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Handout21 - Lecture 21 1 Longitudinal models for mean and...

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Lecture 21 1. Longitudinal models for mean and correlation 2. Correlation and covariance matrices; compound symmetry 3. Proc Mixed : repeated measures 4. Proc Mixed : random effects 1 Longitudinal data: example 1 Family economics data: total family income, expenditures, debt status for 50 families in two groups ( A and B ), annual records from 1990–1995. family_ Obs id income year expenses debt cohort 1 1 66483 1990 49804 no A 2 1 69146 1991 65634 no A 3 1 74643 1992 61820 no A 4 1 79783 1993 68387 no A 5 1 81710 1994 85504 yes A 6 1 86143 1995 75640 no A 7 2 17510 1990 21609 yes B 8 2 19484 1992 18180 no B 9 2 20979 1993 22985 yes B 10 2 21268 1994 11097 no B 11 2 22998 1995 21768 no B 2
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Plots of individual family data by group: 3 Within-subject correlation Repeated observations from the same subject are correlated. Use wide form of data to see within-family correlations: ODS graphics on; Proc Corr data=family_econ_wide plots=matrix ; var income_1990-income_1995; wide form variables run; ODS graphics off; 4
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5 Pearson Correlation Coefficients Prob > |r| under H0: Rho=0 Number of Observations income_ income_ income_ income_ income_ income_ 1990 1991 1992 1993 1994 1995 income_1990 1.00000 0.99817 0.99506 0.99344 0.98895 0.98883 <.0001 <.0001 <.0001 <.0001 <.0001 46 42 40 41 43 44 income_1991 0.99817 1.00000 0.99735 0.99604 0.99282 0.99194 <.0001 <.0001 <.0001 <.0001 <.0001 42 46 39 42 42 44 income_1992 0.99506 0.99735 1.00000 0.99739 0.99195 0.99331 <.0001 <.0001 <.0001 <.0001 <.0001 40 39 43 38 39 41 income_1993 0.99344 0.99604 0.99739 1.00000 0.99766 0.99674 <.0001 <.0001 <.0001 <.0001 <.0001 41 42 38 45 41 43 income_1994 0.98895 0.99282 0.99195 0.99766 1.00000 0.99817 <.0001 <.0001 <.0001 <.0001 <.0001 43 42 39 41 46 44 income_1995 0.98883 0.99194 0.99331 0.99674 0.99817 1.00000 <.0001 <.0001 <.0001 <.0001 <.0001 44 44 41 43 44 47 6
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Longitudinal data: example 2 Alzheimer’s disease trial. Alzheimer’s disease is a progressive incurable deterioration of intellect and memory. A clinical trial compared lecithin (dietary supplement) against placebo, both given as daily for 4 months; 22 patients in lecithin group, 25 in placebo group. Participant took a memory test at baseline (first visit), and end of each month. Score is number of words recalled from a list, so higher scores are better. idno lecithin score1 score2 score3 score4 score5 1 0 20 15 14 13 13 2 0 14 12 12 10 10 3 0 7 5 5 6 5 4 0 6 10 9 8 7 ( Source: Der and Everitt, Ch. 11) 7 Plot individual profiles and correlation scatterplot matrix. Proc SGpanel data=alz_long; PanelBy lecithin / columns=2; series x=visit y=score / group=idno LINEATTRS= (pattern=1 color="black"); ODS graphics on; Proc Corr data=alzheimer plots=matrix; var score1-score5; run; ODS graphics off; 8
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Decrease in correlation over longer time intervals: Pearson Correlation Coefficients, N = 47 Prob > |r| under H0: Rho=0 score1 score2 score3 score4 score5 score1 1.00000 0.66267 0.67951 0.42892 0.30906 <.0001 <.0001 0.0026 0.0345 score2 0.66267 1.00000 0.86712 0.75344 0.66498 <.0001 <.0001 <.0001 <.0001 score3 0.67951 0.86712 1.00000 0.82909 0.76285 <.0001 <.0001 <.0001 <.0001 score4 0.42892 0.75344 0.82909 1.00000 0.95437 0.0026 <.0001 <.0001 <.0001 score5 0.30906 0.66498 0.76285 0.95437 1.00000 0.0345 <.0001 <.0001 <.0001 11 Consequence of within-subject correlation Repeated longitudinal observations from the same subject are correlated = within-subject observations are not independent . ANOVA and regression assume independent observations, hence don’t apply correctly to correlated data. Model for longitudinal observations must include within-subject correlation.
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