sas code and output 1
cf post-treatment cold feet 1-5 , Table 4.5 Treatment effect estimates. Table 4.5 Treatment effect estimates. Table 4.6 Odds ratios calculated from Model 2. Table 4.6 Odds ratios calculated from Model 2. PROC GENMOD DATA b CLASS centre treat cf1 MODEL cf cf1 treat TYPE3 WALD DIST MULT COVB ESTIMATE 'A-B' treat 1 -1 0 ALPHA 0.05 EXP ESTIMATE 'A-C' treat 1 0 -1 ALPHA 0.05 EXP ESTIMATE 'B-C' treat 0 1 -1 ALPHA 0.05 EXP The GENMOD Procedure Model Information Data Set Distribution Link Function...
Selection of covariance pattern
The covariance patterns and measures of model fit resulting from each analysis are shown in Table 6.1. Correlations between visits are positive in all models, indicating that it is important to take account of the correlations between the repeated measurements. Models 1 and 2 are the simplest covariance patterns. Since they each use two covariance parameters, we choose Model 1, which has the highest likelihood. It seems unlikely that the correlation between periods decays exponentially as they...
A polynomial random coefficients model
In this example, antibody levels to a herpes virus were measured in 45 children suffering from one of two types of cancer solid lump tumour 18 or leukaemia 27 . The measurements were taken during hospital visits for courses of chemotherapy treatment. The duration of treatment ranged from one month to three years median 12 months and the intervals between treatments differed between the children. The aim of the study was to establish whether virus antibody levels were affected by chemotherapy...
sas code and output Uxn
centre centre number, treat treatment A, B, C , patient patient number, dbp diastolic blood pressure at last attended visit, dbpl baseline diastolic blood pressure. The SAS code to produce the main results is given at the end of Section 2.5. Here, we give the code for obtaining the shrunken and unshrunken treatment effects at the first eight centres. PROC MIXED is used first to fit Model 3. ESTIMATE statements are included to calculate the shrunken treatment differences at the first eight...
Bias in fixed and random effects standard errors
Fixed and random effects standard errors are calculated using a formula that is based on a known V e.g. var a XV-1X -1 for fixed effects . When data are balanced the standard errors will not be not biased. However, because V is, in fact, estimated, it is known that in most situations we meet in clinical trials there will be some downward bias in the standard errors. Bias will occur when the data are not balanced across random effects and effects are estimated using information from several...
Iterative generalised least squares IGLS
This method can be used iteratively to fit a mixed model and the results will be the same as those obtained using ML. This approach obtains estimates of the fixed effects parameters, a, by minimising the product of the full residuals weighted by the inverse of the variance matrix, V-1. The residual product is given by y Xa 'V 1 y Xa . The variance parameters are obtained by setting the matrix of products of the full residuals y Xa equal to the variance matrix, V, specified in terms of the...
Mixed Ordinal Logistic Regression
The fixed effects ordinal logistic model can be easily extended to a mixed ordinal logistic regression model by adding random effects terms and allowing covariance patterns in the residual matrix. In Section 4.2.1 the ordinal mixed model will be specified. The residual matrix for mixed categorical models has a more complex form than for GLMMs and will be defined in Section 4.2.2. As in GLMMs, there can be benefits in reparameterising random effects models as covariance pattern models and this...
Expressing the model in matrix notation
The ordinal logistic model can alternatively be expressed using matrix notation. However, the multinomial distribution is not a member of the exponential family and cannot be linked to the model parameters using a single link function. This hurdle can be overcome by re-expressing the data in binary form. To do this we allow each observation to become a vector of c 1 correlated binary observations c number of categories . For example, if there are four categories, then we could let y 1 become 1,...
Analysis Of Categorical Data
Apart from the case of binary data, response variables which are purely categorical, without an underlying scale, are extremely rare. We will therefore only consider data on ordinal scales in this section. Variables classified as none, mild, moderate and severe will arise in a variety of contexts. To illustrate techniques, we will again take an example from Jones and Kenward 1989 and delete five observations from the second treatment period. The example is a placebo-controlled trial of a...
Example ABBA crossover design
We illustrate the AB BA cross-over design with results from an unpublished study comparing two diuretics in the treatment of mild to moderate heart failure. After initial screening for suitability, there was a period of not less than one day, and not more than seven days, where diuretic treatment was withheld. Immediately prior to randomisation to either the AB sequence of treatment, or the BA sequence, baseline observations were taken. Each treatment period lasted for five days, with an...
Repeated Measures Data
There were four post-treatment visits in the multi-centre hypertension trial introduced in the previous section. However, so far in this chapter we have chosen only to model measurements made at the final visit, which were of primary interest. An alternative strategy would be to include measurements from all four post-treatment visits in the model. Since measurements are made repeatedly on the same patients, we can describe these types of data as repeated measures data. For illustrative...
The GLMM
In the GLMM the dispersion parameter can be influenced by over- or under-dispersion of the data and by the effects of random effects shrinkage. Random effects shrinkage can cause the predicted residual variance to be greater than that observed particularly when uniform random effects categories are present in binary data, and the dispersion parameter will help to overcome this discrepancy. Thus, interpretation of the dispersion parameter in the GLMM can be difficult, since it is not always...
Some Useful Definitions
We conclude this introductory chapter with some definitions. The terms we are introducing here will recur frequently within subsequent chapters, and the understanding of these definitions and their relevance should increase as their applications are seen in greater detail. The terms we will introduce are containment, balance and error strata. In the analyses we will be presenting, we usually wish to concentrate on estimates of treatment effects. With the help of the definitions we are...
Residual maximum likelihood REML
Residual maximum likelihood sometimes referred to as restricted maximum likelihood was first suggested by Patterson and Thompson 1971 . In this approach, the parameter a is eliminated from the log likelihood so that it is defined only in terms of the variance parameters. We outline the method below. First, we obtain a likelihood function based on the residual terms, y Xa. This contrasts with the likelihood initially defined which is based directly on the observations, y. You will notice that...
Checking model assumptions in Model 6
The code below can be used to obtain the residual and normal plots shown in Figure 6.1 a . These are based on the scaled residuals. When the VCIRY option is used, SAS only scales the marginal residuals produced by the OUTPM option . However, here these are the same as the conditional residuals produced by the OUTP option since no random effects are fitted. PROC MIXED NOCLPRINT CLASS treat visit pat MODEL dbp dbp1 treat visit treat visit DDFM KR VCIRY OUTPM predm REPEATED visit SUBJECT pat TYPE...
Model 2 1
PROC GLIMMIX CLASS centre treat MODEL cf cf1 treat DIST MULT DDFM KR RANDOM centre ESTIMATE 'A-B' treat 1 -1 0 CL OR ESTIMATE 'A-C' treat 1 0 -1 CL OR ESTIMATE 'B-C' treat 0 1 -1 CL OR The output from this analysis has a similar form to that shown for Model 3 below. Model 3 PROC GLIMMIX CLASS centre treat MODEL cf cf1 treat DIST MULT DDFM KR RANDOM centre centre treat ESTIMATE 'A-B' treat 1 -1 0 CL OR ESTIMATE 'A-C' treat 1 0 -1 CL OR ESTIMATE 'B-C' treat 0 1 -1 CL OR Fixed Effects SE...
The likelihood and quasilikelihood functions
As in normal mixed models a popular way of fitting the GLMM is based on maximising the likelihood function for the model parameters. However, a difficulty with this is that true likelihood functions can only be defined for random effects and random coefficients models. A true likelihood function is not available for covariance pattern models since a general multivariate distributional form does not exist for non-normal data for normal data the multivariate normal distribution was used ....
The quasilikelihood function for covariance pattern models
In these models the observations are correlated and the model is parameterised by the fixed effects, a, and the variance parameters used in the R matrix, yR. However, since a general multivariate distributional form is not available for non-normal data we cannot define a true likelihood function. This difficulty is overcome by instead specifying a quasi-likelihood function, QL a, yR y , which has similar properties to a true likelihood. It is defined so that the differential of its log with...
Including a baseline covariate Model B
Model A was a very simple model for assessing the effect of treatment on DBP. It is usually reasonable to assume that there may be some relationship between pre- and post-treatment values on individual patients. Patients with relatively high DBP before treatment are likely to have higher values after treatment, and likewise for patients with relatively low DBPs. We can utilise this information in Table 1.1 Number of patients included in analyses of final visits by treatment and centre. Table...