Repeated Measures Data

Transcription

1 Repeated Measures Data Mixed Models Lecture Notes By Dr. Hanford page 1 Data where subjects are measured repeatedly over time - predetermined intervals (weekly) - uncontrolled variable intervals between measurements Reasons for repeated measures - ensure treatment is effective over a specified period - monitor safety aspects of the treatment over a specified period - measure length of time for treatment to become effective Objectives of repeated measures analysis - measure average treatment effect over time - assess possible time*treatment interactions - assess treatment response profile (e.g. rea Under Curve (UC), tmax, cmax) - identify possible covariance patterns in the repeated measurements - determine an appropriate model to describe the time and measurement relationship. Fixed effects approaches lthough it is possible to treat the patient/animal as a fixed effect in special situations, normally the patient/animal is treated as random. The book briefly covers a few of these situations in Chapter 6 and covers the advantages and disadvantages of these. Most of them require fixed time intervals and no missing data. Mixed model approaches dvantages of using mixed models with repeated measures data - a single model can be used to estimate overall treatment effects and to estimate treatment effects at each time point. - Missing data does not cause problems as long as they can be assumed missing at random. - The covariance pattern of the repeated measurements can be determined and taken into account. Covariance Pattern Models Several ways to analyze repeated measures data using mixed models are available. The first one we will cover is the random effects model with animal effects treated as random. It allows for constant correlation between all observations on the same animal. However,

2 Mixed Models Lecture Notes By Dr. Hanford page 1 correlations between observations on the same animal are not always constant. Observations recorded on consecutive visits may be more highly correlated than observations measured on visits further apart. covariance pattern model can be used to model this. The basic structure of the covariance pattern model is R1 R R R R5 R R6 R7 R8 R9 R1 Where the submatrix R i covariance block correspond to the ith animal where the dimension of the submatrix correspond to the number of repeated measurements for animal i. The s are also block matrices that correspond between observations of different animals and assumes that the correlation between observations among animals are zero. Covariance patterns There are a large number of possible covariance patterns for each of the submatrices R i available for mixed models. We will present several of the more commonly used patterns which are available in SS. Simple covariance patterns For our example, we will include animals, where nimal 1 had visits, nimal had visits and nimal had visits, where the visits are at fixed times. Independent covariance pattern If the within-subjects correlation is zero, the covariance model is called independent and is the simplest model. The number of parameters that need to be estimated is equal to the number of visits and assumes that the visits are independent. This covariance structure is also referred to as heterogeneous uncorrelated because it allows for differing variances at each visit. For our example, the independent covariance pattern would be

3 Mixed Models Lecture Notes By Dr. Hanford page R 1 General or unstructured covariance pattern However, in most experiments there is more than likely some correlation present. If the correlation between each pair of visits (θ ij ) (within-subjects correlation) is allowed to differ and the variance for each time point (σ i ) is allowed to differ, then the covariance pattern is called general or unstructured. This is the most complex of the simple correlation patterns. The number of parameters that need to be estimated is (N*(N+1))/. For example, if there are visits (N=), then the number of parameters would be (*5)/=1. The general covariance pattern for our example follows: R First-order autoregressive covariance pattern When time periods are evenly spaced, the first-order autoregressive model is sometimes appropriate. This model is also referred to as R(1). In this model, the variances are the same for all of the visits and the covariances decrease exponentially depending on their distance (θ ij = ρ i-j σ ). Note that the covariance is a function of the variance so the scalar variance can be pulled out of the matrix and that there are only two parameters that need to be estimated. For our example, the first-order autoregressive would be:

4 Mixed Models Lecture Notes By Dr. Hanford page R Compound symmetry covariance pattern If all of the covariances among the visits are thought to be the same, then the compound symmetry covariance pattern is used. This pattern does not make any assumption about the relationship between the variance and covariance and only has two parameters that need to be estimated. For our example, this would be: R Toeplitz covariance pattern The Toeplitz covariance is similar to the first-order autoregressive covariance pattern in the way that the pairs separated by the same distance share the same correlations. But with the Toeplitz covariance pattern there is no known function relating the ρ values to the distance. This covariance pattern makes sense when visits are equally spaced. The number of parameters that need to be estimated is equal to the maximum number of visits, which is greater than the two parameters needed for R(1). For our example, the Toeplitz covariance pattern would be

5 R Mixed Models Lecture Notes By Dr. Hanford page Heterogeneous variances among visits The first-order autoregressive, compound symmetry and Toeplitz covariance patterns all assumed that the variances for all of the visits were the same. However, sometimes variability will differ between the visits for the trait being measured. In other words, the variances are heterogeneous. The independent and general covariance patterns allowed for heterogeneity among the variances. The R(1), compound symmetry and Toeplitz covariance patterns can also be modified to allow for heterogeneous variances and are presented for our example. Heterogeneous first-order autoregressive The number of parameters needed for the heterogeneous R(1) is the number of visits plus 1. In our example, with visits, we have 5 parameters that need to be estimated. R Heterogeneous compound symmetry The number of parameters needed for the heterogeneous compound symmetry is also the number of visits plus 1.

6 R Mixed Models Lecture Notes By Dr. Hanford page Heterogeneous Toeplitz Just as with the homogeneous covariance patterns, the number of parameters that need to be estimated for the heterogeneous Toeplitz covariance pattern is greater than for heterogeneous R(1) or compound symmetry. gain, this is because there is no known function relating the ρ values to the distance. For our example, the heterogeneous Toeplitz covariance pattern would be R First-order ante dependence covariance pattern If visits are not equally spaced and the correlation structure changes over time, the R(1) and Toeplitz covariance patterns no longer make the most sense. The first-order ante dependence covariance pattern NTE(1) is a more general model that keeps the main features of the R(1) and Toeplitz, but allows for unequal visit spacing and changes in the correlation structure over time.

7 Mixed Models Lecture Notes By Dr. Hanford page R Note that the NTE(1) covariance pattern - permits the variance among observations to change over time - permits the correlations to change over time (correlations between pairs of observations is the product of the correlations between adjacent times between observations) - N-1 parameters estimated. For our example *-1=7. Separate covariance patterns for each treatment Response to treatment may differ in amount of variability, so different sets of covariance parameters may be used for each treatment group. For our example, nimals 1 and received Treatment and nimal received Treatment B. Previously, when we assumed homogeneous treatment variances, the number of parameters that needed to be estimated was two. ssuming heterogeneous treatment variances increases the number of parameters multiplicatively by the number of treatments. For our two treatment example, this would be *= parameters. If the compound symmetry covariance structure is used, the structure we would now have would be R B B B B B B B B B B B B B B B B In our example, we present only animals. If we assume that there are additional animals in the study and that the maximum number of visits for the study is, for the

8 Mixed Models Lecture Notes By Dr. Hanford page 19 general or unstructured pattern, the number of parameters in our example would be *1=.,1,1,1,1,,,1,, B,1 B,1 B,1 B,1 R B,1 B, B,, B B,1 B, B, B, B,1 B, B, B,,1,1,1, Banded covariances With the exception of the independent structure, all of the above covariance pattern structures assumed that there was some correlation among visits. s the distance in visit number between two different visits increases, the correlation between the observations from the visits in most cases decreases. The correlations between widely separated observations is close to zero and banding the covariance matrices by setting correlations between these observations to zero may be appropriate. The advantage of using a banded covariance structure is that it reduces the number of parameters that need to be estimated. Banding is very useful in experiments where there are a large number of visits or measurement time points. For example, if in our example, we used a band size of with a general covariance pattern, we would get the following, where the θ correlation parameter is assumed to be zero and is not estimated: R Selecting an appropriate covariance pattern There are many choices of covariance patterns for modeling repeated measures data. Choosing the most appropriate pattern is important in order to draw accurate conclusions.

9 Mixed Models Lecture Notes By Dr. Hanford page 1 If the covariance pattern chosen is too simple, then you risk increasing Type I error rate and underestimating standard errors. If the covariance pattern chosen is too complex, you sacrifice power and efficiency. s more covariance parameters are included, the chances of over fitting increase. There are tools available to help select a covariance pattern. First, there are graphical tools to help you visualize patterns of correlation between observations at different times. We will go through examples of graphical tools later. Secondly, there are measures of model fit, including information criteria and statistical comparison between models. Information criteria Information criteria measure the relative fit of competing covariance patterns. s more parameters are included in the model, the likelihood statistic is expected to become larger. There are two statistics based on the likelihood that make allowance for the number of covariance parameters fitted and can be used to compare two models which fit the same fixed effects. kaike s information criterion (IC) IC=log(L)-q Where q is the number of covariance parameters and log(l) is the log of the likelihood for the model. SS calculates the IC as -(log(l))+q. The second statistic is Schwarz s information criterion (SIC) SIC=log(L)-(q*log(N-p))/ Where p is the number of fixed effects, N is the number of observations and q is the number of covariance parameters. This is also referred to as the BIC (Bayesian information criterion) or SBC (Schwarz's Bayesian information criterion). SS calculates the BIC as -(log(l))+(q*log(n-p)) where N is the number of subjects. Models with larger values of IC and SIC as presented in the text mean better fit. With SS PROC MIXED, smaller means a better fit. Whether IC or BIC is better is not clear. Statistical comparison between models The likelihood ratio test can also be used to compare models which fit the same fixed effects and whose covariance patterns are nested. Nesting is when the covariance pattern in the simpler model can be obtained by restricting some of the parameters in the more complex model. n example of this would be as follows: R compound symmetry pattern is nested within

10 Mixed Models Lecture Notes By Dr. Hanford page 11 R Toeplitz pattern, because a restriction on the Toeplitz pattern that θ 1 =θ would be the same as the compound symmetry pattern. However, the compound symmetry pattern is not nested within the R(1) R because the off diagonal elements in the R(1) are a function of both the σ and the ρ. The likelihood ratio test statistic is (log(l 1 )-log(l )) ~ χ df where df=difference in number of covariance parameters fitted. Which covariance patterns to consider? good strategy is to start with simple patterns such as compound symmetry or R(1). Once the general covariance pattern is determined, more complex patterns based on the selected general covariance pattern can be investigated. The more complex pattern would be accepted only if using them leads to a significant improvement in the likelihood. If there are only a few repeated measurements, there will be little difference between models using different covariance patterns on overall treatment effects. The compound symmetry pattern is likely to be robust and can be used with reasonable confidence. Example: Covariance pattern models for normal data We will use the blood pressure data again, but this time consider the repeated measures data. The table below summarizes the number of patients that attended each visit. Visit Treatment Treatment B Treatment C Total

11 Mixed Models Lecture Notes By Dr. Hanford page 1 Visualizing correlation patterns graphically The first graphical tool is visualizing the correlation structure by plotting changes in covariance and correlation among residuals on the same subject at different times over distance between times of observations. The following PROC MIXED program allows us to obtain the correlations and covariances among the residuals. proc mixed data=dbp; class trt pat visit; model dbp=trt visit; repeated/type=un subject=pat(trt) sscp rcorr; ods output covparms=cov; ods output rcorr=corr; run; The repeated statement determines the form of the covariance, type=un specifies an unstructured correlation pattern within each pat(trt). subject=pat(trt) specifies that errors are correlated within each pat(trt). In other words, the observations on different pat(trt) levels are independent, but observations on the same pat(trt) are not. The sscp option causes the cov(e ijk,e ijk' ) to be computed directly from the corrected sums of squares and cross products matrix rather than the default REML procedure. The rcorr option causes the correlations to be computed. The two ODS statements create new SS data sets containing the covariances and correlations. The following code sets up the dataset for producing the plot for visualizing correlation patterns data times; do time1=1 to ; do time=1 to time1; dist=time1-time; output; end; end; run; Obs time1 time dist CovParm Subject Estimate UN(1,1) pat(trt) UN(,1) pat(trt) UN(,) pat(trt) UN(,1) pat(trt) UN(,) pat(trt) UN(,) pat(trt) UN(,1) pat(trt) UN(,) pat(trt) UN(,) pat(trt) UN(,) pat(trt) These are the estimated covariances among residuals on the same subject at different times over distance between times of observations. For example UN(,1) is the

12 Mixed Models Lecture Notes By Dr. Hanford page 1 covariance between the first and fourth time points. The following SS code plots the above covariances. data complot; merge times cov; proc print; run; axis1 value=(font=swiss h=) label=(angle=9 f=swiss h= 'Covariance of between Subj effects'); axis value=(font=swiss h=) label=(f=swiss h= 'Distance'); legend1 value=(font=swiss h=) label=(f=swiss h= 'From Time'); symbol1 color=black interpol=join line=1 value=square; symbol color=black interpol=join line= value=circle; symbol color=black interpol=join line= value=triangle; symbol color=black interpol=join line= value=star; proc gplot data=covplot; plot estimate*dist=time/vaxis=axis1 haxis=axis legend=legend1; run; The values plotted at distance= are the variances among the observations at each of the four visits. These range from about 7 to 9. This small range along with no trend of increasing or decreasing variance with visit, suggests that a covariance pattern with a constant variance over time, such as a Toeplitz or R(1) will probably be adequate. s

13 Mixed Models Lecture Notes By Dr. Hanford page 1 the distance between pairs of observations increases, the covariance tends to decrease, again suggesting a Toeplitz or R(1) covariance pattern. Even though the plot narrows the choice of which covariance patterns may be appropriate, we will fit and compare six different covariance patterns to our data. Model Covariance pattern 1 compound symmetry first order autoregressive (R(1)) Toeplitz General 5 Separate compound symmetry for each treatment group 6 Separate Toeplitz pattern for each treatment group Treatment, time, treatment*time and baseline effects will be included as fixed effects. The following SS code fits the compound symmetry model. proc mixed data=dbp; class trt pat visit; model dbp=trt visit dbp; repeated visit/type=cs subject=pat r rcorr; run; The following output is produced. Model Information Data Set Dependent Variable Covariance Structure Subject Effect Estimation Method Residual Variance Method Fixed Effects SE Method Degrees of Freedom Method WORK.DBP dbp Compound Symmetry pat REML Profile Model-Based Between-Within Dimensions Covariance Parameters Columns in X 1 Columns in Z Subjects 88 Max Obs Per Subject Number of Observations Number of Observations Read 19 Number of Observations Used 19 Number of Observations Not Used Iteration History Iteration Evaluations - Res Log Like Criterion

14 Mixed Models Lecture Notes By Dr. Hanford page Convergence criteria met. Estimated R Matrix for pat Estimated R Correlation Matrix for pat Covariance Parameter Estimates Cov Parm Subject Estimate CS pat.169 Residual 6.1 Fit Statistics - Res Log Likelihood 76. IC (smaller is better) 767. ICC (smaller is better) 767. BIC (smaller is better) 77.7 The next three models use the same SS code, except the type= is modified as follows: type=r(1) for R(1) type=toep for Toeplitz type=un for general or unstructured The following are the covariance parameters and information criterion R(1) Residual Estimated R Correlation Matrix for R(1) Res Log Likelihood 785. IC (smaller is better) 789. ICC (smaller is better) 789. BIC (smaller is better) Toeplitz Residual Estimated R Correlation Matrix for Toeplitz

15 Mixed Models Lecture Notes By Dr. Hanford page Res Log Likelihood 75.6 IC (smaller is better) ICC (smaller is better) BIC (smaller is better) 77. General or Unstructured UN(1,1) pat UN(,) pat 7.91 UN(,) pat UN(,) pat 7.58 Estimated R Correlation Matrix for General or Unstructured Res Log Likelihood 7. IC (smaller is better) 76. ICC (smaller is better) 76.5 BIC (smaller is better) The following table summarizes the fit information for the four models Model - Res Log (L) IC BIC 1 compound symmetry first order autoregressive (R(1)) Toeplitz General For all of the models, correlations between visits are positive, indicating that it is important to take into account the correlations between repeated measures. Comparing the fit statistics among these first four covariance patterns, it appears that the compound symmetry (the simplest model with covariance parameters) and the Toeplitz (with covariance parameters) covariance patterns fit the data better than either the R(1) or general covariance patterns. The compound symmetry model is nested within the Toeplitz pattern, so they can be compared statistically using the likelihood ratio test. The likelihood ratio test statistic would be (log(l 1 )-log(l )) ~ χ df = (.5*76.-.5*75.6)= =1.8 =. The p-value associated with this =., indicating that the Toeplitz structure is a significant improvement over the simpler compound symmetry structure. Model 5 s covariance pattern allows for differences in variances and covariances among the treatments. The SS code for fitting this model with a compound symmetry covariance pattern follows proc mixed noclprint data=dbp; class trt pat visit;

16 Mixed Models Lecture Notes By Dr. Hanford page 17 model dbp=trt visit dbp/ddfm=satterth; repeated visit/type=cs subject=pat group=trt r=1,, rcorr=1,,; run; There are a couple of additions/changes in the repeated statement. group=trt allows for estimation of the parameters by treatment. r=1,, rcorr=1,, indicates that the covariance matrices and correlation matrices for Patients 1( Trt C), (Trt B), and (Trt ) be printed out. Following are the results for Model 5 Estimated R Matrix for pat 1 - Treatment C Estimated R Correlation Matrix for pat 1 - Treatment C Estimated R Matrix for pat - Treatment B

17 Mixed Models Lecture Notes By Dr. Hanford page 18 Estimated R Correlation Matrix for pat - Treatment B Estimated R Matrix for pat Treatment Estimated R Correlation Matrix for pat Treatment Covariance Parameter Estimates Cov Parm Subject Group Estimate Variance pat trt 9.9 CS pat trt Variance pat trt B CS pat trt B Variance pat trt C CS pat trt C 7.99 Fit Statistics - Res Log Likelihood 77.5 IC (smaller is better) ICC (smaller is better) BIC (smaller is better) Looking at the results, the pattern for compound symmetry differ among the treatments, indicating that separate covariances for each treatment group may be necessary. Because Model (Toeplitz) fit significantly better than Model 1 (compound symmetry), a Toeplitz model fitting for separate covariances for each treatment group should be fitted next (Model 6) with the following SS code and results. proc mixed data=dbp; class trt pat visit; model dbp=trt visit dbp; repeated visit/type=toep subject=pat group=trt r=1,, rcorr=1,,; lsmeans trt/ diff pdiff cl; Estimated R Matrix for pat 1 - Treatment C Estimated R Correlation Matrix for pat

18 Mixed Models Lecture Notes By Dr. Hanford page Estimated R Matrix for pat - Treatment B Estimated R Correlation Matrix for pat Estimated R Matrix for pat Estimated R Correlation Matrix for pat Covariance Parameter Estimates Cov Parm Subject Group Estimate Variance pat trt TOEP() pat trt TOEP() pat trt 1.77 TOEP() pat trt.17 Variance pat trt B 68.1 TOEP() pat trt B 8.9 TOEP() pat trt B.76 TOEP() pat trt B Variance pat trt C TOEP() pat trt C 5.76 TOEP() pat trt C 6.95 TOEP() pat trt C 5.75 Fit Statistics - Res Log Likelihood 7. IC (smaller is better) 78. ICC (smaller is better) 78. BIC (smaller is better) Both the simple Toeplitz model (Model ), and the compound symmetry with treatment model (Model 5) are nested within Model 6, so we can test to see if Model 6 fits significantly better than the other two models. Model ( parameters) vs Model 6 (1 parameters) (log(l 1 )-log(l )) ~ χ df = =6.6= 8 (p=.8) Model 5 (6 parameters) vs Model 6 (1 parameters)

19 Mixed Models Lecture Notes By Dr. Hanford page 15 (log(l 1 )-log(l )) ~ χ df = =.5= 6 (p=.6) Therefore, we are statistically justified in using the more complex covariance pattern. This may partly be due to the large size of the trial, which allows us to estimate the covariances with reasonable accuracy.