Nonparametric Estimation of Effect Heterogeneity in Rare Events Meta-Analysis: Bivariate, Discrete Mixture Model

Abstract

Meta-analysis provides an integrated analysis and summary of the effects observed in $k$ independent studies. The conventional analysis proceeds by first calculating a study-specific effect estimate, and then provides further analysis on the basis of the available $k$ independent effect estimates associated with their uncertainty measures. Here we consider a setting where counts of events are available from $k$ independent studies for a treatment and a control group. We suggest to model this situation with a study-specific Poisson regression model, and allow the study-specific parameters of the Poisson model to arise from a nonparametric mixture model. This approach then allows the estimation of the heterogeneity variance of the effect measure of interest in a nonparametric manner. A case study is used to illustrate the methodology throughout the paper.

APPENDIX

STATA CODE

All computations were done using the package STATA16 (StataCorp 2019). In the following we demonstrate how to fit a two-component, bivariate mixture model. Different numbers of components can be achieved by replacing 2 with any desired number of components. The log-relative risks for the components are ${-}0.1581758$ and ${-}0.743582$. The associated weights can be found using the command estat lcprob which is provided at the end of the output below. $y$ are the counts of events and $\log n$ is the log-size of the study. $treat$ is a binary treatment indicator with 1 denoting being in the treatment group and 0 otherwise.

. fmm 2 : poisson y treat, exposure(logn)

Finite mixture model Number of obs = 122

Log likelihood = -307.22036

Class : 1

Response : y

Model : poisson

	Coef.	Std. Err.	z	P>|z|	[95% Conf.	Interval]
y	$\,$
treat	-0.1581758	0.1603739	-0.99	0.324	-0.4725029	0.1561514
_cons	1.128934	0.1005598	11.23	0.000	0.9318403	1.326027
ln(logn)	1	(exposure)

Class : 2

Response : y

Model : poisson

	Coef.	Std. Err.	z	P>|z|	[95% Conf.	Interval]
y	$\,$
treat	-0.743582	0.1821602	-4.08	0.000	-1.100609	-0.3865546
_cons	-0.2447636	0.1432094	-1.71	0.087	-0.5254488	0.0359217
ln(logn)	1	(exposure)

. estat lcprob

Latent class marginal probabilities Number of obs = 122

	Delta-method
	Margin	Std. Err.	[95% Conf.	Interval]
Class	$\,$
1	0.2027119	0.0486684	0.1235069	0.3144862
2	0.7972881	0.0486684	0.6855138	0.8764931

The fmm module also has the option to keep certain parameters homogeneous, i.e. not to involve them in the mixture. This can be achieved with the optional parameter coef as shown below. Here we keep the parameter of the log-relative risk estimated constant over the components.. fmm 2, lcinvariant(coef): poisson y treat, exposure(logn)

Finite mixture model Number of obs = 122

Log likelihood = -311.15859=

Class : 1

Response : y

Model : poisson

	Coef.	Std. Err.	z	P>|z|	[95% Conf.	Interval]
y	$\,$
treat	-0.3426311	0.1499642	-2.28	0.022	-0.6365555	-0.0487067
_cons	1.094218	0.0881464	12.41	0.000	0.9214543	1.266982
ln(logn)	1	(exposure)

Class : 2

Response : y

Model : poisson

	Coef.	Std. Err.	z	P>|z|	[95% Conf.	Interval]
y	$\,$
treat	-0.3426311	0.1499642	-2.28	0.022	-0.6365555	-0.0487067
_cons	-0.5363991	0.1513947	-3.54	0.000	-0.8331272	-0.2396709
ln(logn)	1	(exposure)

A conventional two-stage meta-analysis can be accomplished by using the STATA-package metan. The data format needs to be of the so-called $a-b-c-d$-type:

• $a$ number of events in the treatment group

• $b$ number of non-events in the treatment group

• $c$ number of events in the control group

• $d$ number of non-events in the control group.

We yield the following output (the parameter random produces a random effects analysis including an estimte of $\tau^{2}$):

. metan a b c d, random

Study	RR	[95% Conf.	Interval]	% Weight
1	0.650	0.326	1.298	6.18
2	0.146	0.053	0.405	3.27
3	0.760	0.400	1.444	6.89
4	1.647	0.153	17.694	0.67
5	0.625	0.218	1.791	3.09
6	2.356	0.125	44.467	0.44
7	0.321	0.074	1.381	1.71
8	0.309	0.041	2.304	0.93
9	0.440	0.222	0.872	6.28
10	0.132	0.007	2.396	0.46
11	0.395	0.017	9.306	0.39
13	0.111	0.005	2.673	0.38
14	0.111	0.015	0.845	0.92
15	0.167	0.036	0.777	1.55
16	1.065	0.069	16.537	0.51
17	0.314	0.090	1.096	2.28
18	0.095	0.005	1.687	0.46
19	0.077	0.005	1.314	0.48
20	0.200	0.010	4.063	0.42
21	0.156	0.007	3.742	0.38
22	1.093	0.071	16.941	0.51
23	0.143	0.007	2.722	0.44
24	0.138	0.032	0.601	1.69
25	0.468	0.068	3.239	1.00
26	0.264	0.058	1.201	1.59
27	0.671	0.184	2.450	2.13
28	0.228	0.051	1.021	1.63
29	0.204	0.010	4.143	0.42
30	0.324	0.013	7.851	0.38
31	0.100	0.013	0.752	0.92
32	0.168	0.007	4.059	0.38
33	0.073	0.004	1.384	0.44
34	0.694	0.046	10.589	0.52
35	0.493	0.093	2.614	1.33
36	0.814	0.399	1.662	5.87
37	0.358	0.145	0.888	4.00
38	1.396	0.582	3.347	4.25
39	0.565	0.298	1.073	6.90
40	0.250	0.050	1.251	1.42
41	0.200	0.010	4.096	0.42
42	0.099	0.006	1.721	0.47
43	1.000	0.146	6.873	1.01
44	0.301	0.083	1.090	2.16
45	0.455	0.120	1.727	2.02
46	0.145	0.018	1.146	0.88
47	1.018	0.381	2.716	3.49
48	0.333	0.015	7.619	0.39
49	0.398	0.086	1.833	1.57
50	0.250	0.029	2.159	0.81
51	0.490	0.046	5.231	0.68
52	5.185	0.261	103.110	0.43
53	0.124	0.036	0.425	2.34
54	0.169	0.069	0.418	4.02
55	0.587	0.055	6.286	0.68
56	0.543	0.104	2.828	1.36
57	0.143	0.008	2.679	0.45
58	0.625	0.109	3.569	1.22
59	3.000	0.326	27.631	0.77
60	1.000	0.063	15.849	0.50
61	0.250	0.029	2.159	0.81
12	(Excluded)
D+L pooled RR	0.429	0.352	0.523	100.00

Heterogeneity chi-squared = 63.49 (d.f. = 59) p = 0.321

I-squared (variation in RR attributable to heterogeneity) = 7.1%

Estimate of between-study variance Tau-squared = 0.0405"

Test of RR=1 : z= 8.39 p = 0.000