Menu location: Analysis_Meta-Analysis_Proportion.
This function enables you to calculate an overall proportion from a set of proportions, for example from a systematic review of studies of adherence with a particular drug treatment.
This function can also be applied to a review of diagnostic test studies to give an overall sensitivity or specificity as follows:
|
|
|
DISEASE/OUTCOME: |
|
|
|
|
Present |
Absent |
|
TEST: |
+ |
a (true +ve) |
b (false +ve) |
|
|
- |
c (false -ve) |
d (true -ve) |
Sensitivity = a/(a+c)
Specificity = d/(b+d)
Another way to summarise diagnostic test performance is via the diagnostic odds ratio:
Diagnostic odds ratio = true/false = (a * d)/(b * c)
In order to run a meta-analysis of diagnostic odds ratios simply use the odds ratio meta-analysis function with the experimental group as the true (test correct) outcomes and the control group as the false outcomes – enter a as experimental group responders; a+d as experimental group number; b as control group responders; and b+c as control group total.
Many study designs can be expressed as a proportion, and relatively complex statistical models can be explored using sets of proportions – for example a random effects logistic regression. You should seek the assistance of a statistician if you want to pursue analyses deeper than the summaries that this StatsDirect function offers.
The inconsistency of results across studies is summarised in the I² statistic, which is the percentage of variation across studies that is due to heterogeneity rather than chance – see the heterogeneity section for more information.
Beyond this meta-analysis function, logistic regression can be used to compare pooled proportions. Consult with a statistician if you are considering a random effects logistic model.
DATA INPUT:
You enter the number of subjects responding (with the study outcome) and the total number of subjects studied. You may also enter a title for each study.
Technical Validation
Statistically, binomial proportions are not as simple as look – for example there is a lack of clear consensus in the literature over confidence intervals for proportions (Newcome, 1998; Brown, 2001). For the purpose of meta-analysis, StatsDirect first transforms proportions into a quantity (the Freeman-Tukey variant of the arcsine square root transformed proportion - Stuart and Ord, 1994) suitable for the usual fixed and random effects summaries (DerSimonian and Laird, 1986). The pooled proportion is calculated as the back-transform of the weighted mean of the transformed proportions, using inverse arcsine variance weights for the fixed effects model and DerSimonian-Laird (1986) weights for the random effects model:
- where p hat is the fixed effects pooled proportion, x is the Freeman-Tukey transformed proportion, w is the inverse variance weight for the transformed proportion, q is the Cochran q statistic, tau squared is the moment-based estimate of the between-studies variance, w underscore r is the DerSimonian-Laird weight, and p hat underscore r is the random effects estimate of the pooled proportion.
Example
The following data represent adherence with medication from 22 fictitious trials of a class of drug:
|
Trial |
Adherent |
Total |
|
Brown |
214 |
311 |
|
Lamont |
58 |
65 |
|
Lally |
59 |
67 |
|
Orwell |
182 |
285 |
|
Wagner |
65 |
73 |
|
Werner |
66 |
116 |
|
Venner |
99 |
183 |
|
Adams |
600 |
696 |
|
Brenner |
45 |
57 |
|
Borrowdale |
165 |
277 |
|
Byers |
32 |
35 |
|
Daniels |
49 |
60 |
|
Darling |
175 |
199 |
|
Ehert |
155 |
311 |
|
Fern |
64 |
81 |
|
Mullen |
526 |
537 |
|
Orton |
104 |
107 |
|
Jones |
97 |
102 |
|
Ning |
2310 |
4612 |
|
Sherraton |
72 |
91 |
|
Zu |
37 |
37 |
|
Tarone |
31 |
87 |
To analyse these data in StatsDirect first prepare them in three workbook columns and label these columns appropriately. Alternatively, open the test workbook using the file open function of the file menu. Then select proportion from the meta-analysis section of the analysis menu, and then select the columns 'Trial', 'Adherent', and 'Total' as prompted.
For this example:
|
Study |
Proportion |
95% CI (exact) |
% Weight (fixed effects) |
||
|
1 |
0.688103 |
0.633392 |
0.739187 |
3.71 |
Brown |
|
2 |
0.892308 |
0.790618 |
0.955591 |
0.78 |
Lamont |
|
3 |
0.880597 |
0.778215 |
0.947015 |
0.81 |
Lally |
|
4 |
0.638596 |
0.579853 |
0.694424 |
3.4 |
Orwell |
|
5 |
0.890411 |
0.795436 |
0.951484 |
0.88 |
Wagner |
|
6 |
0.568966 |
0.473763 |
0.660561 |
1.39 |
Werner |
|
7 |
0.540984 |
0.465895 |
0.614723 |
2.19 |
Venner |
|
8 |
0.862069 |
0.834196 |
0.886827 |
8.29 |
Adams |
|
9 |
0.789474 |
0.66113 |
0.88621 |
0.69 |
Brenner |
|
10 |
0.595668 |
0.5353 |
0.653966 |
3.31 |
Borrowdale |
|
11 |
0.914286 |
0.769425 |
0.981962 |
0.43 |
Byers |
|
12 |
0.816667 |
0.695604 |
0.904764 |
0.73 |
Daniels |
|
13 |
0.879397 |
0.825884 |
0.921178 |
2.38 |
Darling |
|
14 |
0.498392 |
0.441461 |
0.555354 |
3.71 |
Ehert |
|
15 |
0.790123 |
0.685373 |
0.872724 |
0.97 |
Fern |
|
16 |
0.979516 |
0.963644 |
0.989731 |
6.4 |
Mullen |
|
17 |
0.971963 |
0.920245 |
0.99418 |
1.28 |
Orton |
|
18 |
0.95098 |
0.889304 |
0.983894 |
1.22 |
Jones |
|
19 |
0.500867 |
0.486332 |
0.515402 |
54.84 |
Ning |
|
20 |
0.791209 |
0.693308 |
0.869362 |
1.09 |
Sherraton |
|
21 |
1 |
0.905109 |
1 |
0.45 |
Zu |
|
22 |
0.356322 |
0.256493 |
0.466236 |
1.05 |
Tarone |
Fixed effects (inverse variance)
Pooled proportion = 0.639787 (95% CI = 0.629497 to 0.650014)
Non-combinability of studies
Cochran Q = 1556.452343 (df = 21) P < 0.0001
Moment-based estimate of between studies variance = 0.268056
I² (inconsistency) = 98.7% (95% CI = 98.5% to 98.8%)
Random effects (DerSimonian-Laird)
Pooled proportion = 0.785852 (95% CI = 0.689309 to 0.86858)
The differences between trials are very large (99% inconsistency), therefore a random effects model should be followed. In order to change the weights displayed from fixed to random effects, go to the menu Analysis_Meta-Analysis_Calculation Options and select 'Random effects diagnostics' – this will show you that the weights used to calculate the pooled proportion are similar in the random effects approach, unlike the weights shown above for the fixed effects approach. We conclude that the adherence for this class of drug is approximately 79%, and with 95% confidence at least 69%.