## 4.3 Meta-Analysis with binary outcomes

### 4.3.1 Event rate data

In some cases, you will work with binary outcome data (e.g., dead/alive, Depressive Disorder/no Depressive Disorder) instead of continuous data. In such a case, you will probably be more interested in outcomes like the pooled Odd’s Ratio or the Relative Risk Reduction.

Here, have two options again:

• The effect sizes are already calculated. In this case, we can use the metagen function as we did before (see Chapter 4.1 and Chapter 4.2). The calculated effect TE then describes the Odds Ratio, or whatever binary outcome we calculated previously for our data.
• We only have the raw outcome data. If this is the case, we will have to use the meta::metabin function instead. We’ll show you how to do this now.

For meta-analyses of binary outcomes, we need our data in the following format:

Column Description
Author This signifies the column for the study label (i.e., the first author)
Ee Number of events in the experimental treatment arm
Ne Number of participants in the experimental treatment arm
Ec Number of events in the control arm
Nc Number of participants in the control arm
Subgroup This is the label for one of your subgroup codes. It’s not that important how you name it, so you can give it a more informative name (e.g. population). In this column, each study should then be given a subgroup code, which should be exactly the same for each subgroup, including upper/lowercase letters. Of course, you can also include more than one subgroup column with different subgroup codings, but the column name has to be unique

I’ll use my dataset binarydata, which also has this format

str(binarydata)
## Classes 'tbl_df', 'tbl' and 'data.frame':    11 obs. of  5 variables:
##  $Author: chr "Alcorta-Fleischmann" "Craemer" "Eriksson" "Jones" ... ##$ Ee    : num  2 18 6 3 0 8 12 1 7 17 ...
##  $Ne : num 279 1273 1858 297 300 ... ##$ Ec    : num  1 17 5 6 1 9 12 10 8 21 ...
##  \$ Nc    : num  70 1287 1852 314 295 ...

The other parameters are like the ones we used in the meta-analyses with continuous outcome data, with two exceptions:

• sm: As we want to have a pooled effect for binary data, we have to choose another summary measure now. We can choose from “OR” (Odds Ratio), “RR” (Risk Ratio), or RD (Risk Difference), among other things.
• incr. This lets us define if and how we want conitinuity correction to be performed. Such a correction is necessary in cases where one of the cells in your data is zero (e.g., because no one in the intervention arm died). This can be a frequent phenomenon in some contexts, and distorts our effect size estimates. By default, the metabin function adds the value 0.5 in all cells were N is zero (Gart and Zweifel 1967). This value can be changed using the incr-parameter (e.g., incr=0.1). If your trial arms are very uneven in terms of their total $$n$$, we can also use the treatment arm continuity correction (J. Sweeting, J. Sutton, and C. Lambert 2004). This can be done by using incr="TACC".

Here’s the code for a meta-analysis with raw binary data

I have decided to run a random-effect-model meta-analysis. I want the summary measure to be the Risk Ratio (RR).

m.bin<-metabin(Ee,
Ne,
Ec,
Nc,
data=binarydata,
studlab=paste(Author),
comb.fixed = FALSE,
comb.random = TRUE,
method.tau = "SJ",
hakn = TRUE,
prediction=TRUE,
incr=0.1,
sm="RR")
m.bin
##                         RR            95%-CI %W(random)
## Alcorta-Fleischmann 0.5018 [0.0462;  5.4551]        3.0
## Craemer             1.0705 [0.5542;  2.0676]       14.6
## Eriksson            1.1961 [0.3657;  3.9124]        8.5
## Jones               0.5286 [0.1334;  2.0945]        7.0
## Knauer              0.0894 [0.0001; 57.7924]        0.5
## Kracauer            0.9076 [0.3512;  2.3453]       10.9
## La Sala             0.9394 [0.4233;  2.0847]       12.7
## Maheux              0.0998 [0.0128;  0.7768]        3.9
## Schmidthauer        0.7241 [0.2674;  1.9609]       10.3
## van der Zee         0.8434 [0.4543;  1.5656]       15.1
## Wang                0.5519 [0.2641;  1.1534]       13.5
##
## Number of studies combined: k = 11
##
##                          RR           95%-CI     t p-value
## Random effects model 0.7420 [0.5202; 1.0582] -1.87  0.0905
## Prediction interval         [0.2305; 2.3882]
##
## Quantifying heterogeneity:
## tau^2 = 0.2417; H = 1.00 [1.00; 1.36]; I^2 = 0.0% [0.0%; 45.8%]
##
## Test of heterogeneity:
##     Q d.f. p-value
##  7.34   10  0.6929
##
## Details on meta-analytical method:
## - Mantel-Haenszel method
## - Sidik-Jonkman estimator for tau^2
## - Hartung-Knapp adjustment for random effects model
## - Continuity correction of 0.1 in studies with zero cell frequencies

L’Abbé Plots

So-called L’Abbé plots (L’Abbé, Detsky, and O’Rourke 1987) are a good way to visualize data based on event rates. In a L’Abbé plot, the event rate of a study’s intervention group is plotted against the event rate in the control group, and the $$N$$ of the study is signified by the size of the bubble in the plot. Despite the simplicity in its principles, this plot allows us to check for three important aspects of our meta-analysis with binary outcomes:

• The overall trend of our meta-analysis. If we are expecting a type of intervention to have a protective effect (i.e., making an adverse outcome such as death or depression onset less likely) the studies should mostly lie in the bottom-right corner of the L’Abbé plot, because the control group event rate should be higher than the intervention group event rate. If there’s no effect of the intervention compared to the control group, the event rates are identical and the study is shown on the diagonal of the L’Abbé plot.
• Heterogeneity of effect sizes. The plot also allows us to eyeball for single studies or groups of studies which contribute to the heterogeneity of the effect we found. It could be the case, for example, that most studies lie in the bottom-right part of the plot as they report positive effects, while a few studies lie in the top-left sector indicated negative effects. Especially if such studies have a small precision (i.e., a small $$N$$ of participants, indicated by small bubbles in the plot), they could have distorted our pooled effect and may contribute to the between-study heterogeneity.
• Heterogeneity of event rates. It may also be the case that some of the heterogeneity in our meta-analysis was introduced by the fact that the event rates “per se” are higher or lower in some studies compared to the others. The L’Abbé plot provides as with this information, as studies with higher event rates will naturally tend towards the top-right corner of the plot.

The results of the metabin function can be easily used to generate L’Abbé plots using the labbe.metabin function included in the meta package. We can specify the following parameters:

Parameter Description
x This signifies our metabin meta-analysis output
bg The background color of the studies
col The line color of the studies
studlab Wether the names of the studies should be printed in the plot (TRUE/FALSE)
col.fixed The color of the dashed line symbolizing the pooled effect of the meta-analysis, if the fixed-effect-model was used
col.random The color of the dashed line symbolizing the pooled effect of the meta-analysis, if the random-effects-model was used

For this example, i’ll use the m.bin output i previously generated using the metabin function.

labbe.metabin(x = m.bin,
bg = "blue",
studlab = TRUE,
col.random = "red")

Works like a charm! We see that the dashed red line signifying the pooled effect estimate of my meta-analysis is running trough the bottom-right sector of my L’Abbé plot, meaning that the overall effect size is positive (i.e., that the intervention has a preventive effect).

However, it also becomes clear that all studies clearly follow this trend: we see that most studies lie tightly in the bottom-left corner of the plot, meaning that these studies had small event rates (i.e., the event in these studies was very rare irrespective of group assignment). We also see that two of our included studies don’t fall into this pattern: Schmidthauer and van der Zee. Those two studies have higher event rates, and both favor the intervention group more clearly than the others did.

### 4.3.2 Incidence rates

The previous chapter primarily dealt with raw event data. Such data usually does not contain any information on the time span during which events did or did not occur. Given that studies often have drastically different follow-up times (e.g., 8 weeks vs. 2 years), it often makes sense to also take the time interval during which events occured into account. In clinical epidemiology, incidence rates are often used to signify how many events occured within a standardized timeframe (e.g., one year). The corresponding effect size is the incidence rate ratio (IRR), which compares the incidence rate in the intervention group to the one in the control group.

To conduct meta-analyses using incidence rate data, so-called person-time data has to be collected or calculated by hand. What it basically needed to calculate person-time data is the number of events and the timeframe during which they occurred. You can find a general introduction into this topic here, and this quick course by the Centers for Disease Control and Prevention gives a hands-on introduction on how person-time data is calculated.

As an example, i will use my IRR.data.RData dataset, in which the person-time (in my case, person-year) is stored in the time.e and time.c column for the experimental and control group, respectively. The event.e and event.c column contains the number of events (in my case, depression onsets) for both groups.

To pool the data, we use the metainc function included in the meta package. We can set the following parameters:

Parameter Function
event.e The number of events in the intervention group
time.e The person-time at risk in the intervention group
event.c The number of events in the control group
time.c The person-time at risk in the control group
data= After =, paste the name of your dataset here
studlab=paste() This tells the function were the labels for each study are stored. If you named the spreadsheet columns as advised, this should be studlab=paste(Author)
comb.fixed= Weather to use a fixed-effects-model
comb.random= Weather to use a random-effects-model. This has to be set to TRUE
method.tau= Which estimator to use for the between-study variance
hakn= Weather to use the Knapp-Hartung-Sidik-Jonkman method
prediction= Weather to print a prediction interval for the effect of future studies based on present evidence
sm= The summary measure we want to calculate. We want to calculate the Incidence Rate Ratio (IRR)
metainc(event.e,
time.e,
event.c,
time.c,
studlab = paste(Author),
data = IRR.data,
sm = "IRR",
method.tau = "DL",
comb.random = TRUE,
comb.fixed = FALSE,
hakn = TRUE)
##                                              IRR           95%-CI
## Stice et al., 2013 (b)                    0.3688 [0.1443; 0.9424]
## Stice et al. 2017a (clinician-led)        0.8312 [0.2030; 3.4038]
## Stice et al. 2017a (peer-led)             0.3536 [0.0680; 1.8393]
## Stice et al. 2017a (online)               1.3888 [0.3687; 5.2314]
## Stice et al. 2017b (non dissonance-based) 0.3638 [0.0844; 1.5678]
## Stice et al. 2017b (dissonance-based)     0.3638 [0.0844; 1.5678]
## Taylor et al., 2006                       0.6184 [0.2604; 1.4686]
## Taylor et al., 2016                       0.7539 [0.4332; 1.3121]
##                                           %W(random)
## Stice et al., 2013 (b)                          14.0
## Stice et al. 2017a (clinician-led)               6.2
## Stice et al. 2017a (peer-led)                    4.5
## Stice et al. 2017a (online)                      7.0
## Stice et al. 2017b (non dissonance-based)        5.8
## Stice et al. 2017b (dissonance-based)            5.8
## Taylor et al., 2006                             16.5
## Taylor et al., 2016                             40.2
##
## Number of studies combined: k = 8
##
##                         IRR           95%-CI     t p-value
## Random effects model 0.6157 [0.4349; 0.8716] -3.30  0.0131
##
## Quantifying heterogeneity:
## tau^2 = 0; H = 1.00 [1.00; 1.44]; I^2 = 0.0% [0.0%; 51.8%]
##
## Test of heterogeneity:
##     Q d.f. p-value
##  4.71    7  0.6953
##
## Details on meta-analytical method:
## - Mantel-Haenszel method
## - DerSimonian-Laird estimator for tau^2
## - Hartung-Knapp adjustment for random effects model

### References

Gart, John J, and James R Zweifel. 1967. “On the Bias of Various Estimators of the Logit and Its Variance with Application to Quantal Bioassay.” Biometrika. JSTOR, 181–87.

J. Sweeting, Michael, Alexander J. Sutton, and Paul C. Lambert. 2004. “What to Add to Nothing? Use and Avoidance of Continuity Corrections in Meta-Analysis of Sparse Data.” Statistics in Medicine 23 (9). Wiley Online Library: 1351–75.

L’Abbé, Kristan A, Allan S Detsky, and Keith O’Rourke. 1987. “Meta-Analysis in Clinical Research.” Annals of Internal Medicine 107 (2). Am Coll Physicians: 224–33.