Method Reference:

Wang, Naike. (2018). How to Conduct a Meta-Analysis of Proportions in R: A Comprehensive Tutorial. 10.13140/RG.2.2.27199.00161.

Meta Analysis

METHOD: Individual effect sizes and their corresponding sampling variances are estimated by fitting a meta-analytic model.

## 
##    pred  ci.lb  ci.ub  pi.lb  pi.ub 
##  0.0864 0.0693 0.1072 0.0160 0.3553
## 
## Random-Effects Model (k = 61; tau^2 estimator: REML)
## 
## tau^2 (estimated amount of total heterogeneity): 0.79 (SE = 0.16)
## tau (square root of estimated tau^2 value):      0.89
## I^2 (total heterogeneity / total variability):   99.20%
## H^2 (total variability / sampling variability):  125.67
## 
## Test for Heterogeneity:
## Q(df = 60) = 4266.38, p-val < .01
## 
## Model Results:
## 
## estimate    se    zval  pval  ci.lb  ci.ub      
##    -2.36  0.12  -19.34  <.01  -2.60  -2.12  *** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
##        estimate ci.lb  ci.ub 
## tau^2      0.79  0.56   1.30 
## tau        0.89  0.75   1.14 
## I^2(%)    99.20 98.87  99.51 
## H^2      125.67 88.61 205.46

RESULTS: The summary proportion of pregnancy across studies is 0.0864% (95% CI 0.0693, 0.1072). τ2 is 0.79 (95% CI=0.56, 1.30), I2 is 99.20% (95% CI = 98.87%, 99.51%), and the Q-statistic is 4266.38 (p<0.01), all of which suggests high heterogeneity in the effect sizes. Sub-group analyses and meta-regression were further applied to identify potential moderating factors that can explain the inconsistencies between effect sizes across studies.”

Forest Plot: All studies

Sub-Group Analysis: Contraceptive Requirement

Assumption: We assume a common between-study variance component - we pool within-group estimates of between-study variance because the sample size of each subgroup is small

## 
## Mixed-Effects Model (k = 61; tau^2 estimator: REML)
## 
## tau^2 (estimated amount of residual heterogeneity):     0.6905 (SE = 0.1461)
## tau (square root of estimated tau^2 value):             0.8309
## I^2 (residual heterogeneity / unaccounted variability): 99.00%
## H^2 (unaccounted variability / sampling variability):   100.35
## R^2 (amount of heterogeneity accounted for):            13.09%
## 
## Test for Residual Heterogeneity:
## QE(df = 57) = 3452.6365, p-val < .0001
## 
## Test of Moderators (coefficients 2:4):
## QM(df = 3) = 10.2644, p-val = 0.0164
## 
## Model Results:
## 
##                                         estimate      se     zval    pval 
## intrcpt                                  -1.5417  0.3748  -4.1138  <.0001 
## contraceptiveCondoms encouraged          -0.5320  0.4246  -1.2528  0.2103 
## contraceptiveEffective method required   -1.1192  0.4142  -2.7020  0.0069 
## contraceptiveTwo methods required        -1.0292  0.4719  -2.1811  0.0292 
##                                           ci.lb    ci.ub      
## intrcpt                                 -2.2762  -0.8072  *** 
## contraceptiveCondoms encouraged         -1.3643   0.3003      
## contraceptiveEffective method required  -1.9311  -0.3074   ** 
## contraceptiveTwo methods required       -1.9541  -0.1044    * 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
##    pred  ci.lb  ci.ub  pi.lb  pi.ub 
##  0.0977 0.0616 0.1516 0.0377 0.2306

There are four sub-groups of contraceptive requirements:
(1) Contraceptives not required (reference),
(2) Condoms encouraged,
(3) Effective method required, including hormonal and long-acting methods, and
(4) Two methods required, including condoms and other highly effective method
RESULTS: The amount of within-group heterogeneity across the two subgroups is assumed to be the same (tau-squared = 0.6905 in this case).The moderator contraceptive requirement does have a moderating effect (QM(3)=10.2644, p = 0.0164).
However, this moderator can only explain 13.09% of the between-study variance (R-square=13.09%). There is still significant unexplained heterogeneity left between all effect sizes in the data (QE(57)=3452.6365, p<0.0001.

Forest Plot by Sub-group: Contraceptive Requirement

Meta-regression: Contraceptive Requirement as Moderator

## 
## Mixed-Effects Model (k = 61; tau^2 estimator: REML)
## 
## tau^2 (estimated amount of residual heterogeneity):     0.6905 (SE = 0.1461)
## tau (square root of estimated tau^2 value):             0.8309
## I^2 (residual heterogeneity / unaccounted variability): 99.00%
## H^2 (unaccounted variability / sampling variability):   100.35
## R^2 (amount of heterogeneity accounted for):            13.09%
## 
## Test for Residual Heterogeneity:
## QE(df = 57) = 3452.6365, p-val < .0001
## 
## Test of Moderators (coefficients 2:4):
## QM(df = 3) = 10.2644, p-val = 0.0164
## 
## Model Results:
## 
##                                         estimate      se     zval    pval 
## intrcpt                                  -1.5417  0.3748  -4.1138  <.0001 
## contraceptiveCondoms encouraged          -0.5320  0.4246  -1.2528  0.2103 
## contraceptiveEffective method required   -1.1192  0.4142  -2.7020  0.0069 
## contraceptiveTwo methods required        -1.0292  0.4719  -2.1811  0.0292 
##                                           ci.lb    ci.ub      
## intrcpt                                 -2.2762  -0.8072  *** 
## contraceptiveCondoms encouraged         -1.3643   0.3003      
## contraceptiveEffective method required  -1.9311  -0.3074   ** 
## contraceptiveTwo methods required       -1.9541  -0.1044    * 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##                                        estimate ci.lb ci.ub
## contraceptiveCondoms encouraged            0.59  0.26  1.35
## contraceptiveEffective method required     0.33  0.14  0.74
## contraceptiveTwo methods required          0.36  0.14  0.90

In comparison to the contraceptives not required sub-group, the odds of being pregnant in the condoms encouraged sub-group is 41% lower. However, this difference was not significant (OR = 0.59, 95% CI = 0.26, 1.35). The odds of being pregnant in the one effective method required and the two effective-methods required sub-groups are 67% and 64% lower than in the contraceptives not required group, respectively. These differences were statistically significant (OR = 0.33, 95% CI = 0.14, 0.74; OR = 0.36, 95% CI = 0.14, 0.90)

Visualizing meta-regression

Sub-Group Analysis: SSS sub-region

Assumption: We assume a common between-study variance component - we pool within-group estimates of between-study variance because the sample size of each subgroup is small

## 
## Mixed-Effects Model (k = 61; tau^2 estimator: REML)
## 
## tau^2 (estimated amount of residual heterogeneity):     0.7057 (SE = 0.1492)
## tau (square root of estimated tau^2 value):             0.8400
## I^2 (residual heterogeneity / unaccounted variability): 99.01%
## H^2 (unaccounted variability / sampling variability):   101.38
## R^2 (amount of heterogeneity accounted for):            11.17%
## 
## Test for Residual Heterogeneity:
## QE(df = 57) = 3089.8329, p-val < .0001
## 
## Test of Moderators (coefficients 2:4):
## QM(df = 3) = 8.9504, p-val = 0.0300
## 
## Model Results:
## 
##                 estimate      se      zval    pval    ci.lb    ci.ub      
## intrcpt          -2.4667  0.1921  -12.8392  <.0001  -2.8432  -2.0901  *** 
## ssa_catEastern    0.2293  0.2918    0.7857  0.4320  -0.3427   0.8012      
## ssa_catWestern    1.4298  0.5226    2.7361  0.0062   0.4056   2.4540   ** 
## ssa_catMulti     -0.0941  0.2838   -0.3315  0.7403  -0.6504   0.4622      
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
##    pred  ci.lb  ci.ub  pi.lb  pi.ub 
##  0.1117 0.0604 0.1974 0.0290 0.3459

There are four sub-groups of SSA region:
(1) Southern (ref),
(2) Eastern,
(3) Western, and
(4) Multi
RESULTS: The amount of within-group heterogeneity across the two subgroups is assumed to be the same (tau-squared = 0.7057 in this case).The moderator SSA sub-region does have a moderating effect (QM(3)= 8.95, p = 0.03).
However, this moderator can only explain 11.17% of the between-study variance (R-square=11.17%). There is still significant unexplained heterogeneity left between all effect sizes in the data (QE(57)=, p<0.0001

Forest Plot by Sub-group: SSA Sub-region

Meta-regression: SSA sub-region as Moderator

## 
## Mixed-Effects Model (k = 61; tau^2 estimator: REML)
## 
## tau^2 (estimated amount of residual heterogeneity):     0.7057 (SE = 0.1492)
## tau (square root of estimated tau^2 value):             0.8400
## I^2 (residual heterogeneity / unaccounted variability): 99.01%
## H^2 (unaccounted variability / sampling variability):   101.38
## R^2 (amount of heterogeneity accounted for):            11.17%
## 
## Test for Residual Heterogeneity:
## QE(df = 57) = 3089.8329, p-val < .0001
## 
## Test of Moderators (coefficients 2:4):
## QM(df = 3) = 8.9504, p-val = 0.0300
## 
## Model Results:
## 
##                 estimate      se      zval    pval    ci.lb    ci.ub      
## intrcpt          -2.4667  0.1921  -12.8392  <.0001  -2.8432  -2.0901  *** 
## ssa_catEastern    0.2293  0.2918    0.7857  0.4320  -0.3427   0.8012      
## ssa_catWestern    1.4298  0.5226    2.7361  0.0062   0.4056   2.4540   ** 
## ssa_catMulti     -0.0941  0.2838   -0.3315  0.7403  -0.6504   0.4622      
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##                estimate ci.lb ci.ub
## ssa_catEastern     1.26  0.71  2.23
## ssa_catWestern     4.18  1.50 11.63
## ssa_catMulti       0.91  0.52  1.59

In comparison to the Southern region sub-group, the odds of being pregnant in the Eastern region sub-group is 26% higher (OR = 1.26, 95% CI 0.71, 2.23) and in the multi-region sub-group is 9% lower (OR = 0.91, 95% CI 0.52, 1.59). However, these differences were not statistically significant. The odds of being pregnant in the Western sub-group is approximately 4 times the odds of being pregnant in the Southern region sub-group (OR=4.19, 95% CI 1.50, 11.63) and this difference is statistically significant.

Visualizing meta-regression

Multiple meta regression

## 
## Mixed-Effects Model (k = 61; tau^2 estimator: REML)
## 
## tau^2 (estimated amount of residual heterogeneity):     0.6410 (SE = 0.1407)
## tau (square root of estimated tau^2 value):             0.8006
## I^2 (residual heterogeneity / unaccounted variability): 98.84%
## H^2 (unaccounted variability / sampling variability):   85.89
## R^2 (amount of heterogeneity accounted for):            19.32%
## 
## Test for Residual Heterogeneity:
## QE(df = 54) = 2646.0170, p-val < .0001
## 
## Test of Moderators (coefficients 2:7):
## QM(df = 6) = 17.6850, p-val = 0.0071
## 
## Model Results:
## 
##                                         estimate      se     zval    pval 
## intrcpt                                  -1.4449  0.4693  -3.0790  0.0021 
## contraceptiveCondoms encouraged          -0.7700  0.4358  -1.7670  0.0772 
## contraceptiveEffective method required   -1.1051  0.4094  -2.6993  0.0069 
## contraceptiveTwo methods required        -1.1185  0.5451  -2.0519  0.0402 
## ssa_catEastern                           -0.0155  0.3358  -0.0461  0.9632 
## ssa_catWestern                            1.1781  0.5407   2.1787  0.0294 
## ssa_catMulti                             -0.2250  0.3263  -0.6893  0.4906 
##                                           ci.lb    ci.ub     
## intrcpt                                 -2.3647  -0.5251  ** 
## contraceptiveCondoms encouraged         -1.6241   0.0841   . 
## contraceptiveEffective method required  -1.9074  -0.3027  ** 
## contraceptiveTwo methods required       -2.1869  -0.0501   * 
## ssa_catEastern                          -0.6736   0.6426     
## ssa_catWestern                           0.1183   2.2379   * 
## ssa_catMulti                            -0.8646   0.4147     
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##                                        estimate ci.lb ci.ub
## contraceptiveCondoms encouraged            0.46  0.20  1.09
## contraceptiveEffective method required     0.33  0.15  0.74
## contraceptiveTwo methods required          0.33  0.11  0.95
## ssa_catEastern                             0.98  0.51  1.90
## ssa_catWestern                             3.25  1.13  9.37
## ssa_catMulti                               0.80  0.42  1.51

Results from a multiple meta-regression analysis shows that contraceptive requirement and SSA sub-region as moderators account for 19.32% of heterogeneity. However, there is still a significant amount of residual heterogeneity that was not explained by these moderators.

Time trend


Notes from reference article: “When the distribution of a set of observed proportions is skewed (i.e., the observed proportions are extremely high or low), we usually apply transformations to the data in order to make it conform to the normal distribution as much as possible,
enhancing the validity of the following statistical analyses (Barendregt, et al., 2013). Specifically, after transforming the observed proportions, all analyses are conducted using the transformed proportion as the effect size statistic (e.g., the natural logarithm of proportion) and the inverse of the variance of the transformed proportion as study weight. For reporting, the transformed summary proportion and its confidence interval are converted back to proportions for ease of interpretation (Borenstein et al., 2009). In practice, the approximate likelihood approach (Agresti & Coull, 1998) is arguably the predominant framework in modeling proportional data (Hamza, et al., 2008; Nyaga et al., 2014).

In the logit transformation, observed proportions are first converted to the natural logarithm of the proportions (i.e., the logit). After the transformation, the logit transformed proportions are assumed to follow a normal distribution and all analyses are performed on the logit as the effect size statistic. After the analysis, the logits are converted back into proportions for reporting.

Following the suggestions proposed by Lipsey and Wilson (2001) and Viechtbauer (2010), using direct proportions would be adequate when the observed proportions identified across studies are between 0.2 and 0.8; applying the logit transformation would be acceptable when the observed proportions are less than 0.2 or larger than 0.8”