PEP 6305 Measurement in Health & Physical Education

Topic 10: Repeated Measures

Section 10.1

n This Topic has 2 Sections.

Reading

n Vincent & Weir, Statistics in Kinesiology, 4th ed. Chapter 12 “Analysis of Variance with Repeated Measures”

Purpose

n To demonstrate the comparison of measures repeatedly collected from the same group of subjects using analysis of variance (ANOVA).

Repeated Measures: Within Subjects Factors

n Repeated measures research, in which the same measure is collected several times in a single group of subjects, is very common in studies of kinesiology and health.

¨ Pretest-posttest designs are one example, in which a measure collected before a treatment is compared to the same measure collected after the treatment.

n Repeated measures designs are also called within subjects designs.

¨ The repeated factor is called a “within” subjects factor because comparisons are made multiple times ("repeated") “within” the same subject rather than across ("between") different subjects.

n Recall that simple ANOVA requires that the means being compared are independent of one another. When the subjects are in groups that are independent from one another, then group is a between subjects factor.

n Since repeated measures are collected on the same subjects, the means of those measures study are dependent.

¨ A particular subject’s scores will be more alike than scores collected from multiple subjects, meaning that there is less variability from measure to measure than observed from person to person in simple ANOVA.

¨ Repeated measures ANOVA separates the two sources of variance: measures and persons. This separation of the sources of variance decreases MS_E, the random variation (sampling error) component, because there are now two sources of known variation (subjects and measures) instead of just one (subjects) as in simple ANOVA. The variation in scores due to differences between subjects is separated from variation due to differences from measure to measure within a subject. (A smaller MS_E increases power. Why?)

¨ Instead of comparing treatment effects to a group of different subjects, **treatment effects are compared across multiple measures in the same subjects. Each subject provides their own "control" value for the comparison. Consequently, this type of design is more sensitive** to differences (i.e., requires smaller differences in the dependent variable to reject the null hypothesis) than are between subjects designs.

n Repeated measures ANOVA requires different computation than simple ANOVA.

n The repeated measures ANOVA null hypothesis is that the means of the measures all have the same value. If the number of repeated measures = k , the null hypothesis is:

¨ , or the differences between the means of each repeated measure is equal to 0.

n The statistic used in repeated measures ANOVA is F, the same statistic as in simple ANOVA, but now computed using the ratio of the variation “within” the subjects to the “error” variation.

¨ The observed between measures variance is an estimate of the variation between measures that would be expected in the population under the conditions of the study.

¨ The observed error variance is an estimate of the variation that would be expected to occur as a result of sampling error alone.

¨ If the observed (computed) value for F is significantly higher than the value expected by sampling variation alone, then the variance between groups is larger than would be expected by sampling error alone.

· In other words, at least one mean differs from the others enough to cause large variation between the measures.

Repeated Measures ANOVA Assumptions

n The assumptions of repeated measures ANOVA are similar to simple ANOVA, except that independence is not required and an assumption about the relations among the repeated measures (sphericity) is added.

n Normality

¨ The dependent variable is normally distributed in the population being sampled.

¨ Normality of the dependent variable can be evaluated using a histogram and skewness and kurtosis statistics.

n Homogeneity of variance

¨ If there are separate groups of subjects in addition to the repeated measure (within subjects) factor, then the variance of the dependent variable in each group is equal (in the population).

¨ As in simple ANOVA, homogeneity of variance can be evaluated using a variety of statistical tests, but the most straightforward method is to compare the within-group variances; one or more variances twice as large as other variances may be a problem.

n Sphericity

¨ This means that the variances of the repeated measures are all equal, and the correlations among the repeated measures are all equal.

¨ This assumption is needed to allow for comparing the variances among the repeated measures (within subjects).

¨ When evidence suggests sphericity is violated, the analysis is adjusted. We’ll discuss this briefly in the next section.

“Within Subjects” Variance and “Error” Variance

n Means are computed for each measure (typically the measures are in the columns of the data file) by summing across the subjects and dividing by the number of subjects:

where X_ij is the score of a person (i) for measure j and N is the number of subjects.

n Means are computed for each subject (typically the subjects are the rows of the data file) by summing across the measures and dividing by the number of measures:

where J is the total number of measures in the study.

n The grand mean is computed by summing across all measures and all subjects and dividing by the total number of scores (N × j):

n The between measures variance is the variation of the mean of each measure from the grand mean.

n The between subjects variance is the variation of the mean of each subject from the grand mean.

n The total variance is the variation of each score (each measure for each subject) from the grand mean.

n The error variance is any variation not accounted for by the variation among the subjects and the variation among the measures.

¨ This "residual" variance = total variance – between subjects variance – between measures variance.

Sum of Squares

n Each component (Between Measures, Between Subjects, Error/Residual, and Total) has a SS.

¨ Between Measures:

¨ Between Subjects:

¨ Total:

¨ Error/Residual:

n Each of these SS is a measure of variability.

n Repeated measures ANOVA compares the between measures variability to the error variability using the F statistic.

n Both SS measures are standardized by their respective degrees of freedom, creating Mean Squares.

Mean Squares

n Compute the MS by dividing the SS by its respective degrees of freedom (df).

¨ df = (the number of elements being summed in the SS) – (the number of means subtracted in the SS)

¨ For each SS equation, compute the difference between the number of elements ahead of the subtraction sign from the number of elements behind the subtraction sign.

¨ df_W : How many elements are being summed in the SS_M equation above? The J measures’ means. How many means are subtracted from these J means? 1—the grand mean is subtracted from the J means. So df_M = J – 1.

¨ df_B : How many elements are being summed in the SS_S equation above? The N subjects' means. How many means are subtracted from these N means? The grand mean is subtracted from the n means (in each group). So df_S = N – 1.

¨ df_T : What is the df for SS_T? N × J elements (all subjects and all measures) are summed, and then the grand mean is subtracted, so df_T = NJ – 1.

¨ df_E : What is the df for SS_E? df_E = [df_T – df_M – df_S]= [(NJ – 1) – (J – 1) – (N – 1)] = [NJ – 1 – J + 1 – N + 1] = [NJ – J – N + 1] = (N – 1)(J – 1).

n Each component (Between Measures, Between Subjects, and Error) in ANOVA has a MS : divide each SS by its df.

¨ Between Measures:

¨ Between Subjects:

¨ Error:

n Compute F :

MS_E represents the variation expected as a result of sampling error alone.

n As in simple ANOVA, we compare the observed sample F value to the F distribution to determine the likelihood that the value is due to sampling error alone.

¨ If MS_M is several times larger than MS_E, we conclude that the variation between measures is larger than sampling error. In other words, the error probability (p-value) will be low, and we infer that the measures are significantly different.

¨ The key difference between simple ANOVA and repeated measures ANOVA is that the variation from measure to measure and the variation from subject to subject have been removed from the SS_E in repeated measures, thus **adjusting the SS_E (and MS_E) for the within-subjects dependency of the measures**.

Click to go to the next section (Section 10.2)

PEP 6305 Measurement in Health & Physical Education

Topic 10: Repeated Measures

Section 10.1

n This Topic has 2 Sections.

Reading

n Vincent & Weir, Statistics in Kinesiology, 4th ed. Chapter 12 “Analysis of Variance with Repeated Measures”

Purpose

n To demonstrate the comparison of measures repeatedly collected from the same group of subjects using analysis of variance (ANOVA).

Repeated Measures: Within Subjects Factors

n Repeated measures research, in which the same measure is collected several times in a single group of subjects, is very common in studies of kinesiology and health.

¨ Pretest-posttest designs are one example, in which a measure collected before a treatment is compared to the same measure collected after the treatment.

n Repeated measures designs are also called within subjects designs.

¨ The repeated factor is called a “within” subjects factor because comparisons are made multiple times ("repeated") “within” the same subject rather than across ("between") different subjects.

n Recall that simple ANOVA requires that the means being compared are independent of one another. When the subjects are in groups that are independent from one another, then group is a between subjects factor.

n Since repeated measures are collected on the same subjects, the means of those measures study are dependent.

¨ A particular subject’s scores will be more alike than scores collected from multiple subjects, meaning that there is less variability from measure to measure than observed from person to person in simple ANOVA.

n Repeated measures ANOVA requires different computation than simple ANOVA.

n The repeated measures ANOVA null hypothesis is that the means of the measures all have the same value. If the number of repeated measures = k , the null hypothesis is:

¨ , or the differences between the means of each repeated measure is equal to 0.

n The statistic used in repeated measures ANOVA is F, the same statistic as in simple ANOVA, but now computed using the ratio of the variation “within” the subjects to the “error” variation.

¨ The observed between measures variance is an estimate of the variation between measures that would be expected in the population under the conditions of the study.

¨ The observed error variance is an estimate of the variation that would be expected to occur as a result of sampling error alone.

¨ If the observed (computed) value for F is significantly higher than the value expected by sampling variation alone, then the variance between groups is larger than would be expected by sampling error alone.

· In other words, at least one mean differs from the others enough to cause large variation between the measures.

Repeated Measures ANOVA Assumptions

n The assumptions of repeated measures ANOVA are similar to simple ANOVA, except that independence is not required and an assumption about the relations among the repeated measures (sphericity) is added.

n Normality

¨ The dependent variable is normally distributed in the population being sampled.

¨ Normality of the dependent variable can be evaluated using a histogram and skewness and kurtosis statistics.

n Homogeneity of variance

¨ If there are separate groups of subjects in addition to the repeated measure (within subjects) factor, then the variance of the dependent variable in each group is equal (in the population).

¨ As in simple ANOVA, homogeneity of variance can be evaluated using a variety of statistical tests, but the most straightforward method is to compare the within-group variances; one or more variances twice as large as other variances may be a problem.

n Sphericity

¨ This means that the variances of the repeated measures are all equal, and the correlations among the repeated measures are all equal.

¨ This assumption is needed to allow for comparing the variances among the repeated measures (within subjects).

¨ When evidence suggests sphericity is violated, the analysis is adjusted. We’ll discuss this briefly in the next section.

“Within Subjects” Variance and “Error” Variance

n Means are computed for each measure (typically the measures are in the columns of the data file) by summing across the subjects and dividing by the number of subjects:

where Xij is the score of a person (i) for measure j and N is the number of subjects.

n Means are computed for each subject (typically the subjects are the rows of the data file) by summing across the measures and dividing by the number of measures:

where J is the total number of measures in the study.

n The grand mean is computed by summing across all measures and all subjects and dividing by the total number of scores (N × j):

n The between measures variance is the variation of the mean of each measure from the grand mean.

n The between subjects variance is the variation of the mean of each subject from the grand mean.

n The total variance is the variation of each score (each measure for each subject) from the grand mean.

n The error variance is any variation not accounted for by the variation among the subjects and the variation among the measures.

¨ This "residual" variance = total variance – between subjects variance – between measures variance.

Sum of Squares

n Each component (Between Measures, Between Subjects, Error/Residual, and Total) has a SS.

¨ Between Measures:

¨ Between Subjects:

¨ Total:

¨ Error/Residual:

n Each of these SS is a measure of variability.

n Repeated measures ANOVA compares the between measures variability to the error variability using the F statistic.

n Both SS measures are standardized by their respective degrees of freedom, creating Mean Squares.

Mean Squares

n Compute the MS by dividing the SS by its respective degrees of freedom (df).

¨ df = (the number of elements being summed in the SS) – (the number of means subtracted in the SS)

¨ For each SS equation, compute the difference between the number of elements ahead of the subtraction sign from the number of elements behind the subtraction sign.

¨ dfW : How many elements are being summed in the SSM equation above? The J measures’ means. How many means are subtracted from these J means? 1—the grand mean is subtracted from the J means. So dfM = J – 1.

¨ dfB : How many elements are being summed in the SSS equation above? The N subjects' means. How many means are subtracted from these N means? The grand mean is subtracted from the n means (in each group). So dfS = N – 1.

¨ dfT : What is the df for SST? N × J elements (all subjects and all measures) are summed, and then the grand mean is subtracted, so dfT = NJ – 1.

¨ dfE : What is the df for SSE? dfE = [dfT – dfM – dfS] = [(NJ – 1) – (J – 1) – (N – 1)] = [NJ – 1 – J + 1 – N + 1] = [NJ – J – N + 1] = (N – 1)(J – 1).

n Each component (Between Measures, Between Subjects, and Error) in ANOVA has a MS : divide each SS by its df.

¨ Between Measures:

¨ Between Subjects:

¨ Error:

n Compute F :

MSE represents the variation expected as a result of sampling error alone.

n As in simple ANOVA, we compare the observed sample F value to the F distribution to determine the likelihood that the value is due to sampling error alone.

¨ If MSM is several times larger than MSE, we conclude that the variation between measures is larger than sampling error. In other words, the error probability (p-value) will be low, and we infer that the measures are significantly different.

¨ The key difference between simple ANOVA and repeated measures ANOVA is that the variation from measure to measure and the variation from subject to subject have been removed from the SSE in repeated measures, thus adjusting the SSE (and MSE) for the within-subjects dependency of the measures.

where X_ij is the score of a person (i) for measure j and N is the number of subjects.

¨ df_W : How many elements are being summed in the SS_M equation above? The J measures’ means. How many means are subtracted from these J means? 1—the grand mean is subtracted from the J means. So df_M = J – 1.

¨ df_B : How many elements are being summed in the SS_S equation above? The N subjects' means. How many means are subtracted from these N means? The grand mean is subtracted from the n means (in each group). So df_S = N – 1.

¨ df_T : What is the df for SS_T? N × J elements (all subjects and all measures) are summed, and then the grand mean is subtracted, so df_T = NJ – 1.

¨ df_E : What is the df for SS_E? df_E = [df_T – df_M – df_S]= [(NJ – 1) – (J – 1) – (N – 1)] = [NJ – 1 – J + 1 – N + 1] = [NJ – J – N + 1] = (N – 1)(J – 1).

MS_E represents the variation expected as a result of sampling error alone.

¨ If MS_M is several times larger than MS_E, we conclude that the variation between measures is larger than sampling error. In other words, the error probability (p-value) will be low, and we infer that the measures are significantly different.