Longitudinal Studies. Cross-Sectional Studies. Example: Y i =(Y it1,y it2,...y i,tni ) T. Repeatedly measure individuals followed over time

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1 Longitudinal Studies Cross-Sectional Studies Repeatedly measure individuals followed over time One observation on each subject Sometimes called panel studies (e.g. Economics, Sociology, Food Science) Different subjects are measured at different points in time (e.g. at different ages) Cannot entirely distinguish between cohort and age effects Reading ability example (DHLZ, pages 1-2) 455 Able to distinguish changes over time within individuals (age effects) from differences among individuals (cohort effects) Must account for correlations among measurements taken on the same individual Vector of measurements Y i =(Y it1,y it2,...y i,tni ) T on the ith subject, or experimental unit. 456 Example: Measure muscle strength of elderly subjects at 0, 6, and 12 months (Daniels and Hogan, 2000, Biometrics) Randomized clinical trial Investigate the effects of recombinant human growth hormone (rhgh) therapy for building and maintaining muscle strength in the elderly (Kiel et al, 1998). The trial enrolled 161 subjects and randomized them to one of four treatments: Placebo Growth hormone only (0.015 mg/kg rhgh) Both placebo and growth hormone were administered via daily injections Muscle strength measures were recorded at baseline, six months and twelve months. Strength is measured as the maximum foot-pounds of torque which can be exerted against resistance provided by a mechanical device Exercise plus placebo Exercise plus growth hormone

2 Measure Depression over time (Pourahmadi and Daniels, 2002 Biometrics) Patients were assigned active treatment and measured weekly for 16 weeks Weekly depression scores 549 subjects with no missing baseline covariates. Main questions of interest 1. Is a combined drug/psychotherapy treatment more effective than the only psychotherapy treatment in reducing depression? 2. Is initial severity an important predictor of patient improvement? 3. Do treatment and initial severity interact in their impact on the rate of improvement? 459 Current practices for treatment of major depression emphasizes symptom severity in determining the need for anti-depressant drugs. For these 549 patients, about 30% (2840) of the possible measurements were missing, mostly intermittently. beginitemize Several of the studies measured depression bi-weekly for part or all of the active phase of treatment so we have some observations missing by design. Some subjects dropped out (about 16%). Some were missing completely at random (MCAR) but others were related to side effects of treatment or being so depressed that they were provided an alternative treatment. 460 Classes of Models for Longitudinal Data Reduce the repeated measurements to a single value for each individual and perform a univariate analysis Marginal Models Consider a vector of observations, Y i = (Y i1,...,y ini ) T for i =1,...,m individuals (or experimental units) Expect observations taken on the same individual (experimental unit) to be correlated Marginal models Random effects (hierarchical) models Transition models 461 Consider a linear model for the conditional means, eg, E(Y ij )=β 0 + β 1 (x i )+β 2 t ij + β 3 t 2 ij Model the variances and covariances V i = Var Y i1 Y i2. Y i,ni = σ 2 1 σ 12 σ 1,ni σ 12 σ 2 2 σ 2,ni... σ ni,1 σ ni,2 σ 2 n i 462

3 Consider the model: Y i N(X i β, Σ i ) i = 1,2,...,m Assume independent responses from different individuals Y 1 Y 2. Y m N X 1 β X 2 β. X m β, V V V m The block diagonal covariance structure is important The covariance structure V i can differ across individuals Estimation in Marginal Models Good reference for likelihood estimation: Jennrich and Schluchter (1986) Biometrics Ordinary Least Squares: Minimize (Y Xβ) T (Y Xβ)= m (Y i X i β) T (Y i X i β) i=1 Set partial derivatives with respect to the elements of β equal to zero to obtain the estimating equations m i=1 XT i (Y i X i β)=0 The unique solution is ˆβ = (X T X) 1 X T Y Marginal models are used when inferences about β are of primary interest = m i=1 XT i X i 1 m i=1 XT i Y i 463 Maximum Likelihood Estimation: Generalized Least Squares: Minimize (Y Xβ) T Σ 1 (Y Xβ) = m i=1 (Y i X i β) T Vi 1 (Y i X i β) Set partial derivatives with respect to the elements of β equal to zero to obtain the estimating equations m i=1 XT i V i 1 (Y i X i β)=0 The unique solution is ˆβ = (X T Σ 1 X) 1 X T Σ 1 Y The natural logarithm of the multivariate Gaussian likelihood is log(l(β, Σ)) = 1 ( m 2 i=1 (2π) n i +log( V i )+(Y i X i β) T Vi 1 (Y i X i β) ) Set partial derivatives with respect to the elements of β equal to zero to obtain the estimating equations 1 m 2 i=1 XT i V i 1 (Y i X i β)=0 The unique solution is = m i=1 XT i V i 1 X i 1 m i=1 XT i V i 1 Y i 464 ˆβ = (X T Σ 1 X) 1 X T Σ 1 Y = m i=1 XT i V i 1 X i 1 m i=1 XT i V i 1 Y i 465

4 For given Σ, themleforβ is the generalized least squares estimator You can simultaneously obtain the maximum likelihood estimator ˆΣ for Σ If we plug in ˆβ for β and ˆΣ for Σ we obtain 2log(L(β, Σ)) = ( ) m i=1 (2π) n i T 1 +log( ˆV i )+(Y i X i β) ˆV i (Y i X i ˆβ) Likelihood ratio tests for comparing models Maximum likelihood estimates of variance components tend to be too small Restricted Maximum Likelihood Estimation (REML) Maximizing a Gaussian likelihood that does not depend on E(Y) = Xβ. Maximize a likelihood function for error contrasts linear combinations of observations that do not depend on Xβ will need a set of m n i rank(x) i=1 linearly independent error contrasts Gaussian model: Y N(Xβ,Σ) For a non-random matrix L LY N(L(Xβ,LΣL T ) Consequently, LY does not depend on Xβ if and only if LX =0. But LX =0if and only if L = M(I P X ) for some M with n = ( ) mi=1 n i rows, where P X = X(X T X) X T To avoid losing information we must have row rank(m) = n rank(x) = n p Then a set of n p error contrasts is r = M(I P X )Y N n p (0,M(I P X )Σ 1 (I P X )M T ) call this W, then rank(w )=n p and W 1 exists

5 For any M (n p) n withrowrankequalto The Restricted likelihood is L(Σ; r) = 1 (2π) (n p)/2 W 1/2e 1 2 rt W 1 r n p = n rank(x) the log-likelihood can be expressed in terms of e =(I X(XΣ 1 X T ) 1 X T Σ 1 )Y as The resulting log-likelihood is l(σ; e) = constant 1 2 log( Σ ) l(σ; r) = (n p) log(2π) log W 1 2 rt W 1 r 1 2 log( XT Σ 1 X ) 1 2 et Σ 1 e where X is any set of p =rank(x) linearly independent columns of X. Denote the resulting REML estimator as ˆΣ REML Selecting Covariance Structure Estimation of fixed effects: For any estimable function Cβ, the blue is the generalized least squares estimator Cb GLS = C(X T Σ 1 X) 1 X T Σ 1 Y An approximation is C ˆβ = C(X T ˆΣ 1 REML X) X T ˆΣ 1 REML Y and for large samples: C ˆβ N(Cβ,C(X T Σ 1 X) C T ) if you specified the correct model for Σ AIC: Akaike Information Criterion AIC=-2 loglik + 2*p (p is number of parameters) When n is large, often favors models with too many parameters (penalty does not change with sample size) In SAS, when specify ML, p is the number of fixed effects parameters plus the number of covariance parameters; when you specify REML, p is the number of covariance parameters AICC - continuity corrected version (see Burnham and Anderson, 1998)

6 BIC: Bayesian Information Criterion BIC= -2 loglik + p*log(n) (n is the number of subjects) based on approximation to the Bayes Factor In SAS, when you specify ML, p is the number of fixed effects parameters + number of covariance parameters; in REML, p is the number of covariance parameters In SAS,n is the number of subjects Bayesian Analysis: specify prior distributions for β (often either a Gaussian prior or a non-informative prior) and Σ (inverse Wishart prior); for other choices see Leonard, 1992 Annals; Brown, Le, and Zidek, 1994; Daniels and Kass, 1999 JASA; Barnard, Mc- Culloch, and Meng, 2000, Statistica Sinica Empirical Bayes: estimate hyperparameters of prior distributions from the data What if you Mis-Specify Σ? What if you assumed the wrong structure for Σ? Often assume some parametric structure for Σ Time series structures (AR, MA) Structured antedependence models (SAD), (Zimmerman and Nunez- Anton, 1997) Compound Symmetry Likelihood is a function of fewer paramters L(β, Σ(α)) Often get an estimator for β that is consistent has a large sample normal distribution is not quite efficient A consistent estimator of the large sample covariance matrix for the estimator of β is obtained from a sandwich variance estimator More stable estimator for Σ (fewer parameters) and subsequently, a more stable estimator for β

7 Shrinkage Estimators Chen, 1979; Daniels and Kass (1999, 2001 Biometrics); Daniels and Pourahmadi (2001) Strategy: The default in PROC MIXED in SAS is to take Var(Y )=σ 2 e I You can change this by using the REPEATED statement in PROC MIXED Shrink toward the structure Data determines the amount of shrinkage Many different parameterizations on which to shrink Properties: Consistent Asymptotic normality Asymptotic efficiency 478 REPEATED / type = subject = subj (program) r print the R matrix for one subject variables in the class statement rcorr; print the correlation martix for one subject 479 Compound Symmetry: (type = CS) R = σ σ2 2 σ 2 2 σ 2 2 σ 2 2 σ 2 2 σ σ2 2 σ 2 2 σ 2 2 σ 2 2 σ 2 2 σ σ2 2 σ 2 2 σ 2 2 σ 2 2 σ 2 2 σ σ2 2 Unstructured: (type = UN) R = σ 2 1 σ 12 σ 13 σ 14 σ 12 σ 2 2 σ 23 σ 24 σ 13 σ 23 σ 2 3 σ 34 σ 14 σ 24 σ 34 σ 2 4 Variance components: (type = VC) (default) R = σ σ σ σ 2 4 Toeplitz: (type = TOEP) R = σ 2 σ 1 σ 2 σ 3 σ 1 σ 2 σ 1 σ 2 σ 2 σ 1 σ 2 σ 1 σ 3 σ 2 σ 1 σ

8 Heterogeneous Toeplitz: (type = TOEPH) R = σ 2 1 σ 1 σ 2 ρ 1 σ 1 σ 2 ρ 2 σ 1 σ 4 ρ 3 σ 2 σ 1 ρ 1 σ 2 2 σ 2 σ 3 ρ 1 σ 2 σ 4 ρ 2 σ 3 σ 1 ρ 2 σ 3 σ 2 ρ 1 σ 2 3 σ 3 σ 4 ρ 1 σ 4 σ 1 ρ 3 σ 4 σ 2 ρ 2 σ 4 σ 3 ρ 1 σ 2 4 First Order Autoregressive: (type = AR(1)) R = σ 2 1 ρ ρ 2 ρ 3 ρ 1 ρ ρ 2 ρ 2 ρ 1 ρ ρ 3 ρ 2 ρ 1 First order Ante-dependence: (type = ANTE(1)) R = σ 2 1 σ 1 σ 2 ρ 1 σ 1 σ 3 ρ 1 ρ 2 σ 2 σ 1 ρ 1 σ 2 2 σ 2 σ 3 ρ 2 σ 3 σ 1 ρ 2 ρ 1 σ 3 σ 2 ρ 2 σ 2 3 Heterogeneous AR(1): (type = ARH(1)) R = σ 2 1 σ 1 σ 2 ρ σ 1 σ 3 ρ 2 σ 1 σ 4 ρ 3 σ 2 σ 1 ρ σ 2 2 σ 2 σ 3 ρ σ 2 σ 4 ρ 2 σ 3 σ 1 ρ 2 σ 3 σ 2 ρ σ 2 3 σ 3 σ 4 ρ σ 4 σ 1 ρ 3 σ 4 σ 2 ρ 2 σ 4 σ 3 ρ σ Fitting Marginal Models in SAS and Splus Spatial power: (type = sp(pow)(list)) list of variables defining coordinates R = σ 2 1 ρ d 12 ρ d 13 ρ d 14 ρ d 12 1 ρ d 23 ρ d 24 ρ d 13 ρ d 23 1 ρ d 34 ρ d 14 ρ d 24 ρ d 34 1 where d ij is the Euclidean distance between the i-th and j-th observations provided by one subject (or unit). You can replace pow with a number of other choices. SAS: the MIXED procedure /* Enter the cow protein data */ data set2; infile c:\st565\dhlz.example1_4.data ; input diet cow week protein; proc sort data=set2; by diet week; proc means data=set2 noprint; by diet week; var protein; output out=means mean=pmean; proc print data=means; axis1 label=(f=swiss h=1.8 a=90 r=0 "Protein (percent)") order = 2.5 to 4.5 by 0.5 value=(f=swiss h=1.8) w=3.0 length= 4.0in; axis2 label=(f=swiss h=2.0 "Time(weeks)") order = 0 to 20 by 5 value=(f=swiss h=1.8) w= 3.0 length = 6.5 in;

9 SYMBOL1 V=CIRCLE H=1.7 w=3 l=1 i=join ; SYMBOL2 V=DIAMOND H=1.7 w=3 l=3 i=join ; SYMBOL3 V=square H=1.7 w=3 l=9 i=join ; PROC GPLOT DATA=means; PLOT pmean*week=diet / vaxis=axis1 haxis=axis2; TITLE1 ls=0.01in H=2.0 F=swiss "Protein Content in Milk"; footnote ls=0.01in; RUN; /* perform a one-way anova at each time point */ proc sort data=set2; by week diet; proc glm data=set2; by week; class diet; model protein = diet / ss1 ss3; lsmeans diet / stderr pdiff; week= Dependent Variable: protein The GLM Procedure Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total week= Dependent Variable: protein The GLM Procedure Sum of Source DF Squares Mean Square F Value Pr>F Model Error Corrected Total protein Standard LSMEAN diet LSMEAN Error Pr > t Number < < < Least Squares Means for effect diet Pr > t for H0: LSMean(i)=LSMean(j) Dependent Variable: protein i/j NOTE: To ensure overall protection level, only probabilities associated with pre-planned comparisons should be used. 488 The GLM Procedure Least Squares Means protein Standard LSMEAN diet LSMEAN Error Pr > t Number < < < Least Squares Means for effect diet Pr > t for H0: LSMean(i)=LSMean(j) Dependent Variable: protein i/j NOTE: To ensure overall protection level, only probabilities associated with pre-planned comparisons should be used. 489

10 The Mixed Procedure Model Information Data Set Dependent Variable Covariance Structure Subject Effect Estimation Method Residual Variance Method Fixed Effects SE Method Degrees of Freedom Method WORK.SET2 protein Compound Symmetry cow(diet) REML Profile Model-Based Between-Within /* Fit cubic trends across time with a compound symmetry covariance structure */ proc mixed data=set2; class diet cow; model protein = diet diet*week diet*week*week diet*week*week*week / noint s htype=1 outpm=means; repeated / type=cs sub=cow(diet); Class Level Information Class Levels Values diet cow Iteration History Iteration Evaluations -2 Res Log Like Criterion Convergence criteria met. Covariance Parameter Estimates Cov Parm Subject Estimate CS cow(diet) Residual Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better) Null Model Likelihood Ratio Test DF Chi-Square Pr > ChiSq <.0001 Solution for Fixed Effects Standard Effect diet Estimate Error DF t Value Pr > t diet <.0001 diet <.0001 diet <.0001 week*diet <.0001 week*diet <.0001 week*diet <.0001 week*w*diet <.0001 week*w*diet <.0001 week*w*diet <.0001 week*w*w*diet <.0001 week*w*w*diet <.0001 week*w*w*diet < /* Fit cubic trends across time with a general (unstructured) covariance structure */ proc mixed data=set2; class diet cow; model protein = diet diet*week diet*week*week diet*week*week*week / noint s htype=1 outpm=means; repeated / type=un sub=cow(diet); 493

11 Iteration History Iteration Evaluations -2 Res Log Like Criterion Convergence criteria met. Fit Statistics -2 Res Log Likelihood AIC (smaller is better) 41.4 AICC (smaller is better) BIC (smaller is better) The Mixed Procedure Null Model Likelihood Ratio Test DF Chi-Square Pr > ChiSq <.0001 Solution for Fixed Effects Standard Effect diet Estimate Error DF t Value Covariance Parameter Estimates Cov Parm Subject Estimate UN(1,1) cow(diet) UN(2,1) cow(diet) UN(2,2) cow(diet) UN(3,1) cow(diet) UN(19,16) cow(diet) UN(19,17) cow(diet) UN(19,18) cow(diet) UN(19,19) cow(diet) diet diet diet week*diet week*diet week*diet week*week*diet week*week*diet week*week*diet week*week*week*diet week*week*week*diet week*week*week*diet The Mixed Procedure Type 1 Tests of Fixed Effects /* To test for diet effects fit another form of the same model */ Num Den Effect DF DF F Value Pr > F diet <.0001 week*diet week*week*diet <.0001 week*week*week*diet <.0001 proc mixed data=set2; class diet cow; model protein = diet week diet*week week*week diet*week*week week*week*week diet*week*week*week / s htype=3 outpm=means; repeated / type=un sub=cow(diet);

12 Fit Statistics -2 Res Log Likelihood AIC (smaller is better) 41.4 AICC (smaller is better) BIC (smaller is better) Null Model Likelihood Ratio Test DF Chi-Square Pr > ChiSq <.0001 Solution for Fixed Effects Standard Effect diet Estimate Error DF t Value Intercept diet diet diet week week*diet week*diet week*diet week*week week*week*diet week*week*diet week*week*diet week*week*week week*week*week*diet week*week*week*diet week*week*week*diet /* To test for diet effects fit another form of the same model */ Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F diet week <.0001 week*diet week*week <.0001 week*week*diet week*week*week <.0001 week*week*week*diet proc mixed data=set2; class diet cow; model protein = diet week diet*week week*week diet*week*week week*week*week diet*week*week*week / s htype=3 outpm=means; repeated / type=un sub=cow(diet); /* plot the fitted curves */ axis1 label=(f=swiss h=1.8 a=90 r=0 "Protein (percent)") order = 2.5 to 4.5 by.5 value=(f=swiss h=1.8) w=3.0 length= 4.0in; axis2 label=(f=swiss h=2.0 "Time(weeks)") order = 0 to 20 by 5 value=(f=swiss h=1.8) w= 3.0 length = 6.5 in;

13 SYMBOL1 V=CIRCLE H=1.7 w=3 l=1 i=join ; SYMBOL2 V=DIAMOND H=1.7 w=3 l=3 i=join ; SYMBOL3 V=square H=1.7 w=3 l=9 i=join ; PROC GPLOT DATA=means; PLOT pred*week=diet / vaxis=axis1 haxis=axis2; TITLE1 ls=0.01in H=2.0 F=swiss "Estimated Protein Content"; footnote ls=0.01in; RUN; Solution for Fixed Effects Standard Effect diet Estimate Error DF t Value /* To test for diet effects fit another form of the same model */ proc mixed data=set2; class diet cow; model protein = diet week diet*week week*week diet*week*week week*week*week diet*week*week*week / s htype=3 outpm=means df=kr; repeated / type=un sub=cow(diet); Fit Statistics -2 Res Log Likelihood AIC (smaller is better) 41.4 AICC (smaller is better) BIC (smaller is better) Null Model Likelihood Ratio Test DF Chi-Square Pr > ChiSq <.0001 Intercept diet diet diet week week*diet week*diet week*diet week*week week*week*diet week*week*diet week*week*diet week*week*week week*week*week*diet week*week*week*diet week*week*week*diet Effect diet Pr > t Intercept <.0001 diet diet diet 3. week <.0001 week*diet week*diet week*diet 3. week*week week*week*diet week*week*diet week*week*diet 3. week*week*week week*week*week*diet week*week*week*diet week*week*week*diet

14 # This file posted as milkprotein.ssc # # This code is applied to the milk protein # data from DHLZ, page 8. Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F diet week <.0001 week*diet week*week <.0001 week*week*diet week*week*week <.0001 week*week*week*diet set1 <- read.table("c:/mydocuments/courses/st565/data/dhlz.example1_4.data", col.names=c("diet","cow","week","protein")) set1 # Create factors set1$dietf <- as.factor(set1$diet) set1$weekf <- as.factor(set1$week) set1$cowf <- as.factor(set1$cow) # Sort the data set by subject i <- order(set1$cow,set1$week) set1 <- set1[i,] # Delete the list called i rm(i) # Compute sample means Observed Protein Means means <- tapply(set1$protein, list(set1$week,set1$diet),mean) means # Make a profile plot of the means # Unix users should insert the motif( ) # command x.axis <- unique(set1$week) par(fin=c(6.0,6.0),pch=18,mkh=.1,mex=1.5, cex=1.2,lwd=3) matplot(c(1,19), c(3.0,4.0), type="n", xlab="time(weeks)", ylab="protein (percent)", main= "Observed Protein Means") matlines(x.axis,means,type= l,lty=c(1,3,7)) matpoints(x.axis,means, pch=c(16,17,15)) legend(1,2.45,legend=c("barley diet", Barley+lupins, Lupin diet ),lty=c(1,3,7),bty= n ) Protein (percent) Time(weeks) Barley diet Barley+lupins Lupin diet

15 AIC BIC loglik # Use the gls( ) function to fit a # model where the errors have a # compound symmetry covariance structure # within cows. options(contrasts=c("contr.treatment","contr.poly")) set1.glscs <- gls(protein ~ dietf+ week+ dietf*week+week^2+dietf*week^2 + week^3 + dietf*week^3, data=set1, correlation = corcompsymm(form=~1 cow), method=c("reml")) summary(set1.glscs) anova(set1.glscs) Correlation Structure: Compound symmetry Formula: ~ 1 cow Parameter estimate(s): Rho Coefficients: Value Std.Error t-value p-value (Intercept) <.0001 dietf dietf week <.0001 I(week^2) <.0001 I(week^3) <.0001 dietf2week dietf3week dietf2i(week^2) dietf3i(week^2) dietf2i(week^3) dietf3i(week^3) Standardized residuals: Min Q1 Med Q3 Max Residual standard error: Degrees of freedom: 1337 total; 1325 residual Denom. DF: 1325 numdf F-value p-value (Intercept) <.0001 dietf week <.0001 I(week^2) <.0001 I(week^3) <.0001 dietf:week dietf:i(week^2) dietf:i(week^3) # Try an auto regressive covariance # structures across weeks within cows set1.glsar <- gls(protein ~ dietf+ week+ dietf*week+week^2+dietf*week^2 + week^3 + dietf*week^3, data=set1, correlation = corar1(form=~1 cow), method=c("reml")) summary(set1.glsar) anova(set1.glsar)

16 Generalized least squares fit by REML Model: protein ~ dietf + week + dietf * week + week^2 + dietf * week^2 + week^3 + dietf * week^3 Data: set1 AIC BIC loglik Correlation Structure: AR(1) Formula: ~ 1 cow Parameter estimate(s): Phi Coefficients: Value Std.Error t-value p-value (Intercept) <.0001 dietf dietf week <.0001 I(week^2) <.0001 I(week^3) dietf2week dietf3week dietf2i(week^2) dietf3i(week^2) dietf2i(week^3) dietf3i(week^3) Standardized residuals: Min Q1 Med Q3 Max Residual standard error: Degrees of freedom: 1337 total; 1325 residual Denom. DF: 1325 numdf F-value p-value (Intercept) <.0001 dietf <.0001 week <.0001 I(week^2) <.0001 I(week^3) <.0001 dietf:week dietf:i(week^2) dietf:i(week^3) # Try a general correlation structure set1.glss <- gls(protein ~ dietf + week+ dietf*week +week^2+dietf*week^2 + week^3 + dietf*week^3, data=set1, correlation = corsymm(form=~1 cow), weights = varident(form = ~1 weekf), method=c("reml")) summary(set1.glss) anova(set1.glss) # Try an AR(1) correlation structure # with hterogeneous variances set1.glss <- gls(protein ~ dietf + week+ dietf*week +week^2+dietf*week^2 + week^3 + dietf*week^3, data=set1, correlation = corar1(form=~1 cow), weights = varident(form = ~ 1 week), method=c("reml")) summary(set1.glsarh) anova(set1.glss) # Compare the fit of various covariance # structures. anova(set1.glss, set1.glscs) anova(set1.glss, set1.glsar) anova(set1.glss, set1.glsar, set1.glsarh) # To compare the continuous week model to the # model where we fit a different mean at each # time point, we must compare likelihood values # instead of REML likelihood values. set1.glsarmle <- gls(protein ~ dietf+ weekf+dietf*weekf, data=set1, correlation = corar1(form=~1 cow), method=c("ml")) set1.glscarmle <- gls(protein ~ dietf+ week+ dietf*week + week^2 + dietf*week^2 + week^3 + dietf*week^3, data=set1, correlation = corar1(form=~1 cow), method=c("ml"))

17 corstruct functions anova(set1.glsarmle,set1.glscarmle) Model df AIC BIC loglik set1.glsarmle set1.glscarmle vs <.0001 Test L.Ratio p-value set1.glsarmle set1.glacarmle 1 vs <.0001 corcompsymm compound symmetry corsym general corar1 autoregressive of order 1 corcar1 continous time AR(1) corarm A autoregressive-moving average corexp exponential 1 exp( s/ρ) corgaus expgaus 1 exp[ (s/ρ) 2 ] corlin linear 1 (1 s/ρ)i(s <ρ) corratio rational quadratic (s/ρ) 2 /[1 + (s/ρ) 2 ] corspher spherical 1 [1 1.5(s/ρ) +0.5(sρ) 3 ]I(s <ρ)