Bayesian Data Borrowing

Durbadal Ghosh
Sanofi

Acknowledgements

I thank Dr Xu and Dr Zhang for helping me a lot in this project.

Model

\(i\): Patient index
\(h\): Time Interval
\(T_{ih}\): Amount of time spent in interval \(h\) by patient \(i\)
\(E_{ih}\): Indicator variable stating if patient \(i\) died in interval \(h\)
\(X_{ih}\): Indicator variable indicating treatment or control.

Likelihood

\[E_{ih} \sim Poi(\mu_{ih}) \]

\(\mu_{ih}=T_{ih}\alpha_h exp(\beta_h X_{ih})\)

Priors

\(\alpha_h\) (Borrowing paramater) \(\sim Gamma(a_h,b_h)\) where \(a_h\) and \(b_h\) is estimated from historical data.
\(\beta_h\) (Treatment effect) \(\sim N(0,.001)\)

Simulation

We compare our Bayesian borrowing method with Cox’s proportional hazard model, Bayesian method with non-borrowing and piecewise Cox’s model.

Senarios:

Event rate for historical data for all cases is 0.3
Proportional hazard
Equal Event Rates for Current data and historical Data (0.3)
Event rates for current data is greater than the event rates for true/usual (historical) data. (0.35,0.4)
Event rates for current data is lower than the event rates for true/usual (historical) data. (0.25,0.2)
Non Proportional hazard
Equal Event Rates for Current data and historical Data (0.3)
Event rates for current data is greater than the event rates for true/usual (historical) data. (0.35,0.4)
Event rates for current data is lower than the event rates for true/usual (historical) data. (0.25,0.2)

Results

Setup

Number of time interval is 8
MCMC iterations = 5000, Number of Chains = 2

Proportional hazard assumption

Equal Event Rates for Current data and historical Data

Setup

Event Rate for historical data = 0.3
Event Rate for current data = 0.3
HR for historical data =0.6
HR for current data =0.6
Sample size for historical data =800
Sample size for current data =900

Tables

Int	True HR	Est_HR	5%_est_HR	95%_est_HR	cox_HR	Est_HR_borrow	5%_y_est_HR	95%_y_est_HR
1	0.6	0.626	0.422	0.942	0.627	0.635	0.432	0.917
2	0.6	0.658	0.410	1.031	0.658	0.668	0.420	1.008
3	0.6	0.602	0.366	0.955	0.602	0.613	0.382	0.964
4	0.6	0.641	0.352	1.002	0.642	0.654	0.382	0.999
5	0.6	0.642	0.332	1.051	0.641	0.651	0.349	1.066
6	0.6	0.654	0.316	1.036	0.654	0.667	0.330	0.998
7	0.6	0.649	0.352	1.052	0.651	0.660	0.389	1.006
8	0.6	0.679	0.342	1.212	0.678	0.688	0.369	1.142

Estimation of Cox’s proportional hazard:

## [1] 0.6441812

Bias

HR estimation

Event rates for current data is greater than the event rates for true/usual (historical) data.

Setup

Event Rate for historical data = 0.3
Event Rate for current data = 0.35
HR for historical data =0.6
HR for current data =0.6
Sample size for historical data =800
Sample size for current data =900

Tables

Int	True HR	Est_HR	5%_est_HR	95%_est_HR	cox_HR	Est_HR_borrow	5%_y_est_HR	95%_y_est_HR
1	0.6	0.621	0.417	0.867	0.622	0.673	0.449	0.956
2	0.6	0.648	0.397	0.998	0.648	0.692	0.438	1.065
3	0.6	0.626	0.405	0.923	0.627	0.667	0.435	1.030
4	0.6	0.602	0.369	0.886	0.603	0.641	0.407	0.961
5	0.6	0.643	0.395	0.934	0.643	0.688	0.428	1.025
6	0.6	0.626	0.357	0.949	0.626	0.662	0.382	0.979
7	0.6	0.656	0.364	1.192	0.657	0.700	0.395	1.154
8	0.6	0.629	0.363	0.950	0.630	0.678	0.420	1.016

Estimation of Cox’s proportional hazard:

## [1] 0.631829

Bias

HR estimation

Event rates for current data is greater than the event rates for true/usual (historical) data.

Setup

Event Rate for historical data = 0.3
Event Rate for current data = 0.4
HR for historical data =0.6
HR for current data =0.6
Sample size for historical data =800
Sample size for current data =900

Tables

Int	True HR	Est_HR	5%_est_HR	95%_est_HR	cox_HR	Est_HR_borrow	5%_y_est_HR	95%_y_est_HR
1	0.6	0.623	0.451	0.836	0.623	0.669	0.473	0.889
2	0.6	0.622	0.416	0.929	0.622	0.665	0.433	0.959
3	0.6	0.621	0.387	0.903	0.621	0.669	0.412	1.016
4	0.6	0.616	0.401	0.932	0.616	0.667	0.426	0.965
5	0.6	0.619	0.356	0.898	0.619	0.669	0.371	0.992
6	0.6	0.633	0.392	1.122	0.633	0.692	0.424	1.154
7	0.6	0.635	0.378	0.967	0.635	0.702	0.427	1.070
8	0.6	0.678	0.360	1.087	0.679	0.742	0.390	1.227

Estimation of Cox’s proportional hazard:

## [1] 0.6310383

Bias

HR estimation

Event rates for current data is lower than the event rates for true/usual (historical) data.

Setup

Event Rate for historical data = 0.3
Event Rate for current data = 0.25
HR for historical data =0.6
HR for current data =0.6
Sample size for historical data =800
Sample size for current data =900

Tables

Int	True HR	Est_HR	5%_est_HR	95%_est_HR	cox_HR	Est_HR_borrow	5%_y_est_HR	95%_y_est_HR
1	0.6	0.608	0.395	0.863	0.608	0.579	0.369	0.845
2	0.6	0.657	0.433	0.916	0.657	0.632	0.416	0.883
3	0.6	0.644	0.352	1.011	0.644	0.622	0.350	0.984
4	0.6	0.626	0.375	0.912	0.626	0.620	0.380	0.895
5	0.6	0.617	0.331	1.084	0.617	0.612	0.351	1.044
6	0.6	0.655	0.374	1.087	0.655	0.662	0.392	1.195
7	0.6	0.679	0.323	1.246	0.679	0.695	0.355	1.311
8	0.6	0.653	0.314	1.181	0.653	0.676	0.347	1.254

Estimation of Cox’s proportional hazard:

## [1] 0.6423694

Bias

HR estimation

Event rates for current data is lower than the event rates for true/usual (historical) data.

Setup

Event Rate for historical data = 0.3
Event Rate for current data = 0.2
HR for historical data =0.6
HR for current data =0.6
Sample size for historical data =800
Sample size for current data =900

Tables

Int	True HR	Est_HR	5%_est_HR	95%_est_HR	cox_HR	Est_HR_borrow	5%_y_est_HR	95%_y_est_HR
1	0.6	0.606	0.354	0.901	0.606	0.599	0.336	0.899
2	0.6	0.642	0.374	1.013	0.643	0.639	0.371	1.078
3	0.6	0.666	0.336	1.069	0.665	0.671	0.321	1.166
4	0.6	0.648	0.350	1.114	0.648	0.670	0.357	1.293
5	0.6	0.640	0.347	1.174	0.639	0.647	0.354	1.227
6	0.6	0.631	0.290	1.242	0.633	0.653	0.288	1.352
7	0.6	0.675	0.361	1.224	0.675	0.704	0.359	1.419
8	0.6	0.668	0.322	1.164	0.666	0.702	0.333	1.304

Estimation of Cox’s proportional hazard:

## [1] 0.6468758

Bias

HR estimation

Non-Proportional hazard models

Equal Event Rates for Historical Data

Setup

Event Rate for historical data = 0.3
Event Rate for current data = 0.3
HR for historical data =0.9, 0.6
HR for current data =0.9, 0.6
Sample size for historical data = 800
Sample size for current data = 900

Tables

Int	True HR	Est_HR	5%_est_HR	95%_est_HR	cox_HR	Est_HR_borrow	5%_y_est_HR	95%_y_est_HR
1	0.6	0.939	0.640	1.377	0.940	0.957	0.678	1.376
2	0.6	0.941	0.706	1.405	0.941	0.965	0.707	1.383
3	0.6	0.634	0.387	0.974	0.634	0.641	0.399	1.009
4	0.6	0.621	0.363	0.953	0.621	0.628	0.394	0.942
5	0.6	0.632	0.326	0.937	0.632	0.650	0.328	1.074
6	0.6	0.639	0.366	1.111	0.640	0.650	0.385	1.071
7	0.6	0.653	0.355	1.122	0.653	0.673	0.375	1.179
8	0.6	0.649	0.358	1.096	0.649	0.670	0.380	1.119

Estimation of Cox’s proportional hazard:

## [1] 0.7138587

Bias

HR estimation

Event rates for current data is greater than the event rates for true/usual (historical) data.

Setup

Event Rate for historical data = 0.3
Event Rate for current data = 0.35
HR for historical data =0.9, 0.6
HR for current data =0.9, 0.6
Sample size for historical data = 800
Sample size for current data = 900

Tables

Int	True HR	Est_HR	5%_est_HR	95%_est_HR	cox_HR	Est_HR_borrow	5%_y_est_HR	95%_y_est_HR
1	0.6	0.939	0.657	1.327	0.939	1.016	0.716	1.463
2	0.6	0.942	0.593	1.488	0.942	1.006	0.641	1.582
3	0.6	0.627	0.396	0.917	0.627	0.666	0.423	1.058
4	0.6	0.598	0.375	0.900	0.599	0.637	0.401	0.967
5	0.6	0.643	0.405	0.960	0.643	0.689	0.420	1.000
6	0.6	0.625	0.360	0.958	0.625	0.660	0.401	0.976
7	0.6	0.659	0.363	1.193	0.660	0.703	0.391	1.180
8	0.6	0.629	0.387	0.993	0.629	0.678	0.430	1.064

Estimation of Cox’s proportional hazard:

## [1] 0.7080066

Bias

HR estimation

Event rates for current data is greater than the event rates for true/usual (historical) data.

Setup

Event Rate for historical data = 0.3
Event Rate for current data = 0.4
HR for historical data =0.9, 0.6
HR for current data =0.9, 0.6
Sample size for historical data = 800
Sample size for current data = 900

Tables

Int	True HR	Est_HR	5%_est_HR	95%_est_HR	cox_HR	Est_HR_borrow	5%_y_est_HR	95%_y_est_HR
1	0.6	0.931	0.664	1.229	0.932	1.000	0.698	1.343
2	0.6	0.928	0.611	1.375	0.928	0.994	0.648	1.479
3	0.6	0.625	0.395	0.896	0.624	0.672	0.416	0.995
4	0.6	0.619	0.395	0.956	0.619	0.669	0.432	1.010
5	0.6	0.616	0.353	0.881	0.616	0.666	0.368	0.979
6	0.6	0.634	0.372	1.133	0.635	0.694	0.407	1.132
7	0.6	0.632	0.378	1.001	0.631	0.698	0.424	1.092
8	0.6	0.677	0.348	1.099	0.679	0.742	0.383	1.199

Estimation of Cox’s proportional hazard:

## [1] 0.7079339

Bias

HR estimation

Event rates for current data is lower than the event rates for true/usual (historical) data.

Setup

Event Rate for historical data = 0.3
Event Rate for current data = 0.25
HR for historical data =0.9, 0.6
HR for current data =0.9, 0.6
Sample size for historical data = 800
Sample size for current data = 900

Tables

Int	True HR	Est_HR	5%_est_HR	95%_est_HR	cox_HR	Est_HR_borrow	5%_y_est_HR	95%_y_est_HR
1	0.6	0.941	0.594	1.355	0.941	0.897	0.591	1.340
2	0.6	0.956	0.649	1.434	0.955	0.918	0.636	1.328
3	0.6	0.646	0.360	1.018	0.646	0.624	0.362	0.948
4	0.6	0.629	0.371	0.961	0.629	0.623	0.373	0.941
5	0.6	0.618	0.335	1.105	0.617	0.613	0.353	1.039
6	0.6	0.652	0.372	1.122	0.652	0.659	0.383	1.085
7	0.6	0.684	0.324	1.296	0.685	0.701	0.344	1.386
8	0.6	0.654	0.327	1.195	0.654	0.678	0.350	1.313

Estimation of Cox’s proportional hazard:

## [1] 0.722293

Bias

HR estimation

Event rates for current data is lower than the event rates for true/usual (historical) data.

Setup

Event Rate for historical data = 0.3
Event Rate for current data = 0.2
HR for historical data =0.9, 0.6
HR for current data =0.9, 0.6
Sample size for historical data = 800
Sample size for current data = 900

Tables

Int	True HR	Est_HR	5%_est_HR	95%_est_HR	cox_HR	Est_HR_borrow	5%_y_est_HR	95%_y_est_HR
1	0.6	0.941	0.594	1.355	0.941	0.897	0.591	1.340
2	0.6	0.956	0.649	1.434	0.955	0.918	0.636	1.328
3	0.6	0.646	0.360	1.018	0.646	0.624	0.362	0.948
4	0.6	0.629	0.371	0.961	0.629	0.623	0.373	0.941
5	0.6	0.618	0.335	1.105	0.617	0.613	0.353	1.039
6	0.6	0.652	0.372	1.122	0.652	0.659	0.383	1.085
7	0.6	0.684	0.324	1.296	0.685	0.701	0.344	1.386
8	0.6	0.654	0.327	1.195	0.654	0.678	0.350	1.313

Estimation of Cox’s proportional hazard:

## [1] 0.722293

Bias

HR estimation

Summary Results

Proportional hazard

Int	True HR	Est_HR	HR_no_borrow	cox_HR	Est_HR	HR_no_borrow	cox_HR	Est_HR	HR_no_borrow	cox_HR	Est_HR	HR_no_borrow	cox_HR	Est_HR	HR_no_borrow	cox_HR
1	0.6	0.626	0.635	0.644	0.621	0.673	0.632	0.623	0.669	0.631	0.608	0.579	0.642	0.606	0.599	0.647
2	0.6	0.658	0.668	0.644	0.648	0.692	0.632	0.622	0.665	0.631	0.657	0.632	0.642	0.642	0.639	0.647
3	0.6	0.602	0.613	0.644	0.626	0.667	0.632	0.621	0.669	0.631	0.644	0.622	0.642	0.666	0.671	0.647
4	0.6	0.641	0.654	0.644	0.602	0.641	0.632	0.616	0.667	0.631	0.626	0.620	0.642	0.648	0.670	0.647
5	0.6	0.642	0.651	0.644	0.643	0.688	0.632	0.619	0.669	0.631	0.617	0.612	0.642	0.640	0.647	0.647
6	0.6	0.654	0.667	0.644	0.626	0.662	0.632	0.633	0.692	0.631	0.655	0.662	0.642	0.631	0.653	0.647
7	0.6	0.649	0.660	0.644	0.656	0.700	0.632	0.635	0.702	0.631	0.679	0.695	0.642	0.675	0.704	0.647
8	0.6	0.679	0.688	0.644	0.629	0.678	0.632	0.678	0.742	0.631	0.653	0.676	0.642	0.668	0.702	0.647

Non-Proportional hazard

Int	True HR	Est_HR	HR_no_borrow	cox_HR	Est_HR	HR_no_borrow	cox_HR	Est_HR	HR_no_borrow	cox_HR	Est_HR	HR_no_borrow	cox_HR	Est_HR	HR_no_borrow	cox_HR
1	0.6	0.939	0.957	0.714	0.939	1.016	0.708	0.931	1.000	0.708	0.941	0.897	0.722	0.941	0.897	0.722
2	0.6	0.941	0.965	0.714	0.942	1.006	0.708	0.928	0.994	0.708	0.956	0.918	0.722	0.956	0.918	0.722
3	0.6	0.634	0.641	0.714	0.627	0.666	0.708	0.625	0.672	0.708	0.646	0.624	0.722	0.646	0.624	0.722
4	0.6	0.621	0.628	0.714	0.598	0.637	0.708	0.619	0.669	0.708	0.629	0.623	0.722	0.629	0.623	0.722
5	0.6	0.632	0.650	0.714	0.643	0.689	0.708	0.616	0.666	0.708	0.618	0.613	0.722	0.618	0.613	0.722
6	0.6	0.639	0.650	0.714	0.625	0.660	0.708	0.634	0.694	0.708	0.652	0.659	0.722	0.652	0.659	0.722
7	0.6	0.653	0.673	0.714	0.659	0.703	0.708	0.632	0.698	0.708	0.684	0.701	0.722	0.684	0.701	0.722
8	0.6	0.649	0.670	0.714	0.629	0.678	0.708	0.677	0.742	0.708	0.654	0.678	0.722	0.654	0.678	0.722

Future Works

Looking at more scenarios
Developing more complex non-parametric model

THANK YOU