`rma.mv()`

estimates this model. Therefore, if a particular study had only measured one of the two outcomes, it could still be included in the analysis. The variance-covariance matrix of that study would then simply be a $1 \times 1$ matrix, with the value of that matrix equal to the sampling variance of the observed outcome.### Table of Contents

## Berkey et al. (1998)

### The Methods and Data

Berkey et al. (1998) describe a meta-analytic multivariate model for the analysis of multiple correlated outcomes. The use of the model is illustrated with results from 5 trials comparing surgical and non-surgical treatments for medium-severity periodontal disease. Reported outcomes include the change in probing depth (PD) and attachment level (AL) one year after the treatment. The effect size measure used for this meta-analysis was the (raw) mean difference, calculated in such a way that positive values indicate that surgery was more effective than non-surgical treatment in decreasing the probing depth and increasing the attachment level. The data are provided in Table I in the article and are stored in the dataset `dat.berkey1998`

that comes with the metafor package:

library(metafor) dat <- dat.berkey1998 dat

(I copy the dataset into `dat`

, which is a bit shorter and therefore easier to type further below). The contents of the dataset are:

trial author year ni outcome yi v1i v2i 1 1 Pihlstrom et al. 1983 14 PD 0.47 0.0075 0.0030 2 1 Pihlstrom et al. 1983 14 AL -0.32 0.0030 0.0077 3 2 Lindhe et al. 1982 15 PD 0.20 0.0057 0.0009 4 2 Lindhe et al. 1982 15 AL -0.60 0.0009 0.0008 5 3 Knowles et al. 1979 78 PD 0.40 0.0021 0.0007 6 3 Knowles et al. 1979 78 AL -0.12 0.0007 0.0014 7 4 Ramfjord et al. 1987 89 PD 0.26 0.0029 0.0009 8 4 Ramfjord et al. 1987 89 AL -0.31 0.0009 0.0015 9 5 Becker et al. 1988 16 PD 0.56 0.0148 0.0072 10 5 Becker et al. 1988 16 AL -0.39 0.0072 0.0304

So, the results from the various trials indicate that surgery is preferable for reducing the probing depth, while non-surgical treatment is preferable for increasing the attachment level.

Since each trial provides effect size estimates for both outcomes, the estimates are correlated. The `v1i`

and `v2i`

values are the variances and covariances of the observed effects. In particular, for each study, variables `v1i`

and `v2i`

form a 2×2 variance-covariance matrix of the observed effects, with the diagonal elements corresponding to the sampling variances of the mean differences (the first for probing depth, the second for attachment level) and the off-diagonal value corresponding to the covariance of the two mean differences.

### Variance-Covariance Matrix

Before we can proceed with the model fitting, we need to construct the full (block-diagonal) variance-covariance for all studies from these two variables. We can do this using the `vcalc()`

function in one line of code:

V <- vcalc(vi=1, cluster=author, rvars=c(v1i, v2i), data=dat)

The `V`

matrix is then equal to:

[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [1,] 0.0075 0.0030 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [2,] 0.0030 0.0077 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [3,] 0.0000 0.0000 0.0057 0.0009 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [4,] 0.0000 0.0000 0.0009 0.0008 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 [5,] 0.0000 0.0000 0.0000 0.0000 0.0021 0.0007 0.0000 0.0000 0.0000 0.0000 [6,] 0.0000 0.0000 0.0000 0.0000 0.0007 0.0014 0.0000 0.0000 0.0000 0.0000 [7,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0029 0.0009 0.0000 0.0000 [8,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0009 0.0015 0.0000 0.0000 [9,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0148 0.0072 [10,] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0072 0.0304

### Multivariate Random-Effects Model

A multivariate random-effects model can now be used to meta-analyze the two outcomes simultaneously. The model is given by $$\left[ \begin{array}{c}
y_{PD,i}\\
y_{AL,i}
\end{array} \right]
=
\left[ \begin{array}{c}
\mu_{PD} \\
\mu_{AL}
\end{array} \right]
+
\left[ \begin{array}{c}
u_{PD,i} \\
u_{AL,i}
\end{array} \right]
+
\left[ \begin{array}{c}
\epsilon_{PD,i} \\
\epsilon_{AL,i}
\end{array} \right],$$ where $$G = \mbox{Var}
\left[ \begin{array}{c}
u_{PD,i} \\
u_{AL,i}
\end{array} \right]
=
\left[ \begin{array}{cc}
\tau^2_{PD} & \rho \tau_{PD} \tau_{AL} \\
\rho \tau_{PD} \tau_{AL} & \tau^2_{AL}
\end{array} \right]$$ and $$V_i = \mbox{Var}
\left[ \begin{array}{c}
\epsilon_{PD,i} \\
\epsilon_{AL,i}
\end{array} \right]
=
\left[ \begin{array}{cc}
\mbox{var}_{PD,i} & \mbox{cov}_{PD,i;AL,i} \\
\mbox{cov}_{PD,i;AL,i} & \mbox{var}_{AL,i}
\end{array} \right],$$ where the elements in this second variance-covariance matrix are given by the $2 \times 2$ blocks along the diagonal in the `V`

matrix. Therefore, $$V = \left[ \begin{array}{ccc}
V_1 & & \\
& \ddots & \\
& & V_5
\end{array} \right].$$

We can fit this model with:

res <- rma.mv(yi, V, mods = ~ 0 + outcome, random = ~ outcome | trial, struct="UN", data=dat, method="ML") print(res, digits=3)

Multivariate Meta-Analysis Model (k = 10; method: ML) Variance Components: outer factor: trial (nlvls = 5) inner factor: outcome (nlvls = 2) estim sqrt k.lvl fixed level tau^2.1 0.026 0.162 5 no AL tau^2.2 0.007 0.084 5 no PD rho.AL rho.PD AL PD AL 1 0.699 - no PD 0.699 1 5 - Test for Residual Heterogeneity: QE(df = 8) = 128.227, p-val < .001 Test of Moderators (coefficient(s) 1,2): QM(df = 2) = 155.772, p-val < .001 Model Results: estimate se zval pval ci.lb ci.ub outcomeAL -0.338 0.080 -4.237 <.001 -0.494 -0.182 *** outcomePD 0.345 0.049 6.972 <.001 0.248 0.442 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

This is what Berkey et al. (1998) call a multivariate maximum likelihood (MML) random-effects model.^{1)} Note that `rma.mv()`

uses REML estimation by default, so `method="ML"`

must be explicitly requested. The `random = ~ outcome | trial`

part adds random effects for each outcome within each trial to the model. With `struct="UN"`

, the random effects are allowed to have different variances for each outcome and are allowed to be correlated.

The results show that the amount of heterogeneity in the attachment level (AL) outcome (i.e., $\hat{\tau}_{AL}^2 = .026$) is larger than the amount of heterogeneity in the probing depth (PD) outcome (i.e., $\hat{\tau}_{PD}^2 = .007$). Furthermore, the true outcomes appear to correlate quite strongly (i.e., $\hat{\rho} = .70$). On average, surgery is estimated to lead to significantly greater decreases in probing depth (i.e., $\hat{\mu}_{PB} = .35$), but non-surgery is more effective in increasing the attachment level (i.e., $\hat{\mu}_{AL} = -.34$).

### Multivariate Mixed-Effects Meta-Regression Model

The results given in Table II in the paper actually are based on a meta-regression model, using year of publication as a potential moderator. To replicate those analyses, we use:

res <- rma.mv(yi, V, mods = ~ 0 + outcome + outcome:I(year - 1983), random = ~ outcome | trial, struct="UN", data=dat, method="ML") print(res, digits=3)

Multivariate Meta-Analysis Model (k = 10; method: ML) Variance Components: outer factor: trial (nlvls = 5) inner factor: outcome (nlvls = 2) estim sqrt k.lvl fixed level tau^2.1 0.025 0.158 5 no AL tau^2.2 0.008 0.090 5 no PD rho.AL rho.PD AL PD AL 1 0.659 - no PD 0.659 1 5 - Test for Residual Heterogeneity: QE(df = 6) = 125.756, p-val < .001 Test of Moderators (coefficient(s) 1,2,3,4): QM(df = 4) = 143.439, p-val < .001 Model Results: estimate se zval pval ci.lb ci.ub outcomeAL -0.335 0.079 -4.261 <.001 -0.489 -0.181 *** outcomePD 0.348 0.052 6.694 <.001 0.246 0.450 *** outcomeAL:I(year - 1983) -0.011 0.024 -0.445 0.656 -0.059 0.037 outcomePD:I(year - 1983) 0.001 0.015 0.063 0.950 -0.029 0.031 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Note that publication year was centered at 1983, as was done by the authors. These results correspond to those given in the rightmost column in Table II on page 2545 (column "multiple outcomes MML"). The output above directly provides the correlation among the true effects. We can obtain the covariance with:

round(res$G[1,2], digits=3)

[1] 0.009

To test whether the slope of publication year actually differs for the two outcomes, we can fit the same model with:

res <- rma.mv(yi, V, mods = ~ 0 + outcome*I(year - 1983), random = ~ outcome | trial, struct="UN", data=dat, method="ML") print(res, digits=3)

The output is identical, except for the last part, which is now equal to:

Model Results: estimate se zval pval ci.lb ci.ub outcomeAL -0.335 0.079 -4.261 <.001 -0.489 -0.181 *** outcomePD 0.348 0.052 6.694 <.001 0.246 0.450 *** I(year - 1983) -0.011 0.024 -0.445 0.656 -0.059 0.037 outcomePD:I(year - 1983) 0.012 0.020 0.593 0.553 -0.027 0.051 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Therefore, the slope is actually not significantly different for the two outcomes ($p = .553$). In fact, it does not appear as if publication year is at all related to the two outcomes.

### Comparing Different Random Effects Structures

One could actually consider a simpler model for these data, which assumes a compound symmetry structure for the random effects (this would imply that the amount of heterogeneity is the same for the two outcomes). A formal comparison of the two models can be conducted using a likelihood ratio test:

res1 <- rma.mv(yi, V, mods = ~ 0 + outcome, random = ~ outcome | trial, struct="UN", data=dat, method="ML") res0 <- rma.mv(yi, V, mods = ~ 0 + outcome, random = ~ outcome | trial, struct="CS", data=dat, method="ML") anova(res0, res1)

df AIC BIC AICc logLik LRT pval QE Full 5 -1.6813 -0.1684 13.3187 5.8407 128.2267 Reduced 4 -2.4111 -1.2008 5.5889 5.2056 1.2702 0.2597 128.2267

Since model `res1`

has one more parameter than model `res0`

, the test statistic (`LRT`

) follows (approximately) a chi-square distribution with 1 degree of freedom. The p-value (0.2597) suggests that the more complex model (with `struct="UN"`

) does not actually yield a significantly better fit than the simpler model (with `struct="CS"`

). However, with only 5 studies, this test is likely to have very low power.

### References

Berkey, C. S., Hoaglin, D. C., Antczak-Bouckoms, A., Mosteller, F., & Colditz, G. A. (1998). Meta-analysis of multiple outcomes by regression with random effects. *Statistics in Medicine, 17*(22), 2537–2550.

^{1)}