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Computing Adjusted Effects Based on Meta-Regression Models

After fitting a random-effects model and finding heterogeneity in the effects, meta-analysts often want to examine whether one or multiple moderator variables (i.e., predictors) are able to account for the heterogeneity (or at least part of it). Meta-regression models can be used for this purpose. A question that frequently arises in this context is how to compute an 'adjusted effect' based on such a model. This tutorial describes how to compute such adjusted effects for meta-regression models involving continuous and categorical moderators.

Data Preparation / Inspection

I will use the BCG vaccine (against tuberculosis) dataset for this tutorial. We start by computing the log risk ratios and corresponding sampling variances for the 13 trials included in this dataset.

dat <- dat.bcg
dat <- escalc(measure="RR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat,
              slab=paste0(dat$author, ", ", dat$year))
dat
   trial               author year tpos  tneg cpos  cneg ablat      alloc      yi     vi
1      1              Aronson 1948    4   119   11   128    44     random -0.8893 0.3256
2      2     Ferguson & Simes 1949    6   300   29   274    55     random -1.5854 0.1946
3      3      Rosenthal et al 1960    3   228   11   209    42     random -1.3481 0.4154
4      4    Hart & Sutherland 1977   62 13536  248 12619    52     random -1.4416 0.0200
5      5 Frimodt-Moller et al 1973   33  5036   47  5761    13  alternate -0.2175 0.0512
6      6      Stein & Aronson 1953  180  1361  372  1079    44  alternate -0.7861 0.0069
7      7     Vandiviere et al 1973    8  2537   10   619    19     random -1.6209 0.2230
8      8           TPT Madras 1980  505 87886  499 87892    13     random  0.0120 0.0040
9      9     Coetzee & Berjak 1968   29  7470   45  7232    27     random -0.4694 0.0564
10    10      Rosenthal et al 1961   17  1699   65  1600    42 systematic -1.3713 0.0730
11    11       Comstock et al 1974  186 50448  141 27197    18 systematic -0.3394 0.0124
12    12   Comstock & Webster 1969    5  2493    3  2338    33 systematic  0.4459 0.5325
13    13       Comstock et al 1976   27 16886   29 17825    33 systematic -0.0173 0.0714

The yi values are the observed log risk ratios and variable vi contains the corresponding sampling variances.

Random-Effects Model

To estimate the average log risk ratio, we fit a random-effects model to these data.

res <- rma(yi, vi, data=dat)
res
Random-Effects Model (k = 13; tau^2 estimator: REML)
 
tau^2 (estimated amount of total heterogeneity): 0.3132 (SE = 0.1664)
tau (square root of estimated tau^2 value):      0.5597
I^2 (total heterogeneity / total variability):   92.22%
H^2 (total variability / sampling variability):  12.86
 
Test for Heterogeneity:
Q(df = 12) = 152.2330, p-val < .0001
 
Model Results:
 
estimate      se     zval    pval    ci.lb    ci.ub
 -0.7145  0.1798  -3.9744  <.0001  -1.0669  -0.3622  ***
 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

To make the results easier to interpret, we can exponentiate the estimated average log risk ratio (i.e., $-0.7145$) to obtain an estimate of the average risk ratio. This can be done with the predict() function as follows.

predict(res, transf=exp, digits=2)
 pred ci.lb ci.ub pi.lb pi.ub
 0.49  0.34  0.70  0.15  1.55

Hence, the estimated average risk ratio is $0.49$ (95% CI: $0.35$ to $0.70$). In other words, the data suggest that the vaccine reduces the risk of a tuberculosis infection by about 50% on average (or the risk in vaccinated groups is on average about half of the risk of non-vaccinated groups).

We can draw a forest plot that shows the results of the individual studies and the estimate based on the random-effects model at the bottom (the polygon shape with the center corresponding to the estimate and the ends corresponding to the confidence interval bounds).

forest(res, xlim=c(-6.8,3.8), header=TRUE, atransf=exp,
       at=log(c(1/16, 1/8, 1/4, 1/2, 1, 2, 4, 8)), digits=c(2L,4L))

Meta-Regression with a Continuous Moderator

Variable ablat in the dataset gives the absolute latitude of the study locations. The distance from the equator of the study sites may be a relevant moderator of the effectiveness of the vaccine (Bates, 1982; Colditz et al., 1994; Ginsberg, 1998). So let's fit a meta-regression model that includes this variable as a predictor.

res <- rma(yi, vi, mods = ~ ablat, data=dat)
res
Mixed-Effects Model (k = 13; tau^2 estimator: REML)
 
tau^2 (estimated amount of residual heterogeneity):     0.0764 (SE = 0.0591)
tau (square root of estimated tau^2 value):             0.2763
I^2 (residual heterogeneity / unaccounted variability): 68.39%
H^2 (unaccounted variability / sampling variability):   3.16
R^2 (amount of heterogeneity accounted for):            75.62%
 
Test for Residual Heterogeneity:
QE(df = 11) = 30.7331, p-val = 0.0012
 
Test of Moderators (coefficient 2):
QM(df = 1) = 16.3571, p-val < .0001
 
Model Results:
 
         estimate      se     zval    pval    ci.lb    ci.ub
intrcpt    0.2515  0.2491   1.0095  0.3127  -0.2368   0.7397
ablat     -0.0291  0.0072  -4.0444  <.0001  -0.0432  -0.0150  ***
 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The results indeed suggest that the size of the average log risk ratio depends on absolute latitude.

In such a meta-regression model, there is no longer the average effect, so if we want to estimate the size of the effect, we have to specify a value for ablat. One possibility is to estimate the average risk ratio based on the average absolute latitude of the 13 trials included in the meta-analysis. We can do this as follows.

predict(res, newmods = mean(dat$ablat), transf=exp, digits=2)
 pred ci.lb ci.ub pi.lb pi.ub
 0.49  0.39  0.60  0.27  0.87

Hence, at about 33.5 degrees absolute latitude (see mean(dat$ablat)), the estimated average risk ratio is $0.49$ (95% CI: $0.39$ to $0.60$). There is no guarantee that this predicted value will be equal to the value obtained from the random-effects model, although it often won't be too dissimilar. However, if the moderator is able to account for at least some of the heterogeneity in the effects, then the confidence interval should be narrower. We see this happening in this example (note that about 75% of the heterogeneity is accounted for by this moderator). Although one can debate the terminology, some might call this predicted effect an 'adjusted effect' based on the meta-regression model.

Note that if you draw a forest plot for a meta-regression model, the predicted/fitted effects corresponding to the values of the moderator of the included studies will be indicated in the plot as gray-shaded polygons.

forest(res, xlim=c(-6.8,3.8), header=TRUE, atransf=exp,
       at=log(c(1/16, 1/8, 1/4, 1/2, 1, 2, 4, 8)), digits=c(2L,4L),
       ilab=dat$ablat, ilab.xpos=-3.5, order=order(dat$ablat), ylim=c(-1.5,15))
text(-3.5, 15, "Lattitude", font=2)
abline(h=0)
sav <- predict(res, newmods = mean(dat$ablat))
addpoly(sav$pred, sei=sav$se, atransf=exp, digits=2, mlab="Adjusted Effect")

In the forest plot, the values of the moderator are added as an extra column and the studies are ordered based on their absolute latitude value to make this clearer. Some extra space was also added to the forest plot to add the 'adjusted effect' at the bottom.

Meta-Regression with a Categorical Moderator

Next, let's see how this works when the moderator variable of interest is categorical. For this, we will use variable alloc, which indicates the way participants were allocated to the vaccinated versus the control group (with levels random, alternate, and systematic).

res <- rma(yi, vi, mods = ~ alloc, data=dat)
res
Mixed-Effects Model (k = 13; tau^2 estimator: REML)
 
tau^2 (estimated amount of residual heterogeneity):     0.3615 (SE = 0.2111)
tau (square root of estimated tau^2 value):             0.6013
I^2 (residual heterogeneity / unaccounted variability): 88.77%
H^2 (unaccounted variability / sampling variability):   8.91
R^2 (amount of heterogeneity accounted for):            0.00%
 
Test for Residual Heterogeneity:
QE(df = 10) = 132.3676, p-val < .0001
 
Test of Moderators (coefficients 2:3):
QM(df = 2) = 1.7675, p-val = 0.4132
 
Model Results:
 
                 estimate      se     zval    pval    ci.lb   ci.ub
intrcpt           -0.5180  0.4412  -1.1740  0.2404  -1.3827  0.3468
allocrandom       -0.4478  0.5158  -0.8682  0.3853  -1.4588  0.5632
allocsystematic    0.0890  0.5600   0.1590  0.8737  -1.0086  1.1867
 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The intercept reflects the estimated average log risk ratio for level alternate (the reference level), while the other two coefficients estimate how much the average log risk ratios for levels random and systematic differ from the reference level. Note that the omnibus test of these two coefficients is not significant ($Q_M = 1.77, df = 2, p = .41$), which indicates that there is insufficient evidence that the average log risk ratio actually differs across the three levels. However, for illustration purposes, we'll proceed with our analysis of this moderator.

First, we can compute the estimated average risk ratios for the three levels with:

predict(res, newmods = c(0,0), transf=exp, digits=2)
predict(res, newmods = c(1,0), transf=exp, digits=2)
predict(res, newmods = c(0,1), transf=exp, digits=2)
  pred ci.lb ci.ub pi.lb pi.ub
1 0.60  0.25  1.41  0.14  2.57
2 0.38  0.23  0.64  0.10  1.38
3 0.65  0.33  1.28  0.17  2.53

Note: The output above was obtained with predict(res, newmods = rbind(c(0,0), c(1,0), c(0,1)), transf=exp, digits=2), which provides the three estimates in a single line of code. However, the code above is a bit easier to read and shows how we need to set the two dummy variables (that are created for the random and systematic levels) to 0 or 1 to obtain the estimates for the three levels.

But what should we do if we want to compute an 'adjusted effect' again? In other words, what values should we plug into our model equation (and hence into newmods) to obtain such an estimate? One common approach is to use the mean of the respective dummy variables. We can obtain these values from the 'model matrix' and taking means across columns (leaving out the intercept).

colMeans(model.matrix(res))[-1]
    allocrandom allocsystematic
      0.5384615       0.3076923

Using these means for newmods then yields the 'adjusted estimate'.

predict(res, newmods = colMeans(model.matrix(res))[-1], transf=exp, digits=2)
 pred ci.lb ci.ub pi.lb pi.ub
 0.48  0.33  0.70  0.14  1.66

Again, this estimate is very close to the one obtained from the random-effects model. Also, since this moderator does not actually account for any heterogeneity, the confidence interval is essentially the same width as the one from the random-effects model.

What do these column means above actually represent? They are in fact proportions and indicate that 53.8% of the trials used random allocation (i.e., 7 out of the 13), 30.8% used systematic allocation (i.e., 4 out of the 13), and hence 15.4% used alternating allocation (i.e., 2 out of the 13). The predicted effect computed above is therefore an estimate for a population of studies where the relative frequencies of the different allocation methods are like those observed in the 13 trials included in the meta-analysis.

But we are not restricted to setting the proportions in this way. Another common approach is to assume that the population to which we want to generalize includes studies where each allocation method is used equally often. For a three-level factor, we then need to set the values for the two dummy variables to 1/3. This yields:

predict(res, newmods = c(1/3,1/3), transf=exp, digits=2)
 pred ci.lb ci.ub pi.lb pi.ub
 0.53  0.35  0.79  0.15  1.84

Meta-Regression with Continuous and Categorical Moderators

We can also consider a model that includes a mix of continuous and categorical moderators.

res <- rma(yi, vi, mods = ~ ablat + alloc, data=dat)
res
Mixed-Effects Model (k = 13; tau^2 estimator: REML)
 
tau^2 (estimated amount of residual heterogeneity):     0.1446 (SE = 0.1124)
tau (square root of estimated tau^2 value):             0.3803
I^2 (residual heterogeneity / unaccounted variability): 70.11%
H^2 (unaccounted variability / sampling variability):   3.35
R^2 (amount of heterogeneity accounted for):            53.84%
 
Test for Residual Heterogeneity:
QE(df = 9) = 26.2034, p-val = 0.0019
 
Test of Moderators (coefficients 2:4):
QM(df = 3) = 11.0605, p-val = 0.0114
 
Model Results:
 
                 estimate      se     zval    pval    ci.lb    ci.ub
intrcpt            0.2932  0.4050   0.7239  0.4691  -0.5006   1.0870
ablat             -0.0273  0.0092  -2.9650  0.0030  -0.0453  -0.0092  **
allocrandom       -0.2675  0.3504  -0.7633  0.4453  -0.9543   0.4193
allocsystematic    0.0585  0.3795   0.1540  0.8776  -0.6854   0.8023
 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The same idea as described above can again be applied to obtain an 'adjusted effect', but now adjusted for both moderators. Again, we just need to compute column means over the model matrix and use this for newmods.

predict(res, newmods = colMeans(model.matrix(res))[-1], transf=exp, digits=2)
 pred ci.lb ci.ub pi.lb pi.ub
 0.47  0.36  0.62  0.22  1.05

Final Thoughts

Some may scoff at the idea of computing such an 'adjusted effect', questioning its usefulness and interpretability. I'll leave this discussion for another day. However, the method above is nothing different than what is used to compute so-called 'marginal means' (or 'least squares means', although that term is a bit outdated).

References

Bates, J. H. (1982). Tuberculosis: Susceptibility and resistance. American Review of Respiratory Disease, 125(3 Pt 2), 20-24.

Colditz, G. A., Brewer, T. F., Berkey, C. S., Wilson, M. E., Burdick, E., Fineberg, H. V., & Mosteller, F. (1994). Efficacy of BCG vaccine in the prevention of tuberculosis: Meta-analysis of the published literature. Journal of the American Medical Association, 271(9), 698-702.

Ginsberg, A. M. (1998). The tuberculosis epidemic: Scientific challenges and opportunities. Public Health Reports, 113(2), 128-136.

tips/computing_adjusted_effects.txt · Last modified: 2020/08/18 19:53 by Wolfgang Viechtbauer