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[R-meta] Computing Effect Size for Difference in Differences with Different Populations

5 messages · Mika Manninen, Wolfgang Viechtbauer, Reza Norouzian

Mika Manninen
# 
Dear community,

I am currently working on a meta-analysis that aims to examine the
difference in training effects between two populations. Both
populations underwent the same training, but at pre-test, the groups
have significantly different means and standard deviations (about
1-2sd difference in means).

I am interested in computing the effect size for the difference in
differences between the two groups. Specifically, I would like to know
what is the best way to calculate the effect size given the
significant difference in means and standard deviations at pre-test.

Would the below be roughly accurate (Option 1):

Option 1.

G1 <- escalc(measure="SMCRH", m1i=postm_G1, m2i=prem_G1,
sd1i=postsd_G1,ni=n_G1, sd2i = presd_G1, ri=c(rep(0.7,10)), data=G)
G2 <- escalc(measure="SMCRH", m1i=postm_G2, m2i=prem_G2,
sd1i=postsd_G2, ni=n_G2, sd2i = presd_G2, ri=c(rep(0.7,10)), data=G)
dat <- data.frame(yi = G1$yi - G2$yi, vi = G1$vi + G2$vi)

Option 2.

ES = (G1 post_mean - G2 pre_mean) - (G2 post_mean - G2 pre_mean) / pldpre_sd
pldpre_sd = sqrt((presdG1^2 + presdG2^2) / 2)

### dataset

set.seed(123)

postm_G1 <- rnorm(100, mean = 14, sd = 2.5)
prem_G1 <- rnorm(100, mean = 10, sd = 2)
postsd_G1 <- rnorm(100, mean = 1.4, sd = 0.2)
presd_G1 <- rnorm(100, mean = 1, sd = 0.2)
n_G1 <- rpois(100, lambda = 50)

postm_G2 <- rnorm(100, mean = 7.5, sd = 1.8)
prem_G2 <- rnorm(100, mean = 5, sd = 1.2)
postsd_G2 <- rnorm(100, mean = 0.9, sd = 0.2)
presd_G2 <- rnorm(100, mean = 0.6, sd = 0.2)
n_G2 <- rpois(100, lambda = 50)

G <- data.frame(postm_G1, prem_G1, postsd_G1, n_G1, presd_G1,
postm_G2, prem_G2, postsd_G2, n_G2, presd_G2)

###

Thank you in advance for your time and help.

Best wishes,
Mika
3 days later
Wolfgang Viechtbauer
# 
Hi Mika,

Depends on what you mean by 'best'. But note that escalc(measure="SMCRH, ...) computes (m1i-m2i)/sd1i, so you are using the post-treatment SD to standardize, which is a bit unusual and different from your second approach where you use the average pre-treatment SD to standardize. This aside, the two approaches should lead to rather similar estimates, especially if the pre-treatment SDs (assuming those are used in both approaches) are similar across the two groups.

Best,
Wolfgang
Mika Manninen
# 
Hey Wolfgang,

Thank you very much for the reply.

My mistake with the sd1i argument, I wasn't supposed to be using the post
sds. Thank you for pointing that out.

In my data the pre-treatment SDs are not similar (neither are the
pre-treatment means) between the groups. That is why I am a bit unsure
regarding the most appropriate ES computation method. I have mostly
meta-analysed RCT data in which the two groups are almost identical at
pre-treatment. I could not really find examples in which the difference in
treatment response is being compared between two different populations
receiving the same treatment.

In any case, the following is an okay representation of the actual data and
depending on which ES computation approach I use, the result looks quite
different. So, I was wondering if you could help me in deciding which of
the two is more appropriate or perhaps there is a third option. The data
(or a subsection of it) can also be meta-analysed with raw effect sizes
which does lead to different conclusions as well compared to the SMCRH/SMCC
approach).

Thank you very much in advance,
Mika

### dataset
set.seed(123)
n_G1 <- rpois(50, lambda = 50)
n_G2 <- n_G1

postm_G1 <- rnorm(50, mean = 14, sd = 2.5)
prem_G1 <- rnorm(50, mean = 10, sd = 2)
postsd_G1 <- rnorm(50, mean = 2.4, sd = 0.4)
presd_G1 <- rnorm(50, mean = 1.9, sd = 0.3)

postm_G2 <- rnorm(50, mean = postm_G1 - 7.5, sd = 1.8)
prem_G2 <- rnorm(50, mean = prem_G1 - 5, sd = 1.2)
postsd_G2 <- rnorm(50, mean = 1.2, sd = 0.2)
presd_G2 <- rnorm(50, mean = 1, sd = 0.2)

G <- data.frame(prem_G1,presd_G1, postm_G1, postsd_G1, n_G1,
prem_G2,presd_G2, postm_G2,postsd_G2, n_G2)
G


# Option 1 (could be SMCC as well I suppose)
G1 <- escalc(measure="SMCRH", m1i=postm_G1, m2i=prem_G1, sd1i=presd_G1,
ni=n_G1, sd2i = postsd_G1, ri=c(rep(0.7,50)), data=G)
G2 <- escalc(measure="SMCRH", m1i=postm_G2, m2i=prem_G2, sd1i=presd_G2,
ni=n_G2, sd2i = postsd_G2, ri=c(rep(0.7,50)), data=G)
dat <- data.frame(yi = G1$yi - G2$yi, vi = G1$vi + G2$vi)
dat
# Crude mean of Effect sizes
mean(dat$yi)

# Option 2
pldpre_sd = sqrt((presd_G1^2 + presd_G2^2) / 2)
ES = ((postm_G1 - prem_G1) - (postm_G2 - prem_G2)) / pldpre_sd
ES
# Crude mean of Effect sizes from this formula
mean(ES)


to 16. maalisk. 2023 klo 17.36 Viechtbauer, Wolfgang (NP) (
wolfgang.viechtbauer at maastrichtuniversity.nl) kirjoitti:

  
  


  


      

    

    

      
      

        


  

        

      Reza Norouzian
    

    

      
      #
    

  


  

      
Dear Mika,

You can certainly use both effect size measures and see to what extent the
choice can affect the results.

But if the traditions in your area of research allow, it may also be worth
running your model using the usual standardized mean differences (SMDs) as
the effect size of choice for each group at each time point and then
conduct appropriate contrasts to get the difference in differences (DID)
meta-analytically.

I believe in the **dev. version** of metafor, Wolfgang has recently added a
new function called emmprep() that in conjunction with library emmeans may
be helpful to achieve this relatively easily.

For example, in the data below (consisting of 3 groups and 3 time points),
you can compute the DID across 3 groups for long-term gains (i.e.,
differences in gains between time1 and time3) as follows:

# install dev version of metafor
library(emmeans)

dat$effect <- 1:nrow(dat)

# Model:
a1 <- rma.mv(smd ~group*time, V=vi, random = list(~time|study, ~1|effect),
data=dat,
            dfs = "contain")

# New function in metafor:
sav1 <- emmprep(a1)

# Marginal means:
emms <- emmeans(sav1, ~group*time)

# Desired contrasts among marginal means:
contrast(emms, list("Long-term: Gain(gr2) - Gain(gr1)" =
c(1,-1,0,0,0,0,-1,1,0),  # length must match #of rows in emms
                                "Long-term: Gain(gr2) - Gain(gr3)" =
c(0,-1,1,0,0,0,0,1,-1),
                                 "Long-term: Gain(gr3) - Gain(gr1)" =
c(1,0,-1,0,0,0,-1,0,1)
             ))

The above steps assume that you are interested in comparing the groups at
the final stage of the studies (time3), but see differences in their
initial status (time1) that demand taking those differences into account.
Similar steps can be taken to target differences at time2 taking into
account the initial differences.

The above procedure could be used with other models as well. For instance,
pretending that dat is not multilevel in structure, we could fit the
following model and repeat the above steps.

a2 <- rma(smd ~group*time, vi=vi, data=dat, test = "knha")

Kind regards,
Reza

# Data
dat <- structure(list(study = c(1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3,
                                    3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5,
5, 5, 5, 5, 5, 5, 5, 5,
                                    5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
5, 5, 6, 6, 6, 6, 6, 6,
                                    7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9,
9, 10, 10, 10, 10, 10,10),
                          smd = c(1.681, 4.122, 2.6, 3.457, 1.546, 3.072,
0.22, 3.545,
                                4.454, 0.624, 0.047, 0.306, 1.48, 1.2,
1.483, 1.054, 0.638, 0.998,
                                -0.31, -0.079, 0.955, 0.721, 1.531, 0.974,
-0.104, 0.235, -0.11,
                                -0.005, -0.117, 0.384, -0.194, 0.299,
0.452, 0.66, 0.268, 0.132,
                                0.078, 0.254, -0.339, -0.057, 0.482, 0.278,
0.104, 0.253, 0.107,
                                0.652, 0.386, 0.134, -0.329, -0.366,
-0.011, -0.359, 0, -1.279,
                                -0.178, -0.066, -0.149, -0.547, -0.173,
-0.279, 0.795, 0.172,
                                0.422, 0.461, 1.591, 1.001, -0.352, 0.473,
0.453, 0.3, 0.798,
                                0.868, 0.897, 0.862),
                           vi = c(0.208, 0.481, 0.284, 0.384, 0.2,
                                  0.335, 0.08, 0.206, 0.278, 0.114, 0.106,
0.101, 0.138, 0.125,
                                  0.127, 0.124, 0.111, 0.112, 0.072, 0.073,
0.08, 0.077, 0.092,
                                  0.081, 0.048, 0.049, 0.048, 0.048, 0.047,
0.047, 0.047, 0.047,
                                  0.049, 0.051, 0.049, 0.048, 0.048, 0.049,
0.049, 0.048, 0.048,
                                  0.047, 0.047, 0.047, 0.047, 0.049, 0.047,
0.047, 0.112, 0.112,
                                  0.11, 0.112, 0.11, 0.132, 0.101, 0.099,
0.101, 0.103, 0.101,
                                  0.1, 0.196, 0.168, 0.186, 0.172, 0.239,
0.189, 0.136, 0.138,
                                  0.149, 0.135, 0.157, 0.146, 0.16, 0.146),
                          group = structure(c(1L, 2L, 1L, 2L, 1L, 2L, 1L,
1L, 1L, 1L, 2L, 3L, 1L, 2L, 3L, 1L, 2L,
                                              3L, 1L, 2L, 1L, 2L, 1L, 2L,
1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 1L,
                                              1L, 1L, 1L, 1L, 1L, 1L, 1L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L,
                                              1L, 1L, 1L, 1L, 1L, 1L, 2L,
1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L,
                                              2L, 1L, 1L, 1L, 2L, 1L, 2L,
1L, 2L), levels = c("1", "2", "3"
                                              ), class = "factor"),
                         time = structure(c(1L, 1L, 2L, 2L, 3L, 3L, 1L, 2L,
3L, 1L, 1L, 1L, 2L, 2L, 2L, 3L, 3L, 3L, 1L, 1L, 2L,
                                            2L, 3L, 3L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 2L, 2L, 2L, 2L, 3L,
                                            3L, 3L, 3L, 2L, 2L, 2L, 2L, 3L,
3L, 3L, 3L, 1L, 1L, 2L, 2L, 3L,
                                            3L, 1L, 1L, 2L, 2L, 3L, 3L, 1L,
1L, 2L, 2L, 3L, 3L, 1L, 2L, 1L,
                                            1L, 2L, 2L, 3L, 3L), levels =
c("1", "2", "3"), class = "factor")),
                         class = "data.frame", row.names = c(NA, -74L))


On Thu, Mar 16, 2023 at 5:47?PM Mika Manninen via R-sig-meta-analysis <

        
r-sig-meta-analysis at r-project.org> wrote:

        

            

      
      
      
    

        

      
      
    

        

  
  


  


      

    

    

      
      

        


  

        

      Wolfgang Viechtbauer
    

    

      
      #
    

  


  

      
The difference comes down to how the mean changes are standardized. You will find that

G1 <- escalc(measure="SMCRH", m1i=postm_G1, m2i=prem_G1, sd1i=pldpre_sd, ni=n_G1, sd2i=postsd_G1, ri=c(rep(0.7,50)), data=G)
G2 <- escalc(measure="SMCRH", m1i=postm_G2, m2i=prem_G2, sd1i=pldpre_sd, ni=n_G2, sd2i=postsd_G2, ri=c(rep(0.7,50)), data=G)
dat <- data.frame(yi = G1$yi - G2$yi, vi = G1$vi + G2$vi)

plot(dat$yi, ES, pch=19)
abline(0,1)

gives nearly identical results, the only difference arising due to the bias correction that is not applied to ES. If you have the 'devel' version of metafor installed, you can switch that off with:

G1 <- escalc(measure="SMCRH", m1i=postm_G1, m2i=prem_G1, sd1i=pldpre_sd, ni=n_G1, sd2i=postsd_G1, ri=c(rep(0.7,50)), data=G, correct=FALSE)
G2 <- escalc(measure="SMCRH", m1i=postm_G2, m2i=prem_G2, sd1i=pldpre_sd, ni=n_G2, sd2i=postsd_G2, ri=c(rep(0.7,50)), data=G, correct=FALSE)
dat <- data.frame(yi = G1$yi - G2$yi, vi = G1$vi + G2$vi)

and then the results are identical.

So, in the first approach (using sd1i=presd_G1 and sd1i=presd_G2, respectively, for the two groups), the mean changes are standardized by the group-specific (pre-treatment) SDs. In the 'ES-approach', the mean changes are standardized by the square root of the average (pre-treatment) variances. That is why I wrote that the two approaches should lead to similar estimates if the pre-treatment SDs are similar across the two groups. But

G$presd_G1 / G$presd_G2

shows that this is not really the case, so the two approaches give different estimates.

In the first approach, you are quantifying the effect size as the difference in the amount of change relative to the variability at the pre-treatment assessment within each group individually. In the second approach, you are quantifying the effect size as the difference in the amount of change relative to the average variability at the pre-treatment assessment.

Neither is right or wrong I would say.

One practical issue: With the ES-approach, you would still have the challenge of deriving the appropriate equation for the sampling variance of ES.

Best,
Wolfgang