Lab 11. Multilevel Models. Description of Data

Transcription

1 Lab 11 Multilevel Models Henian Chen, M.D., Ph.D. Description of Data MULTILEVEL.TXT is clustered data for 386 women distributed across 40 groups. ID: 386 women, id from 1 to 386, individual level (level 1). GROUP: 40 groups, group from 1 to 40, group level (level 2). The groups meet regularly to discuss diet and weight control, group size ranges from 5 to 15 women. MOTIVATC: motivation to lose weight, measured on a six-point scale, and centered around the grand mean of the 386 cases. WEIGHTL: weight loss in pounds. WEIGHTG: 0=light weight loss (weight loss<15 pounds) 1=heavy weight loss (weight loss>=15) Linear Regression Model for n=386 (analysis based on individual level--level 1) proc import datafile='a:multilevel.txt' out=multilevel dbms=tab replace; proc print; proc reg; model weightl=; model weightl=motivatc; Dependent Variable: WEIGHTL Model 1 Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept <.0001 Model 2 Intercept <.0001 MOTIVATC <.0001

2 Linear Regression Models by group (analysis based on group level--level 2) proc reg outest=out; model weightl=motivatc; by group; data groups; set out; keep group intercept motivatc; proc print data=groups; proc univariate data=groups; var intercept motivatc; histogram intercept motivatc / normal; Linear Regression Models by group cont. GROUP INTERCEPT MOTIVATC (SLOPE) intercepts: from to slopes: from to Distribution of 40 Intercepts

3 Distribution of 40 Slopes Linear Regression Models by group cont. There is no need to use a two-level model if: 1) all 40 intercepts are identical (no variance among the intercepts), and 2) all 40 slopes are identical (no variance among the slopes). But in this case the intercepts and slopes varied. There is the effect of clustering or group membership on the weight loss. Unconditional Two-Level Linear Regression Model (1) proc mixed data=multilevel covtest noitprint; class group; model weightl= / solution; random intercept / subject=group; PROC MIXED statement calls the procedure. COVEST: tests the variance components (random effects). NOITPRINT statement tells SAS not to print the iteration history. CLASS statement specifies that GROUP is classification variable as opposed to continuous variable. MODEL statement is an equation whose left-side contains the name of the dependent variable, in this case WEIGHTL. The right-hand side contains a list of the fixed-effect variables (predictors). The intercept is contained in all models. We just test the intercept in this unconditional model. RANDOM statement contains a list of the random effects, in this case intercept.

4 Unconditional Two-Level Linear Regression Model (2) The Mixed Procedure Model Information Data Set WORK.MULTILEVEL Dependent Variable WEIGHTL Covariance Structure Variance Components Subject Effect GROUP Estimation Method REML Residual Variance Method Profile Fixed Effects SE Method Model-Based Degrees of Freedom Method Containment Dimensions Covariance Parameters 2 Columns in X 1 Columns in Z Per Subject 1 Subjects 40 Max Obs Per Subject 15 Observations Used 386 Observations Not Used 0 Total Observations 386 Unconditional Two-Level Linear Regression Model (3) Covariance Parameter Estimates (Random Effects) Standard Z Cov Parm Subject Estimate Error Value Pr Z Intercept GROUP Residual < is the variance among the 40 group intercepts (between group variance). This value is significantly greater than zero. It tells us that there is random variation among the intercepts of the individual groups, we should not ignore clustering is the level 1 residual variance (within group variance). This value is significantly greater than zero. The degree of clustering is measured by intraclass correlation (ICC). The ICC measures the proportion of the total variance of a variable that is accounted for by the clustering. The ICC ranges from 0 for complete independence of observations to 1 for complete dependence. One of the linear regression analysis assumption is that ICC=0. ICC = / ( ) = The estimated ICC of 0.24 is very substantial. Unconditional Two-Level Linear Regression Model (4) The Mixed Procedure: Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > t Intercept < tells us the average group-level weight loss in our sample (DF=39). Model 1 (n=386) Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept < tells us the average individual level weight loss. In OLS model, the standard error is , smaller than the standard error of multilevel model (0.4090). This means OLS overestimates the finding.

5 Two-Level Model Including Level-1 Predictor (motivatc) (1) proc mixed data=multilevel covtest noitprint; class group; model weightl=motivatc / solution notest; random intercept motivatc / subject=group; Solution for Fixed Effects Effect Estimate Standard Error DF t Value Pr > t Intercept <.0001 MOTIVATC <.0001 For every one unit increase in motivation (6-point scale), the weight loss increases by 3.08 pounds, with an average weight loss per group of pounds. Two-Level Model Including Level-1 Predictor (motivatc) (2) Covariance Parameter Estimates (Random Effects) Standard Z Cov Parm Subject Estimate Error Value Pr Z Intercept GROUP MOTIVATC GROUP Residual < is the variance among the 40 group intercepts. This value is significantly greater than zero. Significant random variation among the intercepts indicates we need to include clustering in our analysis is the variance among the 40 group slopes. This value is significantly greater than zero. Significant random variation among the slopes indicates we need to include clustering in our analysis is the level 1 residual variance. This value is significantly greater than zero. Two-Level Model Including Level-1 Predictor (motivatc) (3) Unconditional Model Conditional Model ( + motivatc ) Intercept Residual ( ) / = 63.4% 63.4% of the within group variance in weight loss has been accounted for by the level 1 motivation variable. ( ) / = 49.0% 49.0% of the between-group differences in average weight loss has been accounted for by the level 1 motivation variable.

6 Logistic Regression Model proc logistic data=multilevel descending; model weightg=motivate; Total WEIGHTG Frequency Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept <.0001 MOTIVATE <.0001 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits MOTIVATE Two-Level Logistic Regression Model (1) SAS Program %inc 'c:\apps\sas institute \sas\v8\stat\sample \glimmix.sas /nosource; %glimmix(data=multilevel, stmts=%str( class group; model weightg=motivate / notest solution; random intercept / subject=group;), error=binomial) Two-Level Logistic Regression Model (2) Covariance Parameter Estimates (Random Effects) Cov Parm Subject Estimate Intercept GROUP Residual ICC = / ( ) = 64.8%. We have to employ the two-level logistic regression model. Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > t Intercept <.0001 MOTIVATE <.0001 OR=exp(2.0699)=7.92, the odds of weight loss increase by 7.92 for each unit increase in motivate.

7 Multilevel Growth Models Models that represent individual trajectories over age or time A common form of growth trajectory is specified for all individuals, but individuals may vary in the parameters that characterize the growth (e.g., slope and intercept for linear growth). ID AGE WAVE XFIN2 DXPDSUMA SEX Our Transitions data consists of monthly records for 233 subjects ages 17 to 27. Subject is the first subject in our data and has 100 records, only some of which are shown here. Subject is the last subject in our data and here is represented by a few of the 120 months of data. We are looking at the relationship between Finance Transitional Level (XFIN2, from 0 to 100), sex (male=1), and any PD diagnosis in adolescence (DXPDSUMA, yes=1). We centered age by 22. Unconditional Linear Growth Model (1) SAS Program proc mixed method=ml noclprint covtest noitprint; class id wave; model xfin2 = age /solution ddfm=bw notest; random intercept age / type=un subject=id; repeated wave / type=ar(1) subject=id; * RANDOM / TYPE=un, is the structure of the variancecovariance matrix for the intercepts and slopes. ** REPEATED / TYPE=ar(1) is the structure of variancecovariance matrix within person (error-covariance structure)

8 Unconditional Linear Growth Model (2) Covariance Parameter Estimates (Random Effects) Standard Z Cov Parm Subject Estimate Error Value Pr Z Intercept ID < Slope ID Intercept*Slope ID < AR(1) ID < Residual < is the variance among the 233 intercepts. This value is significantly greater than zero; intercepts vary across persons is the variance among the 233 slopes. P= means slopes don t vary across persons. ICC for intercepts = / ( ) = Unconditional Linear Growth Model (3) Solution for Fixed Effects Effect Estimate Standard Error DF t Value Pr > t Intercept <.0001 age E < is our estimate of the average intercept across persons (the average finance transition level when age=22) is our estimate of the average slope across persons. The average young adult has a finance transition level score of at age 22, and increasing about 4 points per year. Multilevel Growth Model including Level-2 Predictors (1) SAS Program proc mixed method=ml noclprint covtest noitprint; class id wave; model xfin2 = age sex dxpdsuma age*sex age*dxpdsuma /solution ddfm=bw notest; random intercept age / type=un subject=id; repeated wave / type=ar(1) subject=id; This model adds sex, PD and their slopes to the previous model.

9 Multilevel Growth Model including Level-2 Predictors (2) Covariance Parameter Estimates (Random Effects) Standard Z Cov Parm Subject Estimate Error Value Pr Z Intercept ID < Slope ID Intercept*Slope ID < AR(1) ID < Residual < The variance for the intercepts changed from to Computing ( ) / = 0.154, we find a 15.4% reduction. In other word, sex, PD, and their interaction with age account for 15.4% of the explained variation in intercepts. The variance for the slopes changed from to Computing ( ) / = 0.192, we find a 19.2% reduction. In other word, sex, PD, and their interaction with age account for 19.2% of the explained variation in slopes. Multilevel Growth Model including Level-2 Predictors (3) Effect Estimate Standard Error DF t Value Pr > t Intercept <.0001 age E <.0001 sex dxpdsuma age*sex E age*dxpdsuma E is our estimate of the average intercept across persons (the average finance transition level for female without PD at their age 22) is our estimate of the average slope across persons, increasing about 3.7 points per year is the difference on finance transition level between male and female. Male is higher than female is the difference on finance transition level between subjects with PD and subjects without PD. Subjects with PD are lower than subjects without PD is the difference on linear slope between male and female. Slope for male is = Female s slope is SEXPNT Age 22, male intercept Age 22, female 30 intercept = AGE Here, graphically, is the difference between males and females SEX Female Male is the difference on finance transition level between male and female. Male is higher than female is the difference on linear slope between male and female. Slope for male is = Female s slope is

10 Finance Transitional Level Here, graphically, is the difference between subjects with adolescent PD and those without DX No diagnosis PD diagnosed in adolescence AGE is the difference on finance transition level between subjects with PD and subjects without PD. Subjects with PD is less than subjects without PD is the difference on linear slope between subjects with PD and subjects without PD. Slope for subjects with PD is = Slope for subjects without PD is