (14-30 Hz) power. We performed a similar regional analysis as for the exponent and offset with a 2×2×2×2 (group, condition, hemisphere, region) repeated measures ANOVA. However, we did not include age as a covariate, as it did not correlate with any of the periodic measures. The sample size was 117 (DYS n = 57, CON n = 60) since in 2 participants the algorithm did not find the beta peak above the aperiodic component in the left frontal electrodes during the task.

The analysis revealed a significant effect of condition ( F (1,115) = 8.58, p = .004, η 2 p = .069, BF incl = 5.82) with post-hoc comparison indicating that the beta power was greater during the language task ( M = 0.53, SD = 0.22) than at rest ( M = 0.50, SD = 0.19, p corr = .004). There were also significant effects of region ( F (1,115) = 10.98, p = .001, η 2 p = .087, BF incl = 23.71), and hemisphere ( F (1,115) = 12.08, p < .001, η 2 p = .095, BF incl = 23.91). For the main effect of region, post-hoc comparisons indicated that the beta power was greater in the temporal ( M = 0.52, SD = 0.21) as compared to the frontal region ( M = 0.50, SD = 0.19, p corr = .001), while for the main effect of hemisphere, post-hoc comparisons indicated that the beta power was greater in the right ( M = 0.52, SD = 0.20) than in the left hemisphere ( M = 0.51, SD = 0.20, p corr < .001). There was a significant interaction between condition and region ( F (1,115) = 12.68, p < .001, η 2 p = .099, BF incl = 55.26) with greater beta power during the language task as compared to rest significant in the temporal ( M rest = 0.50, SD rest = 0.20, M task = 0.55, SD task = 0.24, p corr < .001), while not in the frontal region ( M rest = 0.49, SD rest = 0.18, M task = 0.51, SD task = 0.22, p corr = .077). Also, greater beta power in the temporal as compared to the frontal region was significant during the language task ( p corr < .001), while not at rest ( p corr = .283). The effect of group ( F (1,115) = 0.05, p = .817, η 2 p = .000, BF incl < 0.00001) was not significant and Bayes Factor indicated against including it in the model. Any other interactions were not significant and Bayesian statistics indicated against including them in the model or did not provide evidence for either inclusion or exclusion.

Additionally, building upon previous findings which demonstrated differences in dyslexia in aperiodic and periodic components within the parieto-occipital region ( Turri et al., 2023 ), we have included analyses for the same cluster of electrodes in the Supplementary Material. However, in this region, we also did not find evidence for group differences either in the exponent, offset or beta power.

No evidence for group differences in Glu and GABA concentrations in the left STS

In total, 59 out of 120 participants underwent MRS session at 7T MRI scanner - 29 from the dyslexic group (13 females, 16 males) and 30 from the control group (14 females, 16 males). The MRS voxel was placed in the left STS, in a region showing highest activation for both visual and auditory words (compared to control stimuli) localized individually in each participant, based on an fMRI task (see Figure 1E for overlap of the MRS voxel placement across participants and Figure 1F for MRS spectra). We decided to analyze the neurometabolites’ levels derived from the left STS, as this region is consistently related to functional and structural differences in dyslexia across languages ( Yan et al., 2021 ).

Due to insufficient magnetic homogeneity or interruption of the study by the participants, 5 participants from the dyslexic group had to be excluded. We excluded further 4 participants due to poor quality of the obtained spectra thus the results for Glu are reported for 50 participants - 21 in the dyslexic (12 females, 9 males) and 29 in the control group (13 females, 16 males). In the case of GABA, we additionally excluded 3 participants based on the Cramér-Rao Lower Bounds (CRLB) > 20%. Therefore, the results for GABA, Glu/GABA ratio and Glu/GABA imbalance are reported for 47 participants - 20 in the dyslexic (12 females, 8 males) and 27 in the control group (11 females, 16 males). Demographic and behavioral characteristics for the subsample of 47 participants are provided in the Table S2.

For each metabolite, we performed a separate univariate ANCOVA with the effect of group being tested and voxel’s gray matter volume (GMV) as a covariate (see Figure 1G ). For the Glu analysis, we also included age as a covariate, due to negative correlation between variables ( r = -.35, p = .014, BF 10 = 3.41). The analysis revealed significant effect of GMV ( F (1,46) = 8.18, p = .006, η 2 p = .151, BF incl = 12.54), while the effects of age ( F (1,46) = 3.01, p = .090, η 2 p = .061, BF incl = 1.15) and group ( F (1,46) = 1.94, p = .170, 1 = .040, BF incl = 0.63) were not significant and Bayes Factor did not provide evidence for either inclusion or exclusion.

Conversely, GABA did not correlate with age ( r = -.11, p = .481, BF 10 = 0.23), thus age was not included as a covariate. The analysis revealed a significant effect of GMV ( F (1,44) = 4.39, p = .042, η 2 p = .091, BF incl = 1.64), however Bayes Factor did not provide evidence for either inclusion or exclusion. The effect of group was not significant ( F (1,44) = 0.49, p = .490, η 2 p = .011, BF incl = 0.35) although Bayesian statistics did not provide evidence for either inclusion or exclusion.

Also, Glu/GABA ratio did not correlate with age ( r = -.05, p = .744, BF 10 = 0.19), therefore age was not included as a covariate. The results indicated that the effect of GMV was not significant ( F (1,44) = 0.95, p = .335, η 2 p = .021, BF incl = 0.43) while Bayes Factor did not provide evidence for either inclusion or exclusion. The effect of group was not significant ( F (1,44) = 0.01, p = .933, η 2 p = .000, BF incl = 0.29) and Bayes Factor indicated against including it in the model.

Following a recent study examining developmental changes in both EEG and MRS E/I biomarkers ( McKeon et al., 2024 ), we calculated an additional measure of Glu/GABA imbalance, computed as the absolute residual value from the linear regression of Glu predicted by GABA with greater values indicating greater Glu/GABA imbalance. Alike the previous work ( McKeon et al., 2024 ), we took the square root of this value to ensure a normal distribution of the data. This measure did not correlate with age ( r = -.05, p = .719, BF 10 = 0.19); thus, age was not included as a covariate. The results indicated that the effect of GMV was not significant ( F (1,44) = 0.63, p = .430, η 2 p = .014, BF incl = 0.37) while Bayes Factor did not provide evidence for either inclusion or exclusion. The effect of group was not significant ( F (1,44) = 0.74, p = .396, η 2 p = .016, BF incl = 0.39) although Bayesian statistics did not provide evidence for either inclusion or exclusion.

Correspondence between Glu and GABA concentrations and EEG E/I biomarkers is limited

Next, we investigated correlations between Glu and GABA concentrations in the left STS and EEG markers of E/I balance. Semi-partial correlations were performed ( Table 1 ) to control for confounding variables - for Glu the effects of age and GMV were regressed, for GABA, Glu/GABA ratio and Glu/GABA imbalance the effect of GMV was regressed, while for exponents and offsets the effect of age was regressed. For zero-order correlations between variables see Table S3.

Semi-partial Correlations Between Direct and Indirect Markers of Excitatory-Inhibitory Balance. For Glu the Effects of Age and Gray Matter Volume (GMV) Were Regressed, for GABA, Glu/GABA Ratio and Glu/GABA Imbalance the Effect of GMV was Regressed, While for Exponents and Offsets the Effect of Age was Regressed

Glu negatively correlated with offset in the left STS both at rest ( r = -.38, p = .007, BF 10 = 6.28; Figure 1H ) and during the language task ( r = -.37, p = .009, BF 10 = 5.05), while any other correlations between Glu and EEG markers were not significant and Bayesian statistics indicated in favor of null hypothesis or provided absence of evidence for either hypothesis. Furthermore, Glu/GABA imbalance positively correlated with exponent at rest both averaged across all electrodes ( r = .29, p = .048, BF 10 = 1.21), as well as in the left STS electrodes ( r = .35, p = .017, BF 10 = 2.87) although Bayes Factor provided absence of evidence for either alternative or null hypothesis. Conversely, GABA and Glu/GABA ratio were not significantly correlated with any of the EEG markers and Bayesian statistics indicated in favor of null hypothesis or provided absence of evidence for either hypothesis.

Testing the paths from neural noise to reading

The neural noise hypothesis of dyslexia predicts impact of the neural noise on reading through the impairment of 1) phonological awareness, 2) lexical access and generalization and 3) multisensory integration ( Hancock et al., 2017 ). Therefore, we analyzed correlations between these variables, reading skills and direct and indirect markers of E/I balance. For the composite score of phonological awareness, we averaged z-scores from phoneme deletion, phoneme and syllable spoonerisms tasks. For the composite score of lexical access and generalization we averaged z-scores from objects, colors, letters and digits subtests from rapid automatized naming (RAN) task, while for the composite score of reading we averaged z-scores from words and pseudowords read per minute, and text reading time in reading comprehension task. The outcomes from the RAN and reading comprehension task have been transformed from raw time scores to items/time scores in order to provide the same direction of relationships for all z-scored measures, with greater values indicating better skills. For the multisensory integration score we used results from the redundant target effect task reported in our previous work ( Glica et al., 2024 ), with greater values indicating a greater magnitude of multisensory integration.

Age positively correlated with multisensory integration ( r = .38, p < .001, BF 10 = 87.98), composite scores of reading ( r = .22, p = .014, BF 10 = 2.24) and phonological awareness ( r = .21, p = .021, BF 10 = 1.59), while not with the composite score of RAN ( r = .13, p = .151, BF 10 = 0.32). Hence, we regressed the effect of age from multisensory integration, reading and phonological awareness scores and performed semi-partial correlations ( Table 2 , for zero-order correlations see Table S4).

Semi-partial Correlations Between Reading, Phonological Awareness, Rapid Automatized Naming, Multisensory Integration and Markers of Excitatory-Inhibitory Balance. For Reading, Phonological Awareness and Multisensory Integration the Effect of Age was Regressed, for Glu the Effects of Age and Gray Matter Volume (GMV) Were Regressed, for GABA, Glu/GABA Ratio and Glu/GABA Imbalance the Effect of GMV was Regressed, While for Exponents and Offsets the Effect of Age was Regressed

Phonological awareness positively correlated with offset in the left STS at rest ( r = .18, p = .049, BF 10 = 0.77) and with beta power in the left STS both at rest ( r = .23, p = .011, BF 10 = 2.73; Figure 2A ) and during the language task ( r = .23, p = .011, BF 10 = 2.84; Figure 2B ), although Bayes Factor provided absence of evidence for either alternative or null hypothesis. Furthermore, multisensory integration positively correlated with GABA concentration ( r = .31, p = .034, BF 10 = 1.62) and negatively with Glu/GABA ratio ( r = -.32, p = .029, BF 10 = 1.84), although Bayes Factor provided absence of evidence for either alternative or null hypothesis. Any other correlations between reading skills and E/I balance markers were not significant and Bayesian statistics indicated in favor of null hypothesis or provided absence of evidence for either hypothesis.

Associations between beta power, phonological awareness and reading. (A) Semi-partial correlation between phonological awareness controlling for age and beta power (in the left STS electrodes) at rest and (B) during the language task. (C) Partial correlation between phonological awareness and reading controlling for age. (D) Mediation analysis results. Unstandardized b regression coefficients are presented. Age was included in the analysis as a covariate. 95% CI - 95% confidence intervals. left STS - values averaged across 3 electrodes corresponding to the left superior temporal sulcus (T7, TP7, TP9).

Given that beta power correlated with phonological awareness, and considering the prediction that neural noise impedes reading by affecting phonological awareness — we examined this relationship through a mediation model. Since phonological awareness correlated with beta power in the left STS both at rest and during language task, the outcomes from these two conditions were averaged prior to the mediation analysis. Macro PROCESS v4.2 ( Hayes, 2017 ) on IBM SPSS Statistics v29 with model 4 (simple mediation) with 5000 Bootstrap samples to assess the significance of indirect effect was employed. Since age correlated both with phonological awareness and reading, we also included age as a covariate.

The results indicated that both effects of beta power in the left STS ( b = .96, t (116) = 2.71, p = .008, BF incl = 7.53) and age ( b = .06, t (116) = 2.55, p = .012, BF incl = 5.98) on phonological awareness were significant. The effect of phonological awareness on reading was also significant ( b = .69, t (115) = 8.16, p < .001, BF incl > 10000), while the effects of beta power ( b = -.42, t (115) = -1.25, p = .213, BF incl = 0.52) and age ( b = .03, t (115) = 1.18, p = .241, BF incl = 0.49) on reading were not significant when controlling for phonological awareness. Finally, the indirect effect of beta power on reading through phonological awareness was significant ( b = .66, SE = .24, 95% CI = [.24, 1.18]), while the total effect of beta power was not significant ( b = .24, t (116) = 0.61, p = .546, BF incl = 0.41). The results from the mediation analysis are presented in Figure 2D .

Although similar mediation analysis could have been conducted for the Glu/GABA ratio, multisensory integration, and reading based on the correlations between these variables, we did not test this model due to the small sample size (47 participants), which resulted in insufficient statistical power.

The current study aimed to validate the neural noise hypothesis of dyslexia ( Hancock et al., 2017 ) utilizing E/I balance biomarkers from EEG power spectra and ultra-high-field MRS. Contrary to its predictions, we did not observe differences either in 1/f slope, beta power, or Glu and GABA concentrations in participants with dyslexia. Relations between E/I balance biomarkers were limited to significant correlations between Glu and the offset when controlling for age, and between Glu/GABA imbalance and the exponent.

In terms of indirect markers, our study found no evidence of group differences in the aperiodic components of the EEG signal. In most of the models, we did not find evidence for either including or excluding the effect of the group when Bayesian statistics were evaluated. The only exception was the regional analysis for the offset, where results indicated against including the group factor in the model. These findings diverge from previous research on an Italian cohort, which reported decreased exponent and offset in the dyslexic group at rest, specifically within the parieto-occipital region, but not the frontal region ( Turri et al., 2023 ). Despite our study involving twice the number of participants and utilizing a longer acquisition time, we observed no group differences, even in the same cluster of electrodes (refer to Supplementary Material). The participants in both studies were of similar ages. The only methodological difference – EEG acquisition with eyes open in our study versus both eyes-open and eyes-closed in the work by Turri and colleagues (2023) – cannot fully account for the overall lack of group differences observed. The diverging study outcomes highlight the importance of considering potential inflation of effect sizes in studies with smaller samples.

Although a lower exponent of the EEG power spectrum has been associated with other neurodevelopmental disorders, such as ADHD ( Ostlund et al., 2021 ) or ASD (but only in children with IQ below average) ( Manyukhina et al., 2022 ), our study suggests that this is not the case for dyslexia. Considering the frequent comorbidity of dyslexia and ADHD ( Germanò et al., 2010 ; Langer et al., 2019 ), increased neural noise could serve as a common underlying mechanism for both disorders. However, our specific exclusion of participants with a comorbid ADHD diagnosis indicates that the EEG spectral exponent cannot serve as a neurobiological marker for dyslexia in isolation. No information regarding such exclusion criteria was provided in the study by Turri et al. (2023) ; thus, potential comorbidity with ADHD may explain the positive findings related to dyslexia reported therein.

Regarding the aperiodic-adjusted oscillatory EEG activity, Bayesian statistics for beta power, indicated in favor of excluding the group factor from the model. Non-significant group differences in beta power at rest have been previously reported in studies that did not account for aperiodic components ( Babiloni et al., 2012 ; Fraga González et al., 2018 ; Xue et al., 2020 ). This again contrasts with the study by Turri et al. (2023) , which observed lower aperiodic-adjusted beta power (at 15-25 Hz) in the dyslexic group. Concerning beta power during task, our results also contrast with previous studies which showed either reduced ( Spironelli et al., 2008 ) or increased ( Rippon and Brunswick, 2000 ) beta activity in participants with dyslexia. Nevertheless, since both of these studies employed phonological tasks and involved children’s samples, their relevance to our work is limited.

In terms of direct neurometabolite concentrations derived from the MRS, we found no evidence for group differences in either Glu, GABA or Glu/GABA imbalance in the language-sensitive left STS. Conversely, the Bayes Factor suggested against including the group factor in the model for the Glu/GABA ratio. While no previous study has localized the MRS voxel based on the individual activation levels, nonsignificant group differences in Glu and GABA concentrations within the temporo-parietal and visual cortices have been reported in both children and adults ( Kossowski et al., 2019 ), as well as in the ACC in children ( Horowitz-Kraus et al., 2018 ). Although our MRS sample size was half that of the EEG sample, previous research reporting group differences in Glu concentrations involved an even smaller dyslexic cohort (10 participants with dyslexia and 45 typical readers in Pugh et al., 2014 ). Consistent with earlier studies that identified group differences in Glu and GABA concentrations ( Del Tufo et al., 2018 ; Pugh et al., 2014 ) we reported neurometabolite levels relative to total creatine (tCr), indicating that the absence of corresponding results cannot be ascribed to reference differences. Notably, our analysis of the fMRI localizer task revealed greater activation in the control group as compared to the dyslexic group within the left STS for words than control stimuli (see Figure 1E and the Supplementary Material) in line with previous observations ( Blau et al., 2009 ; Dębska et al., 2021 ; Yan et al., 2021 ).

Irrespective of dyslexia status, we found negative correlations between age and exponent and offset, consistent with previous research ( Cellier et al., 2021 ; McSweeney et al., 2021 ; Schaworonkow and Voytek, 2021 ; Voytek et al., 2015 ) and providing further evidence for maturational changes in the aperiodic components (indicative of increased E/I ratio). At the same time, in line with previous MRS works ( Kossowski et al., 2019 ; Marsman et al., 2013 ), we observed a negative correlation between age and Glu concentrations. This suggests a contrasting pattern to EEG results, indicating a decrease in neuronal excitation with age. We also found a condition-dependent change in offset, with a lower offset observed at rest than during the language task. The offset value represents the uniform shift in power across frequencies ( Donoghue et al., 2020 ), with a higher offset linked to increased neuronal spiking rates ( Manning et al., 2009 ). Change in offset between conditions is consistent with observed increased alpha and beta power during the task, indicating elevated activity in both broadband (offset) and narrowband (alpha and beta oscillations) frequency ranges during the language task.

In regard to relationships between EEG and MRS E/I balance biomarkers, we observed a negative correlation between the offset in the left STS (both at rest and during the task) and Glu levels, after controlling for age and GMV. This correlation was not observed in zero-order correlations (see Supplementary Material). Contrary to our predictions, informed by previous studies linking the exponent to E/I ratio ( Colombo et al., 2019 ; Gao et al., 2017 ; Waschke et al., 2021 ), we found the correlation with Glu levels to involve the offset rather than the exponent. This outcome was unexpected, as none of the referenced studies reported results for the offset. However, given the strong correlation between the exponent and offset observed in our study ( r = .68, p < .001, BF 10 > 10000 and r = .72, p < .001, BF 10 > 10000 at rest and during the task respectively) it is conceivable that similar association might be identified for the offset if it were analyzed.

Nevertheless, previous studies examining relationships between EEG and MRS E/I balance biomarkers ( McKeon et al., 2024 ; van Bueren et al., 2023 ) did not identify a similar negative association between Glu and the offset. Instead, one study noted a positive correlation between the Glu/GABA ratio and the exponent ( van Bueren et al., 2023 ), which was significant in the intraparietal sulcus but not in the middle frontal gyrus. This finding presents counterintuitive evidence, suggesting that an increased E/I balance, as indicated by MRS, is associated with a higher aperiodic exponent, considered indicative of decreased E/I balance. In line with this pattern, another study discovered a positive relationship between the exponent and Glu levels in the dorsolateral prefrontal cortex ( McKeon et al., 2024 ). Furthermore, they observed a positive correlation between the exponent and the Glu/GABA imbalance measure, calculated as the absolute residual value of a linear relationship between Glu and GABA ( McKeon et al., 2024 ), a finding replicated in the current work. This implies that a higher spectral exponent might not be directly linked to MRS-derived Glu or GABA levels, but rather to a greater disproportion (in either direction) between these neurotransmitters. These findings, alongside the contrasting relationships between EEG and MRS biomarkers and age, suggest that these methods may reflect distinct biological mechanisms of E/I balance.

Evidence regarding associations between neurotransmitters levels and oscillatory activity also remains mixed. One study found a positive correlation between gamma peak frequency and GABA concentration in the visual cortex ( Muthukumaraswamy et al., 2009 ), a finding later challenged by a study with a larger sample ( Cousijn et al., 2014 ). Similarly, a different study noted a positive correlation between GABA in the left STS and gamma power ( Balz et al., 2016 ), another study, found non-significant relation between these measures ( Wyss et al., 2017 ). Moreover, in a simultaneous EEG and MRS study, an event-related increase in Glu following visual stimulation was found to correlate with greater gamma power ( Lally et al., 2014 ). We could not investigate such associations, as the algorithm failed to identify a gamma peak above the aperiodic component for the majority of participants. Also, contrary to previous findings showing associations between GABA in the motor and sensorimotor cortices and beta power ( Cheng et al., 2017 ; Gaetz et al., 2011 ) or beta peak frequency ( Baumgarten et al., 2016 ), we observed no correlation between Glu or GABA levels and beta power. However, these studies placed MRS voxels in motor regions which are typically linked to movement-related beta activity ( Baker et al., 1999 ; Rubino et al., 2006 ; Sanes and Donoghue, 1993 ) and did not adjust beta power for aperiodic components, making direct comparisons with our findings limited.

Finally, we examined pathways posited by the neural noise hypothesis of dyslexia, through which increased neural noise may impact reading: phonological awareness, lexical access and generalization, and multisensory integration ( Hancock et al., 2017 ). Phonological awareness was positively correlated with the offset in the left STS at rest, and with beta power in the left STS, both at rest and during the task. Additionally, multisensory integration showed correlations with GABA and the Glu/GABA ratio. Since the Bayes Factor did not provide conclusive evidence supporting either the alternative or null hypothesis, these associations appear rather weak. Nonetheless, given the hypothesis’s prediction of a causal link between these variables, we further examined a mediation model involving beta power, phonological awareness, and reading skills. The results suggested a positive indirect effect of beta power on reading via phonological awareness, whereas both the direct (controlling for phonological awareness and age) and total effects (without controlling for phonological awareness) were not significant. This finding is noteworthy, considering that participants with dyslexia exhibited reduced phonological awareness and reading skills, despite no observed differences in beta power. Given the cross-sectional nature of our study, further longitudinal research is necessary to confirm the causal relation among these variables. The effects of GABA and the Glu/GABA ratio on reading, mediated by multisensory integration, warrant further investigation. Additionally, considering our finding that only males with dyslexia showed deficits in multisensory integration ( Glica et al., 2024 ), sex should be considered as a potential moderating factor in future analyses. We did not test this model here due to the smaller sample size for GABA measurements.

Our findings suggest that the neural noise hypothesis, as proposed by Hancock and colleagues (2017) , does not fully explain the reading difficulties observed in dyslexia. Despite the innovative use of both EEG and MRS biomarkers to assess excitatory-inhibitory (E/I) balance, neither method provided evidence supporting an E/I imbalance in dyslexic individuals. Importantly, our study focused on adolescents and young adults, and the EEG recordings were conducted during rest and a spoken language task. These factors may limit the generalizability of our results. Future research should include younger populations and incorporate a broader array of tasks, such as reading and phonological processing, to provide a more comprehensive evaluation of the E/I balance hypothesis. Additionally, our findings are consistent with another study by Tan et al. (2022) which found no evidence for increased variability (’noise’) in behavioral and fMRI response patterns in dyslexia. Together, these results highlight the need to explore alternative neural mechanisms underlying dyslexia and suggest that cortical hyperexcitability may not be the primary cause of reading difficulties.

In conclusion, while our study challenges the neural noise hypothesis as a sole explanatory framework for dyslexia, it also underscores the complexity of the disorder and the necessity for multifaceted research approaches. By refining our understanding of the neural underpinnings of dyslexia, we can better inform future studies and develop more effective interventions for those affected by this condition.

Materials and methods

Participants.

A total of 120 Polish participants aged between 15.09 and 24.95 years ( M = 19.47, SD = 3.06) took part in the study. This included 60 individuals with a clinical diagnosis of dyslexia performed by the psychological and pedagogical counseling centers (28 females and 32 males) and 60 control participants without a history of reading difficulties (28 females and 32 males). All participants were right-handed, born at term, without any reported neurological/psychiatric diagnosis and treatment (including ADHD), without hearing impairment, with normal or corrected-to-normal vision, and IQ higher than 80 as assessed by the Polish version of the Abbreviated Battery of the Stanford-Binet Intelligence Scale-Fifth Edition (SB5) ( Roid et al., 2017 ).

The study was approved by the institutional review board at the University of Warsaw, Poland (reference number 2N/02/2021). All participants (or their parents in the case of underaged participants) provided written informed consent and received monetary remuneration for taking part in the study.

Reading and Reading-Related Tasks

Participants’ reading skills were assessed by multiple paper-pencil tasks described in detail in our previous work ( Glica et al., 2024 ). Briefly, we evaluated words and pseudowords read in one minute ( Szczerbiński and Pelc-Pękała, 2013 ), rapid automatized naming ( Fecenec et al., 2013 ), and reading comprehension speed. We also assessed phonological awareness by a phoneme deletion task ( Szczerbiński and Pelc-Pękała, 2013 ) and spoonerisms tasks ( Bogdanowicz et al., 2016 ), as well as orthographic awareness (Awramiuk and Krasowicz-Kupis, 2013). Furthermore, we evaluated non-verbal perception speed ( Ciechanowicz and Stańczak, 2006 ) and short-term and working memory by forward and backward conditions from the Digit Span subtest from the WAIS-R ( Wechsler, 1981 ). We also assessed participants’ multisensory audiovisual integration by a redundant target effect task, which results have been reported in our previous work ( Glica et al., 2024 ).

Electroencephalography Acquisition and Procedure

EEG was recorded from 62 scalp and 2 ear electrodes using the Brain Products system (actiCHamp Plus, Brain Products GmbH, Gilching, Germany). Data were recorded in BrainVision Recorder Software (Vers. 1.22.0002, Brain Products GmbH, Gilching, Germany) with a 500 Hz sampling rate. Electrodes were positioned in line with the extended 10-20 system. Electrode Cz served as an online reference, while the Fpz as a ground electrode. All electrodes’ impedances were kept below 10 kΩ. Participants sat in a chair with their heads on a chin-rest in a dark, sound-attenuated, and electrically shielded room while the EEG was recorded during both a 5-minute eyes-open resting state and the spoken language comprehension task. The paradigm was prepared in the Presentation software (Version 20.1, Neurobehavioral Systems, Inc., Berkeley, CA, www.neurobs.com ).

During rest, participants were instructed to relax and fixate their eyes on a white cross presented centrally on a black background. After 5 minutes, the spoken language comprehension task automatically started. The task consisted of 3 to 5 word-long sentences recorded in a speech synthesizer which were presented binaurally through sound-isolating earphones. After hearing a sentence, participants were asked to indicate whether the sentence was true or false by pressing a corresponding button. In total, there were 256 sentences – 128 true (e.g., “Plants need water”) and 128 false (e.g., “Dogs can fly”).

Sentences were presented in a random order in two blocks of 128 trials. At the beginning of each trial, a white fixation cross was presented centrally on a black background for 500 ms, then a blank screen appeared for either 500, 600, 700, or 800 ms (durations set randomly and equiprobably) followed by an auditory sentence presentation. The length of sentences ranged between 1.17 and 2.78 seconds and was balanced between true ( M = 1.82 seconds, SD = 0.29) and false sentences ( M = 1.82 seconds, SD = 0.32; t (254) = -0.21, p = .835; BF 10 = 0.14). After a sentence presentation, a blank screen was displayed for 1000 ms before starting the next trial. To reduce participants’ fatigue, a 1-minute break between two blocks of trials was introduced, and it took approximately 15 minutes to complete the task.

fMRI Acquisition and Procedure

MRI data were acquired using Siemens 3T Trio system with a 32-channel head coil. Structural data were acquired using whole brain 3D T1-weighted image (MP_RAGE, TI = 1100 ms, GRAPPA parallel imaging with acceleration factor PE = 2, voxel resolution = 1mm 3 , dimensions = 256×256×176). Functional data were acquired using whole-brain echo planar imaging sequence (TE = 30ms, TR = 1410 ms, flip angle FA = 90°, FOV = 212 mm, matrix size = 92×92, 60 axial slices 2.3mm thick, 2.3×2.3 mm in-plane resolution, multiband acceleration factor = 3). Due to a technical issue, data from two participants were acquired with a 12-channel coil (see Supplementary Material).

The fMRI task served as a localizer for later MRS voxel placement in language-sensitive left STS. The task was prepared using Presentation software (Version 20.1, Neurobehavioral Systems, Inc., Berkeley, CA, www.neurobs.com ) and consisted of three runs, each lasting 5 minutes and 9 seconds. Two runs involved the presentation of visual stimuli, while the third run of auditory stimuli. In each run, stimuli were presented in 12 blocks, with 14 stimuli per block. In visual runs, there were four blocks from each category: 1) 3 to 4 letters-long words, 2) the same words presented as a false font string (BACS font) ( Vidal et al., 2017 ), and 3) strings of 3 to 4-long consonants. Similarly, in the auditory run, there were four blocks from each category: 1) words recorded in a speech synthesizer, 2) the same words presented backward, and 3) consonant strings recorded in a speech synthesizer. Stimuli within each block were presented for 800 ms with a 400 ms break in between. The duration of each block was 16.8 seconds. Between blocks, a fixation cross was displayed for 8 seconds. Participants performed a 1-back task to maintain focus. The blocks were presented in a pseudorandom order and each block included 2 to 3 repeated stimuli.

MRS Acquisition and Procedure

The GE 7T system with a 32-channel coil was utilized. Structural data were acquired using whole brain 3D T1-weighted image (3D-SPGR BRAVO, TI = 450ms, TE = 2.6ms, TR = 6.6ms, flip angle = 12 deg, bandwidth = ±32.5kHz, ARC acceleration factor PE = 2, voxel resolution = 1mm, dimensions = 256 x 256 x 180). MRS spectra with 320 averages were acquired from the left STS using single-voxel spectroscopy semiLaser sequence ( Deelchand et al., 2021 ) (voxel size = 15 x 15 x 15 mm, TE = 28ms, TR = 4000ms, 4096 data points, water suppressed using VAPOR). Eight averages with unsuppressed water as a reference were collected.

To localize left STS, T1-weighted images from fMRI and MRS sessions were coregistered and fMRI peak coordinates were used as a center of voxel volume for MRS. Voxels were then adjusted to include only the brain tissue. During the acquisition, participants took part in a simple orthographic task.

Statistical Analyses

The continuous EEG signal was preprocessed in the EEGLAB ( Delorme and Makeig, 2004 ). The data were filtered between 0.5 and 45 Hz (Butterworth filter, 4th order) and re-referenced to the average of both ear electrodes. The data recorded during the break between blocks, as well as bad channels, were manually rejected. The number of rejected channels ranged between 0 and 4 ( M = 0.19, SD = 0.63). Next, independent component analysis (ICA) was applied. Components were automatically labeled by ICLabel ( Pion-Tonachini et al., 2019 ), and those classified with 50-100% source probability as eye blinks, muscle activity, heart activity, channel noise, and line noise, or with 0-50% source probability as brain activity, were excluded. Components labeled as “other” were visually inspected, and those identified as eye blinks and muscle activity were also rejected. The number of rejected components ranged between 11 and 46 ( M = 28.43, SD = 7.26). Previously rejected bad channels were interpolated using the nearest neighbor spline ( Perrin et al., 1989 , 1987 ).

The preprocessed data were divided into a 5-minute resting-state signal and a signal recorded during a spoken language comprehension task using MNE ( Gramfort, 2013 ) and custom Python scripts. The signal from the task was cut up based on the event markers indicating the beginning and end of a sentence. Only trials with correct responses given between 0 and 1000 ms after the end of a sentence were included. The signals recorded during every trial were further multiplied by the Tukey window with α = 0.01 in order to normalize signal amplitudes at the beginning and end of every trial. This allowed a smooth concatenation of signals recorded during task trials, resulting in a continuous signal derived only when participants were listening to the sentences.

The continuous signal from the resting state and the language task was epoched into 2-second-long segments. An automatic rejection criterion of +/-200 μV was applied to exclude epochs with excessive amplitudes. The number of epochs retained in the analysis ranged between 140–150 ( M = 149.66, SD = 1.20) in the resting state condition and between 102–226 ( M = 178.24, SD = 28.94) in the spoken language comprehension task.

Power spectral density (PSD) for 0.5-45 Hz in 0.5 Hz increments was calculated for every artifact-free epoch using Welch’s method for 2-second-long data segments windowed with a Hamming window with no overlap. The estimated PSDs were averaged for each participant and each channel separately for the resting state condition and the language task. Aperiodic and periodic (oscillatory) components were parameterized using the FOOOF method ( Donoghue et al., 2020 ). For each PSD, we extracted parameters for the 1-43 Hz frequency range using the following settings: peak_width_limits = [1, 12], max_n_peaks = infinite, peak_threshold = 2.0, mean_peak_height = 0.0, aperiodic_mode = ‘fixed’. Apart from broad-band aperiodic parameters (exponent and offset), we also extracted power, bandwidth, and the center frequency parameters for the theta (4-7 Hz), alpha (7-14 Hz), beta (14-30 Hz) and gamma (30-43 Hz) bands. Since in the majority of participants, the algorithm did not find the peak above the aperiodic component in theta and gamma bands, we calculated the results only for the alpha and beta bands. The results for other periodic parameters than the beta power are reported in Supplementary Material.

Apart from the frequentist statistics, we also performed Bayesian statistics using JASP ( JASP Team, 2023 ). For Bayesian repeated measures ANOVA, we reported the Bayes Factor for the inclusion of a given effect (BF incl ) with the ’across matched model’ option, as suggested by Keysers and colleagues (2020) , calculated as a likelihood ratio of models with a presence of a specific factor to equivalent models differing only in the absence of the specific factor. For Bayesian t -tests and correlations, we reported the BF 10 value, indicating the ratio of the likelihood of an alternative hypothesis to a null hypothesis. We considered BF incl/10 > 3 and BF incl/10 < 1/3 as evidence for alternative and null hypotheses respectively, while 1/3 < BF incl/10 < 3 as the absence of evidence ( Keysers et al., 2020 ).

MRS voxel localization in the native space

The data were analyzed using Statistical Parametric Mapping (SPM12, Wellcome Trust Centre for Neuroimaging, London, UK) run on MATLAB R2020b (The MathWorks Inc., Natick, MA, USA). First, all functional images were realigned to the participant’s mean. Then, T1-weighted images were coregistered to functional images for each subject. Finally, fMRI data were smoothed with a 6mm isotropic Gaussian kernel.

In each subject, the left STS was localized in the native space as a cluster in the middle and posterior left superior temporal sulcus, exhibiting higher activation for visual words versus false font strings and auditory words versus backward words (logical AND conjunction) at p < .01 uncorrected. For 6 participants, the threshold was lowered to p < .05 uncorrected, while for another 6 participants, the contrast from the auditory run was changed to auditory words versus fixation cross due to a lack of activation for other contrasts.

In the Supplementary Material, we also performed the group-level analysis of the fMRI data (Tables S5-S7 and Figure S1).

MRS data were analyzed using fsl-mrs version 2.0.7 ( Clarke et al., 2021 ). Data stored in pfile format were converted into NIfTI-MRS using spec2nii tool. We then used the fsl_mrs_preproc function to automatically perform coil combination, frequency and phase alignment, bad average removal, combination of spectra, eddy current correction, shifting frequency to reference peak and phase correction.

To obtain information about the percentage of WM, GM and CSF in the voxel we used the svs_segmentation with results of fsl_anat as an input. Voxel segmentation was performed on structural images from a 3T scanner, coregistered to 7T structural images in SPM12. Next, quantitative fitting was performed using fsl_mrs function. As a basis set, we utilized a collection of 27 metabolite spectra simulated using FID-A ( Simpson et al., 2017 ) and a script tailored for our experiment. We supplemented this with synthetic macromolecule spectra provided by fsl_mrs . Signals acquired with unsuppressed water served as water reference.

Spectra underwent quantitative assessment and visual inspection and those with linewidth higher than 20Hz, %CRLB higher than 20%, and poor fit to the model were excluded from the analysis (see Table S8 in the Supplementary Material for a detailed checklist). Glu and GABA concentrations were expressed as a ratio to total-creatine (tCr; Creatine + Phosphocreatine).

Data Availability Statement

Behavioral data, raw and preprocessed EEG data, 2 nd level fMRI data, preprocessed MRS data and Python script for the analysis of preprocessed EEG data can be found at OSF: https://osf.io/4e7ps/

Acknowledgements

This study was supported by the National Science Centre grant (2019/35/B/HS6/01763) awarded to Katarzyna Jednoróg.

We gratefully acknowledge valuable discussions with Ralph Noeske from GE Healthcare for his support in setting up the protocol for an ultra-high field MR spectroscopy and sharing the set-up for basis set simulation in FID-A.

• Buitelaar J
• dos Santos FP
• Verschure PFMJ
• McAlonan G.
• Krasowicz-Kupis G
• Albertini G
• Roa Romero Y
• Ittermann B
• Senkowski D
• Baumgarten TJ
• Oeltzschner G
• Hoogenboom N
• Wittsack H-J
• Schnitzler A
• van Atteveldt N
• Bogdanowicz KM
• Bogdanowicz M
• Sajewicz-Radtke U
• Karpińska E
• Łockiewicz M
• Ciechanowicz A
• Napolitani M
• Gosseries O
• Casarotto S
• Brichant J-F
• Massimini M
• Chieregato A
• Harrison PJ
• Dzięgiel-Fivet G
• Łuniewska M
• Grabowska A
• Deelchand DK
• Berrington A
• Seraji-Bozorgzad N
• Del Tufo SN
• Fulbright RK
• Peterson EJ
• Sebastian P
• Jaworowska A
• Yingling CD
• Johnstone J
• Davenport L
• Finkelman T
• Furman-Haran E
• Fraga González G
• van der Molen MJW
• de Geus EJC
• van der Molen MW.
• Roberts TPL
• Giacometti P
• Wasilewska K
• Kossowski B
• Żygierewicz J
• Horowitz-Kraus T
• Ermentrout B
• Wagenmakers E-J
• Bogorodzki P
• Roberts M V.
• Haenschel C
• Lasnick OHM
• MacMaster FP
• Villiermet N
• Manyukhina VO
• Prokofyev AO
• Obukhova TS
• Schneiderman JF
• Altukhov DI
• Stroganova TA
• Orekhova E V
• Marchesotti S
• Donoghue JP
• van den Heuvel MP
• Hilleke E. HP
• Hetherington H
• McSweeney M
• Swerdlow NR
• Muthukumaraswamy SD
• Swettenham JB
• Karalunas SL
• Echallier JF
• Pion-Tonachini L
• Kreutz-Delgado K
• Edenberg HJ
• Chorlian DB
• O’Connor SJ
• Rohrbaugh J
• Schuckit MA
• Hesselbrock V
• Conneally PM
• Tischfield JA
• Begleiter H
• Grigorenko EL
• Seidenberg MS
• Brunswick N
• Hatsopoulos NG
• Salvatore S V.
• Zorumski CF
• Mennerick S
• Schaworonkow N
• Scrivener CL
• Hennessy TJ
• Spironelli C
• Penolazzi B
• Szczerbiński M
• Pelc-Pękała O
• van Bueren NER
• van der Ven SHG
• Cohen Kadosh R.
• Van Hirtum T
• Ghesquière P
• Tempesta ZR
• Achermann R

Article and author information

Katarzyna jednoróg, for correspondence:, version history.

• Sent for peer review : June 11, 2024
• Preprint posted : June 12, 2024
• Reviewed Preprint version 1 : September 5, 2024

© 2024, Glica et al.

This article is distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use and redistribution provided that the original author and source are credited.

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