The number and timing of time -dependent covariate measurements for the Cox regression model

The problem of analyzing data with updated measurements in the time-dependent proportional hazards model arises frequently in practice. One available option is to reduce the number of intervals (or updated measurements) to be included in the Cox regression model. We empirically investigated the bias of the estimator of the time-dependent covariate while varying the effect of failure rate, sample size, true values of the parameters and the number of intervals. We also evaluated how often a time-dependent covariate needs to be collected and assessed the effect of sample size and failure rate on the power of testing a time-dependent effect.^ A time-dependent proportional hazards model with two binary covariates was considered. The time axis was partitioned into k intervals. The baseline hazard was assumed to be 1 so that the failure times were exponentially distributed in the ith interval. A type II censoring model was adopted to characterize the failure rate. The factors of interest were sample size (500, 1000), type II censoring with failure rates of 0.05, 0.10, and 0.20, and three values for each of the non-time-dependent and time-dependent covariates (1/4,1/2,3/4).^ The mean of the bias of the estimator of the coefficient of the time-dependent covariate decreased as sample size and number of intervals increased whereas the mean of the bias increased as failure rate and true values of the covariates increased. The mean of the bias of the estimator of the coefficient was smallest when all of the updated measurements were used in the model compared with two models that used selected measurements of the time-dependent covariate. For the model that included all the measurements, the coverage rates of the estimator of the coefficient of the time-dependent covariate was in most cases 90% or more except when the failure rate was high (0.20). The power associated with testing a time-dependent effect was highest when all of the measurements of the time-dependent covariate were used. An example from the Systolic Hypertension in the Elderly Program Cooperative Research Group is presented. ^

ATXA -dependent gene expression in Bacillus anthracis

The Bacillus anthracis toxin genes, cya, lef , and pag, can be viewed as a regulon, in which transcription of all three genes is activated in trans by the same regulatory gene, atxA, in response to the same signal, CO2. I determined that several phenotypes are associated with the atxA gene. In addition to being toxin-deficient, an atxA -null mutant grows poorly on minimal media and sporulates early compared to the parent strain. Furthermore, an atxA-null mutant has an altered 2-D gel protein profile. I used a genetic approach to find additional atxA-regulated genes. Random transcriptional lacZ fusions were generated in B. anthracis using transposon Tn 917-LTV3. Transposon-insertion libraries were screened for mutants expressing increased β-galactosidase activity in 5% CO2. Introduction of an atxA-null mutation in these mutants revealed that 79% of the CO2-regulated fusions were also atxA-dependent. DNA sequence analysis of transposon insertion sites in mutants carrying CO 2/atxA-regulated fusions revealed ten mutants harboring transposon insertions in loci distinct from the toxin genes. The majority of the tcr (toxin co-regulated) loci mapped within the pXO1 pathogenicity island. These results indicate a clear association of atxA with CO2-enhanced gene expression in B. anthracis and provide evidence that atxA regulates genes other than the structural genes for the anthrax toxin proteins. ^ Characterization of one tcr locus revealed a new regulatory gene, pagR. The pagR gene (300 nt) is located downstream of pag. pagR is cotranscribed with pag and is responsible for autogenous control of the operon. pagR also represses expression of cya and lef. Repression of toxin gene expression by pagR may be mediated by atxA. The steady state level of atxA mRNA is increased in a pagR mutant. Recombinant PagR protein purified from Escherichia coli did not specifically bind the promoter regions of pagA or atxA. An unidentified factor in B. anthracis crude extracts, however, was able to bind the atxA promoter in the absence of PagR or AtxA. These investigations increase our knowledge of virulence regulation in B. anthracis and ultimately will lead to a better understanding of anthrax disease. ^