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Dankmar Böhning, Patarawan Sangnawakij, The identity of two meta-analytic likelihoods and the ignorability of double-zero studies, Biostatistics, Volume 22, Issue 4, October 2021, Pages 890–896, https://doi.org/10.1093/biostatistics/kxaa004
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Summary
In meta-analysis, the conventional two-stage approach computes an effect estimate for each study in the first stage and proceeds with the analysis of effect estimates in the second stage. For counts of events as outcome, the risk ratio is often the effect measure of choice. However, if the meta-analysis includes many studies with no events the conventional method breaks down. As an alternative one-stage approach, a Poisson regression model and a conditional binomial model can be considered where no event studies do not cause problems. The conditional binomial model excludes double-zero studies, whereas this is seemingly not the case for the Poisson regression approach. However, we show here that both models lead to the same likelihood inference and double-zero studies (in contrast to single-zero studies) do not contribute in either case to the likelihood.
1. Introduction
Meta-analysis and data fusion of studies with rare events has become recently a matter of prime interest. An example is the post-admission surveillance of the anti-diabetes drug Rosiglitazone where concern arose in terms of adverse events. A meta-analysis of Nissen and Wolski (2007, 2010) showed an increased risk ratio for myocardial infarction while being on treatment. This meta-analysis started a debate on how to deal with rare event studies in a meta-analysis. For a review on this debate, see Böhning and others (2015). The central problem is the occurrence of single-zero and double-zero trials, former being a trial where exactly in one arm a zero count occurs, the latter being a trial where both arms contain zero counts as study outcome. To demonstrate the problem, we look at a meta-analysis on perinatal death in post-term pregnancy comparing selective induction with routine induction. The data are taken from Piaget-Rossel and Taffé (2019a) (originally stemming from Crowley (2010)) and provided in Table 1 for convenience. The meta-analysis consists only of 0 studies, 11 double-zero studies, and 8 single-zero studies. Note that the studies are not small in arm size, the smallest having 10 at risk while the largest has 1706 at risk. It is clear that a conventional two-stage meta-analysis is not possible for these kind of data. Recall that a two-stage meta-analysis proceeds, in principle, as follows. In the first stage, an effect measure such as the risk ratio is determined for each study which generates a sample of study statistics which are in a second stage further analyzed by computing a summary statistic, for example, and performing heterogeneity analysis. For a review of the approach see, for example, the recent works by Borenstein and others (2009) or Schwarzer and others (2015). The two-stage approach fails for the data of Table 1 as for none of the 19 studies the risk ratio can be reliably estimated. For a discussion and review of rare event meta-analysis including zero counts, see also Lane (2013), and for an alternative approach based on the arcsine-transformation which can cope with zero-event studies as well, see Rücker and others (2009). Other recent contributions include the works of Piaget-Rossel and Taffé (2019a,b).
. | Selective induction . | Routine induction . | ||
---|---|---|---|---|
Study |$i$| . | Deaths . | At risk . | Deaths . | At risk . |
1 | 0 | 55 | 2 | 57 |
2 | 0 | 118 | 0 | 119 |
3 | 0 | 131 | 1 | 134 |
4 | 0 | 57 | 0 | 55 |
5 | 0 | 481 | 0 | 235 |
6 | 1 | 78 | 0 | 78 |
7 | 0 | 66 | 0 | 53 |
8 | 0 | 76 | 0 | 90 |
9 | 0 | 195 | 1 | 207 |
10 | 0 | 214 | 0 | 195 |
11 | 0 | 152 | 1 | 150 |
12 | 0 | 103 | 0 | 97 |
13 | 0 | 94 | 1 | 94 |
14 | 0 | 188 | 1 | 168 |
15 | 0 | 12 | 0 | 10 |
16 | 0 | 109 | 0 | 129 |
17 | 0 | 1701 | 2 | 1706 |
18 | 0 | 57 | 0 | 51 |
19 | 0 | 235 | 0 | 175 |
. | Selective induction . | Routine induction . | ||
---|---|---|---|---|
Study |$i$| . | Deaths . | At risk . | Deaths . | At risk . |
1 | 0 | 55 | 2 | 57 |
2 | 0 | 118 | 0 | 119 |
3 | 0 | 131 | 1 | 134 |
4 | 0 | 57 | 0 | 55 |
5 | 0 | 481 | 0 | 235 |
6 | 1 | 78 | 0 | 78 |
7 | 0 | 66 | 0 | 53 |
8 | 0 | 76 | 0 | 90 |
9 | 0 | 195 | 1 | 207 |
10 | 0 | 214 | 0 | 195 |
11 | 0 | 152 | 1 | 150 |
12 | 0 | 103 | 0 | 97 |
13 | 0 | 94 | 1 | 94 |
14 | 0 | 188 | 1 | 168 |
15 | 0 | 12 | 0 | 10 |
16 | 0 | 109 | 0 | 129 |
17 | 0 | 1701 | 2 | 1706 |
18 | 0 | 57 | 0 | 51 |
19 | 0 | 235 | 0 | 175 |
. | Selective induction . | Routine induction . | ||
---|---|---|---|---|
Study |$i$| . | Deaths . | At risk . | Deaths . | At risk . |
1 | 0 | 55 | 2 | 57 |
2 | 0 | 118 | 0 | 119 |
3 | 0 | 131 | 1 | 134 |
4 | 0 | 57 | 0 | 55 |
5 | 0 | 481 | 0 | 235 |
6 | 1 | 78 | 0 | 78 |
7 | 0 | 66 | 0 | 53 |
8 | 0 | 76 | 0 | 90 |
9 | 0 | 195 | 1 | 207 |
10 | 0 | 214 | 0 | 195 |
11 | 0 | 152 | 1 | 150 |
12 | 0 | 103 | 0 | 97 |
13 | 0 | 94 | 1 | 94 |
14 | 0 | 188 | 1 | 168 |
15 | 0 | 12 | 0 | 10 |
16 | 0 | 109 | 0 | 129 |
17 | 0 | 1701 | 2 | 1706 |
18 | 0 | 57 | 0 | 51 |
19 | 0 | 235 | 0 | 175 |
. | Selective induction . | Routine induction . | ||
---|---|---|---|---|
Study |$i$| . | Deaths . | At risk . | Deaths . | At risk . |
1 | 0 | 55 | 2 | 57 |
2 | 0 | 118 | 0 | 119 |
3 | 0 | 131 | 1 | 134 |
4 | 0 | 57 | 0 | 55 |
5 | 0 | 481 | 0 | 235 |
6 | 1 | 78 | 0 | 78 |
7 | 0 | 66 | 0 | 53 |
8 | 0 | 76 | 0 | 90 |
9 | 0 | 195 | 1 | 207 |
10 | 0 | 214 | 0 | 195 |
11 | 0 | 152 | 1 | 150 |
12 | 0 | 103 | 0 | 97 |
13 | 0 | 94 | 1 | 94 |
14 | 0 | 188 | 1 | 168 |
15 | 0 | 12 | 0 | 10 |
16 | 0 | 109 | 0 | 129 |
17 | 0 | 1701 | 2 | 1706 |
18 | 0 | 57 | 0 | 51 |
19 | 0 | 235 | 0 | 175 |
We consider the one-stage approach based upon generalized linear models. It appears reasonable to use a Poisson regression model for the data of Table 1, treating the count of deaths as Poisson outcome, conditional on a linear predictor which includes a binary covariate for the type of induction, a fixed effect for each study as well as the log-number at risk as offset. This approach has been suggested, for example, in Cai and others (2010).
Yet, another approach (also mentioned in Cai and others (2010), Stijnen and others (2010), or Böhning and others (2015)) considers the fact that the count in the intervention conditional upon the sum of both outcome over the study arms is binomial with the sum of both count outcomes as size parameter and the event parameter only involving the parameter of interest. This is an attractive approach as it does not involve a main effect for each study any more, hence eliminates the baseline parameter and only involves the parameter of interest, |$RR$|. It can be seen as moving from a two-sample problem to an equivalent one-sample problem. However, it is often argued against this approach that double-zero studies do not play any role as for those the binomial denominator is zero, whereas it is felt that they contribute in the Poisson regression approach mentioned in the previous paragraph (see also Kuss (2015) for this aspect). See also Piaget-Rossel and Taffé (2019c) for an investigation of the role of zero-event studies in rare events meta-analysis.
We have implemented both approaches, the Poisson regression and the conditional, binomial model for the data of Table 1 and present the results in Table 2. We find the result, potentially surprising, that the inference based upon the Poisson regression model and the conditional, binomial coincides entirely. Hence also double-zero studies do not contribute to the Poisson regression model, potentially, in contrast to common belief.
Model . | Risk ratio . | SE . | |$z$| . | P-value . | CI . |
---|---|---|---|---|---|
Poisson regression | 0.1114 | 0.1174 | |$-$|2.08 | 0.037 | 0.0141 – 0.8795 |
conditional binomial | 0.1114 | 0.1174 | |$-$|2.08 | 0.037 | 0.0141 – 0.8795 |
Mantel–Haenszel | 0.1113 | 0.0141 – 0.8799 |
Model . | Risk ratio . | SE . | |$z$| . | P-value . | CI . |
---|---|---|---|---|---|
Poisson regression | 0.1114 | 0.1174 | |$-$|2.08 | 0.037 | 0.0141 – 0.8795 |
conditional binomial | 0.1114 | 0.1174 | |$-$|2.08 | 0.037 | 0.0141 – 0.8795 |
Mantel–Haenszel | 0.1113 | 0.0141 – 0.8799 |
Model . | Risk ratio . | SE . | |$z$| . | P-value . | CI . |
---|---|---|---|---|---|
Poisson regression | 0.1114 | 0.1174 | |$-$|2.08 | 0.037 | 0.0141 – 0.8795 |
conditional binomial | 0.1114 | 0.1174 | |$-$|2.08 | 0.037 | 0.0141 – 0.8795 |
Mantel–Haenszel | 0.1113 | 0.0141 – 0.8799 |
Model . | Risk ratio . | SE . | |$z$| . | P-value . | CI . |
---|---|---|---|---|---|
Poisson regression | 0.1114 | 0.1174 | |$-$|2.08 | 0.037 | 0.0141 – 0.8795 |
conditional binomial | 0.1114 | 0.1174 | |$-$|2.08 | 0.037 | 0.0141 – 0.8795 |
Mantel–Haenszel | 0.1113 | 0.0141 – 0.8799 |
This short note is organized as follows: in Section 2, we introduce notations and models in the context of a single study. In Section 3, we discuss the general meta-analytic setting and show the identity of the log-likelihoods for the Poisson regression model with the conditional, binomial model. Section 4 closes with a short discussion including the connection to Mantel–Haenszel estimation.
2. A Poisson likelihood
It is assumed that |$X_1$| and |$X_0$| are independent. We also emphasize that, from our perspective, the Poisson assumption in the rare event situation is reasonable, at least a wide-spread assumption. For an approach utilizing the negative-binomial distribution, see Piaget-Rossel and Taffé (2019b).
3. A conditional likelihood
Hence, the main effect of the factor study drops out. In the case of balanced studies (|$T_1=T_0$| or |$r=1$|,) the binomial event parameter |$q$| is simply |$RR/(1+RR),$| where |$RR$| is the risk ratio.
4. Major result
Study |$i$| . | |$x_{1i}$| . | |$x_i$| . | |$r_i$| . |
---|---|---|---|
1 | 0 | 2 | 0.97 |
2 | 0 | 0 | 0.99 |
3 | 0 | 1 | 0.98 |
4 | 0 | 0 | 1.04 |
5 | 0 | 0 | 2.05 |
6 | 1 | 1 | 1.00 |
7 | 0 | 0 | 1.25 |
8 | 0 | 0 | 0.84 |
9 | 0 | 1 | 0.94 |
10 | 0 | 0 | 1.10 |
11 | 0 | 1 | 1.01 |
12 | 0 | 0 | 1.06 |
13 | 0 | 1 | 1.00 |
14 | 0 | 1 | 1.12 |
15 | 0 | 0 | 1.20 |
16 | 0 | 0 | 0.85 |
17 | 0 | 2 | 1.00 |
18 | 0 | 0 | 1.12 |
19 | 0 | 0 | 1.34 |
Study |$i$| . | |$x_{1i}$| . | |$x_i$| . | |$r_i$| . |
---|---|---|---|
1 | 0 | 2 | 0.97 |
2 | 0 | 0 | 0.99 |
3 | 0 | 1 | 0.98 |
4 | 0 | 0 | 1.04 |
5 | 0 | 0 | 2.05 |
6 | 1 | 1 | 1.00 |
7 | 0 | 0 | 1.25 |
8 | 0 | 0 | 0.84 |
9 | 0 | 1 | 0.94 |
10 | 0 | 0 | 1.10 |
11 | 0 | 1 | 1.01 |
12 | 0 | 0 | 1.06 |
13 | 0 | 1 | 1.00 |
14 | 0 | 1 | 1.12 |
15 | 0 | 0 | 1.20 |
16 | 0 | 0 | 0.85 |
17 | 0 | 2 | 1.00 |
18 | 0 | 0 | 1.12 |
19 | 0 | 0 | 1.34 |
Study |$i$| . | |$x_{1i}$| . | |$x_i$| . | |$r_i$| . |
---|---|---|---|
1 | 0 | 2 | 0.97 |
2 | 0 | 0 | 0.99 |
3 | 0 | 1 | 0.98 |
4 | 0 | 0 | 1.04 |
5 | 0 | 0 | 2.05 |
6 | 1 | 1 | 1.00 |
7 | 0 | 0 | 1.25 |
8 | 0 | 0 | 0.84 |
9 | 0 | 1 | 0.94 |
10 | 0 | 0 | 1.10 |
11 | 0 | 1 | 1.01 |
12 | 0 | 0 | 1.06 |
13 | 0 | 1 | 1.00 |
14 | 0 | 1 | 1.12 |
15 | 0 | 0 | 1.20 |
16 | 0 | 0 | 0.85 |
17 | 0 | 2 | 1.00 |
18 | 0 | 0 | 1.12 |
19 | 0 | 0 | 1.34 |
Study |$i$| . | |$x_{1i}$| . | |$x_i$| . | |$r_i$| . |
---|---|---|---|
1 | 0 | 2 | 0.97 |
2 | 0 | 0 | 0.99 |
3 | 0 | 1 | 0.98 |
4 | 0 | 0 | 1.04 |
5 | 0 | 0 | 2.05 |
6 | 1 | 1 | 1.00 |
7 | 0 | 0 | 1.25 |
8 | 0 | 0 | 0.84 |
9 | 0 | 1 | 0.94 |
10 | 0 | 0 | 1.10 |
11 | 0 | 1 | 1.01 |
12 | 0 | 0 | 1.06 |
13 | 0 | 1 | 1.00 |
14 | 0 | 1 | 1.12 |
15 | 0 | 0 | 1.20 |
16 | 0 | 0 | 0.85 |
17 | 0 | 2 | 1.00 |
18 | 0 | 0 | 1.12 |
19 | 0 | 0 | 1.34 |
5. Discussion
Equation (5.12) cannot only be seen as an implicit characterization of the maximum likelihood estimator, it also provides an iterative scheme for generating it by starting with some value for |$RR$| such as |$RR=1$|, generating a new value by computing the right-hand side of (5.12), plugging that in and so forth. The first iterate when starting with |$RR=1$| is the Mantel–Haenszel estimate |$\sum_i w_i T_{i0} x_{i1}/ \sum_i w_i T_{i1} x_{i0}$|, where |$w_i = 1/(T_{i0}+T_{i1})$|. Note that maximum likelihood and Mantel–Haenszel estimator coincide if all studies are balanced (|$r_i=1$|). This explains why for the data of Table 1 both estimators are very close as the studies are nearly balanced (see also Tables 2 and 3). Note that the Mantel–Haenszel estimator is always defined (unless one arm has only zeros in all studies) and is also invariant to the inclusion or exclusion of double-zero studies (whereas it can change its value when single-zero studies are excluded as pointed out in Böhning and others (2015)). Mantel–Haenszel estimation is popular but has limitations when it comes to the inclusion of further covariates. The latter would be easily possible for the conditional logistic regression model by extending the linear predictor by further covariates such a time of place of study, intervention modification or size of study.
6. Software and data
We have used the package STATA (version 16) for the analysis of the meta-analytic data in this article. Code and data can be found at https://github.com/boehning/meta_likelihood. The provided code will reproduce all analysis in this article and can also be used for similar meta-analytic event data and analysis.
Acknowledgments
All authors would like to thank the reviewers and editors for valuable suggestions and comments which lead to considerable improvements in the manuscript. Conflict of Interest: None declared.