Covariate Testing

There is also the issue of Type I error rate, which is the rate at which a covariate is deemed statistically important when in fact it is not. Hypothesis testing a large number of models is referred to as multiplicity and results in an inflated Type I error rate. Because a large number of models are tested as some level of significance, usually p 0.05, then 5% of those models will be selected as being an improvement over the comparator model based on chance alone. 

In their study, Ribbing and Jonsson found that when only one covariate was in the model, 80% power was achieved with (i) 20 subjects having three samples per subject and high correlation ( 0.85) between covariate and pharmacokinetic parameter, (ii) 100 subjects having three samples per subject and medium correlation ( 0.50) between covariate and pharmacokinetic parameter, and (iii) 300 subjects having three samples per subject and low correlation ( 0.15) between covariate and pharmacokinetic parameter. They also found that selection bias increased when the number of subjects decreased. In other words, the estimated value of the parameter relating the covariate to the pharmacokinetic parameter of interest was overestimated as the 

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The next covariate screening approach would be to use a regression-based method and take a more rigorous statistical approach to the problem. Using the generalized additive model (GAM) procedure in SAS, a LOESS smooth was applied to the continuous covariates wherein the procedure was allowed to identify the optimal smoothing parameter for each covariate tested. Two dependent variables were examined t, and the EBE for CL. To avoid possible skewness in the residuals,... 

TABLE 3 Summary of covariance tests for differences between classes for the structure-activity regressions between log EC50(//M) and several independent variables (see TABLE 1). Lines underclasses indicate no significant differences for intercepts, where Het. N = heterocyclic nitrogens, Aik. X = alkyl halides, Cy. Aik. = cyclic alkanes. 

Some variables often have dependencies, such as reservoir porosity and permeability (a positive correlation) or the capital cost of a specific equipment item and its lifetime maintenance cost (a negative correlation). We can test the linear dependency of two variables (say x and y) by calculating the covariance between the two variables (o ) and the correlation coefficient (r) ... 

Let us now consider models that have only more than one measured variable (w>l). The previously described model adequacy tests have multivariate extensions that can be found in several advanced statistics textbooks. For example, the book Introduction to Applied Multivariate Statistics by Srivastava and Carter (1983) presents several tests on covariance matrices. 

In many engineering applications, however, we can easily reduce the problem to the univariate tests presented in the previous section by assuming that the covariance matrix of the errors can be written as... 

Time-to-event analysis in clinical trials is concerned with comparing the distributions of time to some event for various treatment regimens. The two nonparametric tests used to compare distributions are the log-rank test and the Cox proportional hazards model. The Cox proportional hazards model is more useful when you need to adjust your model for covariates. 

Bertilsson, L. et al. (1994). Clozapine disposition covaries with CYP1A2 activity determined by a caffeine test. Br. J. Clin. Pharmacol, 38,471-3. 

A later analysis (Emhart et al. 1987) related PbB levels obtained at delivery (maternal and cord blood) and at 6 months, 2 years, and 3 years of age to developmental tests (MDI, PDI, Kent Infant Development Scale [KID], and Stanford-Binet IQ) administered at 6 months, 1 year, 2 years, and 3 years of age, as appropriate. After controlling for covariates and confounding risk factors, the only significant associations of blood lead with concurrent or later development were an inverse association between maternal (but not cord) blood lead and MDI, PDI, and KID at 6 months, and a positive association between 6-month PbB and 6-month KID. The investigators concluded that, taken as a whole, the results of the 21 analyses of correlation between blood lead and developmental test scores were "reasonably consistent with what might be expected on the basis of sampling variability," that any association of blood lead level with measures of development was likely to be due to the dependence of both PbB and... 

Table 1. Test of Homogeneity of Within Group Covariance Matrices.

It is important to note that if an indicator has little validity, its covariance with other indicators may not produce a clear peak. When there are reasons to believe that one of the indicators has poor validity—reasons that are more serious than not finding enough peaks in the output—it should not be automatically inferred from this failure of the nose count test that the taxon does not exist. The best way to solve this problem is to evaluate a different set of indicators or to drop the invalid indicator. 

Horan, 2000) that the average of two indicator validity estimations (obtained using group assignment and covariance at the hitmax) may be more accurate, but to the best of our knowledge this has not been tested empirically. 

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