Transform odds ration (OR) in the context of multivariate meta-analysis in R (metafor package)

Question

I would like to conduct a multivariate meta-analysis (multiple treatment arm meta analysis) comparing the effect of different drugs. My outcome measure is discrete and describes the number of occurrences of a specific side effect during the period of observation. The studies that i would like to include report the odds ratio (OR) and their 95%-CI of this side effect for every drug relative to a “baseline drug”. The problem is that studies defined different drugs to be the baseline.

• Is it valid to just transform my OR to bring them to a common baseline drug?
• Is it valid to transform the 95%-CI of the OR in the same way?
• There is typically no 95%-CI reported for the baseline drug. Thus
  when I transform this baseline drug, I have no 95%-CI. Is there a way to derive a 95%-CI in this case?

Here is some dummy data:

structure(list(study = structure(c(2L, 2L, 2L, 2L, 2L, 3L, 3L, 
3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L), .Label = c("Liu", "Mark", "Smith"
), class = "factor"), drug = structure(c(1L, 2L, 3L, 4L, 5L, 
1L, 2L, 3L, 4L, 5L, 1L, 2L, 3L, 4L, 5L), .Label = c("A", "B", 
"C", "D", "E"), class = "factor"), OR = c(1, 1.5, 1.7, 1.8, 2.5, 
2.8, 1.1, 1, 2.3, 1.2, 1.8, 1.2, 2.5, 1, 1.8), ci_lb = structure(c(13L, 
5L, 4L, 9L, 11L, 12L, 2L, 13L, 10L, 1L, 6L, 3L, 8L, 13L, 7L), .Label = c("0.5556", 
"0.6225", "0.7619", "1.2628", "1.365", "1.4212", "1.5758", "1.6208", 
"1.7048", "1.7435", "1.9497", "2.3531", "NA"), class = "factor"), 
    ci_ub = structure(c(13L, 2L, 7L, 5L, 10L, 11L, 1L, 13L, 9L, 
    4L, 8L, 3L, 12L, 13L, 6L), .Label = c("1.5775", "1.635", 
    "1.6381", "1.8444", "1.8952", "2.0242", "2.1372", "2.1788", 
    "2.8565", "3.0503", "3.2469", "3.3792", "NA"), class = "factor")), .Names = c("study", 
"drug", "OR", "ci_lb", "ci_ub"), row.names = c(NA, -15L), class = "data.frame")

> my_data
   study drug  OR  ci_lb  ci_ub
1   Mark    A 1.0     NA     NA
2   Mark    B 1.5  1.365  1.635
3   Mark    C 1.7 1.2628 2.1372
4   Mark    D 1.8 1.7048 1.8952
5   Mark    E 2.5 1.9497 3.0503
6  Smith    A 2.8 2.3531 3.2469
7  Smith    B 1.1 0.6225 1.5775
8  Smith    C 1.0     NA     NA
9  Smith    D 2.3 1.7435 2.8565
10 Smith    E 1.2 0.5556 1.8444
11   Liu    A 1.8 1.4212 2.1788
12   Liu    B 1.2 0.7619 1.6381
13   Liu    C 2.5 1.6208 3.3792
14   Liu    D 1.0     NA     NA
15   Liu    E 1.8 1.5758 2.0242


my_data_1 <- my_data[my_data$study=="Mark",]
    my_data_2 <- my_data[my_data$study=="Smith",]
my_data_3 <- my_data[my_data$study=="Liu",]

my_data_2$OR <- my_data_2$OR * 1/my_data_2$OR[my_data_2$drug=="A"]
my_data_3$OR <- my_data_3$OR * 1/my_data_3$OR[my_data_3$drug=="A"]
my_data_2$ci_lb <- my_data_2$ci_lb * 1/my_data_2$OR[my_data_2$drug=="A"]
my_data_3$ci_lb <- my_data_3$ci_lb * 1/my_data_3$OR[my_data_3$drug=="A"]
my_data_2$ci_ub <- my_data_2$ci_ub * 1/my_data_2$OR[my_data_2$drug=="A"]
my_data_3$ci_ub <- my_data_3$ci_ub * 1/my_data_3$OR[my_data_3$drug=="A"]

my_data_new <- rbind(my_data_1, my_data_2, my_data_3)
my_data_new
my_data_new$OR <- round(my_data_new$OR, 2)
my_data_new$ci_lb <- round(my_data_new$ci_lb, 2)
my_data_new$ci_ub <- round(my_data_new$ci_ub, 2)

> my_data_new
   study drug   OR ci_lb ci_ub
1   Mark    A 1.00    NA    NA
2   Mark    B 1.50  1.36  1.64
3   Mark    C 1.70  1.26  2.14
4   Mark    D 1.80  1.70  1.90
5   Mark    E 2.50  1.95  3.05
6  Smith    A 1.00  2.35  3.25
7  Smith    B 0.39  0.62  1.58
8  Smith    C 0.36    NA    NA
9  Smith    D 0.82  1.74  2.86
10 Smith    E 0.43  0.56  1.84
11   Liu    A 1.00  1.42  2.18
12   Liu    B 0.67  0.76  1.64
13   Liu    C 1.39  1.62  3.38
14   Liu    D 0.56    NA    NA
15   Liu    E 1.00  1.58  2.02

Answer 1

Is it valid to just transform my OR to bring them to a common baseline drug?

Yes, a little algebra will show you that this is ok. I.e. you simply divide by the OR of the drug you want to use as baseline.

There is typically no 95%-CI reported for the baseline drug. Thus when I transform this baseline drug, I have no 95%-CI. Is there a way to derive a 95%-CI in this case?

The baseline drug has $OR = 1$ and $CI = (1,1)$ by definition. As you benchmark all drugs against, say drug $A$, then the $OR$, lower, and upper confidence limits is $1$ for drug $A$. Hence you can substitute 1 where you have NAin your dataset.

Is it valid to transform the 95%-CI of the OR in the same way?

If the studies give you no other information, then I guess, that this is the best you can do. You simple use the $CI = (1,1)$ for the baseline drugs.

I think your example is a bit weird as the CIs are symmetric around the OR on the scale given. If you have a decent number of observations CI limits should be approximated by $\exp(\log(OR) \pm 1.96 SE)$, i.e. the CIs should be symmetric around the $\log(OR)$ on a logarithmic scale. This does not seem to be the case here. I guess you need to know how each study have computed their $OR$s and $CI$s, if you want something more sophisticated.

Here is some simple code that does the above for you. First I clean your data a little bit:

# Convert factors to numeric
my_data$ci_lb <- as.numeric(as.character(my_data$ci_lb))
my_data$ci_ub <- as.numeric(as.character(my_data$ci_ub))
my_data$OR    <- as.numeric(as.character(my_data$OR))
# Substitute NAs with 1
my_data[is.na(my_data)] <- 1

Next, we choose our baseline drug, say $A$, and divide the $OR$s and $CI$s with this:

ref <- my_data[my_data$drug == "A", 3:5] 
tmp <- ref[rep(1:nrow(ref), table(my_data$study)), ]
my_data[,6:8] <- round(my_data[, 3:5]/tmp, 4)
print(my_data)
#   study drug  OR  ci_lb  ci_ub   OR.1 ci_lb.1 ci_ub.1
#1   Mark    A 1.0 1.0000 1.0000 1.0000  1.0000  1.0000
#2   Mark    B 1.5 1.3650 1.6350 1.5000  1.3650  1.6350
#3   Mark    C 1.7 1.2628 2.1372 1.7000  1.2628  2.1372
#4   Mark    D 1.8 1.7048 1.8952 1.8000  1.7048  1.8952
#5   Mark    E 2.5 1.9497 3.0503 2.5000  1.9497  3.0503
#6  Smith    A 2.8 2.3531 3.2469 1.0000  1.0000  1.0000
#7  Smith    B 1.1 0.6225 1.5775 0.3929  0.2645  0.4858
#8  Smith    C 1.0 1.0000 1.0000 0.3571  0.4250  0.3080
#9  Smith    D 2.3 1.7435 2.8565 0.8214  0.7409  0.8798
#10 Smith    E 1.2 0.5556 1.8444 0.4286  0.2361  0.5680
#11   Liu    A 1.8 1.4212 2.1788 1.0000  1.0000  1.0000
#12   Liu    B 1.2 0.7619 1.6381 0.6667  0.5361  0.7518
#13   Liu    C 2.5 1.6208 3.3792 1.3889  1.1404  1.5509
#14   Liu    D 1.0 1.0000 1.0000 0.5556  0.7036  0.4590
#15   Liu    E 1.8 1.5758 2.0242 1.0000  1.1088  0.9290

The last three columns then show you the new $OR$s and $CI$s w.r.t. the new baseline drug.