Recall the linear mixed model (also called growth curve model) we fit in class for children?s bone length data. Briefly, bone length (outcome) was measured at ages 8, 8.5, 9, and 9.5. We fit a linear mixed model to predict bone length using a random intercept and slope (for age), and fixed age, i.e.,

Write out the Xi matrix for this model. Are the random intercept and random age term appropriate for this model? Recall when we added sex to the model in class, the covariate did not significantly predict bone growth. Should we leave sex in the model? Why or why not? data bone; input subject t1 t2 string variable; *t=8,8.5,9,9.5 years; y=t1; age=8; output; y=t2; age=8.5;output; y=t3; age=9;output; y=t4; age=9.5;output; cards; 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 ; run; 47.8 46.4 46.3 45.1 47.6 52.5 51.2 49.8 48.1 45 51.2 48.5 52.1 48.2 49.6 50.7 47.2 53.3 46.2 46.3 45.1 42.9 43.5 40.5 45.1 50.1 47.4 47.9 47 43.2 49.5 45.1 52.2 45.8 47.3 50.4 46.3 51.8 48.8 47.3 46.8 45.3 48.5 53.2 53 50 50.8 47 51.4 49.2 52.8 48.9 50.4 51.7 47.7 54.6 47.5 47.6 46.2 44.1 44.2 43 47.1 50.5 51.5 48.3 48.8 44.5 49.6 47.8 53.5 46.3 48.7 51.1 46.8 53.9 49 47.7 47.8 46.1 48.9 53.3 54.3 50.3 52.3 47.3 51.6 53 53.7 49.3 51.2 52.7 48.4 55.1 48.1 51.3 46.5 46.7 45.3 43.8 48.2 51.4 52.7 48.8 50.4 44.6 50.6 52 53.8 47.8 49.6 51.6 47.4 54.6 49.7 48.4 48.5 47.2 49.3 53.7 54.4 52.7 54.4 48.3 51.9 55.5 55 49.8 51.8 53.3 49.5 55.3 48.4 51.8 46.7 47 49.5 44.9 48.6 52.2 54.8 50.6 51.6 47.1 51.4 53.7 54.6 48.7 50.9 52.1 47.9 56.9 t3 t4 sex $; M M M M M M M M M M M M M M M M M M M M F F F F F F F F F F F F F F F F F F proc mixed data=bone method=reml covtest; class subject; * &allows model y=age/s; *fixed intercept and time effects here; random int age/type=simple subject=subject; *random intercept and time effects for each subject; *type=simple asks for random intercept and time to be *independent; run; Covariance Parameters 3 Columns in X 2 Columns in Z Per Subject 2 Subjects 3 8 Max Obs Per Subject 4 Number of Observations Number of Observations Read 15 2 Number of Observations Used 15 2 Number of Observations Not Used 0 Covariance Parameter Estimates Cov Parm Subjec t Estimat e Intercept subject 5.8302 2.5288 2.31 0.0106 age subject 0.03286 0.03118 1.05 0.1460 0.8002 0.1106 Residual Standard Z Value Pr > Z Error 7.23 <.000 solution for fixed effects effect estimat e intercep t age standard error d value f pr> | t| Type 3 Tests of Fixed Effects Effec t age Num DF 1 Den F Value Pr > F DF 37 263.56 <.000 interpretation of the covariance parameters age effect proc mixed data="bone" method="reml" covtest class subject model y="age/s;" random int intercept and time effects for each asks to be correlated run dimensions columns in x z per subjects max obs number observations read used not parameter estimates cov parm subjec t estimat e un residual standard value error pr null likelihood ratio test df chi-square> ChiSq 3 272.92 <.0001 solution for fixed effects effect estimat e intercep t age standard error d value f type tests of effec num df den pr> F DF 37 109.62 <.000 pr> | t| Type 3 Tests of Fixed Effects Effec t Num DF Den F Value Pr > F DF 1 Use the likelihood ratio test comparing unstructured to simple covariance model, H0: Simple covariance proc mixed data=bone method=reml covtest; class subject; model y=age age*age/s; random int age age*age/type=un subject=subject; run; Dimensions Covariance Parameters 7 Columns in X 3 Columns in Z Per Subject 3 Subjects 3 8 Max Obs Per Subject 4 Number of Observations Number of Observations Read 15 2 Number of Observations Used 15 2 Number of Observations Not Used 0 Covariance Parameter Estimates Cov Parm Subjec t Estimat e UN(1,1) subject 2495.95 2016.51 1.24 0.1079 UN(2,1) subject -560.88 460.45 -1.22 0.2232 UN(2,2) subject 127.89 105.36 1.21 0.1124 UN(3,1) subject 31.1756 26.1739 1.19 0.2336 UN(3,2) subject -7.1915 5.9994 -1.20 0.2306 UN(3,3) subject 0.4090 0.3422 1.20 0.1160 0.2051 0.04705 Residual Standard Z Value Error 4.36 Pr Z <.000 fit statistics res log likelihood aic is better aicc bic null model ratio test df chi-square pr> ChiSq 6 271.56 <.0001 solution for fixed effects effect estimat e intercep t age standard error d value f pr> | t| Type 3 Tests of Fixed Effects Effect Num DF Den F Value Pr > F DF age 1 37 1.36 0.251 5 age*ag e 1 37 0.23 0.632 1 Interpretation of the covariance parameters? Interpretation of the age effect? Interpretation of the age*age effect? Age squared obscures significant age effects. *ADD FIXED COVARIATE SEX; proc mixed data=bone method=reml covtest; class subject sex; model y=age sex/s; random int age/type=un subject=subject; run; Dimensions Covariance Parameters 4 Columns in X 4 Columns in Z Per Subject 2 Subjects 3 8 Max Obs Per Subject 4 Number of Observations Number of Observations Read 15 2 Number of Observations Used 15 2 Number of Observations Not Used 0 Covariance Parameter Estimates Cov Parm Subjec t Estimat e UN(1,1) subject 112.17 29.9188 UN(2,1) subject -12.1613 3.3073 UN(2,2) subject 1.4156 0.3776 3.75 <.000 residual standard z value error pr fit statistics res log likelihood aic is better aicc bic null model ratio test df chi-square> ChiSq 3 267.55 <.0001 solution for fixed effects effect se x intercep t age sex f m estimat e standard error d value pr> | t| 31.2106 1.8829 36 16.58 <.0001 . type tests of fixed effects effec t num df den f value pr> F DF age 1 37 sex 1 76 109.62 <.000>