# Exploring heterogeneity in meta-analysis: Subgroup analysis. Part 2

## Exploring heterogeneity in meta-analysis: Subgroup analysis. Part 2

American Journal of Orthodontics and Dentofacial Orthopedics, 2020-09-01, Volume 158, Issue 3, Pages 462-463, Copyright © 2020

In the previous article, we introduced the Q test based on 1-way analysis of variance —an approach of subgroup analysis to compare more than 2 subgroups of studies as defined on the basis of a categorical characteristic of the trials, and the R 2 index that quantifies the proportion of heterogeneity explained by the covariate of interest. In the present article, we use a real example to illustrate the Q test and the R 2 index. The interested reader can use the Excel file (See Supplementary Appendix , available at www.ajodo.org ) as an exercise to understand the calculations.

## Application to real data

We will use the data from 6 studies that examine the efficiency on the initial orthodontic alignment of a conventional vs a self-ligating system. Half of the studies are randomized controlled studies (subgroup A), and the rest are controlled clinical studies (subgroup B). Our goal is to determine whether the effect sizes differ in these 2 study types (subgroups). To answer this question, we will apply a subgroup analysis. Table I illustrates the dataset. The effect size of interest is the mean difference.

Table I
Initial orthodontic alignment using Damon and conventional appliances
Study Damon Conventional Mean difference
Mean SD n Mean SD n Y i v i T2p ${\text{T}}_{\text{p}}^{2}$ w i w2i ${\text{w}}_{\text{i}}^{2}$ w i Y i wiY2i ${\text{w}}_{\text{i}}{\text{Y}}_{\text{i}}^{2}$
Randomized controlled studies (A)
Scott 1.41 1.41 32 1.68 0.66 28 −0.27 0.08 0.048 8.00 64.00 −2.16 0.58
Pandis 1.98 0.90 25 2.22 0.96 25 −0.24 0.07 0.048 8.70 75.69 −2.09 0.50
Wahab 2.16 1.08 14 3.15 1.17 15 −0.99 0.17 0.048 4.65 21.62 −4.60 4.56
Total A 21.35 161.31 −8.85 5.64
Controlled clinical studies (B)
Ong 1.46 0.65 44 1.57 0.59 40 −0.11 0.02 0.048 15.38 236.54 −1.69 0.19
Miles 0.51 0.66 29 0.60 0.75 29 −0.09 0.03 0.048 13.33 177.69 −1.20 0.11
Pandis 1.81 0.71 27 1.41 0.63 27 0.40 0.03 0.048 13.33 177.69 5.33 2.13
Total B 42.04 591.92 2.44 2.43
Total 63.39 753.49 −6.41 8.07
Note: The weight w i is defined as w i = 1/(v i + T 2 ) for each study under the random-effects model.
SD , standard deviation; Y i , within-study mean difference; v i , within-study variance of mean difference; T2p ${\text{T}}_{\text{p}}^{2}$ , subgroup-specific between-study variance; w i , the random-effects weight of the study.

The Figure illustrates the forest plot for the study and subgroup effect sizes under the random-effects model. The combined effect for the randomized controlled studies (the first diamond) is −0.40 with a 95% confidence interval (CI) of −0.78 to −0.02, and the combined effect for the controlled clinical studies (the second diamond) is 0.06 with a 95% CI of −0.25 to 0.37. These 2 combined effects have different magnitudes and directions, but their CIs seem to overlap, and this indicates that these 2 study types might not be statistically different in terms of the mean effect size. The results from the Q test based on a 1-way analysis of variance are presented in Table II . For Q B = 3.16 with 1 degree of freedom, the P value is equal to 0.07 (>0.05 significance level), and hence, we fail to reject the null hypothesis that the effect sizes may be the same in both subgroups.

Forest plot illustrating the study results by subgroup (study type) under the random-effects model. MD , mean difference.
Table II
Results from preforming the Q test based on 1-way ANOVA
Q measure Q QA ${\text{Q}}_{\text{A}}^{\text{∗}}$ QB ${\text{Q}}_{\text{B}}^{\text{∗}}$ Q w Q B
Value 7.42 1.97 2.29 4.26 3.16
ANOVA , analysis of variance; Q∗ , total variance under the random-effects model; QA ${\text{Q}}_{\text{A}}^{\text{∗}}$ , variance in subgroup A (randomized controlled studies) under the random-effects model; QB ${\text{Q}}_{\text{B}}^{\text{∗}}$ , variance in subgroup B (controlled clinical studies) under the random-effects model; Q w , within-subgroup variance; Q B , between-subgroup variance.

T 2 was estimated to be equal to 0.03 and 0.05 in subgroup A and subgroup B, respectively. A pooled estimate of these 2 heterogeneities is equal to 0.048. The T 2 total , obtained from a conventional meta-analysis, was equal to 0.07. Therefore, the R 2 index is as follows:

R2=1T2unexplainedT2total=10.0480.07=0.314(31.4%). ${R}^{2}=1-\frac{{T}_{unexplained}^{2}}{{T}_{total}^{2}}=1-\frac{0.048}{0.07}=0.314\phantom{\rule{0ex}{0ex}}\left(31.4%\right)\text{.}$

## References

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