Analysis of Longitudinal Data: Comparison Between PROC GLM and PROC MIXED. Maribeth Johnson Medical College of Georgia Augusta, GA

Transcription

1 Analysis of Longitudinal Data: Comparison Between PROC GLM and PROC MIXED Maribeth Johnson Medical College of Georgia Augusta, GA

2 Overview Introduction to longitudinal data Describe the data for examples (WT and HR) Analysis using PROC GLM and the RANDOM statement Analysis using PROC GLM and the REPEATED statement Analysis using PROC MIXED Compare error V-C estimates from each Use data with no missing values Repeat using data with missing values Conclusions and Implications

3 Longitudinal Data Multiple measurements of a response variable On the same experimental unit Made over a period of time Experimental units are considered as random effects

4 Correlated Data Quantify the inter-relationships of the measurements If variances and covariances are constant then the relationship can be thought of as ICC reliability Assumption of compound symmetry can be tested against other variance structures using PROC MIXED

5 Correlated Data The ability to model different covariance structures provides an opportunity to investigate and possibly quantify the tracking of k measurements as well as provide the basis for other relationships y ~ N( μ, Σ) where Σ 1 1 = M 1 κ 1 M κ L L O L 1κ κ M k

6 Data Texas site in the studies of child activity and nutrition (SCAN) program - TXSCAN Children aged 3 or 4 years at the start

7 Data Structure Number of children Number of records WT HR 1 of of of of

8 Data Structure WT HR Year N Mean SD N Mean SD

9 Using GLM With The Random Statement Univariate GLM proc glm; class child year; model depvar = child year; random child; Mixed model univariate analysis of variance Random statement prints table of expected MS

10 Using GLM With The Random Statement Univariate GLM The GLM Procedure Source Type III Expected Mean Square CHILD Var(Error) + 4 Var(CHILD) YEAR Var(Error) + Q(YEAR)

11 Using GLM With The Random Statement Univariate GLM E(MS) Child = Var(Error) + k Var(CHILD) k is the average number of observations per child CHILD is treated as a fixed effect Variance components calculated using Method of Moments Var(CHILD) = MS(CHILD) - MS(Error) k ICC = VAR(CHILD) VAR(CHILD) + VAR(Error)

12 Using GLM With The Random Statement Univariate GLM Reliability of a single year of measurement The individual components of variation are maximum-likelihood estimators when the data are balanced

13 Using GLM With The Repeated Statement MV GLM Can t handle missing data Repeated measurements must appear in a multivariate mode in the dataset, one observation for each experimental unit Allows for the multivariate test of the assumption of compound symmetry (CS) If the CS assumption is rejected it can t help in determining the correct underlying covariance structure

14 Using GLM With The Repeated Statement MV GLM proc glm; model depvar1-depvar4 = /nouni; repeated year / printe; NOUNI option suppresses the univariate analyses of each year PRINTE option outputs the partial correlations computed from residuals after fitting the between-subjects model

15 Test for Sphericity Tests whether a set of orthonormal contrasts of the repeated measures variables are independent and have equal variances, i.e. are the data compound symmetric Significance tells you that this condition is not met The estimate of the correlation between measures from the univariate GLM analysis is not valid

16 Test for Sphericity Correlations from the PRINTE option of this analysis are identical to the correlations computed from the RCORR option in the REPEATED statement in PROC MIXED when TYPE=UN is specified as the covariance structure REPEATED statements perform different functions

17 Using PROC MIXED Data is in the same format as GLM using the Random statement (i.e. Univariate GLM) Multiple observations for each experimental unit

18 Standard Linear Model y = Xβ + ε y X is vector of observed data is the design matrix β is unkown vector ο f fixed effects ε is unknown random error vector, iid N ~ (0, )

19 Mixed Model y= Xβ + Zγ + ε Z is a design matrix γ is unkown vector ο f random effects ε is unknown random error vector, no longer required to be iid

20 Using PROC MIXED proc mixed; class child year; model depvar = year; repeated year / subject=child r rcorr type=cov-structure ; REPEATED statement models the covariance structures in R, the variance-covariance matrix of the vector of errors If no REPEATED statement is specified, R is assumed to be equal to I.

21 Using PROC MIXED proc mixed; class child year; model depvar = year; repeated year / subject=child r rcorr type=cov-structure ; SUBJECT=CHILD option is the mechanism for block diagonalizing R

22 Using PROC MIXED proc mixed; class child year; model depvar = year; repeated year / subject=child r type=cov-structure ; rcorr R option of the REPEATED statement requests that the first block of the R matrix be printed

23 Using PROC MIXED proc mixed; class child year; model depvar = year; repeated year / subject=child r rcorr type=cov-structure ; RCORR options prints the correlation matrix corresponding to R

24 Using PROC MIXED proc mixed; class child year; model depvar = year; repeated year / subject=child r rcorr type=cov-structure ; TYPE= option is what determines the V-C structure

25 Covariance Structures Compound symmetric (CS): Most specific structure Variance within years is constant Common correlation between years Two parameters estimated Assumption of Univariate GLM estimates

26 Covariance Structures Heterogeneous compound symmetric (CSH): Common correlation Different variances along the diagonal Five parameters estimated for these data ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ

27 Covariance Structures First-order autoregressive (AR(1)): Variance within years is constant Estimate of the autoregressive parameter Correlations between years separated by the same amount of time are the same (ρ m ) Two parameters estimated ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ

28 Covariance Structures Heterogeneous first-order autoregressive (ARH(1)): Different variances along the diagonal Estimate of the autoregressive parameter Correlations between years separated by the same amount of time are the same (ρ m ) Five parameters estimated for these data ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ

29 Covariance Structures Unstructured (UN): Estimates of all four variances and six covariances for these data All of the correlations between years may be different Identical to those from the PRINTE option of the REPEATED statement in GLM

30 Determine The Preferred Model Likelihood Ratio Test (LRT) One model is a submodel of another Compute - times the difference between their residual log likelihoods (-RLL) Chi-square distribution with degrees of freedom equal to the difference in the number of parameters for the two models Models are preferred where the -RLL is smaller

31 Determine The Preferred Model Akaike s Information Criterion (AIC) Schwarz s Bayesian Criterion (BIC) Model that has the smallest value is the preferred model BIC penalizes models with more covariance parameters more than AIC

32 Determine The Preferred Model Fit Statistics - Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better)

33 Example WT - Balanced Data WT Year N Mean SD

34 V-C and Correlations WT - Balanced Data (N=98) Type = CS: (Variance and covariances in top line, correlations below) Type = CSH: (Variances on diagonal, covariances above, correlations below) parameters estimated from the CS structure are identical to those calculated using the Univariate GLM mean square estimates

35 V-C and Correlations WT - Balanced Data (N=98) Type = CS: (Variance and covariances in top line, correlations below) Type = CSH: (Variances on diagonal, covariances above, correlations below) The sphericity test from the MV GLM was significant

36 V-C and Correlations WT - Balanced Data (N=98) Type = CS: (Variance and covariances in top line, correlations below) Type = CSH: (Variances on diagonal, covariances above, correlations below) The correlation from the CSH structure is identical to that calculated using the Univariate GLM analysis of WT standardized within year

37 Model Comparisons WT - Balanced Data Type Parameters -RLL Comparison Model Chi-square / df CS CSH CS 88 / 3 * *p<.0005 Heterogeneous variances along the diagonal provide a significantly better fit for the CS model

38 V-C and Correlations WT - Balanced Data (N=98) Type = AR(1): (Variance and covariances in top line, correlations below) Type = ARH(1): (Variances on diagonal, covariances above, correlations below) The correlations do not fall off as quickly when variances are allowed to differ

39 Model Comparisons WT - Balanced Data Type Parameters -RLL Comparison Model Chi-square / df AR(1) ARH(1) 5 19 AR(1) 6 / 3 * *p<.0005 Heterogeneous autoregressive covariance structure (ARH(1)) provides a better fit than the assumption that within year variances are equal (AR(1))

40 V-C and Correlations WT - Balanced Data (N=98) Type = CSH: Type = ARH(1): Type = UN: UN produced very similar results to ARH(1) UN identical to the correlations from the Multivariate GLM analysis

41 Model Comparisons WT - Balanced Data Type Parameters -RLL Comparison Model Chi-square / df CSH ARH(1) UN CSH 86 / 5 * UN ARH(1) 8 / 5 *p<.0005 UN provides a significantly better fit to the data than CSH ARH(1) appears to provide the best fit for these data Improper models may underestimate the correlation between adjacent measurements

42 Example WT - Unbalanced WT Year N Mean SD

43 V-C and Correlations WT - Unbalanced Data (N=57) Type = CS: (Variance and covariances in top line, correlations below) Type = CSH: (Variances on diagonal, covariances above, correlations below) Estimates from Univariate GLM no longer ML estimators, not identical to CS but similar Var: 1. vs 11.5 Cov: 10.1 vs 9.5

44 V-C and Correlations WT - Unbalanced Data (N=57) Type = CSH: Type = ARH(1): Type = UN: UN produced very similar results to ARH(1)

45 V-C and Correlations WT Preferred Models Type = ARH(1): Balanced data Type = ARH(1): Unbalanced data Correlation estimates are similar Anywhere from 37 to 145 more children measured in any one year

46 Example HR - Balanced Data HR Year N Mean SD

47 V-C and Correlations HR Balanced Data (N=60) ANOVA estimates and those using the CS structure are identical (r=0.55) Common correlation estimate is similar using CSH (r=0.56) Heterogenous variances do not provide a better fit for these data The MV GLM s test of sphericity was significant so CS is not the correct structure

48 V-C and Correlations HR - Balanced Data (N=60) Type = UN: (Variances on diagonal, covariances above, correlations below) The correlation between successive years is lower when the children are young

49 Model Comparisons WT - Balanced Data Type Parameters -RLL Comparison Model Chi-square / df CS AR(1) UN CS 17 / 8 * UN AR(1) 1 / 8 * *p<.05 UN provides a significantly better fit to the data than CS UN provides a significantly better fit to the data than AR(1)

50 V-C and Correlations HR Preferred Models Type = UN: Balanced data Type = UN: Unbalanced data Correlation estimates are similar Variance smaller in third year There are from 50 to 189 more children in unbalanced analysis

51 V-C and Correlations HR Preferred Models Type = UN: Balanced data Type = UN: Unbalanced data Correlation higher between year 3 and 4 As children age there may be higher correlation between adjacent years

52 Conclusions Univariate GLM V-C calculations are identical to MIXED estimates using type=cs when there is no missing data (ICC) MV GLM V-C estimates are identical to MIXED estimates using type=un If sphericity assumption violated, MV GLM cannot determine best fit

53 Conclusions The ability to model a broader class of V-C structures yield results that make more sense in the context of the problem

54 Conclusions WT variability increases with age Correlation higher between adjacent years ARH(1) appears useful in explaining how WT tracks in very young children

55 Conclusions The UN nature of the relationship between HR measurements aids in assessment of the quality of this measurement

56 Conclusions PROC MIXED gives an opportunity to understand and quantify the true inter-relationships between repeated measurements

57 Implications Inferences for fixed effects may be impacted by the poor choice of error covariance matrix Those impacted will be the repeated variable and its interactions

58 Implications - Example Group Effect of TM intervention on resting SBP N Obs Variable Time N Mean Std Dev TM 30 SUPSBP_A SUPSBP_C SUPSBP_D Control 3 SUPSBP_A SUPSBP_C SUPSBP_D

59 Implications - Example Effect of TM intervention on resting SBP TM or Control=TM TM or Control=Control sbp 160 sbp time time

60 Implications - Example Univariate GLM with Random statement proc glm data=long; class group time family; model sbp=group family(group) time group*time; random family(group) / test; lsmeans group*time/stderr; run; quit;

61 Implications - Example Univariate GLM with Random statement Source DF Type III SS Mean Square F Value Pr > F GROUP*time Error: MS(Error)

62 Implications - Example Multivariate GLM with Repeated statement proc glm data=wide; class group; model supsbp_a supsbp_c supsbp_d = group /nouni; repeated time 3 (0 48) / printe; lsmeans group/ stderr; run; quit;

63 Implications - Example Multivariate GLM with Repeated statement Univariate Tests of Hypotheses for Within Subject Effects Adj Pr > F Source DF Type III SS Mean Square F Value Pr > F G-G H-F time*group Error(time)

64 Implications - Example Multivariate GLM with Repeated statement Multivariate Tests - identical results for Pillai s Trace, Hotelling-Lawley Trace, and Roy s Greatest Root MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no time*group Effect H = Type III SSCP Matrix for time*group E = Error SSCP Matrix S=1 M=0 N=8.5 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda

65 Implications - Example Multivariate GLM with Repeated statement Partial Correlation Coefficients from the Error SSCP Matrix / Prob > r DF = 60 SUPSBP_A SUPSBP_C SUPSBP_D SUPSBP_A < <.0001 SUPSBP_C < <.0001 SUPSBP_D < < Sphericity Tests Variables DF Mauchly's Criterion Chi-Square Pr > ChiSq Transformed Variates <.0001 Orthogonal Components

66 Implications - Example PROC MIXED with Repeated statement proc mixed data=long; class group time family; model sbp= group time group*time/ddfm=kenwardroger; repeated time / subject= family(group) type=un r rcorr; lsmeans group*time; run; quit;

67 Implications - Example PROC MIXED with Repeated statement Type=UN and DDFM=kenwardroger Type 3 Tests of Fixed Effects Effect Num DF Den DF F Value Pr > F GROUP*time Estimated R Correlation Matrix for FAMILY(GROUP) 1003 Control Row Col1 Col Col

68 Implications - Example PROC MIXED with Repeated statement Type=CS Type 3 Tests of Fixed Effects Effect Num DF Den DF F Value Pr > F GROUP*time

69 Implications - Example PROC MIXED with Repeated statement Least Squares Means type=un Effect Group time Estimate Standard Error GROUP*time Control GROUP*time Control GROUP*time Control GROUP*time TM GROUP*time TM Least Squares Means - type=cs Estimate Standard Error GROUP*time TM

70 Implications - Example PROC MIXED with Repeated statement Diff -RLL = 9. with 4 df, p= Both AIC smaller for UN, BIC is not Fit Statistics UN 6 parameters estimated - Res Log Likelihood Fit Statistics CS parameters estimated - Res Log Likelihood AIC (smaller is better) AIC (smaller is better) AICC (smaller is better) AICC (smaller is better) BIC (smaller is better) BIC (smaller is better)

71 Implications Inferences for fixed effects may be impacted by the poor choice of error covariance matrix Those impacted will be the repeated variable and its interactions It is suggested that if Number of subjects - Number of treatment groups - Number of repeated measures > 30 then UN will usually suffice

72 Implications MV GLM results are identical to PROC MIXED results when ddfm=kenwardrogers is used