# Meta analysis: is there such a thing as too little heterogeneity?

First time I've seen results like these and they make me feel uneasy. Is it even appropriate to do a meta-analysis here? Even my poor forest and funnel plots are messed up.

> results <- rma(yi, vi, data=meta)
> results

Random-Effects Model (k = 13; tau^2 estimator: REML)

tau^2 (estimated amount of total heterogeneity): 0 (SE = 0.0043)
tau (square root of estimated tau^2 value):      0
I^2 (total heterogeneity / total variability):   0.00%
H^2 (total variability / sampling variability):  1.00

Test for Heterogeneity:
Q(df = 12) = 7.0093, p-val = 0.8570

Model Results:

estimate      se     zval    pval   ci.lb   ci.ub
0.8101  0.0318  25.5163  <.0001  0.7479  0.8724  ***

---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> predict(results, digits=3, transf=transf.ztor)
pred ci.lb ci.ub cr.lb cr.ub
0.670 0.634 0.703 0.634 0.703
> confint(results)

estimate  ci.lb   ci.ub
tau^2    0.0000 0.0000  0.0123
tau      0.0000 0.0000  0.1107
I^2(%)   0.0000 0.0000 41.8957
H^2      1.0000 1.0000  1.7210  • Worth noting that the 95% ci for $I^2$ is only about half the width of the 100% ci from 0 to 100 so your estimate of $I^2$ is quite imprecise. – mdewey Aug 21 '18 at 12:47

## 1 Answer

From what I can see, you are meta-analyzing correlation coefficients using Fisher's z-transformation, which is appropriate. I'm assuming that the input data are entered correctly, and the analysis steps you have provided are good.

Each study has a high standard error, translating to wide confidence intervals. This could be due to low sample size or high within-study variability. The results of your meta-analysis indicate no between-study heterogeneity, evidence by the estimate for $\tau^2 \approx 0$. This is sensible given that the within-study variability seems to mask (is much larger than) any apparent between-study variability.

The funnel plot is incorrect if the x-axis is truly the correlation, r. However, I think the label is wrong and the effect estimates have not been transformed from Fisher's z back to the scale of r. This would make sense because the data point with the largest effect estimate in the funnel plot (farthest right point, approximately at 1.05) is approximately equivalent to r=0.78, and this is the largest effect across the included studies.

• *Oops. Thanks for catching the fact that I didn't backtransform Fisher's z. – Rooirokbokkie Aug 19 '18 at 15:23