Title 'Vocabulary Growth Study: Repeated measures analysis'; data vocab; *------------Longitudinal vocabulary study ------------------ | Data from the Laboratory School of Univ. of Chicago. They | consist of scores from a cohort of pupils in grades 8-11 on | the vocabulary section of the Cooperative Reading Test. The | scores are scaled to a common, but arbitrary origin and unit | of measurement, so as to be comparable over the four grades. | | Since these data cover an age range in which physical growth | is beginning to decelerate, it is of interest whether a | similar effect occurs in the acquisition of new vocabulary. | | Source of data: R.D. Bock, "Multivariate statistical | methods", 1975, pp453. See also Keesling, J.W., Bock, R.D. | et al, "The Laboratory School study of vocabulary growth, | University of Chicago, 1975. *-------------------------------------------------------------; input subject grade8-grade11; label grade8 = 'Grade 8 vocabulary score' grade9 = 'Grade 9 vocabulary score' grade10 = 'Grade 10 vocabulary score' grade11 = 'Grade 11 vocabulary score'; datalines; 1 1.75 2.6 3.76 3.68 2 .9 2.47 2.44 3.43 3 .8 .93 .4 2.27 4 2.42 4.15 4.56 4.21 5 -1.31 -1.31 -.66 -2.22 6 -1.56 1.67 .18 2.33 7 1.09 1.50 .52 2.33 8 -1.92 1.03 .50 3.04 9 -1.61 .29 .73 3.24 10 2.47 3.64 2.87 5.38 11 -.95 .41 .21 1.82 12 1.66 2.74 2.40 2.17 13 2.07 4.92 4.46 4.71 14 3.30 6.10 7.19 7.46 15 2.75 2.53 4.28 5.93 16 2.25 3.38 5.79 4.40 17 2.08 1.74 4.12 3.62 18 .14 .01 1.48 2.78 19 .13 3.19 .60 3.14 20 2.19 2.65 3.27 2.73 21 -.64 -1.31 -.37 4.09 22 2.02 3.45 5.32 6.01 23 2.05 1.80 3.91 2.49 24 1.48 .47 3.63 3.88 25 1.97 2.54 3.26 5.62 26 1.35 4.63 3.54 5.24 27 -.56 -.36 1.14 1.34 28 .26 .08 1.17 2.15 29 1.22 1.41 4.66 2.47 30 -1.43 .80 -.03 1.04 31 -1.17 1.66 2.11 1.42 32 1.68 1.71 4.07 3.30 33 -.47 .93 1.30 .76 34 2.18 6.42 4.64 4.82 35 4.21 7.08 6.00 5.65 36 8.28 9.55 10.24 10.58 37 1.24 4.90 2.42 2.54 38 5.94 6.56 9.36 7.72 39 .87 3.36 2.58 1.73 40 -.09 2.29 3.08 3.35 41 3.24 4.78 3.52 4.84 42 1.03 2.10 3.88 2.81 43 3.58 4.67 3.83 5.19 44 1.41 1.75 3.70 3.77 45 -.65 -.11 2.40 3.5 46 1.52 3.04 2.74 2.63 47 .57 2.71 1.90 2.41 48 2.18 2.96 4.78 3.34 49 1.10 2.65 1.72 2.96 50 .15 2.69 2.69 3.50 51 -1.27 1.26 .71 2.68 52 2.81 5.19 6.33 5.93 53 2.62 3.54 4.86 5.80 54 .11 2.25 1.56 3.92 55 .61 1.14 1.35 .53 56 -2.19 -.42 1.54 1.16 57 1.55 2.42 1.11 2.18 58 -.04 .50 2.60 2.61 59 3.10 2.00 3.92 3.91 60 -.29 2.62 1.60 1.86 61 2.28 3.39 4.91 3.89 62 2.57 5.78 5.12 4.98 63 -2.19 .71 1.56 2.31 64 -.04 2.44 1.79 2.64 ; *----------------------------------------------------------* | The REPEATED statement makes the analysis of repeated | | measures simple and efficient. Note that since there is | | no between-S design, there is no independent variable | | specified on the MODEL statement. | *----------------------------------------------------------; proc glm data=vocab; model grade8-grade11 = /nouni; repeated grade 4 (8 9 10 11) polynomial / short summary printh printe; Title2 'Multivariate Repeated Measures Analysis'; run; *-------------------------------------------------------------* | The older approach required putting the data in a univariate | form, strung out so each measure was a separate observation | in a single variable. The rest of the analysis below is for | comparison only. *-------------------------------------------------------------; data vravel; set vocab; keep subject grade vocab; grade=8; vocab=grade8; output; grade=9; vocab=grade9; output; grade=10; vocab=grade10; output; grade=11; vocab=grade11; output; run; title2 'Univariate Mixed Model Analysis'; *--------------------------------------------------------------* | In the mixed model (Subjects random, Grade fixed), the error | term for the Grade effect is the GRADE*SUBJECT interaction. | Since this term is not listed in the MODEL statement, it | goes into the residual. The mixed model, however, required | the assumption of compound symmetry (which IS satisfied | here). This form of analysis is HORRIBLY INEFFFICIENT if | you have large N. CONTRAST statements are used to pull out | the trend components of the GRADE effect. *---------------------------------------------------------------; proc glm data=vravel; classes subject grade; model vocab = grade subject; contrast 'Linear' grade -3 -1 1 3; contrast 'Quadratic' grade 1 -1 -1 1; contrast 'Cubic' grade -1 3 -3 1; estimate 'Slope' grade -3 -1 1 3; estimate 'Curvature' grade 1 -1 -1 1; title2 'Mixed Model Analysis with Trends'; run; *-------------------------------------------------------------- | The first GLM with the REPEATED .... POLYNOMIAL statement | transforms the data to scores on orthogonal polynomials. The | steps below do the same analysis by explicitly calculating | these scores and then performing the multivariate tests on | the transformed scores. *---------------------------------------------------------------; data trends; set vocab; mean = mean(of grade8-grade11); linear = -3*grade8 -1*grade9 +1*grade10 +3*grade11; quad = 1*grade8 -1*grade9 -1*grade10 +1*grade11; cubic = -1*grade8 +3*grade9 -3*grade10 +1*grade11; Title2 'Transformation to orthogonal polynomial scores'; proc print data=trends(obs=40); id subject; proc summary; var grade8-grade11 mean--cubic; output out=means mean=grade8-grade11 mean linear quad cubic; proc print; var grade8-grade11 mean linear quad cubic; *------------------------------------------------------------ | If there were a between-S factor, it would be tested based | on the MEAN over the repeated measures, e.g, with a model | MEAN=GROUP. Here, we just test whether the mean is a | constant, as in a 1-sample t-test. *------------------------------------------------------------; PROC GLM DATA=trends; model MEAN = ; title2 'Test of mean'; *------------------------------------------------------------* | Next, test the trend components using a multivariate test. | Again, if there were between-S factors, they would be | included in the model. The multivariate test of the | intercept gives the within-S effects. The test of GROUP | (if present) would give the interactions of within-S with | between-S effects. *------------------------------------------------------------; PROC GLM DATA=trends; model LINEAR QUAD CUBIC = ; manova h=intercept / short printe printh; TITLE2 'Multivariate Analysis of Trend Scores'; run;