Daily consumption of pro-vitamin A biofortified (yellow) cassava improves serum retinol concentrations in preschool children in Nigeria: a randomized controlled trial.

Mixed-effects models for repeated measures (MMRM) analyses using the Kenward-Roger method for adjusting standard errors and degrees of freedom in an "unstructured" (UN) covariance structure are increasingly becoming common in primary analyses for group comparisons in longitudinal clinical trials. We evaluate the performance of an MMRM-UN analysis using the Kenward-Roger method when the variance of outcome between treatment groups is unequal. In addition, we provide alternative approaches for valid inferences in the MMRM analysis framework. Two simulations are conducted in cases with (1) unequal variance but equal correlation between the treatment groups and (2) unequal variance and unequal correlation between the groups. Our results in the first simulation indicate that MMRM-UN analysis using the Kenward-Roger method based on a common covariance matrix for the groups yields notably poor coverage probability (CP) with confidence intervals for the treatment effect when both the variance and the sample size between the groups are disparate. In addition, even when the randomization ratio is 1:1, the CP will fall seriously below the nominal confidence level if a treatment group with a large dropout proportion has a larger variance. Mixed-effects models for repeated measures analysis with the Mancl and DeRouen covariance estimator shows relatively better performance than the traditional MMRM-UN analysis method. In the second simulation, the traditional MMRM-UN analysis leads to bias of the treatment effect and yields notably poor CP. Mixed-effects models for repeated measures analysis fitting separate UN covariance structures for each group provides an unbiased estimate of the treatment effect and an acceptable CP. We do not recommend MMRM-UN analysis using the Kenward-Roger method based on a common covariance matrix for treatment groups, although it is frequently seen in applications, when heteroscedasticity between the groups is apparent in incomplete longitudinal data.

Effect of heteroscedasticity between treatment groups on mixed-effects models for repeated measures.

There have been considerable advances in the methodology for estimating dynamic treatment regimens, and for the design of sequential trials that can be used to collect unconfounded data to inform such regimens. However, relatively little attention has been paid to how such methodology could be used to advance understanding of optimal treatment strategies in a continuous dose setting, even though it is often the case that considerable patient heterogeneity in drug response along with a narrow therapeutic window may necessitate the tailoring of dosing over time. Such is the case with warfarin, a common oral anticoagulant. We propose novel, realistic simulation models based on pharmacokinetic-pharmacodynamic properties of the drug that can be used to evaluate potentially optimal dosing strategies. Our results suggest that this methodology can lead to a dosing strategy that performs well both within and across populations with different pharmacokinetic characteristics, and may assist in the design of randomized trials by narrowing the list of potential dosing strategies to those which are most promising.

Optimal individualized dosing strategies: A pharmacologic approach to developing dynamic treatment regimens for continuous-valued treatments

In some clinical trials or clinical practice, the therapeutic agent is administered repeatedly, and doses are adjusted in each patient based on repeatedly measured continuous responses, to maintain the response levels in a target range. Because a lower dose tends to be selected for patients with a better outcome, simple summarizations may wrongly show a better outcome for the lower dose, producing an incorrect dose-response relationship. In this study, we consider the dose-response relationship under these situations. We show that maximum-likelihood estimates are consistent without modeling the dose-modification mechanisms when the selection of the dose as a time-dependent covariate is based only on observed, but not on unobserved, responses, and measurements are generated based on administered doses. We confirmed this property by performing simulation studies under several dose-modification mechanisms. We examined an autoregressive linear mixed effects model. The model represents profiles approaching each patient's asymptote when identical doses are repeatedly administered. The model takes into account the previous dose history and provides a dose-response relationship of the asymptote as a summary measure. We also examined a linear mixed effects model assuming all responses are measured at steady state. In the simulation studies, the estimates of both the models were unbiased under the dose modification based on observed responses, but biased under the dose modification based on unobserved responses. In conclusion, the maximum-likelihood estimates of the dose-response relationship are consistent under the dose modification based only on observed responses.

Dose-response relationship from longitudinal data with response-dependent dose modification using likelihood methods.

Impact of short- and long-term exposure to air pollution on blood pressure: A two-decade population-based study in Tehran.

The assessment of the dose-response relationship is important but not straightforward when the therapeutic agent is administered repeatedly with dose-modification in each patient and a continuous response is measured repeatedly. We recently proposed an autoregressive linear mixed effects model for such data in which the current response is regressed on the previous response, fixed effects, and random effects. The model represents profiles approaching each patient's asymptote, takes into account the past dose history, and provides a dose-response relationship of the asymptote as a summary measure. In an autoregressive model, intermittent missing data mean the missing values in previous responses as covariates. We previously provided the marginal (unconditional on the previous response) form of the proposed model to deal with intermittent missing data. Irregular timings of dose-modification or measurement can also be treated as equally spaced data with intermittent missing values by selecting an adequately small unit of time. The likelihood is, however, expressed by matrices whose sizes depend on the number of observations for a patient, and the computational burden is large. In this study, we propose a state space form of the autoregressive linear mixed effects model to calculate the marginal likelihood without using large matrices. The regression coefficients of the fixed effects can be concentrated out of the likelihood in this model by the same way of a linear mixed effects model. As an illustration of the approach, we analyzed immunologic data from a clinical trial for multiple sclerosis patients and estimated the dose-response curves for each patient and the population mean.

An autoregressive linear mixed effects model for the analysis of unequally spaced longitudinal data with dose-modification.

When primary endpoints of randomized trials are continuous variables, the analysis of covariance (ANCOVA) with pre-treatment measurements as a covariate is often used to compare two treatment groups. In the ANCOVA, equal slopes (coefficients of pre-treatment measurements) and equal residual variances are commonly assumed. However, random allocation guarantees only equal variances of pre-treatment measurements. Unequal covariances and variances of post-treatment measurements indicate unequal slopes and, usually, unequal residual variances. For non-normal data with unequal covariances and variances of post-treatment measurements, it is known that the ANCOVA with equal slopes and equal variances using an ordinary least-squares method provides an asymptotically normal estimator for the treatment effect. However, the asymptotic variance of the estimator differs from the variance estimated from a standard formula, and its property is unclear. Furthermore, the asymptotic properties of the ANCOVA with equal slopes and unequal variances using a generalized least-squares method are unclear. In this paper, we consider non-normal data with unequal covariances and variances of post-treatment measurements, and examine the asymptotic properties of the ANCOVA with equal slopes using the variance estimated from a standard formula. Analytically, we show that the actual type I error rate, thus the coverage, of the ANCOVA with equal variances is asymptotically at a nominal level under equal sample sizes. That of the ANCOVA with unequal variances using a generalized least-squares method is asymptotically at a nominal level, even under unequal sample sizes. In conclusion, the ANCOVA with equal slopes can be asymptotically justified under random allocation.

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Analysis of covariance with pre-treatment measurements in randomized trials under the cases that covariances and post-treatment variances differ between groups

In randomized trials, an analysis of covariance (ANCOVA) is often used to analyze post-treatment measurements with pre-treatment measurements as a covariate to compare two treatment groups. Random allocation guarantees only equal variances of pre-treatment measurements. We hence consider data with unequal covariances and variances of post-treatment measurements without assuming normality. Recently, we showed that the actual type I error rate of the usual ANCOVA assuming equal slopes and equal residual variances is asymptotically at a nominal level under equal sample sizes, and that of the ANCOVA with unequal variances is asymptotically at a nominal level, even under unequal sample sizes. In this paper, we investigated the asymptotic properties of the ANCOVA with unequal slopes for such data. The estimators of the treatment effect at the observed mean are identical between equal and unequal variance assumptions, and these are asymptotically normal estimators for the treatment effect at the true mean. However, the variances of these estimators based on standard formulas are biased, and the actual type I error rates are not at a nominal level, irrespective of variance assumptions. In equal sample sizes, the efficiency of the usual ANCOVA assuming equal slopes and equal variances is asymptotically the same as those of the ANCOVA with unequal slopes and higher than that of the ANCOVA with equal slopes and unequal variances. Therefore, the use of the usual ANCOVA is appropriate in equal sample sizes.

Analysis of covariance with pre-treatment measurements in randomized trials: comparison of equal and unequal slopes.

In clinical trials, sometimes only a single drug concentration can be measured from a patient, because of the burden on the patient. From a single concentration, we cannot generally obtain point estimates of each pharmacokinetic parameter in a patient. In this article, we propose a method to estimate the clearance using a one-compartment model of a single-bolus intravenous injection from a single concentration at a sampling point between 1.5 and 2.5 half-lives. This method requires an assumed value for the volume of distribution but is robust to misspecification. This approach is illustrated by simulated concentration data and cadralazine concentration data.

Takashi Funatogawa

Papers

An autoregressive linear mixed effects model for the analysis of unequally spaced longitudinal data with dose-modification.

Dose-response relationship from longitudinal data with response-dependent dose modification using likelihood methods.

Analysis of covariance with pre-treatment measurements in randomized trials under the cases that covariances and post-treatment variances differ between groups

Analysis of covariance with pre-treatment measurements in randomized trials: comparison of equal and unequal slopes.

An Estimation Method of the Clearance for a One-Compartment Model of a Single Bolus Intravenous Injection by a Single Sampling