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## We report the F-statistic from a repeated measures ANOVA as:

Be able to explain the assumptions and the basics of the repeated-measures ANOVA

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# (2010) Repeated Measures ANOVA, Friedman (10 Patients).

Another approach to analysis of repeated measures is via general mixed models. This approach can handle balanced as well as unbalanced or missing within-subject data, and it offers more options for modeling the within-subject covariance. The main drawback of the mixed models approach is that it generally requires iteration and, thus, might be less computationally efficient. For further details on this approach, see Chapter 56, and .

This transformation is useful when the levels of the repeated measure represent quantitative values of a treatment, such as dose or time. If the levels are unequally spaced, level values can be specified in parentheses after the number of levels in the statement. For example, if five levels of a drug corresponding to 1, 2, 5, 10, and 20 milligrams are administered to different treatment groups, represented by the variable , the statements

## A repeated measures ANOVA calculates an F-statistic in a similar way:

When the design specifies more than one repeated measures factor, PROC GLM computes the matrix for a given effect as the direct (Kronecker) product of the matrices defined by the statement if the factor is involved in the effect or as a vector of 1s if the factor is not involved. The test for the main effect of a repeated measures factor is constructed using an matrix that corresponds to a test that the mean of the observation is zero. Thus, the main effect test for repeated measures is a test that the means of the variables defined by the matrix are all equal to zero, while interactions involving repeated measures effects are tests that the between-subjects factors involved in the interaction have no effect on the means of the transformed variables defined by the matrix. In addition, you can specify other matrices to test hypotheses of interest by using the statement, since hypotheses defined by statements are also tested in the analysis. To see which combinations of the original variables the transformed variables represent, you can specify the option in the statement. This option displays the transpose of , which is labeled as M in the PROC GLM results. The tests produced are the same for any choice of transformation matrix specified in the statement; however, depending on the nature of the repeated measurements being studied, a particular choice of transformation matrix, coupled with the or option, can provide additional insight into the data being studied.

When the design specifies more than one repeated measures factor, PROC GLM computes the matrix for a given effect as the direct (Kronecker) product of the matrices defined by the statement if the factor is involved in the effect or as a vector of 1s if the factor is not involved. The test for the main effect of a repeated measures factor is constructed using an matrix that corresponds to a test that the mean of the observation is zero. Thus, the main effect test for repeated measures is a test that the means of the variables defined by the matrix are all equal to zero, while interactions involving repeated measures effects are tests that the between-subjects factors involved in the interaction have no effect on the means of the transformed variables defined by the matrix. In addition, you can specify other matrices to test hypotheses of interest by using the statement, since hypotheses defined by statements are also tested in the analysis. To see which combinations of the original variables the transformed variables represent, you can specify the option in the statement. This option displays the transpose of , which is labeled as M in the PROC GLM results. The tests produced are the same for any choice of transformation matrix specified in the statement; however, depending on the nature of the repeated measurements being studied, a particular choice of transformation matrix, coupled with the or option, can provide additional insight into the data being studied.

## The hypotheses of interest in an ANOVA are as follows:

This module will continue the discussion of hypothesis testing, where a specific statement or hypothesis is generated about a population parameter, and sample statistics are used to assess the likelihood that the hypothesis is true. The hypothesis is based on available information and the investigator's belief about the population parameters. The specific test considered here is called analysis of variance (ANOVA) and is a test of hypothesis that is appropriate to compare means of a continuous variable in two or more independent comparison groups. For example, in some clinical trials there are more than two comparison groups. In a clinical trial to evaluate a new medication for asthma, investigators might compare an experimental medication to a placebo and to a standard treatment (i.e., a medication currently being used). In an observational study such as the Framingham Heart Study, it might be of interest to compare mean blood pressure or mean cholesterol levels in persons who are underweight, normal weight, overweight and obese.

Experimental designs called "repeated measures" designs are characterized by having more than one measurement of at least one given variable for each subject. A well-known repeated measures design is the pretest, posttest experimental design, with intervening treatment; this design measures the same subjects twice on an intervally-scaled variable, and then uses the correlated or dependent samples t test in the analysis (Stevens, 1996). As another example, in a 2__3 repeated measures factorial design, each subject has a score for each of the combinations of the factors, or in each of the six cells of the data matrix (Huck & Cormier, 1996).

• ### The ANOVA table above is organized as follows.

Conducting Repeated Measures Analyses: Experimental Design Considerations

• ### The usual assumptions are made for a one-way MANOVA. In this case:

We will use the following experiment to illustrate the statistical procedures associated with repeated measures data...

• ### The test statistic is the F statistic for ANOVA, F=MSB/MSE.

Figure 2: Null hypothesis of analysis of variance with repeated measures with four measurement times

## Here is the corresponding anova:

Where measurements are made under different conditions, the conditions are the levels (or related groups) of the independent variable (e.g., type of cake is the independent variable with chocolate, caramel, and lemon cake as the levels of the independent variable). A schematic of a different-conditions repeated measures design is shown below. It should be noted that often the levels of the independent variable are not referred to as conditions, but treatments. Which one you want to use is up to you. There is no right or wrong naming convention. You will also see the independent variable more commonly referred to as the within-subjects factor.

## This is a particular case of two-way anova.

The above two schematics have shown an example of each type of repeated measures ANOVA design, but you will also often see these designs expressed in tabular form, such as shown below:

## This is still a double anova, i.e., a model of the form

PROC GLM provides both univariate and multivariate tests for repeated measures for one response. For an overall reference on univariate repeated measures, see . The multivariate approach is covered in . For a discussion of the relative merits of the two approaches, see .

## The test statistic is the F statistic for ANOVA, F=MSB/MSE.

In order to deal efficiently with the correlation of repeated measures, the GLM procedure uses the multivariate method of specifying the model, even if only a univariate analysis is desired. In some cases, data might already be entered in the univariate mode, with each repeated measure listed as a separate observation along with a variable that represents the experimental unit (subject) on which measurement is taken. Consider the following data set :

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