Covariate Modelling in Aggregate Data Meta Analysis: A Simulation Study Evaluating Performance of Model Linearization
Patanjali Ravva (1), Mats O. Karlsson (2), Jonathan L. French (1)
(1)Pharmacometrics, Global Clinical Pharmacology, Pfizer Inc, New London, U.S.A; (2)Department of Pharmaceutical Biosciences, Uppsala University, Sweden
Objectives: This simulation study evaluated a proposed method based on second order Taylor series approximation for fitting covariate models to aggregate data (AD) or combined AD and individual patient level data (IPD), as analyzing AD data with IPD models leads to biased parameter estimates [1,2].
Methods: This simulation study is motivated by the meta-analysis of HbA1c lowering (% change from baseline) at 12 weeks following treatment using DPP4 inhibitors. An Emax model with effects of Baseline HbA1c and Age on Emax parameter described well the individual patient data from an internal compound.
IPD of a continuous clinical endpoint (CFB) for 5 drugs (A to E) of similar class were simulated using an Emax dose response model. The model incorporated effects of two continuous covariates on Emax parameter using a power function. A total of 18 studies (2, Drug A; 4, Drugs B to E) each with 5 dose groups (including placebo) and 50 individuals per group were simulated.
A total of 9 scenarios exploring the combination of effects (small, moderate and large) of varying degree of nonlinearity with respect to covariates, and of between study to within study covariate variability were performed (500 per Scenario). For each simulation run, two additional datasets were created by reducing IPD to AD and by combining AD and IPD from Drug A (ADIPD).
Four models were proposed to analyze these data. IPD was modelled with an IPD model (original model); AD was modelled with an AD model (similar to IPD model, with summary values in place of individual values and residual error appropriately corrected for sample size). Additionally, an aggregate model (AD_Lin) was derived using a second order Taylor series approximation of the IPD model and fit to AD. Finally, ADIPD were modelled using a combination of IPD and AD_Lin models.
The bias and precision in parameter estimates under four models were assessed.
Results: The bias in estimated Emax parameter under various simulation scenarios are presented below
|
|
|
Nonlinearity | |||
|
Model |
Small |
Moderate |
Large | ||
|
Between Study to Within Study Variability |
Small |
IPD |
<1% |
<1% |
<1% |
|
AD |
3% |
24% |
70% | ||
|
AD_Lin |
<1% |
2.04% |
9% | ||
|
ADIPD_Lin |
<1% |
2% |
8% | ||
|
Moderate |
IPD |
<1% |
<1% |
<1% | |
|
AD |
2% |
12% |
32% | ||
|
AD_Lin |
<1% |
2% |
5% | ||
|
ADIPD_Lin |
<1% |
1% |
4% | ||
|
Large |
IPD |
<1% |
<1% |
<1% | |
|
AD |
1% |
6% |
16% | ||
|
AD_Lin |
<1% |
1% |
3% | ||
|
ADIPD_Lin |
<1% |
1% |
3% | ||
Conclusions: Parameter estimates from the AD model can be severely biased. The proposed linearization method based on Taylor series approximation adequately addresses the issue of bias when modelling aggregate data using nonlinear models.
References:
[1] Greenland, S (1992). Divergent biases in ecologic and individual-level studies. Statist. Med. 11, 1209-1223.
[2] Wakefield, J and Salway, R (2001). A statistical framework for ecological and aggregate studies. J.R. Statist. Soc. 164, 119-137.
