# Repeated measures Analysis of Variance (ANOVA)

Repeated measures ANOVA is an adaptation of ANOVA for cases where measures are repeated on the same statistical units. In Excel with the XLSTAT software.

Use this tool to carry out Repeated Measures ANOVA (ANalysis Of VAriance). The advanced options enable you to choose the constraints on the model and to take account of interactions between the factors. Multiple comparison tests can be calculated.

XLSTAT proposes two ways for handling repeated measures ANOVA. The classical way using **least squares estimation (LS) **that is based on the same model as the classical ANOVA and the alternative way that is based on the **maximum likelihood estimation** (**REML **and **ML**). This chapter is devoted to the first method. For details on the second method, please read the chapter on mixed models.

Not sure which statistical model is the appropriate one for your data? Check out our guide to learn more on how to choose a method according to your question, the type of your variables (i.e., categorical variables, binary, continuous) and the distribution of data.

## Principles of Repeated measures Analysis of Variance

The principle of repeated measures ANOVA is simple. For each measure, a classical ANOVA model is estimated, then the sphericity of the covariance matrix between measures is tested using Mauchly’s test, Greenhouse-Geisser epsilon or Huynt-Feldt epsilon. If the sphericity hypothesis is not rejected, between- and within-subject effects can be tested.

## Calculation in Repeated measures Analysis of Variance

Repeated measures Analysis of Variance (ANOVA) uses the same conceptual framework as classical ANOVA. The main difference comes from the nature of the explanatory variables. The exploratory variable is measured at different time or repetition. In ANOVA, explanatory variables are often called factors.

If p is the number of factors, the ANOVA model is written as follows:

**y ^{t}_{i} = β_{0} + ∑_{j=1...p} β_{k(i,j),j} + ε_{i}**

where y^{t}_{i} is the value observed for the dependent variable for observation i for measure t, k_{(i,j)} is the index of the category of factor j for observation i, and ε_{i} is the error of the model.

The hypotheses used in ANOVA are identical to those used in linear regression: the errors ε_{i} follow the same normal distribution N(0,s) and are independent.

However, other assumptions need to be respected in the case of repeated measures ANOVA. As measures are taken from the same subjects at different times, the repetitions are correlated. In repeated measures ANOVA we assume that the covariance matrix between the ys is spherical (for example, compound symmetry is a spherical shape). We can drop this hypothesis when using the mixed model based approach.

## Options for Repeated measures Analysis of Variance in XLSTAT

XLSTAT can include in the model interactions and nested effects.

### Interactions

By interaction is meant an artificial factor (not measured) which reflects the interaction between at least two measured factors. For example, if we carry out treatment on a plant, and tests are carried out under two different light intensities, we will be able to include in the model an interaction factor treatment*light which will be used to identify a possible interaction between the two factors. If there is an interaction between the two factors, we will observe a significantly larger effect on the plants when the light is strong and the treatment is of type 2 while the effect is average for weak light, treatment 2 and strong light, treatment 1 combinations.

### Nested effects

When constraints prevent us from crossing every level of one factor with every level of the other factor, nested factors can be used. We say we have a nested effect when fewer than all levels of one factor occur within each level of the other factor. An example of this might be if we want to study the effects of different machines and different operators on some output characteristic, but we can't have the operators change the machines they run. In this case, each operator is not crossed with each machine but rather only runs one machine. XLSTAT has an automatic device to find nested factors and one nested factor can be included in the model.

### Multiple Comparisons Tests

One of the main applications of ANOVA is multiple comparisons testing whose aim is to check if the parameters for the various categories of a factor differ significantly or not. For example, in the case where four treatments are applied to plants, we want to know not only if the treatments have a significant effect, but also if the treatments have different effects.

Numerous tests have been proposed for comparing the means of categories. The majority of these tests assume that the sample is normally distributed. XLSTAT provides the main tests including:

- Tukey's HSD test,
- Bonferroni's t test,
- Dunn-Sidak's test,
- Newman-Keuls's test (SNK),
- Duncan's test,
- REGWQ test.

## Results for Repeated measures Analysis of Variance in XLSTAT

- Summary Statistics
- Goodness of fit statistics
- Analysis of variance table
- Type I and Type III sum of squares tables for
- Predictions and residuals table
- Standardized coefficients table
- Parameters of the model: estimate of the parameters, the corresponding standard error, the Student’s t, the corresponding probability, as well as the confidence interval
- Regression charts

## Tutorial for for Repeated measures Analysis of Variance in XLSTAT

Read about how repeated measures ANOVA can be used to evaluate the effect of the treatment and the effect of time on the depression score measured on two groups of patients.

### analyze your data with xlstat

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