Background
Methods
Modeling approaches
Normal-normal hierarchical model (NNHM)
Model specification
Inference
Relative risk (RR) | Odds ratio (OR) | ||||||
---|---|---|---|---|---|---|---|
I
2
| Equal | One small | One large | Equal | One small | One large | |
k = 2 | 0.25 | 0.0534 | 0.1254 | 0.0396 | 0.1781 | 0.4179 | 0.1321 |
0.50 | 0.0926 | 0.2171 | 0.0687 | 0.3086 | 0.7237 | 0.2289 | |
0.75 | 0.1604 | 0.3761 | 0.1189 | 0.5345 | 1.2536 | 0.3964 | |
0.90 | 0.2777 | 0.6514 | 0.2060 | 0.9258 | 2.1712 | 0.6866 | |
k = 3 | 0.25 | 0.0534 | 0.1069 | 0.0447 | 0.1781 | 0.3563 | 0.1491 |
0.50 | 0.0926 | 0.1852 | 0.0775 | 0.3086 | 0.6172 | 0.2582 | |
0.75 | 0.1604 | 0.3207 | 0.1342 | 0.5345 | 1.0690 | 0.4472 | |
0.90 | 0.2777 | 0.5549 | 0.2324 | 0.9258 | 1.8516 | 0.7746 | |
k = 5 | 0.25 | 0.0534 | 0.0844 | 0.0484 | 0.1781 | 0.2981 | 0.1613 |
0.50 | 0.0926 | 0.1549 | 0.0838 | 0.3086 | 0.5164 | 0.2795 | |
0.75 | 0.1604 | 0.2683 | 0.1452 | 0.5345 | 0.8944 | 0.4840 | |
0.90 | 0.2777 | 0.4648 | 0.2515 | 0.9258 | 1.5491 | 0.8384 |
Generalized linear mixed models (GLMM)
Models
Model specification and inference
Confidence and credible intervals combined effects
I2 as measure of between-study heterogeneity
Extraction of the empirical data set
Simulation procedure
Parameter | Values |
---|---|
Effect measure (θi) | RR, OR |
Design | Equally sized studies, |
One small study (\(\frac {1}{10}\) size) | |
One large study (10-fold size) | |
Observations per study arm (ni) | 25, 50, 100, 250, 500, 1000 |
Number of studies (k) | 2, 3, 5, 10 |
Event rates (p0) | 0.1, 0.3, 0.5, 0.7, 0.9 |
Level of heterogeneity (I2) | No heterogeneity: 0.00 |
Low heterogeneity: 0.25 | |
Moderate heterogeneity: 0.50 | |
High heterogeneity: 0.75 | |
Very high heterogeneity: 0.90 |
Estimation in R
metafor
[39, 40] and bayesmeta
[25, 41] packages are used with their default options. As no implementation in R was found for the PL estimation of Poisson-normal model we translated the steps described by Böhning et al. [7, Ch. 2] into R code which is shown in the Additional file 1.Results
Empirical study
No. | Identifier | Date | Endpoint | Page | Number of studies (k) | Effect measure |
---|---|---|---|---|---|---|
1 | N15-06 | 2017-03 | Morning pain | 85 | 5 | OR |
2 | N15-11 | 2017-03 | Ear infection | 62 | 2 | OR |
3 | S15-02 | 2017-01 | Mortality | 53 | 2 | OR |
4 | D15-02 | 2017-01 | Mortality | 74 | 2 | OR |
5 | A16-71 | 2016-12 | Morbidity | 5 | 6 | OR |
6 | A16-38 | 2016-12 | Vomiting | 4 | 2 | RR |
7 | P14-03 | 2016-11 | Breast cancer screening | 55 | 3 | RR |
8 | N14-02 | 2016-08 | Remission from anxiety disorder | 127 | 2 | OR |
9 | A16-30 | 2016-08 | AIDS-defining events | 103 | 2 | RR |
10 | N15-07 | 2016-08 | Ejaculation dysfunction | 89 | 4 | OR |
11 | A16-11 | 2016-06 | Serious adverse events | 86 | 2 | RR |
12 | A10-03 | 2016-04 | Serious adverse events | 89 | 2 | OR |
13 | A15-57 | 2016-02 | St. George’s respiratory questionnaire response | 22 | 2 | RR |
14 | A15-45 | 2016-01 | Morbidity | 24 | 2 | OR |
15 | A15-31 | 2015-11 | Mortality | 87 | 2 | RR |
16 | A15-25 | 2015-10 | Serious adverse events | 89 | 2 | RR |
17 | A15-21 | 2015-07 | Mortality | 16 | 2 | RR |
18 | S13-04 | 2015-05 | Screening for abdominal aortic aneurysm | 71 | 4 | OR |
19 | A15-06 | 2015-05 | Morbidity | 96 | 3 | RR |
20 | A15-05 | 2015-03 | Morbidity | 4 | 2 | RR |
21 | A14-38 | 2015-01 | Serious adverse events | 65 | 3 | RR |
22 | A14-25 | 2014-11 | Serious adverse events | 115 | 2 | RR |
23 | A14-22 | 2014-10 | Transition Dyspnea Index responder | 67 | 2 | RR |
24 | A14-19 | 2014-09 | Urge to urinate | 75 | 3 | RR |
25 | A14-18 | 2014-09 | Persistent virological response (SVR24) | 194 | 3 | RR |
26 | S13-03 | 2014-06 | Participants with cervical intraepithelial neoplasia 3+ | 15 | 6 | RR |
27 | A13-29 | 2013-10 | Metformidosis | 15 | 3 | RR |
28 | A10-01 | 2013-08 | Remissions | 1183 | 2 | OR |
29 | A13-20 | 2013-08 | Visual acuity | 28 | 3 | RR |
30 | S11-01 | 2013-07 | Bowel cancer | 61 | 7 | OR |
31 | A13-23 | 2013-06 | Mortality | 15 | 2 | RR |
32 | A13-05 | 2013-04 | Full recovery | 19 | 4 | RR |
33 | A05-10 | 2013-04 | Cardiovascular death | 75 | 3 | RR |
34 | A12-19 | 2013-03 | Ocular adverse event | 17 | 2 | RR |
35 | A05-18 | 2012-08 | Serious adverse events | 67 | 18 | OR |
36 | A12-10 | 2012-07 | Adverse events | 20 | 3 | RR |
37 | A12-03 | 2012-04 | Loss of transplant | 23 | 2 | RR |
38 | A12-04 | 2012-04 | Virus occurrence | 22 | 3 | RR |
39 | A09-05 | 2012-04 | Alzheimer’s disease assessment scale | 51 | 6 | OR |
40 | A11-30 | 2012-03 | Mortality | 24 | 2 | OR |
NN-DL | Normal-normal (NN) model using the DerSimonian-Laird (DL) heterogeneinty estimator |
NN-REML | NN model using the restricted maximum-likelihood (REML) estimator |
NN-EB | NN model using the empirical-Bayes (EB) estimator |
PN-PL | Poisson model using profile likelihood (PL) estimation |
BN-UM.FS | Binomial model using unconditional logistic regression and fixed study (nuisance) parameters |
BN-UM.RS | Binomial model using unconditional logistic regression and random study (nuisance) parameters |
BN-CM.EL | Conditional (hypergeometric) model (exact likelihood) |
BN-CM.AL | Conditional (hypergeometric) model (approximate likelihood) |
NN-Bayes HN(0.5) | NN Bayesian model using a half-normal heterogeneity prior with scale 0.5 |
NN-Bayes HN(1.0) | NN Bayesian model using a half-normal heterogeneity prior with scale 1.0 |