Synthetic data method to incorporate external information into a current study

Abstract

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 for m 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 variances 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. The Canadian Journal of Statistics 47: 580–603; 2019 © 2019 Statistical Society of Canada

A synthetic data integration framework to leverage external summary-level information from heterogeneous populations

“…As shown in Figure 3c, we show the result of the Louis information estimator (StackImpute-Louis), one of the three variance estimators proposed by Beesley and Taylor (2020) as they always have similar performances. Gu et al (2019) has shown that when the synthetic data size goes to infinity, the precision gain we achieve in X covariates will converge to a constant, which is shown as the gradually stable trend of the grey curve (Monte Carlo empirical variance of the point estimates, also serves as the empirical truth here). When the synthetic data size increases from one times the internal data size to 10 times for each external study (i.e., total missing rate increases from 66.6% to 95%), the StackImpute variance estimator and Rubin's rule variance continuously underestimate the empirical truth.…”

“…Step 1: Convert each external summary-level information into a set of synthetic data according to Gu et al (2019) and append each of the synthetic data sets to the internal data, from which we create a longer dataset as illustrated in Figure 1. The synthetic data for external study k constitutes of observed X k and the simulated value of Y. Unmeasured variables in the external populations (all B and some X's) will be treated as missing data.…”

“…Step 1 converts the original problem to regression modeling with missing data using a combined dataset of the internal and the synthetic data. Gu et al (2019) provided theoretical justification in special cases to show that the synthetic data method is equivalent to a constrained semi-parametric maximum likelihood approach proposed by Chatterjee et al (2016), and it assumed the external models were the best-fitted models in the class, but the class may not contain the true distribution (Assumption 1). In finite sample size, the larger the number of replicates in each synthetic dataset (denoted as r k in Figure 1), the more precision gain in the estimated coefficient of X; when r k goes to infinity, the precision gain by incorporating external information will converge to a constant (Gu et al, 2019).…”

“…Recent studies on this topic begin with incorporating external auxiliary information from large data, such as census or population-based biobank data, to improve the statistical inference of the internal study, assuming the model that relates the variables is fully/partially shared between data sources, also known as transportability (Bareinboim and Pearl, 2013). Qin (2000); Han and Lawless (2019) proposed possible solutions using empirical likelihood, Chatterjee et al (2016) demonstrated it from the perspective of constrained maximal likelihood while Gu, Taylor, and Mukherjee (2019) utilized the idea from survey methodology to convert the external information into synthetic data. Estes, Mukherjee, and Taylor (2017) then relaxed the transportability assumption of Chatterjee et al (2016)'s by constructing an empirical Bayes estimator that protected against the potential bias.…”

Improving main analysis by borrowing information from auxiliary data

“…In presence of this type of auxiliary data, methods based on the generalized method of moment, generalized regression, weight calibration, constrained maximum likelihood, empirical likelihood, etc., have been proposed to borrow auxiliary information to power up the main study. [1][2][3][4][5][6][7][8][9][10][11][12][13] In this article, we consider a different type of auxiliary data that is also widely seen in applications, that is, an auxiliary measurement collected in the same study but served as the outcome in a secondary analysis. Usually, such kind of auxiliary measurement is highly associated with the primary outcome, and how to incorporate this secondary information to enhance estimation precision for the main analysis is of high interest.…”