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An Introduction to ANCOVA (Analysis of Variance)

ANCOVA  stands for “analysis of covariance.”  To understand how an ANCOVA works, it helps to first understand the ANOVA.

An   ANOVA   (analysis of variance) is used to determine whether or not there is a statistically significant difference between the means of three or more independent groups. 

For example, suppose we want to know whether or not studying technique has an impact on exam scores for a class of students. We randomly split the class into three groups. Each group uses a different studying technique for one month to prepare for an exam. At the end of the month, all of the students take the same exam.

To find out if studying technique impacts exam scores, we can conduct a one-way ANOVA, which will tell us if if there is a statistically significant difference between the mean scores of the three groups.

An  ANCOVA  is an extension of an ANOVA in which we’d like to determine if there is a statistically significant difference between three or more independent groups  after accounting for one or more covariates .

A  covariate  is a continuous variable that co-varies with the response variable.

For example, suppose we want to know whether or not studying technique has an impact on exam scores,  but we want to account for the grade that the student already has in the class . We can use their current grade as a covariate and conduct an ANCOVA to determine if there is a statistically significant difference between the mean exam scores of the three groups.

This allows us to test whether or not studying technique has an impact on exam scores after the influence of the covariate has been removed.

Thus, if we find that there is a statistically significant difference in exam scores between the three studying techniques, we can be sure that this difference exists  even after accounting for the students current grade in the class (i.e. if they’re already doing well or not in the class) .

Assumptions of ANCOVA

Before performing an ANCOVA, it’s important to make sure the following assumptions are met:

• The covariate(s) and the factor variable(s) are independent  – The covariate and the factor variable should be independent of each other, since adding a covariate term into the model only makes sense if the covariate and the factor variable act independently on the response variable.
• The covariate(s) are continuous data.  The covariates should be continuous (i.e. either  interval or ratio data ).
• Homogeneity of variances  – The variances among the groups should be roughly equal.
• Independence  – The observations in each group should be independent.
• Normality  – The data should be roughly normally distributed in each group.
• No extreme outliers  – There should be no extreme outliers in any of the groups that could significantly affect the results of the ANCOVA.

ANCOVA: Example

A teacher wants to know if three different studying techniques have an impact on exam scores, but she wants to account for the current grade that the student already has in the class.

She will perform an ANCOVA using the following variables:

• Factor variable: studying technique
• Covariate: current grade
• Response variable: exam score

The following table shows the dataset for the 15 students that were recruited to participate in the study:

After performing an ANCOVA on the dataset, the teacher ends up with the following results:

The p-value for study technique is  0.03155 . Since this value is less than 0.05, we can reject the null hypothesis that each of the studying techniques leads to the same average exam score,  even after accounting for the student’s current grade in the class .

To determine exactly which studying techniques produce different average exam scores, the teacher would need to run post-hoc tests .

Additional Resources

How to Perform an ANCOVA in Excel How to Perform an ANCOVA in R How to Perform an ANCOVA in Python The Differences Between ANOVA, ANCOVA, MANOVA, and MANCOVA

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Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

2 Replies to “An Introduction to ANCOVA (Analysis of Variance)”

Greetings Zach Thank u for this helpful information . Is there any way to contact with u ?

it seems some steps are missing for ancova in the article? should it be first examine whether there is a linear relationship between the covariate and the response variable? if it does, then it should review it the slope resulted by each factor variable is significant different or not which will ended up with different conclusion?

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Getting Started with Analysis of Covariance

The Analysis of Covariance, or ANCOVA, is a regression model that includes both categorical and numeric predictors, often just one of each. It is commonly used to analyze a follow-up numeric response after exposure to various treatments, controlling for a baseline measure of that same response. For example, given two subjects with the same baseline value of the study outcome, one in a treated group and the other in a control group, will the subjects have different follow-up outcomes on average?

To demonstrate, we’ll replicate an example presented in chapter 22 of Kutner, et al. (2005). A company proposes to study the effects of different types of promotion tactics on sales of a product. Three different promotions, or treatments, will be considered:

• free samples of product for customers and regular shelf space
• additional shelf space
• additional display at end of aisles

Fifteen stores are selected and five each are randomized to each treatment group. Prior to implementing the different promotion treatments, existing products sales are recorded (i.e., the baseline measure). Then the new promotions are implemented for a period of time and the products sales are recorded again (i.e., the follow-up measure).

Let’s read the data into R and prepare for analysis.

We see that store 1 in treatment group 1 had baseline sales of 21 cases and follow-up sales, or “post” sales, of 38 cases.

Looking at mean “post” sales, it seems treatments 1 and 2 were better than treatment 3

Let’s visualize the raw “post” data along with the means (in red) using ggplot2 (Wickham, 2016).

We see quite a bit of variability around the means.

Now let’s incorporate the baseline sales and plot a straight trendline to summarize the relationship between baseline and post sales for each treatment group. We change the color scale to use the "Set2" palette from the RColorBrewer package, a colorblind friendly palette.

Notice the variability within each group is much lower compared to the variability in the previous plot. This is because follow-up sales are correlated with baseline sales. Incorporating this information allows us to make more precise estimates of treatment effects. Also notice the variability within groups appears to be similar across the groups. This is a key assumption of ANCOVA.

Let’s proceed to analyze this data using ANCOVA. For this we use the workhorse lm() function. Below we model “post” as a function of the “baseline” measure and the “trt” grouping variable. We save the model as “m” and investigate the treatment effect using the anova() function. That’s right, the anova() function, not ancova() ! Remember, ANCOVA is just a name for a special type of regression model. The base R anova() function allows us to investigate which predictors are explaining variability in our response through a series of partial F tests.

The second line of the output tests the null hypothesis that “trt” has no effect on “post” after controlling for “baseline” . The small p-value provides evidence against the null. It appears that even after taking baseline measure into account, additional variability between “post” sales can be explained by treatment groups. This leads us to believe that at least one of the promotional strategies may be better than the others.

It’s worth noting that the anova() function uses Type I sums of squares, which means the order of the variables in the model matters . If we switch the order of the predictors, post ~ trt + baseline, data = d , we get different results. Observe:

The first line now tests the null hypothesis that “trt” has no effect on “post” without accounting for baseline sales . For this model, the appropriate function to use would be the Anova() function from the car package, which uses Type II sums of squares. Order of the variables does not matter in Type II sums of squares. Each predictor is tested assuming the other is in the model . Notice the result for “trt” matches the result in anova(m) .

The same tests can be run using base R’s drop1() function, which compares the full model to a model without the predictor. Notice we need to specify test = "F" .

To learn more about Type I and Type II sums of squares, we recommend the article Anova – Type I/II/III SS explained .

Before we get too excited about our results, we should run some model diagnostics to verify important assumptions: constant variance of residuals within groups and normality of residuals.

To check constant variance, we can look at residuals by treatment group. Below we extract residuals from our model, add them to our data, and create a dot plot of residuals by treatment group. The variability of the residuals looks similar between the groups, which we like to see.

Calling plot() on our model object and specifying which = 2 produces a QQ plot of residuals. If our residuals are normally distributed, we expect to see a scatter plot that forms a straight diagonal line. (See our article, Understanding QQ Plots , for more information.) It appears there are slight departures from normality but nothing too alarming.

When assessing a QQ plot, it can be helpful to compare it to other QQ plots created with normally distributed data. Below we do just that. The top left QQ plot with red dots is our original residual QQ plot. The other QQ plots are for random draws from a standard normal distribution. Our residual QQ plot doesn’t look that different from the other plots, which makes us feel satisfied with the normality assumption.

Another assumption of our ANCOVA model is that the baseline effect is constant regardless of treatment group. In other words, the slopes of the linear trend lines are parallel . This seems to be the case in the second exploratory plot above. We specified this assumption in our model by making the effects additive . We have two main effects : “baseline” and “trt”. The effect of one has no effect on the other according to our model specification. But this may not be correct. We can formally test this assumption by fitting a new model where “baseline” and “trt” interact and then testing whether the interaction is warranted. This is sometimes called a Test for Parallel Slopes .

Below we specify an interaction using the formula syntax baseline * trt and save the model as “m_int”. As before we use the anova() function to test the null hypothesis that there is no interaction between “baseline” and “trt” when it comes to explaining variability in “post”.

The row labeled “baseline:trt” has a large p-value, which leads us to not reject the null of no interaction effect. It seems parallel linear trend lines are a safe assumption.

Now that we’ve determined “trt” seems to have some effect on “post” after controlling for “baseline”, how can we estimate those effects? Which promotion strategy is better, and how much better is it? Just because a treatment effect is “significant” doesn’t necessarily mean it’s important or meaningful.

To estimate treatment effects, we use our model to calculate expected “post” values for given “trt” and “baseline” values. For example, expected sales for all treatments assuming a baseline value of 25 can be calculated using the predict() function. (We chose a baseline of 25 since that’s the overall mean of baseline values.)

The emmeans package (Lenth, 2023) automates calculations such as this and provides facilities for making pairwise comparisons of means with confidence intervals on the difference. Below we specify we want to estimate expected mean sales for each treatment group and make pairwise comparisons of those means using the emmeans() function. We simply give it our model object and use the syntax pairwise ~ trt .

The expected mean values are in the section called “emmeans”, which is short for estimated marginal means. This is another of way saying we calculated expected values holding one or more predictors fixed. In this case the emmeans packaged held “baseline” fixed at 25, the overall mean baseline value. Notice this section includes confidence intervals on the estimated means. It appears promotional strategy one (trt = 1) leads to sales of about 37 to 41 cases.

The “contrast” section presents the pairwise comparisons. The estimates are the difference in means and the tests are for the null hypothesis that the difference is equal to 0. All differences appear reliably different from 0 based on the small p-values. Notice these are adjusted Tukey p-values which are inflated to account for the fact we’re running three tests instead of one. This helps guard against false positives (i.e., rejecting a null hypothesis in error). The chance of this happening increases when performing multiple comparisons. (See our article, Understanding Multiple Comparisons and Simultaneous Inference , for more information.)

To calculate confidence intervals on the differences in means we can pipe the output of the emmeans() function into confint() .

It looks like promotional strategy “trt1” can be expected to sell anywhere from about 1 to 8 more cases than promotional strategy “trt2”. This difference may not be meaningful, especially if the cost of promotional strategy “trt1” is much higher than promotional strategy “trt2”.

So how did we know emmeans was holding “baseline” at 25? We called the ref_grid() function on our model object.

This shows what values emmeans will use to calculate expected means. If we want to hold “baseline” at a different value, say 20, we can use the at argument. It requires values be passed to it as a list object.

Finally we might wish to visualize our ANCOVA model with an effect display. The ggpredict() function from the ggeffects package (Lüdecke, 2018) along with its plot() method make this easy. Once again we use the “Set2” palette from the RColorBrewer package by setting colors = RColorBrewer::brewer.pal(3, "Set2") . The vertical distance between the slopes reveal that promotions 1 and 2 are likely to generate more sales than promotion 3.

It’s worth noting that Kutner et al. define the ANCOVA model a little differently. In their formulation, they use sum contrasts (or deviation coding) for the categorical variables and center the numeric variable. To fully replicate their example we need to change the contrast of “trt” and center “baseline”. Below we use the contr.sum() function to change the contrast definition for the “trt” variable. We set the function argument to 3 since “trt” has three levels. Next we create a centered version of “baseline” by subtracting the mean from all values. Finally we fit the same model. Notice the test for “trt” effect is identical to our original model (F = 59.483, p < 0.00001).

The estimated means and contrasts are identical as well.

Interested readers may also wish to verify the effect display is no different, other than “baseline” being centered at 0.

So how is this new model different from the first one we fit? The changes can be seen by comparing the model coefficients. Below we use the coef() function to extract the model coefficients and then wrap them in a named list to help us identify which coefficients go with which model.

The coefficients in the first section are from a model using treatment contrasts and an uncentered “baseline”. The (Intercept) is the estimated mean sales of “trt1” when baseline = 0. This is probably not meaningful since it’s doubtful a product would have sales of 0 in the previous observation period. The coefficients for “trt2” and “trt3” are the expected differences from “trt1” sales assuming baseline = 0. Hence the name “treatment contrast”. In this case since “trt” and “baseline” do not interact, these coefficients are interpretable. In fact these are the contrasts estimated by emmeans (with a change in sign since the order of subtraction is reversed). The coefficient for “baseline”, 0.89, is almost 1, suggesting a one-to-one correspondence between “baseline” and “post”. That is, a one-unit increase in “baseline” sales leads to about a one-unit increase in “post” sales, all else held constant.

The coefficients in the second section are from a model using sum contrasts and a centered “baseline”. The (Intercept) is the estimated mean of the “trt” means when “baseline_c” = 0. This comes out to 33.8. Since “baseline_c” = 0 is the mean value of baseline, the (Intercept) is interpretable. The coefficients for “trt1” and “trt2” are the expected differences between the means of those groups and the mean of the “trt” means, assuming baseline is held at the mean level. The “baseline_c” coefficient is the only similarity to the previous model, since centering a variable does not change its estimated coefficient in additive models such as these. This all may be a bit confusing, so let’s show this using predicted means.

Below we use model “m3” to calculate expected means at all treatment levels assuming “baseline_c” is set to 0.

The mean of those means is the intercept in model “m3”.

And subtracting the mean of the means from the model predicted means of trt1 and trt2 give us the “trt” coefficients.

Notice in a sum contrast the final level is never compared to the other levels. For more information on sum contrasts and other types of coding schemes we recommend the article R Library Contrast Coding Systems for categorical variables .

Since a model with sum contrasts is not making simple comparisons between group means, we should technically not use the Tukey adjustment when making multiple comparisons. The Tukey procedure is only appropriate when directly comparing group means. With a sum contrast we are comparing group means to a mean of group means , a subtle distinction. A general procedure for any type of contrasts involving group means is the Scheffe Multiple Comparison procedure. This is the approach presented in Kutner et al. Fortunately this is easy to implement using emmeans. Simply add the argument adjust = "scheffe" . Notice that although the estimates are the same, the confidence intervals are slightly wider.

The choice of contrast and whether or not to center predictors is subjective. Unless you have a good reason to switch, we recommend sticking with treatment contrasts. These are the default contrasts in R, Stata, SAS, and SPSS. Centering predictors makes intercepts interpretable in models without non-linear effects and come sometimes help with convergence issues in complex models. But when it comes to post-hoc analyses such as making pairwise comparisons between group means, it doesn’t matter whether your numeric data is centered or not.

Why not use differences?

Instead of ANCOVA, wouldn’t it be easier to just take the difference between the follow-up measure and baseline measure and analyze the change in sales using a one-way Analysis of Variance (ANOVA)? Let’s try that. Below we derive a new variable, “diff”, by subtracting “baseline” from “post”, and then model “diff” as a function of “trt”.

It seems like “trt” explains a lot of variability in the mean differences.

Let’s estimate treatment effects for the differences and make pairwise comparisons.

The “emmeans” section presents estimated mean differences and the “contrasts” section presents pairwise comparisons of those differences. The results in the “contrasts” section are actually not much different than what we obtained using the ANCOVA model. It turns out that if the slope on the baseline covariate is close to 1, then ANCOVA and ANOVA are basically the same. Recall above that the slope coefficient for “baseline” was 0.89, which is close to 1.

However, using change as a dependent variable can be problematic, especially if the baseline measure is used to exclude subjects from your study. For example, if subjects need to have a baseline measure higher than some threshold to be included in your study, then there’s good a chance that any change from baseline will be due to regression to the mean rather than the experimental condition. In his article, Statistical Errors in the Medical Literature , Frank Harrell lists other reasons why using “change from baseline” as a dependent variable can lead to problems. Long story short, using change from baseline requires a number of additional assumptions to be met that ANCOVA does not require.

Power and sample size considerations

ANCOVA is often used to analyze experimental data, just like the example we presented above. When it comes to designing an experiment, one of the key questions to consider is, “how much data do we need?” Experiments often cost a lot of money and/or require invasive procedures involving humans or animals. It’s in everyone’s best interest to collect just enough data to reliably answer our research question. Estimating sample sizes for ANCOVA models can be a little challenging. Fortunately, the Superpower package in R (Lakens and Caldwell, 2021) provides the power_oneway_ancova() function to help guide us.

Let’s imagine we’re designing an experiment to assess which of three different training programs best improve the length of time someone can hang from a pull-up bar. Before we begin the experiment, we’ll measure everyone’s baseline time of how long they can hang before exhaustion. We’ll then randomize participants to one of the three training programs, have them follow the program, and then measure how long they can hang at the end of the training program. How many subjects should we recruit?

To analyze this using the power_oneway_ancova() function, we need to hypothesize some quantities:

• the hypothesized mean follow-up hang times in the three groups: mu
• the number of covariates (i.e., numeric predictors in our model): n_cov
• the estimated squared correlation between baseline and follow-up measures: r2
• the significance level at which we’ll reject the null hypothesis of no treatment effect: alpha_level
• the Type II error we’re willing to accept: beta_level

Perhaps we think the mean follow-up hang times for the three training programs will be something like 30 seconds, 40 seconds, and 50 seconds. So we set mu = c(30, 40, 50) . Next we will only use one covariate, the baseline measure, so we set n_cov = 1 . We hypothesize that baseline and follow-up hang times will have a mild positive correlation of about 0.3, so we set r2 = 0.3^2 . (“r2”, or R-squared, is correlation squared.) We imagine our ANCOVA model will have a residual standard error of about 15 seconds, so we set sd = 15 . (This is basically how precise we think our model will be; e.g., estimated mean plus/minus 15 seconds.) Our significance level to reject the null hypothesis of no treatment effect will be 0.05, so we enter alpha_level = 0.05 . Finally we want to have 0.9 probability (i.e., power) of correctly rejecting the null assuming the null truly is false, so we set our desired Type II error to 1 - 0.9: beta_level = 0.10 .

The result says we need about 45 subjects, or 15 in each group, if our hypothesized means, R-squared, and residual standard error are correct. How do we know if they’re correct? We don’t. We just have to do our best to estimate realistic and important values, perhaps using previous studies or pilot data. For more details on power and sample size analysis of ANCOVA models see Shieh (2020) and the vignettes that accompany the Superpower package.

Hopefully you now have a better understanding of how to plan, execute, and interpret an ANCOVA model.

• Fox J, Weisberg S (2019). An R Companion to Applied Regression , Third edition. Sage, Thousand Oaks CA. https://socialsciences.mcmaster.ca/jfox/Books/Companion/ .
• Harrell F (2017). Statistical Errors in the Medical Literature. (n.d.). https://www.fharrell.com/post/errmed/ (accessed April 14, 2023).
• Kutner et al (2005). Applied Linear Statistical Models . McGraw-Hill. (Chapter 22)
• Neuwirth E (2022). RColorBrewer: ColorBrewer Palettes . R package version 1.1-3, https://CRAN.R-project.org/package=RColorBrewer .
• Lakens D & Caldwell AR. (2021). Simulation-Based Power Analysis for Factorial Analysis of Variance Designs. Advances in Methods and Practices in Psychological Science , 4(1), 251524592095150. https://doi.org/10.1177/2515245920951503 (version 0.2.0)
• Lenth R (2023). emmeans: Estimated Marginal Means, aka Least-Squares Means . R package version 1.8.5, https://CRAN.R-project.org/package=emmeans .
• Lüdecke D (2018). “ggeffects: Tidy Data Frames of Marginal Effects from Regression Models.” Journal of Open Source Software , 3 (26), 772. doi:10.21105/joss.00772 https://doi.org/10.21105/joss.00772 . (version 1.2.1)
• R Library Contrast Coding Systems for categorical variables. UCLA: Statistical Consulting Group. from https://stats.oarc.ucla.edu/r/library/r-library-contrast-coding-systems-for-categorical-variables/ (accessed April 17, 2023)
• R Core Team (2023). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/ . (version 4.2.3)
• Shieh G. (2020). Power analysis and sample size planning in ANCOVA designs. Psychometrika , 85(1), 101-120
• Wickham H. ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York, 2016. (version 3.4.2)

Clay Ford Statistical Research Consultant University of Virginia Library April 17, 2023

For questions or clarifications regarding this article, contact  [email protected] .

View the entire collection  of UVA Library StatLab articles, or learn how to cite .

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9: ANCOVA Part I

An analysis of covariance (ANCOVA) procedure is used when the statistical model has both quantitative and qualitative predictors, and is based on the concepts of the General Linear Model (GLM). In ANCOVA, we combine the concepts we have learned so far in this course (applicable to categorical factors) with the principles of regression (applicable to continuous predictors, learned in STAT 501).

In this lesson, we will address the classic case of ANCOVA where the ANOVA model is extended to include the linear effect of a continuous variable, known as the covariate. In the next lesson, we will generalize the ANCOVA model to include the quadratic and cubic effects of the covariate as well.

Fun Facts : When SAS first came out they had only PROC ANOVA and PROC REGRESSION. Then people asked, "What about the case when want to do an ANOVA but have another continuous variable that you suspect will account for extraneous variability in the response?" In response, SAS came out with PROC GLM which is the general linear model. With PROC GLM you could place the continuous regression variable in the ANOVA model and it would run. Or, if you were running a regression, you could include a categorical variable in the regression model and it would also run. The GLM can handle both the regression and the categorical variables in the same model. Note, there is no PROC ANCOVA in SAS, but there is PROC MIXED. PROC GLM had problems when it came to random effects and was effectively replaced by PROC MIXED. The same sort of process was seen in Minitab and accounts for the multiple tabs under Stat > ANOVA and Stat > Regression, and eventually, Stat > General Linear Model (which works for random effects as well). So, we now have the capacity to include covariates and correctly work with random effects via SAS PROC MIXED or Minitab Stat > General Linear Model. But, enough history, let us get to this lesson.

Introduction to Analysis of Covariance (ANCOVA)

A ‘classic’ ANOVA tests for differences in mean responses to categorical factor (treatment) levels. When there is heterogeneity in experimental units, sometimes restrictions on the randomization (blocking) can improve the accuracy of significance testing results. In some situations, however, the opportunity to construct blocks may not exist, but there may exist a continuous variable causing the heterogeneity. Such sources of extraneous variability are referred to as ‘covariates’, and historically have also been termed ‘nuisance’ or ‘concomitant’ variables.

Note that an ANCOVA model is formed by including a continuous covariate in an ANOVA model. As the continuous covariate enters the model as a regression variable, an ANCOVA requires a few additional steps that should be combined with the ANOVA procedure.

• Be familiar with the basics of the General Linear Model (GLM) necessary for ANCOVA implementation.
• Develop the ANCOVA procedure by extending the ANOVA methodology to include a continuous predictor.
• Carry out the testing sequences for ANCOVA with equal and unequal slopes.

9.1 - ANCOVA in the GLM Setting: The Covariate as a Regression Variable

The statistical ANCOVA by definition is a general linear model that includes both ANOVA (categorical) predictors and regression (continuous) predictors. The simple linear regression model is:

\(Y_i=\beta_0+\beta_1 X_i+ \epsilon_i\)

Here, \(\beta_0\) and \(\beta_1\) are the intercept and the slope of the line, respectively. The significance of a regression is equivalent to testing  \(H_0 \colon \beta_1=0 \text{ vs. } H_1\colon \beta_1 \neq 0\) using the \(\text{F-statistic}  = \frac{MS(Regr)}{MSE}\) where \(MS(Regr)\) is the mean sum of squares for regression and MSE is the mean squared error. In this case of a simple linear regression, the F -test is equivalent to a t -test. Now, in adding the regression variable to our one-way ANOVA model, we can envision a notational problem. In the balanced one-way ANOVA, we have the grand mean (\(\mu\)), but now we also have the intercept \(\beta_0\). To get around this, we can use

\(X^*=X_{ij}-\bar{X}_{..}\)

and get the following as an expression of our covariance model:

\(Y_{ij}=\mu+\tau_i+\gamma X^* +\epsilon_{ij}\)

Note, in a GLM, the Type III (model fit) sums of squares for the treatment levels are being corrected (or adjusted) for the regression relationship. This has the effect of evaluating the treatment levels ‘on the same playing field’. That is, comparing the means of the treatment levels at the mean value of the covariate. This process effectively removes the variation due to the covariate that may otherwise be attributed to treatment level differences.

9.2 - Steps in ANCOVA

First, we need to confirm that for at least one of the treatment groups there is a significant regression relationship with the covariate. Otherwise, including the covariate in the model won’t improve the estimation of treatment means.

Once this is confirmed, we need to examine whether the regression relationship between the response and the covariate has the same slope for each treatment group. Graphically, this means that the regression line at each factor level has the same slope and therefore the lines are all parallel.

Depending on the outcome of the test for equal slopes, we have two alternative ways to finish up the ANCOVA:

• Fit a common slope model and adjust the treatment SS for the presence of the covariate.
• Evaluate the differences in means for at least three levels of the covariate.

These steps are illustrated in the following two sections and are diagrammed below:

9.3 - Comparison to ANOVA: Salary Example

To illustrate the role the covariate has in the ANCOVA, let’s look at a hypothetical situation wherein investigators are comparing the salaries of male vs. female college graduates. A random sample of 5 individuals for each gender is compiled, and a simple one-way ANOVA is performed:

\(H_0 \colon \mu_{\text{ Males}}=\mu_{\text{ Females}}\)

•   Example

SAS coding for the one-way ANOVA:

Here is the output we get:

To perform a one-way ANOVA test in Minitab, you can first open the data ( ANCOVA Example Minitab Data ) and then select Stat > ANOVA > One Way…

In the pop-up window that appears, select salary as the Response and gender as the Factor.

Click OK , and the output is as follows.

Analysis of Variance

Model summary.

First, we can input the data manually for such a small dataset.

We then apply a simple one-way ANOVA and display the ANOVA table.

9.4 - Equal Slopes Model: Salary Example

Using technology.

Using our Salary example and the data in the table below, we can run through the steps for the ANCOVA.

Step 1: Are all regression slopes = 0?

A simple linear regression can be run for each treatment group, Males and Females. Running these procedures using statistical software we get the following:

Use the following SAS code:

And here is the output that you get:

And here is the output for this run:

In both cases, the simple linear regressions are significant so the slopes are not = 0.

Step 2: Are the slopes equal?

We can test for this using our statistical software.

In SAS we now use proc mixed and include the covariate in the model.

We will also include a ‘treatment × covariate’ interaction term and the significance of this term answers our question. If the slopes differ significantly among treatment levels, the interaction p -value will be < 0.05.

  To obtain the plot in SAS, we can use the following SAS code:

Step 3: Fit an Equal Slopes Model

We can now proceed to fit an Equal Slopes model by removing the interaction term. Again, we will use our statistical software SAS.

We obtain the following results:

Note! In SAS, the model statement automatically creates an intercept. This produces the correct table for testing the effects (using the adjusted sum of squares). However, including the intercept technically over-parameterizes the ANCOVA model resulting in an additional calculation step to obtain the regression equations. To get the intercepts for the covariate directly, we can re-parameterize the model by suppressing the intercept ( noint ) and then specifying that we want the solutions ( solution ) to the model. However, this should only be done to find the equation estimates, not to test effects. 

Here is what the SAS code looks like for this:

Here is the output:

In the first section of the output above a separate intercept is reported for each gender, the ‘Estimate’ value for each gender, and a common slope for both genders, labeled ‘Years’.

Thus, the estimated regression equation for Females is \(\hat{y} = 2.7 + 15.1 \text{(Years)}\), and for males it is \(\hat{y} = 25.1 + 15.1 \text{(Years)}\)

To this point in this analysis, we can see that 'gender' is now significant. By removing the impact of the covariate, we went from

(without covariate consideration)

(adjusting for the covariate)

Using our Salary example and the data in the table below, we can run through the steps for the ANCOVA. On this page, we will go through the steps using Minitab.

A simple linear regression can be run for each treatment group, Males and Females. To perform regression analysis on each gender group in Minitab, we will have to subdivide the salary data manually and separately, saving the male data into the Male Salary Dataset and the female data into the Female Salary dataset .

Running these procedures using statistical software we get the following:

Open the Male dataset in the Minitab project file ( Male Salary Dataset ).

Then, from the menu bar, select Stat > Regression > Regression > Fit Regression Model

In the pop-up window, select salary into Response and years into Predictors as shown below.

Click OK , and Minitab will output the following.

Regression Analysis: Salary versus years

Regression equation.

salary = 24.8 + 15.2 years

Coefficients

Open Minitab dataset Female Salary Dataset . Follow the same procedure as was done for the Male dataset and Minitab will output the following:

salary = 3.00 + 15.00 years

In both cases, the simple linear regressions are significant, so the slopes are not = 0.

We can test for this using our statistical software. In Minitab, we must now use GLM (general linear model) and be sure to include the covariate in the model. We will also include a ‘treatment x covariate’ interaction term and the significance of this term is what answers our question. If the slopes differ significantly among treatment levels, the interaction p -value will be < 0.05.

First, open the dataset in the Minitab project file Salary Dataset . Then, from the menu select Stat > ANOVA > General Linear Model > Fit General Linear Model

In the dialog box, select salary into Responses, gender into Factors, and years into Covariates.

To add the interaction term, first click Model …. Then, use the shift key to highlight gender and years, and click Add . Click OK , then OK again, and Minitab will display the following output.

It is clear the interaction term is not significant. This suggests the slopes are equal. In a plot of the regressions, we can also see that the lines are parallel.

We can now proceed to fit an Equal Slopes model by removing the interaction term. This can be easily accomplished by starting again with STAT > ANOVA > General Linear Model > Fit General Linear Model 

Click OK , then OK again, and Minitab will display the following output.

To generate the mean comparisons select  STAT > ANOVA > General Linear Model > Comparisons... and fill in the dialog box as seen below.

Click OK and Minitab will produce the following output.

Comparison of salary

Tukey pairwise comparisons: gender, grouping information using the tukey method and 95% confidence.

Means that do not share a letter are significantly different.

First, we can input the data manually.

We can apply a simple linear regression for the male subset of the data and display the results using summary .

Next, we apply a simple linear regression for the female subset.

It is clear the regression for both treatments is significant. We continue to test for unequal slopes in the full dataset using an interaction term.

The interaction term is not significant, suggesting the slopes do not differ significantly. We simplify the model to the equal-slope model without an interaction term.

This is the final model since all terms are significant. We can then produce the LS means for the gender levels.

We can also find the regression equation coefficients. Note the female level was used as the reference level by default.

9.5 - Unequal Slopes Model: Salary Example

If the data collected in the example study were instead as follows:

We would see in Step 2 that we do have a significant 'treatment × covariate' interaction. Using SAS we can run the unequal slope model.

We get the following output:

To generate the covariate regression slopes and intercepts, we can do the following. 

This produces the following output:

Here the intercepts are the estimates for effects labeled 'gender' and the slopes can be derived from the estimates of the effects labeled 'years' and 'years*gender'. Thus, the regression equations for this unequal slopes model are:

\(\text{Females}\;\;\; \hat{y} = 3.0 + 15(Years)\)

\(\text{Males}\;\;\; \hat{y} = 15 + 25(Years)\)

The slopes of the regression lines differ significantly and are not parallel.

The code above also outputs the following:

In this case, we see a significant difference at each level of the covariate specified in the lsmeans statement. The magnitude of the difference between males and females differs, giving rise to the significant interaction. In more realistic situations, a significant 'treatment × covariate' interaction often results in significant treatment level differences at certain points along the covariate axis.

When we re-run the program with the new dataset Salary-new Data , we find a significant interaction between gender and years.

To do this, open the Minitab dataset Salary-new Data .

Go to Stat > ANOVA > General Linear model > Fit General Linear Model and follow the same sequence of steps as in the previous section. In Step 2, Minitab will display the following output.

It is clear the interaction term is significant and should not be removed. This suggests the slopes are not equal. Thus, the magnitude of the difference between males and females differs (giving rise to the interaction significance).

If we were to fit regression models for both gender treatments we would see both regressions are significant. We then test for unequal slopes in the full dataset using an interaction term.

The interaction term is significant, suggesting the slopes differ significantly. This is the final model since all terms are significant. We can then produce regression equation coefficients. To match SAS, we utilize dummy coding. Note the female level was used as the reference level by default.

We can also plot the regression lines for both treatments.

Finally, we can find the differences in the treatment LS means for various years (1, 3, and 5).

9.6 - Lesson 9 Summary

This lesson introduced us to ANCOVA methodology, which accommodates both continuous and categorical predictors. The model discussed in this lesson had one categorical factor and a linear effect of one covariate, the continuous predictor. We noted that the fitted linear relationship between the response and the covariate resulted in a straight line for each factor level, and the ANCOVA procedure depended on the condition of equal (or unequal) slopes. We saw one advantage of ANCOVA was the ability to examine the differences among the factor levels after adjusting for the impact of the covariate on the response.

The salary example comparing males and females after accounting for their years after college illustrated how software can be utilized when using the ANCOVA procedure. In the next lesson, the ANCOVA topic will be extended to include in the model up to a cubic polynomial relationship between response vs covariate.

Advanced ANOVA/ANCOVA

This tutorial teaches use of analysis of covariance (ANCOVA) techniques, with practical exercises based on using SPSS .

• 2.1 Example
• 3 Assumptions
• 4 Example write-up
• 5 Descriptives
• 6.1 Teaching method
• 6.2 Positive effect
• 6.3 Vitamin C
• 6.4 Therapy and depression

Overview [ edit | edit source ]

• ANCOVA evaluates whether population means on the DV , adjusted for differences on the covariate(s) (or 'nuisance variables'), differ across the levels of the IVs . Thus, the question being tested is whether the adjusted group means vary significantly from each other.
• ANCOVA is exactly like ANOVA, except the effects of a third variable are statistically “controlled out” (similar to use of hierarchical multiple linear regression).
• Any number of IVs and CVs can be used to create one-way, two-way, and multivariate ANCOVA designs.

Covariates [ edit | edit source ]

• Typically included to remove extraneous influences from the DV , thus decreasing the within-group variance. Particularly with small sample sizes, well-chosen and reliably measured CVs can markedly improve the power of inferential statistical tests.
• Eliminate some systematic variance outside the control of the researcher that can bias the results.
• Account for differences in response due to unique characteristics of the respondents.
• Try to minimise the number of CVs; too many covariates will reduce the statistical efficiency of the analysis - rule of thumb is that the number of CVs < (.10 x sample size) - (number of groups - 1).

Example [ edit | edit source ]

If you are interested in testing the effect of computer experience on the attitude towards use of internet shopping, and you suspect that those with more positive attitudes toward shopping in general are more likely to have positive attitudes towards internet shopping, you may include attitude toward shopping as a covariate so as to remove its influence from the attitude towards internet shopping measure.

Assumptions [ edit | edit source ]

Assumptions to be met are those for ANOVA , plus:

• Covariates must be linearly related to the DV . The stronger the correlation, the more useful the CV will be. If there is no correlation, then the inclusion of the CV will slightly weaken the power of the test by needlessly consuming a degree of freedom.
• Covariates must have a homogeneous effect on the DV across the IV groups (i.e., homogeneity of slopes or equal effects on the DV across the IV groups). If there is a significant interaction between the covariate and the IV, do not use an ANCOVA.
• The covariate should be unrelated to the IV.
• Covariates should not be overly correlated with one another.

The first two criteria can be checked via a scatterplot of the DV and the CV, with the IV as a control variable (to check for equal slopes).

The third criteria depends on experimental design (e.g., if the CV is measured prior to the IV, then it cannot be affected by the IV).

The fourth criteria can be checked via correlations and scatterplots between the CVs.

Recall that the main ANOVA assumptions are that:

• Each of the observations are independent
• The DV (and the CV) must be interval level of measurement
• The underlying populations (of adjusted scores) must be normally distributed
• Each of the underlying populations (of adjusted scores) must have the same variance

The first two assumptions are a function of experimental design. The third and fourth assumptions are more difficult to test because we do not have the "adjusted scores", so we cannot compare the variances of the adjusted scores across the IV groups, but SPSS provides a Levene's test for the homogeneity of variance for the adjusted scores. Fortunately, provided the samples are sufficiently large, the test is robust to violations of the normality assumption.

Example write-up [ edit | edit source ]

A one-way analysis of covariance (ANCOVA) was conducted. The independent variable, vitamin C, involved three levels: placebo, low dose, and high dose. The dependent variable was the number of days with cold symptoms during treatment and the covariate was the number of days with cold symptoms before treatment. The assumptions for ANCOVA were met. In particular, the homogeneity of the regression effect was evident for the covariate, and the covariate was linearly related to the dependent measure.

Descriptives [ edit | edit source ]

In presenting ANCOVA results, provide a table of means for each of the groups. If the same scale is used to measure the DV and the CV, then the unadjusted group means and SD (from Descriptives) can be presented. If a different scale is used for the DV and CV, then provide both the unadjusted mean (and SD ) and the adjusted mean (and SE ). The adjusted mean (controlling for the CV) is provided in the estimated marginal means table.

Exercises [ edit | edit source ]

Teaching method [ edit | edit source ].

In a hypothetical educational psychology experiment, participants were randomly divided into two groups. One group was taught conventionally, and the other were taught using an innovative method. Prior to allocation to the groups, learning motivation was assessed. Improvements in academic achievement were measured as the DV.

• Achievement (achieve) (DV)
• Teaching Method (teach) (IV - 2 levels - "conservative" and "innovative")
• Motivation (motiv) (CV)

Positive effect [ edit | edit source ]

Conduct an ANCOVA to test for differences in positive effect between males and females, adjusting for differences in age.

• Positive effect (DV)
• Gender (IV)

Vitamin C [ edit | edit source ]

Do low and high doses of Vitamin C reduce incidences of days suffered with a cold? Conduct an ANOVA using:

• Group (IV; Vitamin C: placebo, low, high)
• Days (DV; days with a cold)
• Pre-days (CV; previous days with a cold)

Therapy and depression [ edit | edit source ]

What is the effect of three different therapy types on depression, taking into account pre-existing depression levels? Use:

• Group (IV; Counseling and journal therapy; Journal therapy only; Counseling only)
• Depression prior (CV)
• Depression after (DV)

See also [ edit | edit source ]

• ANCOVA (Wikipedia)

• Advanced ANOVA/Tutorials

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In This Article Expand or collapse the "in this article" section Analysis of Covariance (ANCOVA)

Introduction.

• Encyclopedia Articles
• Chapters in General Statistics Textbooks
• Chapters in Experimental Design Textbooks
• Journal Articles
• Reference Works
• Assumptions and Interpretation Issues
• Early Controversy Surrounding ANCOVA in Nonrandomized Studies
• ANCOVA in Observational Studies Post-Rosenbaum and Rubin 1983 and Rosenbaum and Rubin 1984
• Multiple ANCOVA
• Multiple Factor ANCOVA
• Multivariate ANCOVA
• Alternatives Accommodating Certain Assumption Violations and Design Flaws
• ANOVA on the Y/X Ratio as Outcome
• ANOVA on the Residual Y - bwX
• Alternative Analyses for Observational Studies

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Analysis of Covariance (ANCOVA) by Bradley E. Huitema LAST REVIEWED: 15 January 2020 LAST MODIFIED: 15 January 2020 DOI: 10.1093/obo/9780199828340-0256

The analysis of covariance (ANCOVA) is a method for testing the hypothesis of the equality of two or more population means, ideally in the context of a designed experiment. It is similar in purpose to the analysis of variance (ANOVA), but it differs in that an adjustment is made to both the dependent variable means and the error term to provide both descriptive and inferential advantages. The adjustments are made on the basis of information on one or more variables (called covariates) that are measured on each participant before treatments are applied. The advantages of incorporating the covariate information are typically (1) more meaningful outcome means and (2) a smaller error term than is associated with ANOVA. These adjustments result in more interpretable effects, narrower confidence intervals, and an increase in the statistical power of the analysis. Suppose an experiment is carried out to evaluate effects of two treatments. The randomly assigned treatment groups differ somewhat in average age, and age is correlated with the achievement measure used as the dependent variable. Differences between groups on achievement will be somewhat ambiguous to interpret because the groups differ in terms of both age and treatment condition. The analysis of covariance will provide “adjusted means” that estimate the value the outcome means would have been if the groups had been exactly the same with respect to age. At the same time, within-group variation in achievement scores predictable from the covariate (age) will be removed from the error variation to increase the precision of the test for differences between the adjusted means. The application of ANCOVA in some observational studies (rather than randomized experiments) is controversial and has led to a large literature that explores the concerns surrounding the adequacy of the analysis when used in this context. The label “analysis of covariance” is now viewed as anachronistic by some research methodologists and statisticians because this analysis can be both conceptualized and computed as a variant of the general linear model (GLM). But the term remains useful because it immediately conveys to most researchers the notion that a categorical variable (the treatment conditions) and two continuous variables (the covariate and the dependent variable) are involved in a single analysis. Researchers should be warned, however, that ANCOVA is not the same as the “analysis of covariance structures,” a term that was frequently used in the 1970s and 1980s to refer to what is currently known as a “structural equation model.” Additionally, some sources of information regarding ANCOVA subsume several analyses related to (but different from) ANCOVA under this general heading. Examples of these related analyses include the test of the significance of the covariate, the test for homogeneous regression slopes, and the Johnson-Neyman technique.

General Overviews

The following three subsections list sources containing general overviews and introductions to analysis of covariance (ANCOVA). This list begins with the most elementary sources, progresses through those that are of intermediate length and sophistication, and ends with advanced treatments in the form of journal articles and comprehensive reference works. Short elementary presentations designed for readers interested in only the general ideas on ANCOVA are found in encyclopedia articles written for beginning researchers. Several intermediate and advanced level general statistics texts also provide solid introductions to ANCOVA. More extensive coverage is presented in full chapters on the topic found in several textbooks on experimental design. These textbooks provide the main exposure to ANCOVA for most researchers in the behavioral sciences. More technical presentations are available in articles published in methodology and statistics journals.

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One-way ANCOVA in SPSS Statistics

Introduction.

The one-way ANCOVA ( analysis of covariance ) can be thought of as an extension of the one-way ANOVA to incorporate a covariate . Like the one-way ANOVA, the one-way ANCOVA is used to determine whether there are any significant differences between two or more independent (unrelated) groups on a dependent variable. However, whereas the ANOVA looks for differences in the group means, the ANCOVA looks for differences in adjusted means (i.e., adjusted for the covariate). As such, compared to the one-way ANOVA, the one-way ANCOVA has the additional benefit of allowing you to "statistically control" for a third variable (sometimes known as a "confounding variable"), which you believe will affect your results. This third variable that could be confounding your results is called the covariate and you include it in your one-way ANCOVA analysis.

Note: You can have more than one covariate and although covariates are traditionally measured on a continuous scale, they can also be categorical. However, when the covariates are categorical, the analysis is not often called ANCOVA. In addition, the "one-way" part of one-way ANCOVA refers to the number of independent variables. If you have two independent variables rather than one, you could run a two-way ANCOVA .

If you are familiar with the one-way ANCOVA, you can skip to the Assumptions section. On the other hand, if you are not familiar with the one-way ANCOVA, the example below should help provide some clarity.

Researchers wanted to investigate the effect of three different types of exercise intervention on systolic blood pressure. To do this, they recruited 60 participants to their study. They randomly allocated 20 participants to each of three interventions: a "low-intensity exercise intervention", a "moderate-intensity exercise intervention" and a "high-intensity exercise intervention". The exercise in all interventions burned the same number of calories. Each participant had their "systolic blood pressure" measured before the intervention and immediately after the intervention. The researcher wanted to know if the different exercise interventions had different effects on systolic blood pressure. To answer this question, the researchers wanted to determine whether there were any differences in mean systolic blood pressure after the exercise interventions (i.e., whether post-intervention mean systolic blood pressure different between the different interventions). However, the researchers expected that the impact of the three different exercise interventions on mean systolic blood pressure would be affected by the participants' starting systolic blood pressure (i.e., their systolic blood pressure before the interventions). To control the post-intervention systolic blood pressure for the differences in pre-intervention systolic blood pressure, you can run a one-way ANCOVA with pre-intervention systolic blood pressure as the covariate, intervention as the independent variable and post-intervention systolic blood pressure as the dependent variable. If you find a statistically significant difference between interventions, you can follow up a one-way ANCOVA with a post hoc test to determine which specific exercise interventions differed in terms of their effect on systolic blood pressure (e.g., whether the high-intensity exercise intervention had a greater effect on systolic blood pressure than the low-intensity exercise intervention).

This "quick start" guide shows you how to carry out a one-way ANCOVA (with one covariate) using SPSS Statistics, as well as interpret and report the results from this test. Since the one-way ANCOVA is often followed up with a post hoc test, we also show you how to carry out a post hoc test using SPSS Statistics. However, before we introduce you to this procedure, you need to understand the different assumptions that your data must meet in order for a one-way ANCOVA to give you a valid result. We discuss these assumptions next.

SPSS Statistics

Assumptions.

When you choose to analyse your data using a one-way ANCOVA, part of the process involves checking to make sure that the data you want to analyse can actually be analysed using a one-way ANCOVA. You need to do this because it is only appropriate to use a one-way ANCOVA if your data "passes" nine assumptions that are required for a one-way ANCOVA to give you a valid result. In practice, checking for these nine assumptions just adds a little bit more time to your analysis, requiring you to click a few more buttons in SPSS Statistics when performing your analysis, as well as think a little bit more about your data, but it is not a difficult task.

Before we introduce you to these nine assumptions, do not be surprised if, when analysing your own data using SPSS Statistics, one or more of these assumptions is violated (i.e., is not met). This is not uncommon when working with real-world data rather than textbook examples, which often only show you how to carry out a one-way ANCOVA when everything goes well! However, don’t worry. Even when your data fails certain assumptions, there is often a solution to overcome this. First, let’s take a look at these nine assumptions:

• Assumption #1: Your dependent variable and covariate variable(s) should be measured on a continuous scale (i.e., they are measured at the interval or ratio level). Examples of variables that meet this criterion include revision time (measured in hours), intelligence (measured using IQ score), exam performance (measured from 0 to 100), weight (measured in kg), and so forth. You can learn more about continuous variables in our article: Types of Variable . As stated earlier, you can have categorical covariates (e.g., a categorical variables such as "gender", which has two categories: "males" and "females"), but the analysis is not usually referred to as an ANCOVA in this situation.
• Assumption #2: Your independent variable should consist of two or more categorical , independent groups . Example independent variables that meet this criterion include gender (e.g., two groups: male and female), ethnicity (e.g., three groups: Caucasian, African American and Hispanic), physical activity level (e.g., four groups: sedentary, low, moderate and high), profession (e.g., five groups: surgeon, doctor, nurse, dentist, therapist), and so forth.
• Assumption #3: You should have independence of observations , which means that there is no relationship between the observations in each group or between the groups themselves. For example, there must be different participants in each group with no participant being in more than one group. This is more of a study design issue than something you can test for, but it is an important assumption of a one-way ANCOVA. If your study fails this assumption, you will need to use another statistical test instead of a one-way ANCOVA (e.g., a repeated measures design). If you are unsure whether your study meets this assumption, you can use our Statistical Test Selector , which is part of our enhanced guides.
• Assumption #4: There should be no significant outliers . Outliers are simply data points within your data that do not follow the usual pattern (e.g., in a study of 100 students' IQ scores, where the mean score was 108 with only a small variation between students, one student had a score of 156, which is very unusual, and may even put her in the top 1% of IQ scores globally). The problem with outliers is that they can have a negative effect on the one-way ANCOVA, reducing the validity of your results. Fortunately, when using SPSS Statistics to run a one-way ANCOVA on your data, you can easily detect possible outliers. In our enhanced one-way ANCOVA guide, we: (a) show you how to detect outliers using SPSS Statistics; and (b) discuss some of the options you have in order to deal with outliers. You can learn more about our enhanced content on our Features: Overview page.
• Assumption #5: Your residuals should be approximately normally distributed for each category of the independent variable . We talk about the ANCOVA only requiring approximately normal residuals because it is quite "robust" to violations of normality, meaning that the assumption can be violated to a degree and still provide valid results. You can test for normality using two Shapiro-Wilk tests of normality: one to test the within-group residuals and one to test the overall model fit. Both of these are easily tested for using SPSS Statistics. In addition to showing you how to carry out these tests in our enhanced one-way ANCOVA guide, we also explain what you can do if your data fails this assumption (i.e., if it fails it by more than a little bit).
• Assumption #6: There needs to be homogeneity of variances . You can test this assumption in SPSS Statistics using Levene's test for homogeneity of variances. In our enhanced one-way ANCOVA guide, we (a) show you how to perform Levene’s test for homogeneity of variances in SPSS Statistics, (b) explain some of the things you will need to consider when interpreting your data, and (c) present possible ways to continue with your analysis if your data fails to meet this assumption.
• Assumption #7: The covariate should be linearly related to the dependent variable at each level of the independent variable . You can test this assumption in SPSS Statistics by plotting a grouped scatterplot of the covariate, post-test scores of the dependent variable and independent variable. In our enhanced one-way ANCOVA guide, we show you how to (a) produce this grouped scatterplot in SPSS Statistics, (b) interpret the grouped scatterplot, and (c) present possible ways to continue with your analysis if your data fails to meet this assumption.
• Assumption #8: There needs to be homoscedasticity . You can test this assumption in SPSS Statistics by plotting a scatterplot of the standardized residuals against the predicted values. In our enhanced one-way ANCOVA guide, we (a) show you how to produce a scatterplot in SPSS Statistics to test for homoscedasticity, (b) explain some of the things you will need to consider when interpreting your data, and (c) present possible ways to continue with your analysis if your data fails to meet this assumption.
• Assumption #9: There needs to be homogeneity of regression slopes , which means that there is no interaction between the covariate and the independent variable. By default, SPSS Statistics does not include an interaction term between a covariate and an independent in its GLM procedure so that you can test this. Therefore, in our enhanced one-way ANCOVA guide, we (a) show you how to test for homogeneity of regression slopes separately from the main one-way ANCOVA procedure using SPSS Statistics, (b) interpret the output SPSS Statistics produces, and (c) explain the implications of meeting or violating this assumption.

You can check assumptions #4, #5, #6, #7, #8 and #9 using SPSS Statistics. Before doing this, you should make sure that your data meets assumptions #1, #2 and #3, although you don't need SPSS Statistics to do this. Remember that if you do not run the statistical tests on these assumptions correctly, the results you get when running a one-way ANCOVA might not be valid. This is why we dedicate a number of sections of our enhanced one-way ANCOVA guide to help you get this right. You can find out about our enhanced content on our Features: Overview page, or more specifically, learn how we help with testing assumptions on our Features: Assumptions page.

In the section, Test Procedure in SPSS Statistics , we illustrate the SPSS Statistics procedure to perform a one-way ANCOVA, assuming that no assumptions have been violated. First, we set out the example we use to explain the one-way ANCOVA procedure in SPSS Statistics.

A researcher was interested in determining whether a six-week low- or high-intensity exercise-training programme was best at reducing blood cholesterol concentrations in middle-aged men. Both exercise programmes were designed so that the same number of calories was expended in the low- and high-intensity groups. As such, the duration of exercise differed between groups. The researcher expected that any reduction in cholesterol concentration elicited by the interventions would also depend on the participant's initial cholesterol concentration. As such, the researcher wanted to use pre-intervention cholesterol concentration as a covariate when comparing the post-intervention cholesterol concentrations between the interventions and a control group. Therefore, the researcher ran a one-way ANCOVA with: (a) post-intervention cholesterol concentration ( post ) as the dependent variable; (b) the control and two intervention groups as levels of the independent variable, group ; and (c) the pre-intervention cholesterol concentrations as the covariate, pre .

Setup in SPSS Statistics

In SPSS Statistics, we entered three variables: (1) the dependent variable, post , which is the post-intervention cholesterol concentration; (2) the independent variable, group , which has three categories: "control", "Int_1" (representing the low-intensity exercise intervention), and "Int_2" (representing the high-intensity exercise intervention); and (3) pre , which represents the pre-intervention cholesterol concentrations. In our enhanced one-way ANCOVA guide, we show you how to correctly enter data in SPSS Statistics to run a one-way ANCOVA. You can learn about our enhanced data setup content on our Features: Data Setup page. Alternately, see our generic, "quick start" guide: Entering Data in SPSS Statistics .

Test Procedure in SPSS Statistics

In the General Linear Model > Univariate... procedure below, we show you how to analyse your data using a one-way ANCOVA in SPSS Statistics when the nine assumptions in the Assumptions section have not been violated. At the end of this procedure, we show you how to interpret the results from this test. If you are looking for help to make sure your data meets assumptions #4, #5, #6, #7, #8 and #9, which are required when using a one-way ANCOVA and can be tested using SPSS Statistics, you can learn more about our enhanced content on our Features: Overview page.

Since the steps you need to follow differ based on your version of SPSS Statistics, we set out the General Linear Model > Univariate... procedure based on whether you have version 25, 26, 27 or 28 (or the subscription version ) of SPSS Statistics or version 24 or earlier versions of SPSS Statistics. The latest versions of SPSS Statistics are version 28 and the subscription version . If you are unsure which version of SPSS Statistics you are using, see our guide: Identifying your version of SPSS Statistics .

SPSS Statistics versions 25, 26, 27 and 28 (and the subscription version of SPSS Statistics)

Note: In version 27 and the subscription version , SPSS Statistics introduced a new look to their interface called " SPSS Light ", replacing the previous look for versions 26 and earlier versions , which was called " SPSS Standard ". Therefore, if you have SPSS Statistics version 27 or 28 (or the subscription version of SPSS Statistics), the images that follow will be light grey rather than blue. However, the procedure is identical .

Published with written permission from SPSS Statistics, IBM Corporation.

Now that you have run the General Linear Model > Univariate... procedure to carry out a one-way ANCOVA, go to the Interpreting Results section on the next page. You can ignore the section below, which shows you how to carry out a one-way ANCOVA if you have SPSS Statistics version 24 or an earlier version of SPSS Statistics.

SPSS Statistics version 24 and earlier versions of SPSS Statistics

Go to the next page for the SPSS Statistics output and an explanation of the output.

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• Ann Card Anaesth
• v.22(4); Oct-Dec 2019

Application of Student's t -test, Analysis of Variance, and Covariance

Prabhaker mishra.

Department of Biostatistics and Health Informatics, Sanjay Gandhi Post Graduate Institute of Medical Sciences, Lucknow, Uttar Pradesh, India

Uttam Singh

Chandra m pandey, priyadarshni mishra.

1 Department of Ophthalmology, Sanjay Gandhi Post Graduate Institute of Medical Sciences, Lucknow, Uttar Pradesh, India

Gaurav Pandey

2 Department of Gastroenterology, Sanjay Gandhi Post Graduate Institute of Medical Sciences, Lucknow, Uttar Pradesh, India

Student's t test ( t test), analysis of variance (ANOVA), and analysis of covariance (ANCOVA) are statistical methods used in the testing of hypothesis for comparison of means between the groups. The Student's t test is used to compare the means between two groups, whereas ANOVA is used to compare the means among three or more groups. In ANOVA, first gets a common P value. A significant P value of the ANOVA test indicates for at least one pair, between which the mean difference was statistically significant. To identify that significant pair(s), we use multiple comparisons. In ANOVA, when using one categorical independent variable, it is called one-way ANOVA, whereas for two categorical independent variables, it is called two-way ANOVA. When using at least one covariate to adjust with dependent variable, ANOVA becomes ANCOVA. When the size of the sample is small, mean is very much affected by the outliers, so it is necessary to keep sufficient sample size while using these methods.

Introduction

Student's t test ( t test), analysis of variance (ANOVA), and analysis of covariance (ANCOVA) are statistical methods used in the testing of hypothesis for comparison of means between the groups. For these methods, testing variable (dependent variable) should be in continuous scale and approximate normally distributed. Mean is the representative measure for normally distributed continuous variable and statistical methods used to compare between the means are called parametric methods. For non-normal continuous variable, median is representative measure, and in this situation, comparison between the groups is performed using non-parametric methods. Most parametric test has an alternative nonparametric test.[ 1 , 2 , 3 ]

There are many statistical tests within Student's t test ( t test), ANOVA and ANCOVA, and each test has its own assumptions. Although not every method is popular, some of them can be managed from other available methods. The aim of the present article is to discuss the assumptions, application, and interpretation of the some popular T, ANOVA, and ANCOVA methods i.e., one sample t test, independent samples t test, paired samples t test, one-way ANOVA, two-ways ANOVA, one-way repeated measures ANOVA, two-ways repeated measures ANOVA, one-way ANCOVA, and One-way repeated measures ANCOVA. To understand the above statistical methods, an example [ Table 1 ] with a data set of 20 patients whose age groups, gender, body mass index (BMI), and diastolic blood pressure (DBP) measured at baseline (B/L), 30 min and 60 min are given below. Further, examples related to the above statistical methods are discussed from the given data.

Data of the 20 patients

Age groups: 1 (<30 years), 2 (30-50 years), 3 (>50 years). Gender: M=Male, F=Female, BMI=Body mass index, DBP=Diastolic blood pressure, B/L=Baseline, min=Minute

T test, ANOVA, and ANCOVA

Basic concepts.

The Student's t test (also called T test) is used to compare the means between two groups and there is no need of multiple comparisons as unique P value is observed, whereas ANOVA is used to compare the means among three or more groups.[ 4 , 5 ] In ANOVA, the first gets a common P value. A significant P value of ANOVA test indicates for at least one pair, between which the mean difference was statistically significant.[ 6 ] To identify that significant pair(s), post-hoc test (multiple comparisons) is used. In ANOVA test, when at least one covariate (continuous variable) is adjusted to remove the confounding effect from the result called ANCOVA. ANOVA test (F test) is called “Analysis of Variance” rather than “Analysis of Means” because inferences about means are made by analyzing variance.[ 7 , 8 , 9 ]

Steps in hypothesis testing

Hypothesis building.

Like other tests, there are two kinds of hypotheses; null hypothesis and alternative hypothesis. The alternative hypothesis assumes that there is a statistically significant difference exists between the means, whereas the null hypothesis assumes that there is no statistically significant difference exists between the means.

Computation of test statistics

In these test, first step is to calculate test statistics (called t value in student's t test and F value in ANOVA test) also called calculated value. It is calculated after putting inputs (from the samples) in statistical test formula. In student's t test, calculated t value is ratio of mean difference and standard error, whereas in the ANOVA test, calculated F value is ratio of the variability between groups with the variability of the observations within the groups.[ 1 , 4 ]

Tabulated value

At degree of freedom of the given observations and desired level of the confidence (usually at two-sided test, which is more powerful than one-sided test), corresponding tabulated value of the T test or F test is selected (from the statistical table).[ 1 , 4 ]

Comparison of calculated value with tabulated value and null hypothesis

If the calculated value is greater than the tabulated value, then reject the null hypothesis where null hypothesis states that means are statistically same between the groups.[ 1 , 4 ] As the sample size increases corresponding degree of freedom also increases. For a given level of confidence, higher degree of freedom has lower tabulated value. That's the reason, when the sample size increases, its significance level also improves (i.e., P value is decreasing).

It is one of the most popular statistical techniques used to test whether mean difference between two groups is statistically significant. Null hypothesis stated that both means are statistically equal, whereas alternative hypothesis stated that both means are not statistically equal i.e., they are statistically different to each other.[ 1 , 3 , 7 ] T test are three types i.e., one sample t test, independent samples t test, and paired samples t test.

One-sample t test

The one sample t test is a statistical procedure used to determine whether mean value of a sample is statistically same or different with mean value of its parent population from which sample was drawn. To apply this test, mean, standard deviation (SD), size of the sample (Test variable), and population mean or hypothetical mean value (Test value) are used. Sample should be continuous variable and normally distributed.[ 1 , 9 , 10 , 11 ] One-sample t test is used when sample size is <30. In case sample size is ≥30 used to prefer one sample z test over one sample t test although for one sample z test, population SD must be known. If population SD is not known, one sample t test can be used at any sample size. In one sample Z test, tabulated value is z value (instead of t value in one sample t test). To apply this test through popular statistical software i.e., statistical package for social sciences (SPSS), option can be found in the following menu [Analyze – compare means – one-sample t test].

Example : From Table 1 , BMI (mean ± SD) was given 24.45 ± 2.19, whereas population mean was assumed to be 25.5. One sample t test indicated that mean difference between sample mean and population mean was statistically significantly different to each other ( P = 0.045).

Independent samples t test

The independent t test, also called unpaired t test, is an inferential statistical test that determines whether there is a statistically significant difference between the means in two unrelated (independent) groups?

To apply this test, a continuous normally distributed variable (Test variable) and a categorical variable with two categories (Grouping variable) are used. Further mean, SD, and number of observations of the group 1 and group 2 would be used to compute significance level. In this procedure, first significance level of Levene's test is computed and when it is insignificant ( P > 0.05), equal variances otherwise ( P < 0.05), unequal variances are assumed between the groups and according P value is selected for independent samples t test.[ 1 , 10 , 11 , 12 ] In SPSS [Analyze – compare means – independent samples t test].

Example : From Table 1 , mean BMI of the male ( n = 10) and female ( n = 10) were 24.80 ± 2.20 and 24.10 ± 2.23, respectively. Levene's test ( p = 0.832) indicated that variances between the groups were statistically equal. At equal variances assumed, independent samples t test ( p = 0.489) indicated that mean BMI of the male and female was statistically equal.

Paired samples t test

The paired samples t test, sometimes called the dependent samples t -test, is used to determine whether the change in means between two paired observations is statistically significant? In this test, same subjects are measured at two time points or observed by two different methods.[ 4 ] To apply this test, paired variables (pre-post observations of same subjects) are used where paired variables should be continuous and normally distributed. Further mean and SD of the paired differences and sample size (i.e., no. of pairs) would be used to calculate significance level.[ 1 , 11 , 13 ] In SPSS [Analyze – compare means – paired samples t test].

Example : From Table 1 , DBP of the 20 patients (mean ± SD); at baseline, 30 min and paired differences (difference between baselines and 30 min) were 79.55 ± 4.87, 83.90 ± 5.58, and 4.35 ± 4.16. Paired samples t test indicated that mean difference of paired observations of DBP between baseline and 30 min was statistically significant ( P < 0.001).

ANOVA test (F test)

A statistical technique used to compare the means between three or more groups is known as ANOVA or F test. It is important that ANOVA is an omnibus test statistic. Its significant P value indicates that there is at least one pair in which the mean difference is statistically significant. To determine the specific pair's, post hoc tests (multiple comparisons) are used. There are various ANOVAs test, and their objectives are varying from one test to another. There are two main types of ANOVA i.e., one-way ANOVA and one-way repeated measures ANOVA. First is used for independent observations and later for dependent observations. When used one categorical independent variable called one-way ANOVA, whereas for two categorical independent variables called two-way ANOVA. When used at least one covariate to adjust with dependent variable, ANOVA becomes ANCOVA.[ 1 , 11 , 14 ]

Post-hoc test (multiple comparisons): Post hoc tests (pair-wise multiple comparisons) used to determine the significant pair(s) after ANOVA was found significant. Before applying post-hoc test (in between subjects factors), first need to test the homogeneity of the variances among the groups (Levene's test). If variances are homogeneous ( P ≥ 0.05), select any multiple comparison methods from least significant difference (LSD), Bonferroni, Tukey's, etc.[ 15 , 16 ] If variances are not homogeneous ( P < 0.05), used to select any multiple comparison methods from Games-Howell, Tamhane's T2, etc.[ 15 , 16 ] Bonferroni is a good method for equal variances, whereas Tamhane's T2 for unequal variances as both calculate significance level by controlling error rate. Similarly, for repeated measures ANOVA (RMA) (in within subjects factors), select any method from LSD, Boneferroni, Sidak although Bonferroni might be a better choice. The significance level of each of the multiple comparison method is varying from other methods as each used for a particular situation.

One-way ANOVA

The One-way ANOVA is extension of independent samples t test (In independent samples t test used to compare the means between two independent groups, whereas in one-way ANOVA, means are compared among three or more independent groups). A significant P value of this test refers to multiple comparisons test to identify the significant pair(s).[ 17 ] In this test, one continuous dependent variable and one categorical independent variable are used, where categorical variable has at least three categories. In SPSS [Analyze–compare means–one-way ANOVA].

Example : From Table 1 , 20 patient's DBP (at 30 min) are given. One-way ANOVA test was used to compare the mean DBP in three age groups (independent variable), which was found statistically significant ( p = 0.002). Levene test for homogeneity was insignificant ( p = 0.231), as a result Bonferroni test was used for multiple comparisons, which showed that DBP was significantly different between two pairs i.e., age group of <30 to 30–50 and <30 to >50 ( P < 0.05) but insignificant between one pair i.e., 30–50 to >50 ( P > 0.05).

Two-way ANOVA

The two-way ANOVA is extension of one-way ANOVA [In one-way ANOVA, only one independent variable, whereas in two-way ANOVA, two independent variables are used]. The primary purpose of a two-way ANOVA is to understand whether there is any interrelationship between two independent variables on a dependent variable.[ 18 ] In this test, a continuous dependent variable (approximately normally distributed) and two categorical independent variables are used. In SPSS [Analyze –General Linear Model –Univariate].

Example : From Table 1 , 20 patient's DBP (at 30 min) are given. Two-way ANOVA test was used to compare the mean DBP between age groups (independent variable_1) and gender (independent variable_2), which indicated that there was no significant interaction of DBP with age groups and gender (tests of Between-Subjects effects in age groups*gender; P = 0.626) with effect size (Partial Eta Squared) of 0.065. The result also showed that there was significant difference in estimated marginal means (adjusted mean) of DBP between age groups ( P = 0.005) but insignificant in gender ( P = 0.662), where sex and age groups was adjusted.

One-way repeated measures ANOVA

Repeated Measures ANOVA (RMA) is the extension of the paired t test. RMA is also referred to as within-subjects ANOVA or ANOVA for paired samples. Repeated measures design is a research design that involves multiple measures of the same variable taken on the same or matched subjects either under different conditions or more than two time periods. (In paired samples t test, compared the means between two dependent groups, whereas in RMA, compared the means between three or more dependent groups). Before calculating the significance level, Mauchly's test is used to assess the homogeneity of the variance (also called sphericity) within all possible pairs. When P value of Mauchly's test is insignificant ( P ≥ 0.05), equal variances are assumed and P value for RMA would be taken from sphericity assumed test (Tests of Within-Subjects effects). In case variances are not homogeneous (Mauchly's test: P < 0.05), epsilon (ε) value (which shows the departure of the sphericity, 1 shows perfect sphericity) decides the statistical method to calculate P value for RMA. When ε≥0.75 Huynh-Feldt while for ε< 0.75, Greenhouse-Geisser method (univariate method) or Wilks' lambda (multivariate method) is used to calculate P value for the RMA.[ 19 ] When the RMA is significant, pair-wise comparison contains multiple paired t tests with a Bonferroni correction is used.[ 20 ] In SPSS [Analyze –General Linear Model – Repeated Measures ANOVA].

Example : From Table 1 , 20 patient's DBP were at baseline (79.55 ± 4.87), at 30 min (83.90 ± 5.58), and at 60 min (79.25 ± 5.68). The Mauchly's test of sphericity indicated that variances were equal ( P = 0.099) between the pairs. RMA tests (i.e., Within-Subjects effects) was assessed using sphericity assumed test ( P value = 0.001), which indicated that change in DBP over the time was statistically significant. Bonferroni multiple comparisons indicated that mean difference was statistically significant between DBP_B/l to DBP_30 min and DBP_30 min to DBP_60 min ( P < 0.05) but insignificant between DBP_B/l to DBP_60 min ( P > 0.05).

Two-way repeated measures ANOVA

Two-way Repeated Measures ANOVA is combination of between-subject and within-subject factors. A two-way RMA (also known as a two-factor RMA or a two-way “Mixed ANOVA”) is extension of one-way RMA [In one-way RMA, use one dependent variable under repeated observations (normally distributed continuous variable) and one categorical independent variable (i.e., time points), whereas in two-way RMA; one additional categorical independent variable is used]. The primary purpose of two-way RMA is to understand if there is an interaction between these two categorical independent variables on the dependent variable (continuous variable). The distribution of the dependent variable in each combination of the related groups should be approximately normally distributed.[ 21 ] In SPSS [Analyze–General Linear Model – Repeated Measures], where second independent variable will be included as between subjects factor.

Example : From Table 1 , 20 patient's DBP were at baseline (79.55 ± 4.87), at 30 min (83.90 ± 5.58), and at 60 min (79.25 ± 5.68). The Mauchly's test of sphericity ( P = 0.138) indicated that variances were equal between the pairs. Two-way RMA tests for interaction (i.e., Within-Subjects effects) were assessed using sphericity assumed test (DBP*gender: P value = 0.214), which indicated that there was no interaction of gender with time and associated change in DBP over the time was statistically insignificant.

One-way ANCOVA

One-way ANCOVA is extension of one-way ANOVA [In one-way ANOVA, do not adjust the covariate, whereas in the one-way ANCOVA; adjust at least one covariate]. Thus, the one-way ANCOVA tests find out whether the independent variable still influences the dependent variable after the influence of the covariate(s) has been removed (i.e., adjusted). In this test, one continuous dependent variable, one categorical independent variable, and at least one continuous covariate for removing its effect/adjustment are used.[ 8 , 22 ] In SPSS [Analyze - General Linear Model – Univariate].

Example : From Table 1 , 20 patient's DBP at 30 min are given. One-way ANCOVA test was used to compare the mean DBP in three age groups (independent variable) after adjusting the effect of baseline DBP, which was found to be statistically significant ( P = 0.021). As Levene test for homogeneity was insignificant ( P = 0.601), resultant Bonferroni test was used for multiple comparisons, which showed that DBP was significantly different between one pair i.e., age group of <30 to >50 ( P = 0.031) and insignificant between rest two pairs i.e., <30 to 30–50 and 30–50 to >50 ( P > 0.05).

One-way repeated measures ANOCOVA

One-way repeated measures ANCOVA is the extension of the One-way RMA. [In one-way RMA, we do not adjust the covariate, whereas in the one-way repeated measures ANCOVA, we adjust at least one covariate]. Thus, the One-way repeated Measures ANCOVA is used to test whether means are still statistically equal or different after adjusting the effect of the covariate(s).[ 23 , 24 ] In SPSS [Analyze –General Linear Model – Repeated Measures ANOVA].

Example : From Table 1 , 20 patient's DBP were at baseline (79.55 ± 4.87), at 30 min (83.90 ± 5.58), and at 60 min (79.25 ± 5.68). The Mauchly's test of sphericity indicated that variances were equal ( P = 0.093) between the pairs. RMA tests (i.e., Within-Subjects effects) were assessed using sphericity assumed test (DBP*BMI: P value = 0.011), which indicated that change in DBP over the time was statistically significant after adjusting BMI. Bonferroni multiple comparisons indicated that mean difference was statistically significant between DBP_B/l to DBP_30 min and DBP_30 min to DBP_60 min but insignificant between DBP_B/l to DBP_60 min after adjusting BMI.

Conclusions

Student's t test, ANOVA, and ANCOVA are the statistical methods frequently used to analyze the data. Two common things among these methods are dependent variable must be in continuous scale and normally distributed, and comparisons are made between the means. All above methods are parametric method.[ 2 ] When the size of the sample is small, mean is very much affected by the outliers, so it is necessary to keep sufficient sample size while using these methods.

Financial support and sponsorship

Conflicts of interest.

There are no conflicts of interest.

Acknowledgments

Authors would like to express their deep and sincere gratitude to Dr. Prabhat Tiwari, Professor, Department of Anaesthesiology, Sanjay Gandhi Postgraduate Institute of Medical Sciences, Lucknow, for his encouragement to write this article. His critical reviews and suggestions were very useful for improvement in the article.

• Open access
• Published: 26 May 2024

The sense of coherence scale: psychometric properties in a representative sample of the Czech adult population

• Martin Tušl 1 ,
• Ivana Šípová 2 ,
• Martin Máčel 2 ,
• Kristýna Cetkovská 2 &
• Georg F. Bauer 1  

BMC Psychology volume  12 , Article number:  293 ( 2024 ) Cite this article

Metrics details

Sense of coherence (SOC) is a personal resource that reflects the extent to which one perceives the world as comprehensible, manageable, and meaningful. Decades of empirical research consistently show that SOC is an important protective resource for health and well-being. Despite the extensive use of the 13-item measure of SOC, there remains uncertainty regarding its factorial structure. Additionally, a valid and reliable Czech version of the scale is lacking. Therefore, the present study aims to examine the psychometric properties of the SOC-13 scale in a representative sample of Czech adults.

An online survey was completed by 498 Czech adults (18–86 years old) between November 2021 and December 2021. We used confirmatory factor analysis to examine the factorial structure of the scale. Further, we examined the variations in SOC based on age and gender, and we tested the criterion validity of the scale using the short form of the Mental Health Continuum (MHC) scale and the Generalized Anxiety Disorder (GAD) scale as mental health outcomes.

SOC-13 showed an acceptable one- and three-factor fit only with specified residual covariance between items 2 and 3. We tested alternative short versions by systematically removing poorly performing items. The fit significantly improved for all shorter versions with SOC-9 having the best psychometric properties with a clear one-factorialstructure. We found that SOC increases with age and males score higher than females. SOC showed a moderately strong positive correlation with MHC, and a moderately strong negative correlation with GAD. These findings were similar for all tested versions supporting the criterion validity of the SOC scale.

Our findings suggest that shortened versions of the SOC-13 scale have better psychometric properties than the original 13-item version in the Czech adult population. Particularly, SOC-9 emerges as a viable alternative, showing comparable reliability and validity as the 13-item version and a clear one-factorial structure in our sample.

Peer Review reports

Sense of coherence (SOC) was introduced by the sociologist Aaron Antonovsky as the main pillar of his salutogenic theory, which explains how individuals cope with stressors and stay healthy even in case of adverse life situations [ 1 ]. SOC is a personal resource defined as a global orientation to life determining the degree to which one perceives life as comprehensible, manageable, and meaningful [ 2 ]. A strong SOC enables individuals to cope with stressors and manage tension, thus moving to the ease-end of the ease/disease continuum [ 2 , 3 ]. A person’s strength of SOC can be measured with the Orientation to Life Questionnaire commonly referred to as the SOC scale [ 4 ]. The original version is composed of 29 items (SOC-29) and Antonovsky recommended 13 items for the short version of the scale (SOC-13). To date, both versions of the scale have been used across diverse populations in at least 51 languages and 51 countries [ 5 ]. Studies have consistently shown that SOC correlates strongly with different health and well-being outcomes [ 6 , 7 ] and quality of life measures [ 8 ]. In the context of the recent COVID-19 pandemic, SOC has been identified as the most important protective resource in relation to mental health [ 9 ]. Regarding individual differences, SOC has been shown to strengthen over the life course [ 10 ], males usually score higher than females [ 11 ], and some studies indicate that SOC increases with the level of education [ 12 ]. However, despite the extensive evidence on the criterion validity of the scale, there is still a lack of clarity about its underlying factor structure and dimensionality.

The SOC scale was conceptualized as unidimensional suggesting that SOC in its totality, as a global orientation, influences the movement along the ease/dis-ease continuum [ 2 ]. However, the structure of the scale is rather multidimensional as each item is composed of multiple elements. Antonovsky developed the scale according to the facet theory [ 13 , 14 ] which assumes that social phenomena are best understood when they are seen as multidimensional. Facet theory involves the construction of a mapping sentence which consists of the facets and the sentence linking the facets together [ 15 ]. The SOC scale is composed of five facets: (i) the response mode (comprehensibility, manageability, meaningfulness); (ii) the modality of stimulus (instrumental, cognitive, affective), (iii) its source (internal, external, both), (iv) the nature of the demand it poses (concrete, diffuse, affective), (v) and its time reference (past, present, future). For example, item 3 “Has it happened that people whom you counted on disappointed you?” is a manageability item that can be described with the mapping sentence as follows: "Respondent X responds to an instrumental stimulus (“counted on”), which originated from the external environment (“people”), and which poses a diffuse demand (“disappointed”) being in the past (“has it happened”)." Although each item can be categorized along the SOC component comprehensibility, manageability, or meaningfulness, the items also share elements from the other four facets with items within the same, but also within the other SOC components (see 2, Chap. 4 for details). As Antonovsky states [ 2 , p. 87]: “The SOC facet pulls the items apart; the other facets push them together.”

Thus, the multi-facet nature of the scale can create difficulties in identifying the three theorized SOC components using statistical methods such as factor analysis. In fact, both the unidimensional and the three-dimensional SOC-13 rarely yield an acceptable fit without specifying residual covariance between single items (see 5 for an overview). This has been further exemplified in a recent study which examined the dimensionality of SOC-13 using a network perspective. The authors were unable to identify a clear structure and concluded that SOC is composed of multiple elements that are deeply linked and not necessarily distinct [ 16 ]. As a result, several researchers have suggested modified [ 17 ] or abbreviated versions of the scale, such as SOC-12 [ 18 , 19 ], SOC-11 [ 20 , 21 , 22 ], or SOC-9 [ 23 ], which have empirically shown a better factorial structure. This prompts the general question, whether an alternative short version should be preferred over the 13-item version. In fact, looking into the original literature [ 2 ], it is not clear why Antonovsky chose specifically these 13 items from the 29-item scale. We will address this question with the Czech version of the SOC-13 scale.

Salutogenesis in the Czech Republic

Salutogenesis and the SOC scale were introduced to the Czech audience in the early 90s by a Czech psychologist Jaro Křivohlavý. His work included the Czech translation of the SOC-29 scale [ 24 ] and the application of the concept in research on resilience [ 25 ] and behavioral medicine [ 26 ]. Unfortunately, the early Czech translation of the scale by Křivohlavý is not available electronically, nor could we locate it in library repositories. Later studies examined SOC-29 in relation to resilience [ 27 , 28 ] and self-reported health [ 29 , 30 ], however, it is not clear which translation of SOC-29 the authors used in the studies. A new Czech translation of the SOC-13 scale has recently been developed by the authors of this paper to examine the protective role of SOC for mental health during the COVID-19 crisis [ 31 ]. In line with earlier studies [ 9 ], SOC was identified as an important protective resource for individual mental health. This recent Czech translation of the SOC-13 scale [ 31 ] is the subject of the present study.

Present study

Our study aims to investigate the psychometric properties of the SOC-13 scale within a representative sample of the Czech adult population. Specifically, we will examine the factorial structure of the SOC-13 scale to understand its underlying dimensions and evaluate its internal consistency to ensure its reliability as a measure of SOC. Additionally, we aim to assess criterion validity by examining the scale’s association with established measures of positive and negative mental health outcomes - the Mental Health Continuum [ 32 ] and Generalized Anxiety Disorder [ 33 ]. We anticipate a strong correlation between these measures and the SOC construct [ 6 ]. Furthermore, we will investigate demographic variations in SOC, considering factors such as age, gender, and education. Understanding these variations will provide valuable insights into the applicability of the SOC-13 scale across different population subgroups. Finally, we will explore whether alternative short versions of the SOC scale should be preferred over the 13-item version. This analysis will help determine the most efficient version of the SOC scale for future research.

Study design and data collection

Our study design is a cross-sectional online survey of the Czech adult population. We contracted a professional agency DataCollect ( www.datacollect.cz ) to collect data from a representative sample for our study. Participants were recruited using quota sampling. The inclusion criteria were: being of adult age (18+), speaking the Czech language, and having permanent residence in the Czech Republic. Exclusion criteria related to study participation were predetermined to minimize the risk of biases in the collected data. The order of items in all measures was randomized and we implemented two attention checks in the questionnaire (e.g. “Please, choose option number 2”). Participants were excluded if they did not finish the survey, completed the survey in less than five minutes, did not pass the attention checks, or gave the same answer to more than 10 consecutive items. Data collection was conducted via the online platform Survey Monkey between November 2021 and December 2021.

Translation into the Czech language

Translation of the SOC scale was carried out by the authors of the paper with the help of a qualified translator. We followed the translation guidelines provided on the website of the Society for Research and Theory on Salutogenesis ( www.stars-society.org ), where the original English version of the SOC scale is available for download. Two translations were conducted independently, then compared and checked for differences. Based on this comparison, the agreed version of the scale was back translated into English by a Czech-English translator. The final version was checked for resemblance to the original version in content and in form. Although we used only the short version of the scale in our study (i.e., SOC-13), the translation included the full SOC-29 scale. The Czech translation of the full SOC scale is available as supplementary material.

Sense of coherence. We used the short version of the Orientation to Life Questionnaire [ 3 ] to assess SOC. The measure consists of 13 items evaluated on a 7-point Likert-type scale with different response options. Five items measure comprehensibility (e.g., “Does it happen that you experience feelings that you would rather not have to endure?”), four items measure manageability (e.g., “Has it happened that people whom you counted on disappointed you?”), and four items measure meaningfulness (e.g., “Do you have the feeling that you really don’t care about what is going on around you?”). In our sample, Cronbach’s alpha for the full scale was α = 0.88, for comprehensibility α = 0.76, manageability α = 0.72, and meaningfulness α = 0.70.

Mental health continuum - short form (MHC-SF; 32). This scale consists of 14 items that capture three dimensions of well-being: (i) emotional (e.g. “During the past month, how often did you feel interested in life?”); (ii) social (e.g. “During the past month, how often did you feel that the way our society works makes sense to you?”); (iii) psychological (e.g. “During the past month, how often did you feel confident to think or express your own ideas and opinions?”). The items assess the experiences the participants had over the past two weeks, the response options ranged from 1 (never) to 6 (every day). Internal consistency of the scale was α = 0.90.

Generalized anxiety disorder (GAD; 33). The scale consists of seven items that measure symptoms of anxiety over the past two weeks. Sample items include, e.g. “Over the past two weeks, how often have you been bothered by the following problems?” (i) “feeling nervous, anxious, or on edge”, (ii) “worrying too much about different things”, (iii) “becoming easily annoyed or irritable”. The response options ranged from 0 (not at all) to 3 (almost every day). Internal consistency of the scale was α = 0.92.

Sociodemographic characteristics included age, gender, and level of education (i.e., primary/vocational, secondary, tertiary).

Analytical procedure

Data analysis was conducted in R [ 34 ]. For confirmatory factor analysis, we used the cfa function of the lavaan package 0.6–16 [ 35 ]. We compared a one-factor model of SOC-13 to a correlated three-factor model (correlated latent factors comprehensibility, manageability, and meaningfulness) and a bi-factor model (general SOC dimension and specific dimensions comprehensibility, manageability, meaningfulness). Based on the empirical findings we further assessed the fit of alternative shorter versions of the SOC scale. We assessed the model fit using the comparative-fit index (CFI), Tucker-Lewis index (TLI), root mean square error of approximation (RMSEA), and standardized root mean square residual (SRMR) with the conventional cut-off values. The goodness-of-fit values for CFI and TLI surpassing 0.90 indicate an acceptable fit and exceeding 0.95 a good fit [ 36 ]. A value under 0.08 for RMSEA and SRMR indicates a good fit [ 37 ]. Nested models were compared using chi-square difference tests and the Bayesian Information Criterion (BIC). Models with lower BIC values should be preferred over models with higher BIC values [ 38 ]. All models were fitted using maximum likelihood estimation.

Further, we used the cor function of the stats package 4.3.2 [ 34 ] for Pearson correlation analysis to explore the association between SOC-13 and age, the t.test function of the same package for between groups t-test for differences based on gender, and the aov function with posthoc tests of the same package for one-way between-subjects ANOVA to test for differences based on level of education. To examine the criterion validity of the scale, we used the cor function for Pearson correlation analysis to examine the associations between SOC-13, MHC-SF, and GAD. We conducted the same analyses for the alternative short versions of the scale.

Participants

The median survey completion time was 11 min. In total, 676 participants started the survey and 557 completed it. Of those, 56 were excluded due to exclusion criteria. One additional respondent was excluded because of dubious responses on demographic items (e.g., 100 years old and a student), and two respondents were excluded for not meeting the inclusion criteria (under 18 years old). The final sample included N  = 498 participants. Of those, 53.4% were female, the average age was 49 years ( SD  = 16.6; range = 18–86), 43% had completed primary, 35% secondary, and 22% tertiary education. The sample is a good representation of the Czech adult population Footnote 1 with regard to gender (51% females), age ( M  = 50 years), and education level (44% primary, 33% secondary, 18% tertiary). Representativeness was tested using chi-squared test which yielded non-significant results for all domains.

Descriptive statistics

In Table  1 , we present an inter-item correlation matrix along with skewness, kurtosis, means and standard deviations of single items for SOC-13. Item correlations ranged from r  = 0.07 (items 2 and 4) to r  = 0.67 (items 8 and 9). Strong and moderately strong correlations were found also across the three SOC dimensions (e.g., r  = 0.77 comprehensibility and manageability).

• Confirmatory factor analysis

A one-factor model showed inadequate fit to the data [χ2(65) = 338.2, CFI = 0.889, TLI = 0.867, RMSEA = 0.092, SRMR = 0.062]. Based on existing evidence [ 6 ], we specified residual covariance between items 2 and 3 and tested a modified one-factor model. The model showed an acceptable fit to the data [χ2(64) = 242.6, CFI = 0.927, TLI = 0.911, RMSEA = 0.075, SRMR = 0.050], and it was superior to the one-factor model (Δχ2 = 95.5, Δ df  = 1, p  < 0.001).

A correlated three-factor model showed an acceptable fit considering CFI and SRMR [χ2(63) = 286.6, CFI = 0.909, TLI = 0.885, RMSEA = 0.085, SRMR = 0.058]. The model was superior to the one-factor model (Δχ2 = 51.5, Δ df  = 2, p  < 0.001), however, it was inferior to the modified one-factor model (ΔBIC = -56). We further tested a modified three-factor model with residual covariance between items 2 and 3 which showed an acceptable fit to the data based on CFI and TLI and a good fit based on RMSEA and SRMR [χ2(62) = 191.7, CFI = 0.947, TLI = 0.932, RMSEA = 0.066, SRMR = 0.046]. The model was superior to the three-factor model (Δχ2 = 97.1, Δ df  = 1, p  < 0.001) as well as to the modified one-factor model (Δχ2 = 50.9, Δ df  = 3, p  < 0.001). See Fig.  1 for a detailed illustration of the model.

Finally, we tested a bi-factor model with one general SOC factor and three specific factors (comprehensibility, manageability, meaningfulness), however, the model was not identified.

Correlated three-factor model of SOC-13 with residual covariance between item 2 and item 3

Alternative short versions of the SOC scale

We further tested the fit of alternative shorter versions of the SOC scale by systematically removing poorly performing items. In SOC-12, item 2 was excluded (“Has it happened in the past that you were surprised by the behavior of people whom you thought you knew well?”). This item measures comprehensibility, hence SOC-12 has even distribution of items for each dimension (i.e., comprehensibility, manageability, meaningfulness). Item 2 has previously been identified as problematic [ 6 ] and also in our sample it did not perform well in any of the fitted SOC-13 models (i.e., low factor loading and explained variance). A one-factor SOC-12 model showed an acceptable fit to the data based on CFI and TLI and a good fit based on RMSEA and SRMR [χ2(54) = 221.1, CFI = 0.927, RMSEA = 0.079, SRMR = 0.048]. A correlated three-factor model showed an acceptable fit based on CFI and TLI and a good fit based on RMSEA and SRMR [χ2(52) = 171.1, CFI = 0.948, TLI = 0.932, RMSEA = 0.069 SRMR = 0.043]. The model was superior to the one-factor model (Δχ2 = 50, Δ df  = 3, p  < 0.001). Bi-factor model was not identified.

In SOC-11, we removed item 3 (“Has it happened that people whom you counted on disappointed you?”), which measures manageability. The item had the lowest factor loading and the lowest explained variance in the one-factor SOC-12. A one-factor SOC-11 model showed a good fit to the data [χ2 (44) = 138.5, CFI = 0.955, TLI = 0.944, RMSEA = 0.066, SRMR = 0.038]. A correlated three-factor model was identified but not acceptable due to covariance between comprehensibility and manageability higher than 1 (i.e., Heywood case; 39).

In SOC-10, we removed item 1 (“Do you have the feeling that you don’t really care about what goes on around you?”), which measures meaningfulness. The item had the lowest factor loading and the lowest explained variance in one-factor SOC-11. A one-factor SOC-10 model showed a good fit to the data [χ2 (35) = 126.6, CFI = 0.956, TLI = 0.943, RMSEA = 0.072, SRMR = 0.039]. As in the case of SOC-11, a correlated three-factor model was identified but not acceptable due to covariance between comprehensibility and manageability higher than 1.

Finally, in SOC-9, we removed item 11 (“When something happened, have you generally found that… you overestimated or underestimated its importance / you saw the things in the right proportion”), which measures comprehensibility. The item had the lowest factor loading and the lowest explained variance in one-factor SOC-10. SOC-9 has an even distribution of three items for each dimension. A one-factor model showed a good fit to the data [χ2 (27) = 105.6, CFI = 0.959, TLI = 0.946, RMSEA = 0.076, SRMR = 0.038]. As in the previous models, a correlated three-factor model was identified but not acceptable due to covariance between comprehensibility and manageability higher than 1. See Fig.  2 for an illustration of one-factor SOC-9 model. Detailed results of the confirmatory factor analysis are shown in Table  2 . In Table 3 , we present the items of the SOC-13 (and SOC-9) scale with details about their facet structure.

One-factor model of SOC-9

Differences by gender, age, and education

Correlation analysis indicated that SOC-13 increases with age ( r  = 0.32, p  < 0.001), this finding was identical for all alternative short versions of the SOC scale (see Table  2 ). Further, the results of the two-tailed t-test showed that males ( M  = 4.8, SD  = 1.08) had a significantly higher SOC-13 score [ t (497) = 3.06, p  = 0.002, d  = 0.27] than females ( M  = 4.5, SD  = 1.07). A one-way between-subjects ANOVA did not show any significant effect of level of education on SOC-13 score [F(2, 497) = 1.78, p  = 0.169, η p 2  = 0.022]. These results were similar for all alternative short versions of the SOC scale.

Criterion validity

We found a moderately strong positive correlation ( r  = 0.61, p  < 0.001) between SOC-13 and the positive mental health measure MHC, and a moderately strong negative correlation between SOC-13 and the negative mental health measure GAD ( r = -0.68, p  < 0.001). These findings were similar for all alternative short versions of the SOC scale (see Table  4 ).

Our study examined the psychometric properties of the SOC-13 scale and its alternative short versions SOC-12, SOC-11, SOC-10, and SOC-9 in a representative sample of the Czech adult population. In line with existing studies [ 40 ], we found that SOC increases with age and that males score higher than females. In contrast to some prior findings [ 12 ], we did not find any significant differences in SOC based on the level of education. Further, we tested criterion validity using both positive and negative mental health outcomes (i.e., MHC and GAD). SOC had a strong positive correlation with MHC and a strong negative correlation with GAD, thus adding to the evidence about the criterion validity of the scale [ 6 , 40 ].

Analysis of the factor structure showed that a one-factor SOC-13 had an inadequate fit to our data, however, an acceptable fit was achieved for a modified one-factor model with specified residual covariance between item 2 (“Has it happened in the past that you were surprised by the behavior of people whom you thought you knew well?”) and item 3 (“Has it happened that people whom you counted on disappointed you?”). A correlated three factor model with latent factors comprehensibility, manageability, and meaningfulness showed a better fit than the one factor-model. However, it was also necessary to specify residual covariance between item 2 and item 3 to reach an acceptable fit for all fit indices. A recent Slovenian study [ 41 ] found a similar result and several prior studies (see 6 for an overview) have noted that items 2 and 3 of the SOC-13 scale are problematic. Although the items pertain to different SOC dimensions (item 2 to comprehensibility, item 3 to manageability), multiple studies [e.g., 20 , 42 , 43 ] have reported moderately strong correlation between them and this is also the case in our study ( r  = 0.5, p  < 0.001). The two items aptly illustrate the facet theory behind the scale construction as the SOC component represents only one building block of each item. Although items 2 and 3 theoretically pertain to different SOC components, they share the same elements from the other four facets (i.e., modality, source, demand, and time) which is reflected in the similarity of their wording. Therefore, they will necessarily share residual variance and this needs to be specified to achieve a good model fit. Drageset and Haugan [ 18 ] explain this similarity in that the people whom we know well are usually the ones that we count on, and feeling disappointed and surprised by the behavior of people we know well is closely related. Therefore, it should be theoretically justifiable to specify residual covariance between item 2 and item 3 as a possible solution to improve the fit. As we could show in our sample, the model fit significantly improved for both one-factor and three-factor solutions.

In addition, we examined the fit of alternative short versions of the SOC scale by systematically removing single items that performed poorly. First, in line with previous studies [ 6 ], we addressed the issue of residual covariance in SOC-13 by removing item 2, examining the factor structure of SOC-12. The remaining 12 items were equally distributed within the three SOC components with four items per each component. Interestingly, a one-factor model reached an acceptable fit and the fit further improved for a correlated three-factor model with latent factors of comprehensibility, manageability, and meaningfulness. Although correlated three-factor models were superior to one-factor models, we observed extreme covariances between latent variables, especially in case of comprehensibility and manageability (cov = 0.98). This suggests that the SOC components are not empirically separable and that, indeed, SOC is rather a one-dimensional global orientation with multiple components that are dynamically interrelated as Antonovsky proposed [ 2 ]. This notion was supported in a recent study that explored the dimensionality of the scale using a network perspective [ 16 ]. Our examination of SOC-11, SOC-10 and SOC-9 provided further support for a one-factor structure of the scale. All shorter versions yielded a good one-dimensional fit, however, we could not identify a correlated three-factor model fit due to the Heywood case. This refers to the situation when a solution that otherwise is satisfactory produces communality greater than one explained by the latent factor, which implies that the residual variance of the variable is negative [ 39 ]. In our case, this was true for the latent factors comprehensibility and manageability. However, we demonstrated that we could attain a good one-dimensional fit for all alternative short versions of SOC, and, importantly, they all showed comparable reliability and validity metrics to their longer counterpart SOC-13. In particular, SOC-9 shows very good fit indices and it performs equally well in validity analyses as SOC-13. Given these findings and existing evidence [ 5 ], we propose that future investigations may consider utilizing the SOC-9 scale instead of the SOC-13. It is interesting to point out that the majority of items that were removed for the shorter versions of the scale are negatively worded or reverse-scored (expect for item 11). This is in line with the latest research suggesting that such items can cause problems in model identification as they create additional method factors [ 44 , 45 , 46 ].

Finally, it is important to highlight that Antonovsky did not provide any information about the selection of the 13 items for the short version of the SOC scale [ 2 ]. For example, a detailed examination of the facet structure reveals that none of the items included in SOC-13 refers to future which is part of facet referring to time (i.e., past, present, future). Hence, considering the absence of explicit criteria for item selection in the SOC-13 scale, it would be interesting to gather data from diverse populations utilizing the full SOC-29 scale. Subsequently, through exploratory factor analysis, researchers could derive a new, theory- and empirical-driven, short version of the SOC scale.

Strengths and limitations

A clear strength of our study is that our findings are based on a representative sample that accurately reflects the Czech adult population. Moreover, we implemented rigorous data cleaning procedures, meticulously excluding participants who provided potentially careless or low-quality responses. By doing so, we ensured that our conclusions are based on high-quality data and that they are generalizable to our target population of Czech adults. Finally, we conducted a thorough back-translation procedure to achieve an accurate Czech version of the SOC scale and we carried out systematic testing of different short versions of the SOC scale.

However, our study also has some limitations. First, our conclusions are based on data from a culturally specific country and they may not be generalizable to other populations. It is important to note, however, that most of our findings are in line with multiple existing studies which supports the validity of our conclusions. Second, the data were collected during a later stage of the COVID-19 pandemic, which may have impacted particularly the mental health outcomes we used for criterion validity. It would be worthwhile to investigate whether the data replicate in our population outside of this exceptional situation. Third, it should be noted that we did not examine test-retest reliability of the scale due to the cross-sectional design of our study. Finally, self-reported data are subject to common method biases such as social desirability, recall bias, or consistency motive [ 47 ]. We aimed to minimize this risk by implementing various strategies in the questionnaire, such as randomization of items and the use of disqualifying items (e.g. “Please, choose option number 2”) to disqualify careless answers.

Our study contributes to decades of ongoing research on SOC, the main pillar of the theory of salutogenesis. In line with existing research, we found evidence for the validity of the SOC as a construct, but we could not identify a clear factorial structure of the SOC-13 scale. However, following Antonovsky’s conception of the scale, we believe it is theoretically sound to aim for a one-factor solution of the scale and we could show that this is possible with shorter versions of the SOC scale. We particularly recommend using the SOC-9 scale in future research which shows an excellent one-factor fit and validity indices comparable to SOC-13. Finally, since Antonovsky does not explain how he selected the items of the SOC-13 scale, it would be interesting to examine the possibility of developing a new one-dimensional short version based on exploratory factor analysis of the original SOC-29 scale.

Data availability

The datasets used and analyzed during the current study and the R code used for the statistical analysis are available as supplementary material.

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Acknowledgements

The authors would like to thank to the team of Center of Salutogenesis at the University of Zurich for their helpful comments on the adapted version of the SOC scale.

MT received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 801076, through the SSPH + Global PhD Fellowship Program in Public Health Sciences (GlobalP3HS) of the Swiss School of Public Health. Data collection was supported by the Charles University Strategic Partnerships Fund 2021. The University of Zurich Foundation supported the contribution of GB.

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All authors contributed to the conception and design of the study. MT wrote the manuscript, conducted data analysis, and contributed to data collection. MM and IS conducted data collection, contributed to data analysis, interpretation of results, edited and commented on the manuscript. KC and GB contributed to interpretation of results, edited and commented on the manuscript. All authors have read and approved the final manuscript.

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The study was conducted in accordance with the general principles of the Declaration of Helsinki and with the ethical principles defined by the university and by the national law ( https://cuni.cz/UK-5317.html ). Informed consent was obtained from all participants prior to the completion of the survey. Participation was voluntary and participants could withdraw from the study at any time without any consequences. For anonymous online surveys in adult population no ethical review by an ethics committee was necessary under national law and university rules. See: https://www.muni.cz/en/about-us/organizational-structure/boards-and-committees/research-ethics-committee/evaluation-request .

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Tušl, M., Šípová, I., Máčel, M. et al. The sense of coherence scale: psychometric properties in a representative sample of the Czech adult population. BMC Psychol 12 , 293 (2024). https://doi.org/10.1186/s40359-024-01805-7

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DOI : https://doi.org/10.1186/s40359-024-01805-7

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Analysis of Covariance (ANCOVA)

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• Dionisis Philippas 2  

The analysis of covariance (ANCOVA) is a technique that merges the analysis of variance (ANOVA) and the linear regression. The ANCOVA analyzes grouped data having a response (the dependent variable) and two or more predictor variables (called covariates) where at least one of them is continuous (quantitative, scaled) and one of them is categorical (nominal, non-scaled).

The ANCOVA technique allows analysts to model the response of a variable as a linear function of predictor(s), with the coefficients of the line varying among different groups. Briefly, the main idea is the inclusion of additional factors (covariates) as a statistical control to explain variation on the dependent variable, reduce the error variation, and increase the statistical power (sensitivity) of the underlying design. Thus, it differs from the analysis of variance (ANOVA) which is used to determine whether differences among test samples might be caused by random variation.

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Philippas, D. (2023). Analysis of Covariance (ANCOVA). In: Maggino, F. (eds) Encyclopedia of Quality of Life and Well-Being Research. Springer, Cham. https://doi.org/10.1007/978-3-031-17299-1_82

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