Selecting the Optimal Transformation of a Continuous Covariate in Cox's Regression: Implications for Hypothesis Testing

ABSTRACT

Background: Many exposures in epidemiological studies have nonlinear effects and the problem is to choose an appropriate functional relationship between such exposures and the outcome. One common approach is to investigate several parametric transformations of the covariate of interest, and to select a posteriori the function that fits the data the best. However, such approach may result in an inflated Type I error. Methods: Through a simulation study, we generated data from Cox's models with different transformations of a single continuous covariate. We investigated the Type I error rate and the power of the likelihood ratio test (LRT) corresponding to three different procedures that considered the same set of parametric dose-response functions. The first unconditional approach did not involve any model selection, while the second conditional approach was based on a posteriori selection of the parametric function. The proposed third approach was similar to the second except that it used a corrected critical value for the LRT to ensure a correct Type I error. Results: The Type I error rate of the second approach was two times higher than the nominal size. For simple monotone dose-response, the corrected test had similar power as the unconditional approach, while for non monotone, dose-response, it had a higher power. A real-life application that focused on the effect of body mass index on the risk of coronary heart disease death, illustrated the advantage of the proposed approach. Conclusion: Our results confirm that a posteriori selecting the functional form of the dose-response induces a Type I error inflation. The corrected procedure, which can be applied in a wide range of situations, may provide a good trade-off between Type I error and power.     

Covariate handling approaches in combination with dynamic borrowing for hybrid control studies

Borrowing data from external control has been an appealing strategy for evidence synthesis when conducting randomized controlled trials (RCTs). Often named hybrid control trials, they leverage existing control data from clinical trials or potentially real-world data (RWD), enable trial designs to allocate more patients to the novel intervention arm, and improve the efficiency or lower the cost of the primary RCT. Several methods have been established and developed to borrow external control data, among which the propensity score methods and Bayesian dynamic borrowing framework play essential roles. Noticing the unique strengths of propensity score methods and Bayesian hierarchical models, we utilize both methods in a complementary manner to analyze hybrid control studies. In this article, we review methods including covariate adjustments, propensity score matching and weighting in combination with dynamic borrowing and compare the performance of these methods through comprehensive simulations. Different degrees of covariate imbalance and confounding are examined. Our findings suggested that the conventional covariate adjustment in combination with the Bayesian commensurate prior model provides the highest power with good type I error control under the investigated settings. It has desired performance especially under scenarios of different degrees of confounding. To estimate efficacy signals in the exploratory setting, the covariate adjustment method in combination with the Bayesian commensurate prior is recommended.     

Using marginal structural models to analyze the impact of subsequent therapy on the treatment effect in survival data: Simulations and clinical trial examples

We explore the impact of time-varying subsequent therapy on the statistical power and treatment effects in survival analysis. The marginal structural model (MSM) with stabilized inverse probability treatment weights (sIPTW) was used to account for the effects due to the subsequent therapy. Simulations were performed to compare the MSM-sIPTW method with the conventional method without accounting for the time-varying covariate such as subsequent therapy that is dependent on the initial response of the treatment effect. The results of the simulations indicated that the statistical power, thereby the Type I error, of the trials to detect the frontline treatment effect could be inflated if no appropriate adjustment was made for the impact due to the add-on effects of the subsequent therapy. Correspondingly, the hazard ratio between the treatment groups may be overestimated by the conventional analysis methods. In contrast, MSM-sIPTW can maintain the Type I error rate and gave unbiased estimates of the hazard ratio for the treatment. Two real examples were used to discuss the potential clinical implications. The study demonstrated the importance of accounting for time-varying subsequent therapy for obtaining unbiased interpretation of data.     

A fixed sequence Bonferroni procedure for testing multiple endpoints

A method for controlling the familywise error rate combining the Bonferroni adjustment and fixed testing sequence procedures is proposed. This procedure allots Type I error like the Bonferroni adjustment, but allows the Type I error to accumulate whenever a null hypothesis is rejected. In this manner, power for hypotheses tested later in a prespecified order will be increased. The order of the hypothesis tests needs to be prespecified as in a fixed sequence testing procedure, but unlike the fixed sequence testing procedure all hypotheses can always be tested, allowing for an a priori method of concluding a difference in the various endpoints. An application will be in clinical trials in which mortality is a concern, but it is expected that power to distinguish a difference in mortality will be low. If the effect on mortality is larger than anticipated, this method allows a test with a prespecified method of controlling the Type I error rate. Copyright © 2003 John Wiley & Sons, Ltd.     

Type I error rates from likelihood‐based repeated measures analyses of incomplete longitudinal data

The last observation carried forward (LOCF) approach is commonly utilized to handle missing values in the primary analysis of clinical trials. However, recent evidence suggests that likelihood‐based analyses developed under the missing at random (MAR) framework are sensible alternatives. The objective of this study was to assess the Type I error rates from a likelihood‐based MAR approach – mixed‐model repeated measures (MMRM) – compared with LOCF when estimating treatment contrasts for mean change from baseline to endpoint (Δ). Data emulating neuropsychiatric clinical trials were simulated in a 4 × 4 factorial arrangement of scenarios, using four patterns of mean changes over time and four strategies for deleting data to generate subject dropout via an MAR mechanism. In data with no dropout, estimates of Δ and SE_Δ from MMRM and LOCF were identical. In data with dropout, the Type I error rates (averaged across all scenarios) for MMRM and LOCF were 5.49% and 16.76%, respectively. In 11 of the 16 scenarios, the Type I error rate from MMRM was at least 1.00% closer to the expected rate of 5.00% than the corresponding rate from LOCF. In no scenario did LOCF yield a Type I error rate that was at least 1.00% closer to the expected rate than the corresponding rate from MMRM. The average estimate of SE_Δ from MMRM was greater in data with dropout than in complete data, whereas the average estimate of SE_Δ from LOCF was smaller in data with dropout than in complete data, suggesting that standard errors from MMRM better reflected the uncertainty in the data. The results from this investigation support those from previous studies, which found that MMRM provided reasonable control of Type I error even in the presence of MNAR missingness. No universally best approach to analysis of longitudinal data exists. However, likelihood‐based MAR approaches have been shown to perform well in a variety of situations and are a sensible alternative to the LOCF approach. MNAR methods can be used within a sensitivity analysis framework to test the potential presence and impact of MNAR data, thereby assessing robustness of results from an MAR method. Copyright © 2004 John Wiley & Sons, Ltd.     

Latent class based multiple imputation approach for missing categorical data

In this paper we propose a latent class based multiple imputation approach for analyzing missing categorical covariate data in a highly stratified data model. In this approach, we impute the missing data assuming a latent class imputation model and we use likelihood methods to analyze the imputed data. Via extensive simulations, we study its statistical properties and make comparisons with complete case analysis, multiple imputation, saturated log-linear multiple imputation and the Expectation–Maximization approach under seven missing data mechanisms (including missing completely at random, missing at random and not missing at random). These methods are compared with respect to bias, asymptotic standard error, type I error, and 95% coverage probabilities of parameter estimates. Simulations show that, under many missingness scenarios, latent class multiple imputation performs favorably when jointly considering these criteria. A data example from a matched case–control study of the association between multiple myeloma and polymorphisms of the Inter-Leukin 6 genes is considered.     

Covariate‐adjusted borrowing of historical control data in randomized clinical trials

The borrowing of historical control data can be an efficient way to improve the treatment effect estimate of the current control group in a randomized clinical trial. When the historical and current control data are consistent, the borrowing of historical data can increase power and reduce Type I error rate. However, when these 2 sources of data are inconsistent, it may result in a combination of biased estimates, reduced power, and inflation of Type I error rate. In some situations, inconsistency between historical and current control data may be caused by a systematic variation in the measured baseline prognostic factors, which can be appropriately addressed through statistical modeling. In this paper, we propose a Bayesian hierarchical model that can incorporate patient‐level baseline covariates to enhance the appropriateness of the exchangeability assumption between current and historical control data. The performance of the proposed method is shown through simulation studies, and its application to a clinical trial design for amyotrophic lateral sclerosis is described. The proposed method is developed for scenarios involving multiple imbalanced prognostic factors and thus has meaningful implications for clinical trials evaluating new treatments for heterogeneous diseases such as amyotrophic lateral sclerosis.     

Properties of the rank transformation in factorial analysis of covariance

Real world data often fail to meet the underlying assumption of population normality. The Rank Transformation (RT) procedure has been recommended as an alternative to the parametric factorial analysis of covariance (ANCOVA). The purpose of this study was to compare the Type I error and power properties of the RT ANCOVA to the parametric procedure in the context of a completely randomized balanced 3 × 4 factorial layout with one covariate. This study was concerned with tests of homogeneity of regression coefficients and interaction under conditional (non)normality. Both procedures displayed erratic Type I error rates for the test of homogeneity of regression coefficients under conditional nonnormality. With all parametric assumptions valid, the simulation results demonstrated that the RT ANCOVA failed as a test for either homogeneity of regression coefficients or interaction due to severe Type I error inflation. The error inflation was most severe when departures from conditional normality were extreme. Also associated with the RT procedure was a loss of power. It is recommended that the RT procedure not be used as an alternative to factorial ANCOVA despite its encouragement from SAS, IMSL, and other respected sources.     

To adjust or not to adjust for baseline when analyzing repeated binary responses? The case of complete data when treatment comparison at study end is of interest

The benefits of adjusting for baseline covariates are not as straightforward with repeated binary responses as with continuous response variables. Therefore, in this study, we compared different methods for analyzing repeated binary data through simulations when the outcome at the study endpoint is of interest. Methods compared included chi‐square, Fisher's exact test, covariate adjusted/unadjusted logistic regression (Adj.logit/Unadj.logit), covariate adjusted/unadjusted generalized estimating equations (Adj.GEE/Unadj.GEE), covariate adjusted/unadjusted generalized linear mixed model (Adj.GLMM/Unadj.GLMM). All these methods preserved the type I error close to the nominal level. Covariate adjusted methods improved power compared with the unadjusted methods because of the increased treatment effect estimates, especially when the correlation between the baseline and outcome was strong, even though there was an apparent increase in standard errors. Results of the Chi‐squared test were identical to those for the unadjusted logistic regression. Fisher's exact test was the most conservative test regarding the type I error rate and also with the lowest power. Without missing data, there was no gain in using a repeated measures approach over a simple logistic regression at the final time point. Analysis of results from five phase III diabetes trials of the same compound was consistent with the simulation findings. Therefore, covariate adjusted analysis is recommended for repeated binary data when the study endpoint is of interest. Copyright © 2015 John Wiley & Sons, Ltd.     

Sample Size Re-Estimation Based on Two-Stage Analysis of Variance: Interim Analysis of Clinical Trials

Planning and conducting interim analysis are important steps for long-term clinical trials. In this article, the concept of conditional power is combined with the classic analysis of variance (ANOVA) for a study of two-stage sample size re-estimation based on interim analysis. The overall Type I and Type II errors would be inflated by interim analysis. We compared the effects on re-estimating sample sizes with and without the adjustment of Type I and Type II error rates due to interim analysis.     

Practical Methods for Bounding Type I Error Rate with an Internal Pilot Design

New analytic forms for distributions at the heart of internal pilot theory solve many problems inherent to current techniques for linear models with Gaussian errors. Internal pilot designs use a fraction of the data to re-estimate the error variance and modify the final sample size. Too small or too large a sample size caused by an incorrect planning variance can be avoided. However, the usual hypothesis test may need adjustment to control the Type I error rate. A bounding test achieves control of Type I error rate while providing most of the advantages of the unadjusted test. Unfortunately, the presence of both a doubly truncated and an untruncated chi-square random variable complicates the theory and computations. An expression for the density of the sum of the two chi-squares gives a simple form for the test statistic density. Examples illustrate that the new results make the bounding test practical by providing very stable, convergent, and much more accurate computations. Furthermore, the new computational methods are effectively never slower and usually much faster. All results apply to any univariate linear model with fixed predictors and Gaussian errors, with the t-test a special case.     

Exact methods of testing the homogeneity of prevalences for correlated binary data

Correlated binary data arise in many ophthalmological and otolaryngological clinical trials. To test the homogeneity of prevalences among different groups is an important issue when conducting these trials. The equal correlation coefficients model proposed by Donner in 1989 is a popular model handling correlated binary data. The asymptotic chi-square test works well when the sample size is large. However, it would fail to maintain the type I error rate when the sample size is relatively small. In this paper, we propose several exact methods to deal with small sample scenarios. Their performances are compared with respect to type I error rate and power. The ‘M approach’ and the ‘E + M approach’ seem to outperform the others. A real work example is given to further explain how these approaches work. Finally, the computational efficiency of the exact methods is discussed as a pressing issue of future work.     

Information Criteria for the Paired-Comparisons Problem

We contrast comparisons of several treatments to control in a single experiment versus separate experiments in terms of Type I error rate and power. It is shown that if no Dunnett correction is applied in the single experiment case with relatively few treatments, the distribution of the number of Type I errors is not that different from what it would be in separate experiments with the same number of subjects in each treatment. The difference becomes more pronounced with a larger number of treatments. Extreme outcomes (either very few or very many rejections) are more likely when comparisons are made in a single experiment. When the total number of subjects is the same in a single versus separate experiments, power is generally higher in a single experiment even if a Dunnett adjustment is made.     

Combining an Internal Pilot with an Interim Analysis for Single Degree of Freedom Tests

An internal pilot with interim analysis (IPIA) design combines interim power analysis (an internal pilot) with interim data analysis (two-stage group sequential). We provide IPIA methods for single df hypotheses within the Gaussian general linear model, including one and two group t tests. The design allows early stopping for efficacy and futility while also re-estimating sample size based on an interim variance estimate. Study planning in small samples requires the exact and computable forms reported here. The formulation gives fast and accurate calculations of power, Type I error rate, and expected sample size.     

Two-sample tests for survival data from observational studies

When observational data are used to compare treatment-specific survivals, regular two-sample tests, such as the log-rank test, need to be adjusted for the imbalance between treatments with respect to baseline covariate distributions. Besides, the standard assumption that survival time and censoring time are conditionally independent given the treatment, required for the regular two-sample tests, may not be realistic in observational studies. Moreover, treatment-specific hazards are often non-proportional, resulting in small power for the log-rank test. In this paper, we propose a set of adjusted weighted log-rank tests and their supremum versions by inverse probability of treatment and censoring weighting to compare treatment-specific survivals based on data from observational studies. These tests are proven to be asymptotically correct. Simulation studies show that with realistic sample sizes and censoring rates, the proposed tests have the desired Type I error probabilities and are more powerful than the adjusted log-rank test when the treatment-specific hazards differ in non-proportional ways. A real data example illustrates the practical utility of the new methods.     

Blinded sample size recalculation for clinical trials with normal data and baseline adjusted analysis

Baseline adjusted analyses are commonly encountered in practice, and regulatory guidelines endorse this practice. Sample size calculations for this kind of analyses require knowledge of the magnitude of nuisance parameters that are usually not given when the results of clinical trials are reported in the literature. It is therefore quite natural to start with a preliminary calculated sample size based on the sparse information available in the planning phase and to re‐estimate the value of the nuisance parameters (and with it the sample size) when a portion of the planned number of patients have completed the study. We investigate the characteristics of this internal pilot study design when an analysis of covariance with normally distributed outcome and one random covariate is applied. For this purpose we first assess the accuracy of four approximate sample size formulae within the fixed sample size design. Then the performance of the recalculation procedure with respect to its actual Type I error rate and power characteristics is examined. The results of simulation studies show that this approach has favorable properties with respect to the Type I error rate and power. Together with its simplicity, these features should make it attractive for practical application. Copyright © 2009 John Wiley & Sons, Ltd.     

Evaluation of alternative confidence intervals to address non-inferiority through the stratified difference between proportions

The confidence interval (CI) for the difference between two proportions has been an important and active research topic, especially in the context of non-inferiority hypothesis testing. Issues concerning the Type 1 error rate, power, coverage rate and aberrations have been extensively studied for non-stratified cases. However, stratified confidence intervals are frequently used in non-inferiority trials and similar settings. In this paper, several methods for stratified confidence intervals for the difference between two proportions, including existing methods and novel extensions from unstratified CIs, are evaluated across different scenarios. When sparsity across the strata is not a concern, adding imputed observations to the stratification analysis can strengthen Type-1 error control without substantial loss of power. When sparseness of data is a concern, most of the evaluated methods fail to control Type-1 error; the modified stratified t-test CI is an exception. We recommend the modified stratified t-test CI as the most useful and flexible method across the respective scenarios; the modified stratified Wald CI may be useful in settings where sparsity is unlikely. These findings substantially contribute to the application of stratified CIs for non-inferiority testing of differences between two proportions.     

Some robustness issues for comparing multiple logistic regression slopes to a control for small samples

For comparing several logistic regression slopes to that of a control for small sample sizes, Dasgupta et al. (2001) proposed an "asymptotic" small-sample test and a "pivoted" version of that test statistic. Their results show both methods perform well in terms of Type I error control and marginal power when the response is related to the explanatory variable via a logistic regression model. This study finds, via Monte Carlo simulations, that when the underlying relationship is probit, complementary log-log, linear, or even non-monotonic, the "asymptotic" and the "pivoted" small-sample methods perform fairly well in terms of Type I error control and marginal power. Unlike their large sample competitors, they are generally robust to departures from the logistic regression model.     

Two-Stage Bayesian Model Averaging in Endogenous Variable Models

Economic modeling in the presence of endogeneity is subject to model uncertainty at both the instrument and covariate level. We propose a Two-Stage Bayesian Model Averaging (2SBMA) methodology that extends the Two-Stage Least Squares (2SLS) estimator. By constructing a Two-Stage Unit Information Prior in the endogenous variable model, we are able to efficiently combine established methods for addressing model uncertainty in regression models with the classic technique of 2SLS. To assess the validity of instruments in the 2SBMA context, we develop Bayesian tests of the identification restriction that are based on model averaged posterior predictive p-values. A simulation study showed that 2SBMA has the ability to recover structure in both the instrument and covariate set, and substantially improves the sharpness of resulting coefficient estimates in comparison to 2SLS using the full specification in an automatic fashion. Due to the increased parsimony of the 2SBMA estimate, the Bayesian Sargan test had a power of 50% in detecting a violation of the exogeneity assumption, while the method based on 2SLS using the full specification had negligible power. We apply our approach to the problem of development accounting, and find support not only for institutions, but also for geography and integration as development determinants, once both model uncertainty and endogeneity have been jointly addressed.     

Nonparametric Covariate-Adjusted Association Tests Based on the Generalized Kendall's Tau()

Identifying the risk factors for comorbidity is important in psychiatric research. Empirically, studies have shown that testing multiple, correlated traits simultaneously is more powerful than testing a single trait at a time in association analysis. Furthermore, for complex diseases, especially mental illnesses and behavioral disorders, the traits are often recorded in different scales such as dichotomous, ordinal and quantitative. In the absence of covariates, nonparametric association tests have been developed for multiple complex traits to study comorbidity. However, genetic studies generally contain measurements of some covariates that may affect the relationship between the risk factors of major interest (such as genes) and the outcomes. While it is relatively easy to adjust these covariates in a parametric model for quantitative traits, it is challenging for multiple complex traits with possibly different scales. In this article, we propose a nonparametric test for multiple complex traits that can adjust for covariate effects. The test aims to achieve an optimal scheme of adjustment by using a maximum statistic calculated from multiple adjusted test statistics. We derive the asymptotic null distribution of the maximum test statistic, and also propose a resampling approach, both of which can be used to assess the significance of our test. Simulations are conducted to compare the type I error and power of the nonparametric adjusted test to the unadjusted test and other existing adjusted tests. The empirical results suggest that our proposed test increases the power through adjustment for covariates when there exist environmental effects, and is more robust to model misspecifications than some existing parametric adjusted tests. We further demonstrate the advantage of our test by analyzing a data set on genetics of alcoholism.