A novel multivariable Mendelian randomization framework to disentangle highly correlated exposures with application to metabolomics.

Mendelian randomization (MR) utilizes genome-wide association study (GWAS) summary data to infer causal relationships between exposures and outcomes, offering a valuable tool for identifying disease risk factors. Multivariable MR (MVMR) estimates the direct effects of multiple exposures on an outcome. This study tackles the issue of highly correlated exposures commonly observed in metabolomic data, a situation where existing MVMR methods often face reduced statistical power due to multicollinearity. We propose a robust extension of the MVMR framework that leverages constrained maximum likelihood (cML) and employs a Bayesian approach for identifying independent clusters of exposure signals. Applying our method to the UK Biobank metabolomic data for the largest Alzheimer disease (AD) cohort through a two-sample MR approach, we identified two independent signal clusters for AD: glutamine and lipids, with posterior inclusion probabilities (PIPs) of 95.0% and 81.5%, respectively. Our findings corroborate the hypothesized roles of glutamate and lipids in AD, providing quantitative support for their potential involvement.

A constrained maximum likelihood approach to developing well-calibrated models for predicting binary outcomes.

The added value of candidate predictors for risk modeling is routinely evaluated by comparing the performance of models with or without including candidate predictors. Such comparison is most meaningful when the estimated risk by the two models are both unbiased in the target population. Very often data for candidate predictors are sourced from nonrepresentative convenience samples. Updating the base model using the study data without acknowledging the discrepancy between the underlying distribution of the study data and that in the target population can lead to biased risk estimates and therefore an unfair evaluation of candidate predictors. To address this issue assuming access to a well-calibrated base model, we propose a semiparametric method for model fitting that enforces good calibration. The central idea is to calibrate the fitted model against the base model by enforcing suitable constraints in maximizing the likelihood function. This approach enables unbiased assessment of model improvement offered by candidate predictors without requiring a representative sample from the target population, thus overcoming a significant practical challenge. We study theoretical properties for model parameter estimates, and demonstrate improvement in model calibration via extensive simulation studies. Finally, we apply the proposed method to data extracted from Penn Medicine Biobank to inform the added value of breast density for breast cancer risk assessment in the Caucasian woman population.

Confidence intervals for the cumulative incidence function via constrained NPMLE.

The cumulative incidence function (CIF) displays key information in the competing risks setting, which is common in medical research. In this article, we introduce two new methods to compute non-parametric confidence intervals for the CIF. First, we introduce non-parametric profile-likelihood confidence intervals. The method builds on constrained non-parametric maximum likelihood estimation (NPMLE), for which we derive closed-form formulas. This method can be seen as an extension of that of Thomas and Grunkemeier (J Am Stat Assoc 70:865-871, 1975) to the competing risks setting, when the CIF is of interest instead of the survival function. Second, we build on constrained NPMLE to introduce constrained bootstrap confidence intervals. This extends an interesting approach introduced by Barber and Jennison (Biometrics 52:430-436, 1999) to the competing risks setting. A simulation study illustrates how these methods can perform as compared to benchmarks implemented in popular software. The results suggest that more accurate confidence intervals than usual Wald-type ones can be obtained in the case of small to moderate sample sizes and few observed events. An application to melanoma data is provided for illustration purpose.

A modified Wald test for reference scaled assessment of analytical equivalence.

For the reference scaled equivalence hypothesis to reduce the deficiency of the current practice in analytical equivalence assessment, the Wald test with Constrained Maximum Likelihood Estimate (CMLE) of the standard error was proposed to improve the efficiency when the sample sizes of test and reference product lots are small, and variances are unequal. However, by using the Wald test with CMLE standard error, simulations show that the type I error rate is below the nominal significance level. We proposed the Modified Wald test with CMLE standard error by replacing the maximum likelihood estimate of reference standard deviation with the sample estimate (MWCMLE), resulting in further improvement of type I error rate and power over the Wald test with CMLE standard error. In this paper, we further compare the proposed MWCMLE method to the Exact-test-Based (EB) method and the Generalized Pivotal Quantity (GPQ) method with equal or unequal variances, or equal or unequal sample sizes of both product lots. The simulations show that the proposed MWCMLE method outperforms the other two methods in type I error rate control and power improvement.

Synthetic data method to incorporate external information into a current study.

We consider the situation where there is a known regression model that can be used to predict an outcome, Y, from a set of predictor variables X. A new variable B is expected to enhance the prediction of Y. A dataset of size n containing Y, X and B is available, and the challenge is to build an improved model for Y|X,B that uses both the available individual level data and some summary information obtained from the known model for Y|X. We propose a synthetic data approach, which consists of creating m additional synthetic data observations, and then analyzing the combined dataset of size n+m to estimate the parameters of the Y|X, B model. This combined dataset of size n+m now has missing values of B form of the observations, and is analyzed using methods that can handle missing data (e.g. multiple imputation). We present simulation studies and illustrate the method using data from the Prostate Cancer Prevention Trial. Though the synthetic data method is applicable to a general regression context, to provide some justification, we show in two special cases that the asymptotic variance of the parameter estimates in the Y|X, B model are identical to those from an alternative constrained maximum likelihood estimation approach. This correspondence in special cases and the method's broad applicability makes it appealing for use across diverse scenarios.

Wald tests for variance-adjusted equivalence assessment with normal endpoints.

Equivalence tests may be tested with mean difference against a margin adjusted for variance. The justification of using variance adjusted non-inferiority or equivalence margin is for the consideration that a larger margin should be used with large measurement variability. However, under the null hypothesis, the test statistic does not follow a t-distribution or any well-known distribution even when the measurement is normally distributed. In this study, we investigate asymptotic tests for testing the equivalence hypothesis. We apply the Wald test statistic and construct three Wald tests that differ in their estimates of variances. These estimates of variances include the maximum likelihood estimate (MLE), the uniformly minimum variance unbiased estimate (UMVUE), and the constrained maximum likelihood estimate (CMLE). We evaluate the performance of these three tests in terms of type I error rate control and power using simulations under a variety of settings. Our empirical results show that the asymptotic normalized tests are conservative in most settings, while the Wald tests based on ML- and UMVU-method could produce inflated significance levels when group sizes are unequal. However, the Wald test based on CML-method provides an improvement in power over the other two Wald tests for medium and small sample size studies.

A Comparison of Penalized Maximum Likelihood Estimation and Markov Chain Monte Carlo Techniques for Estimating Confirmatory Factor Analysis Models With Small Sample Sizes.

With small to modest sample sizes and complex models, maximum likelihood (ML) estimation of confirmatory factor analysis (CFA) models can show serious estimation problems such as non-convergence or parameter estimates outside the admissible parameter space. In this article, we distinguish different Bayesian estimators that can be used to stabilize the parameter estimates of a CFA: the mode of the joint posterior distribution that is obtained from penalized maximum likelihood (PML) estimation, and the mean (EAP), median (Med), or mode (MAP) of the marginal posterior distribution that are calculated by using Markov Chain Monte Carlo (MCMC) methods. In two simulation studies, we evaluated the performance of the Bayesian estimators from a frequentist point of view. The results show that the EAP produced more accurate estimates of the latent correlation in many conditions and outperformed the other Bayesian estimators in terms of root mean squared error (RMSE). We also argue that it is often advantageous to choose a parameterization in which the main parameters of interest are bounded, and we suggest the four-parameter beta distribution as a prior distribution for loadings and correlations. Using simulated data, we show that selecting weakly informative four-parameter beta priors can further stabilize parameter estimates, even in cases when the priors were mildly misspecified. Finally, we derive recommendations and propose directions for further research.

Learning from Binary Multiway Data: Probabilistic Tensor Decomposition and its Statistical Optimality.

We consider the problem of decomposing a higher-order tensor with binary entries. Such data problems arise frequently in applications such as neuroimaging, recommendation system, topic modeling, and sensor network localization. We propose a multilinear Bernoulli model, develop a rank-constrained likelihood-based estimation method, and obtain the theoretical accuracy guarantees. In contrast to continuous-valued problems, the binary tensor problem exhibits an interesting phase transition phenomenon according to the signal-to-noise ratio. The error bound for the parameter tensor estimation is established, and we show that the obtained rate is minimax optimal under the considered model. Furthermore, we develop an alternating optimization algorithm with convergence guarantees. The efficacy of our approach is demonstrated through both simulations and analyses of multiple data sets on the tasks of tensor completion and clustering.

Empirical Bayes Estimation and Prediction Using Summary-Level Information From External Big Data Sources Adjusting for Violations of Transportability.

Large external data sources may be available to augment studies that collect data to address a specific research objective. In this article we consider the problem of building regression models for prediction based on individual-level data from an "internal" study while incorporating summary information from an "external" big data source. We extend the work of Chatterjee et al (2016a) by introducing an adaptive empirical Bayes shrinkage estimator that uses the external summary-level information and the internal data to trade bias with variance for protection against departures in the conditional probability distribution of the outcome given a set of covariates between the two populations. We use simulation studies and a real data application using external summary information from the Prostate Cancer Prevention Trial to assess the performance of the proposed methods in contrast to maximum likelihood estimation and the constrained maximum likelihood (CML) method developed by Chatterjee et al (2016a). Our simulation studies show that the CML method can be biased and inefficient when the assumption of a transportable covariate distribution between the external and internal populations is violated, and our empirical Bayes estimator provides protection against bias and loss of efficiency.

Closed-form fiducial confidence intervals for some functions of independent binomial parameters with comparisons.

Approximate closed-form confidence intervals (CIs) for estimating the difference, relative risk, odds ratio, and linear combination of proportions are proposed. These CIs are developed using the fiducial approach and the modified normal-based approximation to the percentiles of a linear combination of independent random variables. These confidence intervals are easy to calculate as the computation requires only the percentiles of beta distributions. The proposed confidence intervals are compared with the popular score confidence intervals with respect to coverage probabilities and expected widths. Comparison studies indicate that the proposed confidence intervals are comparable with the corresponding score confidence intervals, and better in some cases, for all the problems considered. The methods are illustrated using several examples.

Standardized likelihood ratio test for comparing several log-normal means and confidence interval for the common mean.

Standardized likelihood ratio test (SLRT) for testing the equality of means of several log-normal distributions is proposed. The properties of the SLRT and an available modified likelihood ratio test (MLRT) and a generalized variable (GV) test are evaluated by Monte Carlo simulation and compared. Evaluation studies indicate that the SLRT is accurate even for small samples, whereas the MLRT could be quite liberal for some parameter values, and the GV test is in general conservative and less powerful than the SLRT. Furthermore, a closed-form approximate confidence interval for the common mean of several log-normal distributions is developed using the method of variance estimate recovery, and compared with the generalized confidence interval with respect to coverage probabilities and precision. Simulation studies indicate that the proposed confidence interval is accurate and better than the generalized confidence interval in terms of coverage probabilities. The methods are illustrated using two examples.