Handbook_of_statistical_analysis_using_SAS

Convergence criteria met.

Covariance Parameter Estimates

Type 3 Tests of Fixed Effects

Display 11.5

The predicted values for both groups under the random intercept model indicate a rise in cognitive score with time, contrary to the pattern in the observed scores (see Displays 11.2 and 11.3), in which there appears to be a decline in cognitive score in the placebo group and a rise in the lecithin group.

We can now see if the random intercepts and slopes model specified in Eq. (11.3) improve the situation. The model can be fitted as follows:

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Display 11.6

Display 11.7

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proc mixed data=alzheim method=ml covtest; class group idno;

model score=group visit /s outpred=mixout; random int visit /subject=idno type=un;

run;

Random slopes are specified by including visit on the random statement. There are two further changes. The covtest option in the proc statement requests significance tests for the random effects.

The type option in the random statement specifies the structure of the covariance matrix of the parameter estimates for the random effects. The default structure is type=vc (variance components), which models a different variance component for each random effect, but constrains the covariances to zero. Unstructured covariances, type=un, allow a separate estimation of each element of the covariance matrix. In this example, it allows an intercept-slope covariance to be estimated as a random effect, whereas the default would constrain this to be zero.

The results are shown in Display 11.8. First, we see that σ a2 , σ b2 , σ ab, and σ 2 are estimated to be 38.7228, 2.0570, –6.8253, and 3.1036, respectively. All are significantly different from zero. The estimated correlation between intercepts and slopes resulting from these values is –0.76. Again, both fixed effects are found to be significant. The estimated treatment effect, –3.77, is very similar to the value obtained with the random- intercepts-only model. Comparing the AIC values for the random intercepts model (1281.7) and the random intercepts and slopes model (1197.4) indicates that the latter provides a better fit for these data. The predicted values for this second model, plotted exactly as before and shown in Displays 11.9 and 11.10, confirm this because they reflect far more accurately the plots of the observed data in Displays 11.2 and 11.3.

The Mixed Procedure

Model Information

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Convergence criteria met.

Covariance Parameter Estimates

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The Mixed Procedure

Null Model Likelihood Ratio Test

Type 3 Tests of Fixed Effects

Display 11.8

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Display 11.9

Display 11.10

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Exercises

11.1Investigate the effect of adding a fixed effect for the Group × Visit interaction to the models specified in Eqs. (11.1) and (11.2).

11.2Regress each subject’s cognitive score on time and plot the estimated slopes against the estimated intercepts, differentiating the observations by P and L, depending on the group from which they arise.

11.3Fit both a random intercepts and a random intercepts and random slope model to the data on postnatal depression used in Chapter 10. Include the pretreatment values in the model. Find a confidence interval for the treatment effect.

11.4Apply a response feature analysis to the data in this chapter using both the mean and the maximum cognitive score as summary measures. Compare your results with those given in this chapter.

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Chapter 12

Survival Analysis: Gastric

Cancer and Methadone

Treatment of Heroin

Addicts

12.1Description of Data

In this chapter we analyse two data sets. The first, shown in Display 12.1, involves the survival times of two groups of 45 patients suffering from gastric cancer. Group 1 received chemotherapy and radiation, group 2 only chemotherapy. An asterisk denotes censoring, that is, the patient was still alive at the time the study ended. Interest lies in comparing the survival times of the two groups. (These data are given in Table 467 of

SDS.)

However, “survival times” do not always involve the endpoint death. This is so for the second data set considered in this chapter and shown in Display 12.2. Given in this display are the times that heroin addicts remained in a clinic for methadone maintenance treatment. Here, the endpoint of interest is not death, but termination of treatment. Some subjects were still in the clinic at the time these data were recorded and this is indicated by the variable status, which is equal to 1 if the person had departed the clinic on completion of treatment and 0 otherwise.

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Possible explanatory variables for time to complete treatment are maximum methadone dose, whether the addict had a criminal record, and the clinic in which the addict was being treated. (These data are given in Table 354 of SDS.)

For the gastric cancer data, the primary question of interest is whether or not the survival time differs in the two treatment groups; and for the methadone data, the possible effects of the explanatory variables on time to completion of treatment are of concern. It might be thought that such questions could be addressed by some of the techniques covered in previous chapters (e.g., t-tests or multiple regression). Survival times, however, require special methods of analysis for two reasons:

1.They are restricted to being positive so that familiar parametric assumptions (e.g., normality) may not be justifiable.

2.The data often contain censored observations, that is, observations for which, at the end of the study, the event of interest (death in the first data set, completion of treatment in the second) has not occurred; all that can be said about a censored survival time is that the unobserved, uncensored value would have been greater than the value recorded.

Display 12.1

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