Results from 20 hypothetical randomized clinical trials examining the effectiveness of a medication for treating some disease.

dat.viechtbauer2021

Format

The data frame contains the following columns:

trialnumerictrial number
nTinumericnumber of patients in the treatment group
nCinumericnumber of patients in the control group
xTinumericnumber of patients in the treatment group with remission
xCinumericnumber of patients in the control group with remission
dosenumericdosage of the medication provided to patients in the treatment group (in milligrams per day)

Details

The dataset was constructed for the purposes of illustrating the model checking and diagnostic methods described in Viechtbauer (2021). The code below provides the results for many of the analyses and plots discussed in the book chapter.

Source

Viechtbauer, W. (2021). Model checking in meta-analysis. In C. H. Schmid, T. Stijnen, & I. R. White (Eds.), Handbook of meta-analysis (pp. 219-254). Boca Raton, FL: CRC Press. https://doi.org/10.1201/9781315119403

Concepts

medicine, odds ratios, outliers, model checks

Examples

### copy data into 'dat' and examine data
dat <- dat.viechtbauer2021
dat
#>    trial nTi nCi xTi xCi dose
#> 1      1  66  59  42  24  100
#> 2      2  59  65  42  34  200
#> 3      3 253 257  96  32  250
#> 4      4 137 144  51  44  125
#> 5      5 327 326  47  39   50
#> 6      6 584 588  38  87   25
#> 7      7 526 532 390 323  125
#> 8      8  28  30  10   3  125
#> 9      9 191 201 165 126  125
#> 10    10  86  94  58  39  150
#> 11    11 229 221  72  60  100
#> 12    12 153 144  79  56  150
#> 13    13  93  95  48  35  200
#> 14    14  40  40   8   4   25
#> 15    15  85  88  44  21  175
#> 16    16 100 107  10  13   25
#> 17    17  72  64  11   9   25
#> 18    18  80  74  47  23  200
#> 19    19 191 195 144 116  100
#> 20    20  85  85  48  49   75

# \dontrun{

### load metafor package
library(metafor)

### calculate log odds ratios and corresponding sampling variances

dat <- escalc(measure="OR", ai=xTi, n1i=nTi, ci=xCi, n2i=nCi, add=1/2, to="all", data=dat)
dat
#> 
#>    trial nTi nCi xTi xCi dose      yi     vi 
#> 1      1  66  59  42  24  100  0.9217 0.1333 
#> 2      2  59  65  42  34  200  0.7963 0.1414 
#> 3      3 253 257  96  32  250  1.4472 0.0519 
#> 4      4 137 144  51  44  125  0.2961 0.0634 
#> 5      5 327 326  47  39   50  0.2091 0.0534 
#> 6      6 584 588  38  87   25 -0.9069 0.0412 
#> 7      7 526 532 390 323  125  0.6166 0.0178 
#> 8      8  28  30  10   3  125  1.4950 0.4714 
#> 9      9 191 201 165 126  125  1.3157 0.0649 
#> 10    10  86  94  58  39  150  1.0592 0.0955 
#> 11    11 229 221  72  60  100  0.2060 0.0429 
#> 12    12 153 144  79  56  150  0.5137 0.0550 
#> 13    13  93  95  48  35  200  0.5970 0.0873 
#> 14    14  40  40   8   4   25  0.7521 0.3980 
#> 15    15  85  88  44  21  175  1.2139 0.1079 
#> 16    16 100 107  10  13   25 -0.2081 0.1909 
#> 17    17  72  64  11   9   25  0.0884 0.2265 
#> 18    18  80  74  47  23  200  1.1338 0.1129 
#> 19    19 191 195 144 116  100  0.7304 0.0491 
#> 20    20  85  85  48  49   75 -0.0474 0.0949 
#> 

### number of studies

k <- nrow(dat)

### fit models

res.CE <- rma(yi, vi, data=dat, method="CE") # same as method="EE"
res.CE
#> 
#> Common-Effects Model (k = 20)
#> 
#> I^2 (total heterogeneity / total variability):   81.70%
#> H^2 (total variability / sampling variability):  5.46
#> 
#> Test for Heterogeneity:
#> Q(df = 19) = 103.8068, p-val < .0001
#> 
#> Model Results:
#> 
#> estimate      se    zval    pval   ci.lb   ci.ub      
#>   0.5295  0.0586  9.0328  <.0001  0.4146  0.6443  *** 
#> 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 

res.RE <- rma(yi, vi, data=dat, method="DL")
res.RE
#> 
#> Random-Effects Model (k = 20; tau^2 estimator: DL)
#> 
#> tau^2 (estimated amount of total heterogeneity): 0.3174 (SE = 0.1472)
#> tau (square root of estimated tau^2 value):      0.5634
#> I^2 (total heterogeneity / total variability):   81.70%
#> H^2 (total variability / sampling variability):  5.46
#> 
#> Test for Heterogeneity:
#> Q(df = 19) = 103.8068, p-val < .0001
#> 
#> Model Results:
#> 
#> estimate      se    zval    pval   ci.lb   ci.ub      
#>   0.5867  0.1451  4.0425  <.0001  0.3022  0.8712  *** 
#> 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 

res.MR <- rma(yi, vi, mods = ~ dose, data=dat, method="FE")
res.MR
#> 
#> Fixed-Effects with Moderators Model (k = 20)
#> 
#> I^2 (residual heterogeneity / unaccounted variability): 55.49%
#> H^2 (unaccounted variability / sampling variability):   2.25
#> R^2 (amount of heterogeneity accounted for):            58.88%
#> 
#> Test for Residual Heterogeneity:
#> QE(df = 18) = 40.4392, p-val = 0.0018
#> 
#> Test of Moderators (coefficient 2):
#> QM(df = 1) = 63.3676, p-val < .0001
#> 
#> Model Results:
#> 
#>          estimate      se     zval    pval    ci.lb    ci.ub      
#> intrcpt   -0.4235  0.1333  -3.1772  0.0015  -0.6847  -0.1622   ** 
#> dose       0.0079  0.0010   7.9604  <.0001   0.0059   0.0098  *** 
#> 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 

res.ME <- rma(yi, vi, mods = ~ dose, data=dat, method="DL")
res.ME
#> 
#> Mixed-Effects Model (k = 20; tau^2 estimator: DL)
#> 
#> tau^2 (estimated amount of residual heterogeneity):     0.0895 (SE = 0.0582)
#> tau (square root of estimated tau^2 value):             0.2992
#> I^2 (residual heterogeneity / unaccounted variability): 55.49%
#> H^2 (unaccounted variability / sampling variability):   2.25
#> R^2 (amount of heterogeneity accounted for):            71.80%
#> 
#> Test for Residual Heterogeneity:
#> QE(df = 18) = 40.4392, p-val = 0.0018
#> 
#> Test of Moderators (coefficient 2):
#> QM(df = 1) = 22.6429, p-val < .0001
#> 
#> Model Results:
#> 
#>          estimate      se     zval    pval    ci.lb   ci.ub      
#> intrcpt   -0.3041  0.2061  -1.4758  0.1400  -0.7080  0.0998      
#> dose       0.0071  0.0015   4.7585  <.0001   0.0042  0.0101  *** 
#> 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 

### forest and bubble plot

par(mar=c(5,4,1,2))

forest(dat$yi, dat$vi, psize=0.8, efac=0, xlim=c(-4,6), ylim=c(-3,23),
       cex=1, width=c(5,5,5), xlab="Log Odds Ratio (LnOR)")
addpoly(res.CE, row=-1.5, mlab="CE Model")
addpoly(res.RE, row=-2.5, mlab="RE Model")
text(-4, 22, "Trial",         pos=4, font=2)
text( 6, 22, "LnOR [95% CI]", pos=2, font=2)
abline(h=0)


tmp <- regplot(res.ME, xlim=c(0,250), ylim=c(-1,1.5), predlim=c(0,250), shade=FALSE, digits=1,
               xlab="Dosage (mg per day)", psize="seinv", plim=c(NA,5), bty="l", las=1,
               lty=c("solid", "dashed"), label=TRUE, labsize=0.8, offset=c(1,0.7))
res.sub <- rma(yi, vi, mods = ~ dose, data=dat, method="DL", subset=-6)
abline(res.sub, lty="dotted")
points(tmp$xi, tmp$yi, pch=21, cex=tmp$psize, col="black", bg="darkgray")


par(mar=c(5,4,4,2))

### number of standardized deleted residuals larger than +-1.96 in each model

sum(abs(rstudent(res.CE)$z) >= qnorm(.975))
#> [1] 4
sum(abs(rstudent(res.MR)$z) >= qnorm(.975))
#> [1] 3
sum(abs(rstudent(res.RE)$z) >= qnorm(.975))
#> [1] 1
sum(abs(rstudent(res.ME)$z) >= qnorm(.975))
#> [1] 2

### plot of the standardized deleted residuals for the RE and ME models

plot(NA, NA, xlim=c(1,20), ylim=c(-4,4), xlab="Study", ylab="Standardized (Deleted) Residual",
     xaxt="n", main="Random-Effects Model", las=1)
axis(side=1, at=1:20)
abline(h=c(-1.96,1.96), lty="dotted")
abline(h=0)
points(1:20, rstandard(res.RE)$z, type="o", pch=19, col="gray70")
points(1:20, rstudent(res.RE)$z,  type="o", pch=19)
legend("top", pch=19, col=c("gray70","black"), lty="solid",
       legend=c("Standardized Residuals","Standardized Deleted Residuals"), bty="n")


plot(NA, NA, xlim=c(1,20), ylim=c(-4,4), xlab="Study", ylab="Standardized (Deleted) Residual",
     xaxt="n", main="Mixed-Effects Model", las=1)
axis(side=1, at=1:20)
abline(h=c(-1.96,1.96), lty="dotted")
abline(h=0)
points(1:20, rstandard(res.ME)$z, type="o", pch=19, col="gray70")
points(1:20, rstudent(res.ME)$z,  type="o", pch=19)
legend("top", pch=19, col=c("gray70","black"), lty="solid",
       legend=c("Standardized Residuals","Standardized Deleted Residuals"), bty="n")


### Baujat plots

baujat(res.CE, main="Common-Effects Model", xlab="Squared Pearson Residual", ylim=c(0,5), las=1)

baujat(res.ME, main="Mixed-Effects Model", ylim=c(0,2), las=1)


### GOSH plots (skipped because this takes quite some time to run)

if (FALSE) {

res.GOSH.CE <- gosh(res.CE, subsets=10^7)
plot(res.GOSH.CE, cex=0.2, out=6, xlim=c(-0.25,1.25), breaks=c(200,100))

res.GOSH.ME <- gosh(res.ME, subsets=10^7)
plot(res.GOSH.ME, het="tau2", out=6, breaks=50, adjust=0.6, las=1)

}

### plot of treatment dosage against the standardized residuals

plot(dat$dose, rstandard(res.ME)$z, pch=19, xlab="Dosage (mg per day)",
     ylab="Standardized Residual", xlim=c(0,250), ylim=c(-2.5,2.5), las=1)
abline(h=c(-1.96,1.96), lty="dotted", lwd=2)
abline(h=0)
title("Standardized Residual Plot")
text(dat$dose[6], rstandard(res.ME)$z[6], "6", pos=4, offset=0.4)


### quadratic polynomial model

rma(yi, vi, mods = ~ dose + I(dose^2), data=dat, method="DL")
#> 
#> Mixed-Effects Model (k = 20; tau^2 estimator: DL)
#> 
#> tau^2 (estimated amount of residual heterogeneity):     0.0807 (SE = 0.0568)
#> tau (square root of estimated tau^2 value):             0.2840
#> I^2 (residual heterogeneity / unaccounted variability): 52.00%
#> H^2 (unaccounted variability / sampling variability):   2.08
#> R^2 (amount of heterogeneity accounted for):            74.58%
#> 
#> Test for Residual Heterogeneity:
#> QE(df = 17) = 35.4193, p-val = 0.0055
#> 
#> Test of Moderators (coefficients 2:3):
#> QM(df = 2) = 25.7249, p-val < .0001
#> 
#> Model Results:
#> 
#>            estimate      se     zval    pval    ci.lb   ci.ub    
#> intrcpt     -0.6191  0.3163  -1.9571  0.0503  -1.2390  0.0009  . 
#> dose         0.0136  0.0053   2.5694  0.0102   0.0032  0.0240  * 
#> I(dose^2)   -0.0000  0.0000  -1.2610  0.2073  -0.0001  0.0000    
#> 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 

### lack-of-fit model

resLOF <- rma(yi, vi, mods = ~ dose + factor(dose), data=dat, method="DL", btt=3:9)
#> Warning: Redundant predictors dropped from the model.
resLOF
#> 
#> Mixed-Effects Model (k = 20; tau^2 estimator: DL)
#> 
#> tau^2 (estimated amount of residual heterogeneity):     0.1285 (SE = 0.0981)
#> tau (square root of estimated tau^2 value):             0.3585
#> I^2 (residual heterogeneity / unaccounted variability): 60.32%
#> H^2 (unaccounted variability / sampling variability):   2.52
#> R^2 (amount of heterogeneity accounted for):            59.50%
#> 
#> Test for Residual Heterogeneity:
#> QE(df = 11) = 27.7226, p-val = 0.0036
#> 
#> Test of Moderators (coefficients 3:9):
#> QM(df = 7) = 3.5712, p-val = 0.8276
#> 
#> Model Results:
#> 
#>                  estimate      se     zval    pval    ci.lb   ci.ub      
#> intrcpt           -0.5099  0.3035  -1.6803  0.0929  -1.1048  0.0849    . 
#> dose               0.0078  0.0022   3.5002  0.0005   0.0034  0.0122  *** 
#> factor(dose)50     0.3276  0.4916   0.6663  0.5052  -0.6360  1.2912      
#> factor(dose)75    -0.1246  0.5257  -0.2371  0.8126  -1.1550  0.9057      
#> factor(dose)100    0.3051  0.3433   0.8887  0.3741  -0.3677  0.9779      
#> factor(dose)125    0.3285  0.3333   0.9856  0.3243  -0.3247  0.9817      
#> factor(dose)150    0.0950  0.4135   0.2298  0.8183  -0.7154  0.9054      
#> factor(dose)175    0.3538  0.5698   0.6209  0.5347  -0.7631  1.4707      
#> factor(dose)200   -0.2215  0.4392  -0.5042  0.6141  -1.0822  0.6393      
#> 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 

### scatter plot to illustrate the lack-of-fit model

regplot(res.ME, xlim=c(0,250), ylim=c(-1.0,1.5), xlab="Dosage (mg per day)", ci=FALSE,
        predlim=c(0,250), psize=1, pch=19, col="gray60", digits=1, lwd=1, bty="l", las=1)
dosages <- sort(unique(dat$dose))
lines(dosages, fitted(resLOF)[match(dosages, dat$dose)], type="o", pch=19, cex=2, lwd=2)
points(dat$dose, dat$yi, pch=19, col="gray60")
legend("bottomright", legend=c("Linear Model", "Lack-of-Fit Model"), pch=c(NA,19), col="black",
       lty="solid", lwd=c(1,2), pt.cex=c(1,2), seg.len=4, bty="n")


### checking normality of the standardized deleted residuals

qqnorm(res.ME, type="rstudent", main="Standardized Deleted Residuals", pch=19, label="out",
       lwd=2, pos=24, ylim=c(-4,3), lty=c("solid", "dotted"), las=1)


### checking normality of the random effects

sav <- qqnorm(ranef(res.ME)$pred, main="BLUPs of the Random Effects", cex=1, pch=19,
              xlim=c(-2.2,2.2), ylim=c(-0.6,0.6), las=1)
abline(a=0, b=sd(ranef(res.ME)$pred), lwd=2)
text(sav$x[6], sav$y[6], "6", pos=4, offset=0.4)


### hat values for the CE and RE models

plot(NA, NA, xlim=c(1,20), ylim=c(0,0.21), xaxt="n", las=1, xlab="Study", ylab="Hat Value")
axis(1, 1:20, cex.axis=1)
points(hatvalues(res.CE), type="o", pch=19, col="gray70")
points(hatvalues(res.RE), type="o", pch=19)
abline(h=1/20, lty="dotted", lwd=2)
title("Hat Values for the CE/RE Models")
legend("topright", pch=19, col=c("gray70","black"), lty="solid",
       legend=c("Common-Effects Model", "Random-Effects Model"), bty="n")


### heatmap of the hat matrix for the ME model

cols <- colorRampPalette(c("blue", "white", "red"))(101)
h <- hatvalues(res.ME, type="matrix")
image(1:nrow(h), 1:ncol(h), t(h[nrow(h):1,]), axes=FALSE,
      xlab="Influence of the Observed Effect of Study ...", ylab="On the Fitted Value of Study ...",
      col=cols, zlim=c(-max(abs(h)),max(abs(h))))
axis(1, 1:20, tick=FALSE)
axis(2, 1:20, labels=20:1, las=1, tick=FALSE)
abline(h=seq(0.5,20.5,by=1), col="white")
abline(v=seq(0.5,20.5,by=1), col="white")
points(1:20, 20:1, pch=19, cex=0.4)
title("Heatmap for the Mixed-Effects Model")


### plot of leverages versus standardized residuals for the ME model

plot(hatvalues(res.ME), rstudent(res.ME)$z, pch=19, cex=0.2+3*sqrt(cooks.distance(res.ME)),
     las=1, xlab="Leverage (Hat Value)", ylab="Standardized Deleted Residual",
     xlim=c(0,0.35), ylim=c(-3.5,2.5))
abline(h=c(-1.96,1.96), lty="dotted", lwd=2)
abline(h=0, lwd=2)
ids <- c(3,6,9)
text(hatvalues(res.ME)[ids] + c(0,0.013,0.010), rstudent(res.ME)$z[ids] - c(0.18,0,0), ids)
title("Leverage vs. Standardized Deleted Residuals")


### plot of the Cook's distances for the ME model

plot(1:20, cooks.distance(res.ME), ylim=c(0,1.6), type="o", pch=19, las=1, xaxt="n", yaxt="n",
     xlab="Study", ylab="Cook's Distance")
axis(1, 1:20, cex.axis=1)
axis(2, seq(0,1.6,by=0.4), las=1)
title("Cook's Distances")


### plot of the leave-one-out estimates of tau^2 for the ME model

x <- influence(res.ME)

plot(1:20, x$inf$tau2.del,  ylim=c(0,0.15), type="o", pch=19, las=1, xaxt="n", xlab="Study",
     ylab=expression(paste("Estimate of ", tau^2, " without the ", italic(i), "th study")))
abline(h=res.ME$tau2, lty="dashed")
axis(1, 1:20)
title("Residual Heterogeneity Estimates")


### plot of the covariance ratios for the ME model

plot(1:20, x$inf$cov.r,  ylim=c(0,2.0), type="o", pch=19, las=1, xaxt="n",
     xlab="Study", ylab="Covariance Ratio")
abline(h=1, lty="dashed")
axis(1, 1:20)
title("Covariance Ratios")


### fit mixed-effects model without studies 3 and/or 6

rma(yi, vi, mods = ~ dose, data=dat, method="DL", subset=-3)
#> 
#> Mixed-Effects Model (k = 19; tau^2 estimator: DL)
#> 
#> tau^2 (estimated amount of residual heterogeneity):     0.0994 (SE = 0.0645)
#> tau (square root of estimated tau^2 value):             0.3153
#> I^2 (residual heterogeneity / unaccounted variability): 57.63%
#> H^2 (unaccounted variability / sampling variability):   2.36
#> R^2 (amount of heterogeneity accounted for):            64.04%
#> 
#> Test for Residual Heterogeneity:
#> QE(df = 17) = 40.1225, p-val = 0.0012
#> 
#> Test of Moderators (coefficient 2):
#> QM(df = 1) = 15.7509, p-val < .0001
#> 
#> Model Results:
#> 
#>          estimate      se     zval    pval    ci.lb   ci.ub      
#> intrcpt   -0.3064  0.2290  -1.3381  0.1809  -0.7553  0.1424      
#> dose       0.0072  0.0018   3.9687  <.0001   0.0037  0.0108  *** 
#> 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
rma(yi, vi, mods = ~ dose, data=dat, method="DL", subset=-6)
#> 
#> Mixed-Effects Model (k = 19; tau^2 estimator: DL)
#> 
#> tau^2 (estimated amount of residual heterogeneity):     0.0321 (SE = 0.0375)
#> tau (square root of estimated tau^2 value):             0.1792
#> I^2 (residual heterogeneity / unaccounted variability): 30.17%
#> H^2 (unaccounted variability / sampling variability):   1.43
#> R^2 (amount of heterogeneity accounted for):            75.00%
#> 
#> Test for Residual Heterogeneity:
#> QE(df = 17) = 24.3434, p-val = 0.1104
#> 
#> Test of Moderators (coefficient 2):
#> QM(df = 1) = 16.7472, p-val < .0001
#> 
#> Model Results:
#> 
#>          estimate      se     zval    pval    ci.lb   ci.ub      
#> intrcpt   -0.0504  0.1926  -0.2615  0.7937  -0.4279  0.3272      
#> dose       0.0056  0.0014   4.0923  <.0001   0.0029  0.0082  *** 
#> 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
rma(yi, vi, mods = ~ dose, data=dat, method="DL", subset=-c(3,6))
#> 
#> Mixed-Effects Model (k = 18; tau^2 estimator: DL)
#> 
#> tau^2 (estimated amount of residual heterogeneity):     0.0375 (SE = 0.0412)
#> tau (square root of estimated tau^2 value):             0.1937
#> I^2 (residual heterogeneity / unaccounted variability): 33.24%
#> H^2 (unaccounted variability / sampling variability):   1.50
#> R^2 (amount of heterogeneity accounted for):            56.68%
#> 
#> Test for Residual Heterogeneity:
#> QE(df = 16) = 23.9677, p-val = 0.0902
#> 
#> Test of Moderators (coefficient 2):
#> QM(df = 1) = 8.8711, p-val = 0.0029
#> 
#> Model Results:
#> 
#>          estimate      se     zval    pval    ci.lb   ci.ub     
#> intrcpt   -0.0007  0.2216  -0.0030  0.9976  -0.4350  0.4337     
#> dose       0.0051  0.0017   2.9784  0.0029   0.0017  0.0084  ** 
#> 
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 

# }