Assessing the improved discriminatory power of a new biomarker in prognostic models

Although the area under the receiver operating characteristic (AUC) is the most popular measure of the performance of prediction models, it has limitations, especially when it is used to evaluate the added discrimination of a new biomarker in the model. Pencina et al. (2008) proposed two indices, the net reclassification improvement (NRI) and integrated discrimination improvement (IDI), to supplement the improvement in the AUC (IAUC). Their NRI and IDI are based on binary outcomes in case-control settings, which do not involve time-to-event outcome. However, many disease outcomes are time-dependent and the onset time can be censored. Measuring discrimination potential of a prognostic marker without considering time to event can lead to biased estimates. In this dissertation, we have extended the NRI and IDI to survival analysis settings and derived the corresponding sample estimators and asymptotic tests. Simulation studies were conducted to compare the performance of the time-dependent NRI and IDI with Pencina’s NRI and IDI. For illustration, we have applied the proposed method to a breast cancer study.^ Key words: Prognostic model, Discrimination, Time-dependent NRI and IDI ^

Assessment of the effect on statistical power of regression model misspecification by using techniques of mathematical statistics and simulation study

Objectives. This paper seeks to assess the effect on statistical power of regression model misspecification in a variety of situations. ^ Methods and results. The effect of misspecification in regression can be approximated by evaluating the correlation between the correct specification and the misspecification of the outcome variable (Harris 2010).In this paper, three misspecified models (linear, categorical and fractional polynomial) were considered. In the first section, the mathematical method of calculating the correlation between correct and misspecified models with simple mathematical forms was derived and demonstrated. In the second section, data from the National Health and Nutrition Examination Survey (NHANES 2007-2008) were used to examine such correlations. Our study shows that comparing to linear or categorical models, the fractional polynomial models, with the higher correlations, provided a better approximation of the true relationship, which was illustrated by LOESS regression. In the third section, we present the results of simulation studies that demonstrate overall misspecification in regression can produce marked decreases in power with small sample sizes. However, the categorical model had greatest power, ranging from 0.877 to 0.936 depending on sample size and outcome variable used. The power of fractional polynomial model was close to that of linear model, which ranged from 0.69 to 0.83, and appeared to be affected by the increased degrees of freedom of this model.^ Conclusion. Correlations between alternative model specifications can be used to provide a good approximation of the effect on statistical power of misspecification when the sample size is large. When model specifications have known simple mathematical forms, such correlations can be calculated mathematically. Actual public health data from NHANES 2007-2008 were used as examples to demonstrate the situations with unknown or complex correct model specification. Simulation of power for misspecified models confirmed the results based on correlation methods but also illustrated the effect of model degrees of freedom on power.^

A SIMULATION STUDY OF THE SIZE AND POWER OF SOME TWO-SAMPLE TESTS FOR UNEQUALLY CENSORED SURVIVAL DATA (RANK TESTS, LOGRANK, PROPORTIONAL HAZARDS, WILCOXON TESTS, EXPONENTIAL DISTRIBUTION)

Sizes and power of selected two-sample tests of the equality of survival distributions are compared by simulation for small samples from unequally, randomly-censored exponential distributions. The tests investigated include parametric tests (F, Score, Likelihood, Asymptotic), logrank tests (Mantel, Peto-Peto), and Wilcoxon-Type tests (Gehan, Prentice). Equal sized samples, n = 18, 16, 32 with 1000 (size) and 500 (power) simulation trials, are compared for 16 combinations of the censoring proportions 0%, 20%, 40%, and 60%. For n = 8 and 16, the Asymptotic, Peto-Peto, and Wilcoxon tests perform at nominal 5% size expectations, but the F, Score and Mantel tests exceeded 5% size confidence limits for 1/3 of the censoring combinations. For n = 32, all tests showed proper size, with the Peto-Peto test most conservative in the presence of unequal censoring. Powers of all tests are compared for exponential hazard ratios of 1.4 and 2.0. There is little difference in power characteristics of the tests within the classes of tests considered. The Mantel test showed 90% to 95% power efficiency relative to parametric tests. Wilcoxon-type tests have the lowest relative power but are robust to differential censoring patterns. A modified Peto-Peto test shows power comparable to the Mantel test. For n = 32, a specific Weibull-exponential comparison of crossing survival curves suggests that the relative powers of logrank and Wilcoxon-type tests are dependent on the scale parameter of the Weibull distribution. Wilcoxon-type tests appear more powerful than logrank tests in the case of late-crossing and less powerful for early-crossing survival curves. Guidelines for the appropriate selection of two-sample tests are given. ^

Cost and power analysis of linkage disequilibrium methods for locating genes influencing quantitative traits in humans

Linkage disequilibrium methods can be used to find genes influencing quantitative trait variation in humans. Linkage disequilibrium methods can require smaller sample sizes than linkage equilibrium methods, such as the variance component approach to find loci with a specific effect size. The increase in power is at the expense of requiring more markers to be typed to scan the entire genome. This thesis compares different linkage disequilibrium methods to determine which factors influence the power to detect disequilibrium. The costs of disequilibrium and equilibrium tests were compared to determine whether the savings in phenotyping costs when using disequilibrium methods outweigh the additional genotyping costs.^ Nine linkage disequilibrium tests were examined by simulation. Five tests involve selecting isolated unrelated individuals while four involved the selection of parent child trios (TDT). All nine tests were found to be able to identify disequilibrium with the correct significance level in Hardy-Weinberg populations. Increasing linked genetic variance and trait allele frequency were found to increase the power to detect disequilibrium, while increasing the number of generations and distance between marker and trait loci decreased the power to detect disequilibrium. Discordant sampling was used for several of the tests. It was found that the more stringent the sampling, the greater the power to detect disequilibrium in a sample of given size. The power to detect disequilibrium was not affected by the presence of polygenic effects.^ When the trait locus had more than two trait alleles, the power of the tests maximized to less than one. For the simulation methods used here, when there were more than two-trait alleles there was a probability equal to 1-heterozygosity of the marker locus that both trait alleles were in disequilibrium with the same marker allele, resulting in the marker being uninformative for disequilibrium.^ The five tests using isolated unrelated individuals were found to have excess error rates when there was disequilibrium due to population admixture. Increased error rates also resulted from increased unlinked major gene effects, discordant trait allele frequency, and increased disequilibrium. Polygenic effects did not affect the error rates. The TDT, Transmission Disequilibrium Test, based tests were not liable to any increase in error rates.^ For all sample ascertainment costs, for recent mutations ($<$100 generations) linkage disequilibrium tests were less expensive than the variance component test to carry out. Candidate gene scans saved even more money. The use of recently admixed populations also decreased the cost of performing a linkage disequilibrium test. ^

Effect of patient accrual patterns on power and size of log-rank test for time-to-event outcome in clinical trials

The determination of size as well as power of a test is a vital part of a Clinical Trial Design. This research focuses on the simulation of clinical trial data with time-to-event as the primary outcome. It investigates the impact of different recruitment patterns, and time dependent hazard structures on size and power of the log-rank test. A non-homogeneous Poisson process is used to simulate entry times according to the different accrual patterns. A Weibull distribution is employed to simulate survival times according to the different hazard structures. The current study utilizes simulation methods to evaluate the effect of different recruitment patterns on size and power estimates of the log-rank test. The size of the log-rank test is estimated by simulating survival times with identical hazard rates between the treatment and the control arm of the study resulting in a hazard ratio of one. Powers of the log-rank test at specific values of hazard ratio (≠1) are estimated by simulating survival times with different, but proportional hazard rates for the two arms of the study. Different shapes (constant, decreasing, or increasing) of the hazard function of the Weibull distribution are also considered to assess the effect of hazard structure on the size and power of the log-rank test. ^

Parentage analysis with genetic markers in natural populations. I. The expected proportion of offspring with unambiguous paternity.

Recent studies indicate that polymorphic genetic markers are potentially helpful in resolving genealogical relationships among individuals in a natural population. Genetic data provide opportunities for paternity exclusion when genotypic incompatibilities are observed among individuals, and the present investigation examines the resolving power of genetic markers in unambiguous positive determination of paternity. Under the assumption that the mother for each offspring in a population is unambiguously known, an analytical expression for the fraction of males excluded from paternity is derived for the case where males and females may be derived from two different gene pools. This theoretical formulation can also be used to predict the fraction of births for each of which all but one male can be excluded from paternity. We show that even when the average probability of exclusion approaches unity, a substantial fraction of births yield equivocal mother-father-offspring determinations. The number of loci needed to increase the frequency of unambiguous determinations to a high level is beyond the scope of current electrophoretic studies in most species. Applications of this theory to electrophoretic data on Chamaelirium luteum (L.) shows that in 2255 offspring derived from 273 males and 70 females, only 57 triplets could be unequivocally determined with eight polymorphic protein loci, even though the average combined exclusionary power of these loci was 73%. The distribution of potentially compatible male parents, based on multilocus genotypes, was reasonably well predicted from the allele frequency data available for these loci. We demonstrate that genetic paternity analysis in natural populations cannot be reliably based on exclusionary principles alone. In order to measure the reproductive contributions of individuals in natural populations, more elaborate likelihood principles must be deployed.

Molluscan memory of injury: evolutionary insights into chronic pain and neurological disorders.

Molluscan preparations have yielded seminal discoveries in neuroscience, but the experimental advantages of this group have not, until now, been complemented by adequate molecular or genomic information for comparisons to genetically defined model organisms in other phyla. The recent sequencing of the transcriptome and genome of Aplysia californica, however, will enable extensive comparative studies at the molecular level. Among other benefits, this will bring the power of individually identifiable and manipulable neurons to bear upon questions of cellular function for evolutionarily conserved genes associated with clinically important neural dysfunction. Because of the slower rate of gene evolution in this molluscan lineage, more homologs of genes associated with human disease are present in Aplysia than in leading model organisms from Arthropoda (Drosophila) or Nematoda (Caenorhabditis elegans). Research has hardly begun in molluscs on the cellular functions of gene products that in humans are associated with neurological diseases. On the other hand, much is known about molecular and cellular mechanisms of long-term neuronal plasticity. Persistent nociceptive sensitization of nociceptors in Aplysia displays many functional similarities to alterations in mammalian nociceptors associated with the clinical problem of chronic pain. Moreover, in Aplysia and mammals the same cell signaling pathways trigger persistent enhancement of excitability and synaptic transmission following noxious stimulation, and these highly conserved pathways are also used to induce memory traces in neural circuits of diverse species. This functional and molecular overlap in distantly related lineages and neuronal types supports the proposal that fundamental plasticity mechanisms important for memory, chronic pain, and other lasting alterations evolved from adaptive responses to peripheral injury in the earliest neurons. Molluscan preparations should become increasingly useful for comparative studies across phyla that can provide insight into cellular functions of clinically important genes.

The transformation of a health care system: An ecology perspective

Beginning in the early 1980s, the health care system experienced momentous realignments. Fundamental changes in structures of traditional health care organizations, shifts in authority and relationships of professionals and institutions, and the increasing influence of managed care contributed to a relatively stable industry entering into a state of turbulence. The dynamics of these changes are recurring themes in the health services literature. The purpose of this dissertation was to examine the content of this literature over a defined time period and within the perspective of a theory of organizational change. ^ Using a theoretical framework based upon the organizational theory known as Organizational Ecology, secondary data from the period between 1983 and 1994 was reviewed. Analysis of the literature identified through a defined search methodology was focused upon determining the manner in which the literature characterized changes that were described. Using a model constructed from fundamentals of Organizational Ecology with which to structure an assessment of content, literature was summarized for the manner and extent of change in specific organizational forms and for the changes in emphasis by the environmental dynamics directing changes in the population of organizations. Although it was not the intent of the analysis to substantiate causal relationships between environmental resources selected as the determinants of organizational change and the observed changes in organizational forms, the structured review of content of the literature established a strong basis for inferring such a relationship. ^ The results of the integrative review of the literature and the power of the appraisal achieved through the theoretical framework constructed for the analysis indicate that there is considerable value in such an approach. An historical perspective on changes which have transformed the health care system developed within a defined organizational theory provide a unique insight into these changes and indicate the need for further development of such an analytical model. ^

For-profit business safety interventions: A systematic, descriptive review of the literature

The purpose of this research was to determine if principles from organizational theory could be used as a framework to compare and contrast safety interventions developed by for-profit industry for the time period 1986–1996. A literature search of electronic databases and manual search of journals and local university libraries' book stacks was conducted for safety interventions developed by for-profit businesses. To maintain a constant regulatory environment, the business sectors of nuclear power, aviation and non-profits were excluded. Safety intervention evaluations were screened for scientific merit. Leavitt's model from organization theory was updated to include safety climate and renamed the Updated Leavitt's Model. In all, 8000 safety citations were retrieved, 525 met the inclusion criteria, 255 met the organizational safety intervention criteria, and 50 met the scientific merit criteria. Most came from non-public health journals. These 50 were categorized by the Updated Leavitt's Model according to where within the organizational structure the intervention took place. Evidence tables were constructed for descriptive comparison. The interventions clustered in the areas of social structure, safety climate, the interaction between social structure and participants, and the interaction between technology and participants. No interventions were found in the interactions between social structure and technology, goals and technology, or participants and goals. Despite the scientific merit criteria, many still had significant study design weaknesses. Five interventions tested for statistical significance but none of the interventions commented on the power of their study. Empiric studies based on safety climate theorems had the most rigorous designs. There was an attempt in these studies to address randomization amongst subjects to avoid bias. This work highlights the utility of using the Updated Leavitt's Model, a model from organizational theory, as a framework when comparing safety interventions. This work also highlights the need for better study design of future trials of safety interventions. ^

Estimated current patterns of food consumption and perceived changes among patients with lymphoma

Usual food choices during the past year, self-reported changes in consumption of three important food groups, and weight changes or stability were the questions addressed in this cross-sectional survey and retrospective review. The subjects were 141 patients with Hodgkin's disease or other B-cell types of lymphoma within their first three years following completion of initial treatments for lymphoma at the University of Texas M. D. Anderson Cancer Center in Houston, Texas. ^ The previously validated Block-98 Food Frequency Questionnaire was used to estimate usual food choices during the past year. Supplementary questions asked about changes breads and cereals (white or whole grain) and relative amounts of fruits and vegetables compared with before diagnosis and treatment. Over half of the subjects reported consuming more whole grains, fruits, and/or vegetables and almost three quarters of those not reporting such changes had been consuming whole grains before diagnosis and treatment. ^ Various dietary patterns were defined in order to learn whether proportionately more patients who changed in healthy directions fulfilled recognized nutritional guidelines such as 5-A-day fruits and vegetables and Dietary Reference Intakes (DRIB) for selected nutrients. ^ Small sizes of dietary pattern sub-groups limited the power of this study to detect differences in meeting recommended dietary guidelines. Nevertheless, insufficient and excessive intakes were detected among individuals with respect to fruits and vegetables, fats, calcium, selenium, iron, folate, and Vitamin A. The prevalence of inadequate or excess intakes of foods or nutrients even among those who perceived that they had increased or continued to eat whole grains and/or fruits and vegetables is of concern because of recognized effects upon general health and potential cancer related effects. ^ Over half of the subjects were overweight or obese (by BMI category) on their first visit to this cancer center and that proportion increased to almost three-quarters by their last follow-up visits. Men were significantly heavier than women, but no other significant differences in BMI measures were found even after accounting for prescribed steroids and dietary patterns. ^

The role of workplace absence as an organizational mechanism in predicting employee alcohol and drug disorders

Background. EAP programs for airline pilots in companies with a well developed recovery management program are known to reduce pilot absenteeism following treatment. Given the costs and safety consequences to society, it is important to identify pilots who may be experiencing an AOD disorder to get them into treatment. ^ Hypotheses. This study investigated the predictive power of workplace absenteeism in identifying alcohol or drug disorders (AOD). The first hypothesis was that higher absenteeism in a 12-month period is associated with higher risk that an employee is experiencing AOD. The second hypothesis was that AOD treatment would reduce subsequent absence rates and the costs of replacing pilots on missed flights. ^ Methods. A case control design using eight years (time period) of monthly archival absence data (53,000 pay records) was conducted with a sample of (N = 76) employees having an AOD diagnosis (cases) matched 1:4 with (N = 304) non-diagnosed employees (controls) of the same profession and company (male commercial airline pilots). Cases and controls were matched on the variables age, rank and date of hire. Absence rate was defined as sick time hours used over the sum of the minimum guarantee pay hours annualized using the months the pilot worked for the year. Conditional logistic regression was used to determine if absence predicts employees experiencing an AOD disorder, starting 3 years prior to the cases receiving the AOD diagnosis. A repeated measures ANOVA, t tests and rate ratios (with 95% confidence intervals) were conducted to determine differences between cases and controls in absence usage for 3 years pre and 5 years post treatment. Mean replacement costs were calculated for sick leave usage 3 years pre and 5 years post treatment to estimate the cost of sick leave from the perspective of the company. ^ Results. Sick leave, as measured by absence rate, predicted the risk of being diagnosed with an AOD disorder (OR 1.10, 95% CI = 1.06, 1.15) during the 12 months prior to receiving the diagnosis. Mean absence rates for diagnosed employees increased over the three years before treatment, particularly in the year before treatment, whereas the controls’ did not (three years, x = 6.80 vs. 5.52; two years, x = 7.81 vs. 6.30, and one year, x = 11.00cases vs. 5.51controls. In the first year post treatment compared to the year prior to treatment, rate ratios indicated a significant (60%) post treatment reduction in absence rates (OR = 0.40, CI = 0.28, 0.57). Absence rates for cases remained lower than controls for the first three years after completion of treatment. Upon discharge from the FAA and company’s three year AOD monitoring program, case’s absence rates increased slightly during the fourth year (controls, x = 0.09, SD = 0.14, cases, x = 0.12, SD = 0.21). However, the following year, their mean absence rates were again below those of the controls (controls, x = 0.08, SD = 0.12, cases, x¯ = 0.06, SD = 0.07). Significant reductions in costs associated with replacing pilots calling in sick, were found to be 60% less, between the year of diagnosis for the cases and the first year after returning to work. A reduction in replacement costs continued over the next two years for the treated employees. ^ Conclusions. This research demonstrates the potential for workplace absences as an active organizational surveillance mechanism to assist managers and supervisors in identifying employees who may be experiencing or at risk of experiencing an alcohol/drug disorder. Currently, many workplaces use only performance problems and ignore the employee’s absence record. A referral to an EAP or alcohol/drug evaluation based on the employee’s absence/sick leave record as incorporated into company policy can provide another useful indicator that may also carry less stigma, thus reducing barriers to seeking help. This research also confirms two conclusions heretofore based only on cross-sectional studies: (1) higher absence rates are associated with employees experiencing an AOD disorder; (2) treatment is associated with lower costs for replacing absent pilots. Due to the uniqueness of the employee population studied (commercial airline pilots) and the organizational documentation of absence, the generalizability of this study to other professions and occupations should be considered limited. ^ Transition to Practice. The odds ratios for the relationship between absence rates and an AOD diagnosis are precise; the OR for year of diagnosis indicates the likelihood of being diagnosed increases 10% for every hour change in sick leave taken. In practice, however, a pilot uses approximately 20 hours of sick leave for one trip, because the replacement will have to be paid the guaranteed minimum of 20 hour. Thus, the rate based on hourly changes is precise but not practical. ^ To provide the organization with practical recommendations the yearly mean absence rates were used. A pilot flies on average, 90 hours a month, 1080 annually. Cases used almost twice the mean rate of sick time the year prior to diagnosis (T-1) compared to controls (cases, x = .11, controls, x = .06). Cases are expected to use on average 119 hours annually (total annual hours*mean annual absence rate), while controls will use 60 hours. The cases’ 60 hours could translate to 3 trips of 20 hours each. Management could use a standard of 80 hours or more of sick time claimed in a year as the threshold for unacceptable absence, a 25% increase over the controls (a cost to the company of approximately of $4000). At the 80-hour mark, the Chief Pilot would be able to call the pilot in for a routine check as to the nature of the pilot’s excessive absence. This management action would be based on a company standard, rather than a behavioral or performance issue. Using absence data in this fashion would make it an active surveillance mechanism. ^

COSTS AND EFFECTS OF ARKANSAS' LONG-TERM CARE DEMONSTRATION PROJECT (MEDICAID, CHANELLING, COMMUNITY ASSESSMENT)

The purpose of this study was to assess the impact of the Arkansas Long-Term Care Demonstration Project upon Arkansas' Medicaid expenditures and upon the clients it serves. A Retrospective Medicaid expenditure study component used analyses of variance techniques to test for the Project's effects upon aggregated expenditures for 28 demonstration and control counties representing 25 percent of the State's population over four years, 1979-1982.^ A second approach to the study question utilized a 1982 prospective sample of 458 demonstration and control clients from the same 28 counties. The disability level or need for care of each patient was established a priori. The extent to which an individual's variation in Medicaid utilization and costs was explained by patient need, presence or absence of the channeling project's placement decision or some other patient characteristic was examined by multiple regression analysis. Long-term and acute care Medicaid, Medicare, third party, self-pay and the grand total of all Medicaid claims were analyzed for project effects and explanatory relationships.^ The main project effect was to increase personal care costs without reducing nursing home or acute care costs (Prospective Study). Expansion of clients appeared to occur in personal care (Prospective Study) and minimum care nursing home (Retrospective Study) for the project areas. Cost-shifting between Medicaid and Medicare in the project areas and two different patterns of utilization in the North and South projects tended to offset each other such that no differences in total costs between the project areas and demonstration areas occurred. The project was significant ((beta) = .22, p < .001) only for personal care costs. The explanatory power of this personal care regression model (R('2) = .36) was comparable to other reported health services utilization models. Other variables (Medicare buy-in, level of disability, Social Security Supplemental Income (SSI), net monthly income, North/South areas and age) explained more variation in the other twelve cost regression models. ^

Network-based genome-wide association study of Crohn's disease

Genome-wide association studies (GWAS) have rapidly become a standard method for disease gene discovery. Many recent GWAS indicate that for most disorders, only a few common variants are implicated and the associated SNPs explain only a small fraction of the genetic risk. The current study incorporated gene network information into gene-based analysis of GWAS data for Crohn's disease (CD). The purpose was to develop statistical models to boost the power of identifying disease-associated genes and gene subnetworks by maximizing the use of existing biological knowledge from multiple sources. The results revealed that Markov random field (MRF) based mixture model incorporating direct neighborhood information from a single gene network is not efficient in identifying CD-related genes based on the GWAS data. The incorporation of solely direct neighborhood information might lead to the low efficiency of these models. Alternative MRF models looking beyond direct neighboring information are necessary to be developed in the future for the purpose of this study.^

Network-based analysis of genome-wide scans of natural selection

The genomic era brought by recent advances in the next-generation sequencing technology makes the genome-wide scans of natural selection a reality. Currently, almost all the statistical tests and analytical methods for identifying genes under selection was performed on the individual gene basis. Although these methods have the power of identifying gene subject to strong selection, they have limited power in discovering genes targeted by moderate or weak selection forces, which are crucial for understanding the molecular mechanisms of complex phenotypes and diseases. Recent availability and rapid completeness of many gene network and protein-protein interaction databases accompanying the genomic era open the avenues of exploring the possibility of enhancing the power of discovering genes under natural selection. The aim of the thesis is to explore and develop normal mixture model based methods for leveraging gene network information to enhance the power of natural selection target gene discovery. The results show that the developed statistical method, which combines the posterior log odds of the standard normal mixture model and the Guilt-By-Association score of the gene network in a naïve Bayes framework, has the power to discover moderate/weak selection gene which bridges the genes under strong selection and it helps our understanding the biology under complex diseases and related natural selection phenotypes.^

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. ^