Description:

The course contains two parts.

1. Introducing the b-value: combining unbiased and biased estimators from a sensitivity analysis perspective

by Zhexiao Lin, Peter J. Bickel, Peng Ding

In empirical research, when we have multiple estimators for the same parameter of interest, a central question arises: how do we combine unbiased but less precise estimators with biased but more precise ones to improve the inference? Under this setting, the point estimation problem has attracted considerable attention. In this paper, we focus on a less studied inference question: how can we conduct valid statistical inference in such settings with unknown bias? We propose a strategy to combine unbiased and biased estimators from a sensitivity analysis perspective. We derive a sequence of confidence intervals indexed by the magnitude of the bias, which enable researchers to assess how conclusions vary with the bias levels. Importantly, we introduce the notion of the b-value, a critical value of the unknown maximum relative bias at which combining estimators does not yield a significant result. We apply this strategy to three canonical combined estimators: the precision-weighted estimator, the pretest estimator, and the soft-thresholding estimator. For each estimator, we characterize the sequence of confidence intervals and determine the bias threshold at which the conclusion changes. Based on the theory, we recommend reporting the b-value based on the soft-thresholding estimator and its associated confidence intervals, which are robust to unknown bias and achieve the lowest worst-case risk among the alternatives.

2. Bias-aware meta-analysis: beyond fixed-effect and random-effects meta-analyses

Mingxuan Zou, Peng Ding

Meta-analysis aims to synthesize multiple studies to obtain a summary estimate of a population parameter of interest. However, existing meta-analysis methods require strong assumptions that render them not robust to bias and heterogeneity of the studies. In particular, fixed-effect meta-analysis assumes a common parameter across the studies, whereas random-effects meta-analysis models heterogeneity as independent and identically distributed random effects, which still implies a common parameter marginally over the random effects. Due to their limitations, we propose a bias-aware meta-analysis framework that allows for heterogeneous studies with fixed, unknown biases. With no bias, our bias-aware meta-analysis reduces to fixed-effect meta-analysis; with a given bound on the bias, we can still use the fixed-effect meta-analysis point estimator and standard error, but we must inflate the critical value for confidence intervals from the standard normal quantile to a folded normal quantile. We recommend reporting the maximum bias for significance, the maximum value of the bias such that the confidence intervals exclude the null value. We propose a formal model checking procedure and recommend reporting the minimum bias for compatibility, the minimum value of the bias such that the bias-aware meta-analysis model is compatible with the observed data. We discuss the practical implementation of our method and, in particular, the approaches to calibrating the maximum bias for significance. We also compare our method with traditional fixed-effect and random-effects meta-analyses.

Prerequisite: Basic probability and statistics

Reference:

Lin, Z., Bickel, P. and Ding, P. (2026+). Introducing the b-value: combining unbiased and biased estimators from a sensitivity analysis perspective.

Zou, M. and Ding, P. (2026+). Bias-aware meta-analysis: beyond fixed-effect and random-effects meta-analyses.

Target Audience: Undergraduate students, Graduate students

Teaching Language: Chinese & English

Registration: https://www.wjx.top/vm/rc5ALqE.aspx#