GLM Repeated Measures

Overview GLM repeated measures is a procedure used to model dependent variables measured at multiple times using analysis of variance. Each dependent variable is represented by as many variables as there are measurement times (ex., achieve1, achieve2, achieve3, ..., achieven) for n time periods. Predictor variables may be categorical factors or continuous covariates. Normally factors define subgroups in the population and covariates are conceived as control variables. The GLM repeated measures model can test the main effects on repeated measures of between-suibjects (grouping) factors, the main effects of within-subjects factors like measurement times, interaction effects between factors, covariate effects. and effects of interactions between covariates and between-subjects factors. Key assumptions are linear relationships, normal distribution of the dependents, and fixed effects. Interaction effects are modeled by default, as in other analysis of variance procedures.

Contents Key concepts and terms Models Options Statistics Post hoc tests Profile plots Contrast analysis Saving variables Assumptions Frequently asked questions Bibliography

Key Terms and Concepts

• Within-subjects factor. The within-subjects factor is the basis for the repeated measurements. Typically it carries labels such as trials, days, weeks, or years. These labels refer to the nature of the repetition. The within-subjects factor will have as many levels as there are repetitions. In the example below, which is the initial GLM Repeated Measures screen in SPSS, weeks is the within-subject factor and it has 4 levels because measurements were taken in each of four weeks. There can be multiple within-subjects factors, as when measurements are taken 3 days a week for each of 4 weeks, for a total of 12 measurements per subject of analysis. Had that been the case, the within-subjects box below would have the factors day(3) and week(4).
  

• Repeated measures dependent variables. For each measurement taken there should be one variable per repetition. In the example above, response on an opinion poll is the measurement and must appear in the dataset four times, such as poll1, poll2, poll3, and poll4 - corresponding to the four weeks of measurements in the study. The "Measurename" label given in the Repeated Measures Define Factors dialog box above, which was "poll" in our example, then corresponds to the actual measures (poll1, poll2, poll3, and poll4) entered in the Repeated Measures dialog box shown below. (Example data is adapted from SPSS sample data). The Repeated Measures dialog is invoked by clicking the Define button.
  

• Between-subjects factors. There may be one or more categorical variables which group the subjects. Gender would be an example of a common between-subjects factor. In SPSS, between-subjects factors can have string as well as numeric coding.

• Covariates are quantitative independent variables. For any level of the within-subjects factor (or level formed by the intersection of levels of multiple within-subjects factors), a covariate should have the same value (be a constant).

  
  

• Models. Clicking the Model button leads to the Model dialog illustrated below. Note that if Custom is selected, one may build separate models for the within-subjects factors and the between-subjects factors. In this example, however, there is only one within-subjects factor: week. The default is a full factorial model, which is the model with all factor main effects, all covariate main effects, and all factor-by-factor interactions, but with no covariate interactions. There may be factor-by-covariate interactions if Custom is selected. Note that the interaction of each between-subjects term with each within-subjects term is automatically included in the model under GLM repeated measures.
  
  • Type of sum of squares. SPSS supports Types I. II, III, and IV sums of squares. Type III is by far the most common type. Type IV is appropriate if one has cells with no cases ("missing cells"). See the discussion under univariate GLM.

  • Balanced vs. unbalanced models. The GLM repeated measures procedures supports both balanced and unbalanced designs (models). a "balanced design" is one in which each cell formed by the intersection of the between-subjects grouping factors has the same n (sample size).

  
  

• Output options. Clicking the Options button leads to the dialog illustrated below, used mainly to specify output. Note that all effects are assumed to be fixed effects, not random (see below).
  
  • Estimated marginal means may be requested in the upper portion of the Options dialog in SPSS. Output will give the predicted marginal means on the dependent variables for levels of within-subjects and/or between-subjects factors, controlling for any covariates in the model. The univariate test of significance below shows that the repeated measures dependent variables (or at least one of them) differs significantly by stateid, for example.
    

  • Compare main effects. This option, also in the upper portion of the Options dialog. If selected, this output gives uncorrected pairwise comparisons among estimated marginal means for any main effect listed in the "Display means for" list. The output illustrated below is only a small part of the whole pairwise comparison table.
    

  • Confidence interval adjustment.. If "Compare main effects" is selected., the researcher may select among least significant difference (LSD), Bonferroni, or Sidak adjustments to significance levels and confidence intervals. LSD(none) is the default. and significance. T

  • Statistics. The Options button allows the researcher to select a variety of statistical outputs, discussed below.

  • Significance level. At the bottom of the Options dialog, the researcher may set the alpha significance cut-off level.

  
  

• Statistics
  • Descriptive statstics. One may check to request a variety of descriptive statistics (means, standard deviations, counts), as illustrated in the very partial table below. (St is stateid, Bud is the budget factor).
    

  • Hypothesis SSCP matrices. The "Between-Subjects SSCP Matrix" table contains the sums of squares attributable to model effects. These values are used in estimates of effect size, as illustrated below.

  • Estimates of effect size generates partial eta-squared, which is an effect size coefficient associated with each factor. In the illustration below, the "Tests of Between-Subjects Effects" table shows levels of stateid and budget to significantly affect the repeated dependent measure, poll. By partial eta-squared, the stateid effect appears stronger than the budget effect. The stateid effect is strong and the budget effect would be classed as moderate to strong. The interaction of stateid and budget is not significant, however.
    

  • Within-subjects SSCP matrix and within-subjects contrast effects. As illustrated below, the within-subjects SSCP matrix gives sums of squares associated with the within-subjects variable associated with measurement times (week in the example). These sums of squares are the basis for the tests of within-subjects effects. Note that the "Intercept" row in the SSCP matrix table corresponds to the "week" main effect row in the "Within-Subjects Contrasts" table. Since a full factorial model was specified in this example, not only the main effect of week but also all its interactions are tested. Since the default polynomial contrasts was selected, tests are for the linear, quadratic, and cubic effects. In this example, none of the effects of week are significant. We conclude that the between-subject effects of stateid and budget, noted above, do not interact with week of measurement. Therefore we do not interpret the effect size measure, partial eta squared.
    

  • Power tests are included in the effects tables illustrated above. By convention, a power of .80 or higher is considered acceptable. Power is above this level for the stateid*budget interaction in the between-subjects tests above, so therefore we assume that in failing to reject the null hypothesis that this interaction effect is not different from zero, we have not made a Type II error (a false negative). For the tests for the within-subjects factor week, none but one have a power of .80 or higher. While concluding that all of the week effects are non-significant, we do not have adequate power to assume no Type II error was made in this set of conclusions except for the three-way cubic interaction effect.

  • Multivariate tests. The "Multivariate Tests" table in SPSS provides F tests of the within-subjects factor (week) and its interactions with between-subjects grouping factors. Four variants of the F test are provided (Pillai's trace, Wilks' lambda, Hotelling's trace, and Roy's largest root). These are discussed in the section on multivariate GLM. Wilks' lambda is the most commonly reported. Usually the same substantive conclusion emerges from any variant. For these data, as with the within-subjects contrast effects above, we conclude tha none of the main or interaction effects of week are significant. Again, power is acceptable (>.80) only for the three-way interaction.
    
    • Univariate vs. multivariate models. The multivariate tests assume a multivariate model (the particular model is shown in a footnote to the table). The within-subjects contrasts table assumes a univariate model. While both test the effect of the within-subjects variable (week) and its interactions, the assumptions are different. Multivariate models assume that the variance-covariance matrices are the same for each cell formed by the between-subjects grouping variables. Box's M test tests this assumption of homogeneity of variance-covariance matrices. Univariate repeated measures models, also called split plot designs or mixed model designs, assume that the dependent variables are responses to levels of the within-subject factor(s). In the example, it is assumed that poll1, poll2, poll3, and poll4 are responses to the four levels of the week variable. The overall F test of the model is valid in univariate repeated measures models if the variance-covariance matrix can be shown to be circular in form, which is demonstrated by Mauchly's test of sphericity (see Huynh and Mandeville, 1979). Mauchly's test is displayed by default. However, Mauchly's test has problems: (1) for small samples it lacks power; and (2) for large samples, even very small departures from sphericity may be found to be significant.
      • Box's M test. Box's M tests the assumption of multivariate models that the dependent variables (poll1, ...,poll4 in the example) are drawn from a multivariate normal distribution and that there is homogeneity of the covariance matrices of the dependent variables across all level combinations of the between-subjects factors. A finding of nonsignificance upholds the multivariate tests (the researcher fails to reject the null hypothesis that the covariance matrices are equal across design cells). For this example, Box's M is nonsignificant so the researcher concludes the assumptions of the multivariate model (and multivariate tests) are not violated.
        

      • Mauchly's test of sphericity. Mauchly's test of sphericity is used in relation to meeting the assumptions of univariate models (and tests of within-subject effects, illustrated below). A finding of non-significance corresponds to concluding that assumptions are met. For the example below, there is a finding of significance, meaning that there is a violation of sphericity.
        
        • Epsilon adjustment.If Mauchly's test shows violation of sphericity, this may be compensated by an epsilon adjustment. Specifically, the numerator and denominator degrees of freedom in the F test are multiplied by epsion. SPSS repeated measures offers three epsilon estimates. shown above: Greenhouse-Geisser, Huynh-Feldt, and lower bound. Greenhouse-Geisser is considered to be a conservative adjustment, especially when sample size is low. Huynh-Feldt is considered less conservative but may assume values greater than 1.0, in which case it is set to 1.0 (as here). The lower bound method is the most conservative epsilon adjustment.

      • Univariate tests of within-subjects effects. The "Tests of Within-Subjects Effects" table displays univariate tests under each of the three possible epsilon adjustments discussed above. For this example, under any epsilon adjustment, using a univariate model, the researcher still concludes that none of the effects of week or its interactions are significant. Again, power is acceptable only for judgments about the three-way interactions. Of course, if Mauchly's test showed no violation of sphericity, this table would be interpreted in terms of the "Sphericity assumed" rows.
        

    • Bartlett's test of sphericity of the residual covariance matrix is printed when the residual SSCP matrix is requested. Bartlett's test tests if the effects are uncorrelated (that is, proportional to an identity matrix where each effect correlates with itself at the 1.0 level but with others at the 0.0 level). That is, Bartlett's test tests if the repeated dependent measures are independent of each other (here, if poll1, poll2, poll3, and poll4 may be considered independent measures). Nonsignificance means the researcher fails to reject the null hypothesis that all correlations are zero, meaning the variables are independent; A finding of nonsignificance, such as that illustrated below, also means sphericity is not violated - that the variance-covariance matrix is circular in form, meeting the assumptions of univariate models and tests. However, Bartlett's test is compromised for large samples, which cause a finding of significance even for very small departures from independence.
      

    • Parameter estimates with standard errors, t tests, confidence intervals, effect size (partial eta-squared), and observed power are available in the "Parameter Estimates" table of SPSS, shown for the ongoing example only in small part below. The effect size measure, partial eta-squared, may be used to asses which levels of which factors are contributing the most and least to the model.
      

    • Levene's test tests homogeneity of variance. In a well-fitting model, error variance of each repeated measures dependent variable should be the same across groups formed by the between-subjects (grouping) factors. If the Levene statistic is significant at the .05 level or better, the researcher rejects the null hypothesis that the groups have equal variances. That is, a model meeting the assumptions of GLM repeated measures will have nonsignificant findings by Levene's test, as in the illustration below. This test is robust in the face of departures from normality. Note, however, that failure to meet the assumption of homogeneity of variances is not fatal to ANOVA models, which are relatively robust, particularly when groups are of equal sample size.
      

    • Spread-versus-level plots. These plots display eans on the X axis against standard deviations or variances on the Y axis. Each point shows the value of a factor design matrix cell (a cell formed by the intersection of the between-subjects factors) on the mean of the dependent (here, poll1 - there are similar tables for poll2, poll3, and poll4) and on the standad deviation or variance. This is useful in testing the homogeneity of variances assumption and in identifying cells which deviate substantially from the assumption. In the illustration below, the lack of pattern graphically confirms the finding of Levene's test, above, that the assumption of homogeneity of variances is adequately met.
      

    • Residual plots. The more the observed vs. predicted cells display a linear pattern, the better the model predicts the dependent variable. For well-fiutting models, observed and predicted vs. standardized residuals should display no pattern, which in the example below is true of predicted values. However, there is a slight but discernable tendency for residuals to increase as the values of observed poll1 values increase. That is, the model performs less well when predicting high observed values of poll1.
      

    • Lack of fit test. The lack of fit test is discussed in the section on univariate GLM.

    • General estimable function. This table is useful for custom hypothesis tests.

    
    

  • Post hoc tests
    • Post hoc tests. After the overall F tests has shown a model to be significant, the researcher may use post hoc tests to examine whether estimated marginal means differ for levels of specific factors in the model. Post hoc range tests and multiple comparisons for between-subjects factors include least significant difference, Bonferroni, Sidak, Scheffé, Ryan-Einot-Gabriel-Welsch multiple F, Ryan-Einot-Gabriel-Welsch multiple range, Student-Newman-Keuls, Tukey's honestly significant difference, Tukey's b, Duncan, Hochberg's GT2, Gabriel, Waller Duncan t test, Dunnett (one-sided and two-sided), Tamhane's T2, Dunnett's T3, Games-Howell, and Dunnett's C.
      

      In this example post hoc tests were requested for the budget between-subjects factor. The illustration below displays only some of the many tests available. The researcher would not in practice generate such a large table, but rather would select specific tests as appropriate. See the discussion of post-hoc tests in the section on univariate GLM. In this example, as often happens, all tests come to the same finding: for any pairwise comparison between two of the three levels of the between-subjects factor budget, the levels are significantly different on the repeated dependent variable, poll.

      

    
    

  • Profile plots. Profile plots, also called interaction plots, of estimated means of levels of specific factors serve as the graphic equivalent of post hoc tests. Clicking the Plots button leads to the Profile Plots dialog illustrated below. In this example, a week*budget plot has already been added, and when the Add button is pressed, a week*stateid plot will also be selected. The profile plots will have week (4 levels = 4 weeks in this example) on the X axis and estimated marginal means for each week for either budget or stateid on the Y axis.
    
    • Example. The profile plot below shows the budget levels for each of the four weeks of measurement of poll, the repeated dependent measure. Budget category 1 (the blue line) is associated with the highest predicted poll results for all 4 weeks.
      

      A profile plot of stateid by week shows between-state variance in predicted mean poll results, with state 1 returning clearly higher predicted poll measures. However, in GLM between-subjects factors are modeled as fixed effects. That is, the model assumes the researcher is interested in these 10 specific states. If the researcher wishes to model states as a random effect, viewing the 10 states as a sample of the 50 states and enabling generalization to all states, then the researcher should use the linear mixed models procedure instead.

      

    
    
  • Contrast analysis
    • Contrast analysis compares levels of a within-subjects factor (ex., week with 4 levels, corresponding the the repeated measures dependent variables poll1, poll2, poll3, and poll4).

      

      Available contrasts are:

      • Deviation. All levels of a factor except one (the last by default) are compared to the grand mean of repeated measures. This is the default for between-subjects factors.

      • Polynomial. Polynomial contrasts are the default for within-subjects factors. The first degree of freedom reflects the linear effect, the second degrees of freedom reflects the quadratic effect, etc. If the design is balanced (same n in all cells), polynomial contrasts are othogonal. By default, the levels of the within-subjects factor (ex., week with 4 levels for the 4 weeks of measurement) are assumed to be equally spaced, but the researcher may specify other spacings.

      • Difference. Also called the reverse Helmert contrast, comparison is to the mean of prior levels.

      • Helmert. Comparison is to the mean of subsequent levels.

      • Simple. Comparison is to the last level, which is the reference level. The reference level is the last one by default, but the researcher may override this.

      • Repeated. Comparison is to the next level in sequence.

      • Special. It is possible to have researcher-defined contrasts.

      • Example. In the illustration below, simple contrasts were requested for the between-subjects (grouping) variable, budget. Because budget had three levels and the default was taken, the last (highest) budget level (3) became the reference category. The contrast table below shows that both levels 1 and 2 significantly differ from level 3 on the repeated dependent variable, poll. The "Test Results" table gives an overall contrast assessment (contrasts are significant) but perhaps more usefully, give an effect size summary measure, partial eta-squared, which, at .683, shows the magnitude of contrasts to be moderate to strong in size.
        

      
      

    • Saving variables
      • The Save button in SPSS GLM repeated measures supports saving standardized (Pearson), unstandardized, and studentized residuals, predicted values, the residual SSCP matrix, the residual covariance matrix, and the residual correlation matrix (standardized residual covariance matrix), and other measures below.

      • Cook's distance.. A measure of influential (outlying) cases, estimating how much residuals would change if the given case were removed from the dataset.. The larger the Cook's D, the more removal of that case would alter parameter estimates.

      • Leverage values. . Leverage is a measure of the relative influence of the given observation on model fit.
        

      
      

    Assumptions

    • Linearity. As in all general linear models, it is assumed that factors and covariates are linearly related to the dependent variable(s).

    • Normality. The dependent variables are assumed to take on a multivariate normal distribution.

    • Homogeneity of covariances. For purposes of multivariate tests, it is assumed that the covariance matrix formed by the dependent variables is the same for each combination of levels of between-subject effects.

    • Sphericity. For purposes of univariate tests, it is assumed that the covariance matrix formed by the dependent variables is also spherical (circular) in form, which means that the covariance between any two variables is equal to the average of their variances minus a constant.

    • Fixed effects. All effects are calculated on the assumption they are fixed. Other SPSS modules, notably linear mixed models, support random effects.

    • Homogeneity of variances and other assumptions of ANOVA models apply. The homogeneity of variances assumption is assessed with Levene's test, discussed above.

    
    

    Frequently Asked Questions

    • What is the SPSS syntax for GLM repeated measures?

      GLM dependent varlist [BY factor list [WITH covariate list]]
      /WSFACTOR=name levels [{DEVIATION [(refcat)]       }] name...
                             {SIMPLE [(refcat)]          }
                             {DIFFERENCE                 }
                             {HELMERT                    } 
                             {REPEATED                   }
                             {POLYNOMIAL [({1,2,3...})]**}
                             {             {metric  }    }
                             {SPECIAL (matrix)           }
      [/MEASURE=newname newname...]
      [/WSDESIGN=effect effect...]
      [/REGWGT=varname]
      [/METHOD=SSTYPE({1  })]
                      {2  }
                      {3**}
                      {4  }
      [/INTERCEPT=[INCLUDE**] [EXCLUDE]]
      [/MISSING=[INCLUDE] [EXCLUDE**]]
      [/PRINT  = [DESCRIPTIVE] [HOMOGENEITY] [PARAMETER][ETASQ] [RSSCP]
                 [GEF] [LOF] [OPOWER] [TEST [([SSCP] [LMATRIX] [MMATRIX])]]
      [/SAVE=[tempvar [(list of names)]] [tempvar [(list of names)]]...]
             [DESIGN]
      [/EMMEANS=TABLES({OVERALL             })] [COMPARE ADJ(LSD)(BONFERRONI)(SIDAK)]
                       {factor                }           
                       {factor*factor...      }                 
                       {wsfactor              }
                       {wsfactor*wsfactor...  }
                       {factor*...wsfactor*...]
                       {factor*factor...      }
      [/DESIGN={[INTERCEPT...]    }]*
               {[effect effect...]}
      
      * The DESIGN subcommand has the same syntax as is described in GLM: Univariate.
      ** Default if the subcommand or keyword is omitted. 
      
      Example
      GLM Y1 TO Y4 BY GROUP
        /WSFACTOR=YEAR 4.
      

    • What is a "doubly repeated measures design"?
      This simply refers to the fact that there can be more than one set of dependent variables. For instance, one may measure both achievement and endurance in four successive weeks. As usual, there in each set there is one dependent variable per level of the within-subject factor(s), which here would be week. The dependent variables would then be achieve1, achieve2, achieve3, and achieve4 for the achieve variable and would be endure1, endure2, endure3, and endure4 for the endurance variable.

  
  

  Bibliography

  • Huynh, H. & Mandeville, G. K. (1979). Validity conditions in repeated measures designs . Psychological Bulletin, 86, 964-973.