[R-meta] Standard Error of an effect size for use in longitudinal meta-analysis

[Viechtbauer, Wolfgang (SP)]

Dear Simon,

Based on what you describe, dppc and dppt appear to be standardized mean changes using what I would call "change-score standardization" (i.e., (m1 - m2)/SDd, where SDd is the standard deviation of the change scores).

I haven't really tried to figure out the details of what you are doing (and I am not familiar with the distr package), but the large-sample sampling variances of dppc and dppt are:

vc = 1/nc + dppc^2/(2*nc)

and

vt = 1/nt + dppt^2/(2*nt)

(or to be precise, these are the estimates of the sampling variances, since dppc and dppt have been plugged in for the unknown true values).

Hence, the SE of dppt - dppc is simply:

sqrt(vt + vc)

For the "example use" data, this is:

sqrt((1/40 + 0.2^2 / (2*40)) + (1/40 + 0.4^2 / (2*40)))

which yields 0.2291288.

Running your code yields 0.2293698. The former is a large-sample approximation while you seem to be using an 'exact' approach, so they are not expected to coincide, but should be close, which they are.

Best,
Wolfgang

>-----Original Message-----
>From: Simon Harmel [mailto:sim.harmel using gmail.com]
>Sent: Tuesday, 11 August, 2020 14:59
>To: R meta
>Cc: Viechtbauer, Wolfgang (SP)
>Subject: Standard Error of an effect size for use in longitudinal meta-
>analysis
>
>Dear All,
>
>Suppose I know that the likelihood function for an estimate of effect size
>(called `dppc`) measuring the change in a "control" group from pre-test to
>post-test in R language is given by:
>
>    like1 <- function(x) dt(dppc*sqrt(nc), df = nc - 1, ncp = x*sqrt(nc))
>
>where `dppc` is the observed estimate of effect size, and `nc` is the
>"control" group's sample size.
>
>Similarly, the likelihood function for an estimate of effect size (called
>`dppt`) measuring the change in a "treatment" group from pre-test to post-
>test in `R` language is given by:
>
>    like2 <- function(x) dt(dppt*sqrt(nt), df = nt - 1, ncp = x*sqrt(nt))
>
>where `dppt` is the observed estimate of effect size, and `nt` is the
>"treatment" group's sample size.
>
>>>>"Question:" Is there any way to find the "Standard Error (SE)" of the
>`d_dif = dppt - dppc` (in `R`)?
>
>Below, I tried to first get the likelihood function of `d_dif` and then get
>the *Standard Deviation* of that likelihood function. In a sense, I assumed
>I have a Bayesian problem with a "flat prior" and thus "SE" is the standard
>deviation of the likelihood of `d_dif`.
>
>>>> But I am not sure if my work below is at least approximately correct? --
>Thank you, Simon
>
>    library(distr)
>
>    d_dif <- Vectorize(function(dppc, dppt, nc, nt){
>
>     like1 <- function(x) dt(dppc*sqrt(nc), df = nc - 1, ncp = x*sqrt(nc))
>     like2 <- function(x) dt(dppt*sqrt(nt), df = nt - 1, ncp = x*sqrt(nt))
>
>      d1 <- distr::AbscontDistribution(d = like1, low1 = -15, up1 = 15,
>withStand = TRUE)
>      d2 <- distr::AbscontDistribution(d = like2, low1 = -15, up1 = 15,
>withStand = TRUE)
>
>     like.dif <- function(x) distr::d(d2 - d1)(x) ## Gives likelihood of the
>difference i.e., `d_dif`
>
>     Mean <- integrate(function(x) x*like.dif(x), -Inf, Inf)[[1]]
>       SE <- sqrt(integrate(function(x) x^2*like.dif(x), -Inf, Inf)[[1]] -
>Mean^2)
>
>     return(c(SE = SE))
>    })
>
>     # EXAMPLE OF USE:
>     d_dif(dppc = .2, dppt = .4, nc = 40, nt = 40)