1. Longitudinal data: negative correlation within-person example
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1 Lecture Longitudinal data: negative correlation within-person example 2. Mixed model example: assay plates 1 Negative correlation within-person Study to compare bioimpedance methods of estimating total body fluid to the reference method, dilution. Participants were women who had received gastric bypass surgery, and they were measured at post-surgery weeks 2, 6, 24 (6 months), and 52 (1 year). Response is difference = (bioimpedance estimate reference value). Obs ID _NAME_ diff_2 diff_6 diff_24 diff_ d_bs_tbf d_bs_tbf d_bs_tbf d_bs_tbf d_bs_tbf
2 3 Pearson Correlation Coefficients Prob > r under H0: Rho=0 Number of Observations diff_2 diff_6 diff_24 diff_52 diff_ diff_ diff_ diff_
3 Longitudinal plot: individuals alternate high/low values 5 Compare two specifications of the random intercept/compound symmetry model: proc mixed data=tbf_diffs1; class weeks id; model d_bs_tbf = weeks; repeated weeks / subject=id type=cs r rcorr; lsmeans weeks; proc mixed data=tbf_diffs1; class weeks id; model d_bs_tbf = weeks; random intercept / subject=id type=cs v vcorr; lsmeans weeks; 6
4 Repeated-model: Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F weeks Least Squares Means Standard Effect weeks Estimate Error DF t Value Pr > t weeks weeks weeks <.0001 weeks Repeated-model: Estimated R Matrix for ID Row Col1 Col2 Col3 Col Estimated R Correlation Matrix for ID Row Col1 Col2 Col3 Col Covariance Parameter Estimates Cov Parm Subject Estimate CS ID Residual
5 Using the Random specification does not give F-tests nor LSmeans: The Mixed Procedure WARNING: Did not converge. Covariance Parameter Values At Last Iteration Cov Parm Subject Estimate Variance ID 0 CS ID Residual Random specification cannot fit negative correlation 9 Mixed Model example: Rat Diets Earlier we fit a linear model to data from a study by Sabrina Peterson (Food Science and Nutrition) which examined effects on MROD (liver enzyme) over time from diets containing: cruciferous (C) vegetables : broccoli, cabbage, and watercress apiaceous (A) vegetables: parsnips and celery 2 2 factorial in diets: Basal (control), A, C, A+C, with 30 rats assigned to each At three times (7, 30, 60 days), they sacrificed 10 rats from each diet group to measure MROD factorial : A C day (time) 10
6 day diet Frequency Control A C AC Total Total Slightly unbalanced because 2 rats not measured. 11 Part of the data: Obs Plate animal Liver_wt MROD Api Cru day diet C C AC AC Two different ways to specify the 4 diets: by diet (0, A, C, AC) used to make interaction plot with time by combinations of Cru and Api used in model 12
7 Fixed-effects model: Proc GLM Preliminary model: factorial: A C day Proc GLM data=ph6470.rat_diets; class day api cru ; model MROD = liver_wt api cru api*cru day day*api day*cru day*api*cru; lsmeans day*api*cru / slice=day; All 2-factor and 3-factor interactions between experimental factors. 13 Adjust for weight of liver, which was strongly associated with MROD level, but not balanced across days. 14
8 Source DF Type III SS Mean Square F Value Pr > F Liver_wt Api Cru <.0001 Api*Cru day day*api day*cru day*api*cru Which factors affect response MROD? 15 LSmeans adjusted for liver weight: Least Squares Means Standard day Api Cru MROD LSMEAN Error Pr > t < < < < < < < < < < < <
9 Interaction plot: 17 Two main reasons for adjusting comparison of groups: For balance, when groups have different distributions of the predictor Z Adjust for liver weight, which is associated with response and not balanced between time groups To reduce error variance, when predictor Z is associated with response and known to vary across subjects Liver enzyme MROD assayed in plates, 8 samples at a time. Adjusting for plate will remove variability between plates from the error term. Why is it good to reduce error variance? 18
10 Adjust for weight of liver, which was strongly associated with MROD level, but not balanced across days. 19 Adjust for assay plate: 20
11 Fixed-effects model 2 Problem: 10 rats per treatment group, 8 rats per plate Proc GLM data=ph6470.rat_diets; class day api cru plate ; model mrod = plate liver_wt api cru api*cru day day*api day*cru day*api*cru / solution; lsmeans day*api*cru / stderr slice=day; 21 Source DF Type III SS Mean Square F Value Pr > F Plate Liver_wt Api Cru <.0001 Api*Cru day day*api day*cru day*api*cru Plate is significant, and requires estimating 12 fixed-effect parameters. 22
12 Regression coefficients: Standard Parameter Estimate Error t Value Pr > t Plate B Plate B Plate B Plate B Plate B Plate B Plate B Plate B Plate B Plate B Plate B... Liver_wt Api B Api B... Cru B Cru B... Api*Cru B Api*Cru B... Api*Cru B... Api*Cru B... day B... day B day B... day*api B day*api B... day*api B day*api B... day*api B... day*api B... day*cru B day*cru B... day*cru B day*cru B... day*cru B... day*cru B... day*api*cru B day*api*cru B... day*api*cru B... day*api*cru B... day*api*cru B day*api*cru B... day*api*cru B... day*api*cru B... day*api*cru B... day*api*cru B... day*api*cru B... 24
13 When we adjust for plate, we cannot get estimates of the means: Least Squares Means day Api Cru MROD LSMEAN Non-est Non-est Non-est Non-est Non-est Non-est Non-est Non-est Non-est Non-est Non-est Non-est Non-est means non-estimable. 25 Mixed model approach Treat plates as a random effect: Proc Mixed data=ph6470.rat_diets; class day api cru plate ; model mrod = liver_wt api cru api*cru day day*api day*cru day*api*cru; only fixed effects in model random plate / subject=plate v vcorr; lsmeans day*api*cru ; 26
14 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F Liver_wt Api Cru <.0001 Api*Cru day day*api day*cru day*api*cru F-statistic for Cru is a little larger than in the model without plate: Source DF Type III SS Mean Square F Value Pr > F Cru < Now we have LSmeans, adjusted for liver weight and correlations within plates: Least Squares Means Standard Effect day Api Cru Estimate Error day*api*cru day*api*cru day*api*cru day*api*cru day*api*cru day*api*cru day*api*cru day*api*cru day*api*cru day*api*cru day*api*cru day*api*cru
15 How does this work? Proc GLM tries to estimate a regression coefficient (fixed effect) for each plate: Ø 1,...,Ø 15 and spends 12 parameters on this. Proc Mixed assumes plates p j are Normal(0, æ 2 ) and estimates the single Plates parameter that specifies this distribution: æ 2 Plates Cov Parm Subject Estimate Plate Plate Residual The random intercepts for plates ˆp j are residuals 29 Code for interaction plot must include the random effect: ods graphics on; ods select lsmeans meanplot; Proc Glimmix data=ph6470.rat_diets; class diet day plate; model mrod = liver_wt diet day day*diet ; random plate / subject=plate; lsmeans day*diet/ alpha=.32 plots=(meanplot(cl join sliceby=diet)); run; ods graphics off; 30
16 31
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