## Comparing Minimum Hellinger Distance Estimation (MHDE) and Hypothesis Testing to Traditional Statistical Analyses – a Simulation Study

Matthew M Hutmacher, Anand Vidyashankar, Debu Mukherhjee

Pfizer

Background:  MHDE is accomplished by optimizing the Hellinger Distance (HD) objective function (HD2), which is constructed using the HD metric, over the parameter space.  Essentially, the estimates are parameter values that yield the smallest distance (HD2) between the specified model density and an empirical density function of the data.  The HD2 metric also provides an absolute estimate of the lack-of-fit between the model and empirical densities.  MHDE is a consistent estimator and achieves asymptotic normality of the estimates.

MHDE is theoretically asymptotically efficient and a robust estimator.  The efficiency derives from its similarity to maximum likelihood (ML) when the model density is correctly specified.  The robustness results from the down-weighting of the individual observations in the objective function.  The ‘conceptual' weight for MHDE is roughly 1/n, where n is the number of observations.  For maximum likelihood, assuming normally, the weighting is based on the squared error, which is known to be sensitive to outlying (non-normal) observations.

Objectives: Compare the MHDE to traditional (maximum likelihood) estimates under ANOVA and ANCOVA type models for efficiency, bias, type 1 error rate, and power.

Methods: The simulation study was performed for ANOVA and ANCOVA models using three outlier generation scenarios: the first considered no outliers, with residuals generated from a normal distribution, the second used a mixture of normal random variables with 10% outliers, the third consisted of 10% outliers assigned systematically to the second treatment group.  The third scenario represents trial issues such as the breaking of the clinical trial blind, where some patients realized they were on treatment (e.g., injection site reactions).  For the simulation structure, two treatment groups were simulated with 20 subjects per group.  The data were generated under no treatment effect (null hypothesis) and under a treatment effect, which maintained an approximate theoretical power of 60% (the alternative hypothesis), for each scenario.  Hypothesis tests for the ML estimates were performed using a two-sample T-test, and type 1 error rates and powers were computed.  For the MHDE models, an approximate T-test was constructed.  Distributions of the parameter estimates were constructed and the bias and efficiency of the ML and MHDE estimators were compared.

Results: Little bias and similar efficiencies were observed for the scenario with no outliers (first scenario) as expected.  The type 1 error rate and powers were also similar.  For the second scenario, the ML estimates were more variable and the residual variable estimate was overestimated.  The type 1 error rate was maintained for both methods.  However, the ML estimates demonstrated a decreased power compared to MHDE, which was closer to the anticipated 60%.  The ML estimates of the two treatment group effects and the residual error were biased for the third scenario.  The type 1 error and power were also inflated.  In contrast, the MHDE estimates more closely resembled those observed in the first scenario.  Additionally, the type 1 error and power were less affected by the outlying data.

Conclusions: MHDE provided robust estimates and inference when the model was incorrect and efficient estimates when the model was correct.  In contrast, the ML estimates and inferences were affected when the likelihood was misspecified.  Additionally, MHDE maintained greater precision in the estimates when the model was misspecified compared to ML.

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