[R-meta] Converting between SMDs and ORs for meta-analysis of prognostic fators

[James Pustejovsky]

Hi Scott,
Responses below.
James

On Thu, May 9, 2024 at 8:27 PM Scott Tagliaferri <
scott.tagliaferri using gmail.com> wrote:

>
> 1) Just to confirm, based on your example, are you suggesting that the
> SMD^2 is equal to the log odds? Would you have any other
> references/examples for this?
>

Not quite: the log odds is equal to the SMD divided by the within-group
standard deviation of the outcome (or equivalently, the raw mean difference
divided by the within-group sample variance). Unfortunately I am coming up
dry on supporting citations for this relationship--I think I must have run
across it as an exercise from a course in distribution theory back in grad
school, and I am having trouble locating a good reference now. Perhaps
others on the listserv know of something.


> 2) Do you know of any methods to calculate the variance of this log odds?
>

My first thought would be to just use the variance of the SMD, divided by
the within-group sample variance:
Eq. 1: Var(LO) = Var(SMD / SD) ~= Var(SMD) / SD^2
Perhaps a little bit better would be to use the delta method approximation,
which in this case gives:
Eq. 2: Var(LO) = Var(D / S^2) ~= [1 / nA + 1 / nB + 2 SMD^2 / (nA + nB -
2)] / SD^2
The final term of the delta method approximation is a bit different (by a
factor of 4) than what you get by using Eq 1.

A potential critique of the above is that these sampling variances treat
the number of observations in each group as fixed and treat the scores on
the continuous variable as random. In contrast, the usual variance of the
log odds (i.e., what comes out of a logistic regression) treats the
predictor as fixed and group membership as random. I'm not sure how much of
a difference this makes, but it seems like you could check by simulating
some data, computing the SMD and computing the logistic regression
directly, and comparing the standard errors.


> 3) If not, would simulating the data (as per the code example but rather
> across multiple simulations/bootstraps) using the mean, sd, N, min and max
> of the data and running simple logistic regression, be a potential option
> to acquire an approximate odds ratio and confidence interval?
>
> I would venture that the simulation approach you describe would be very
similar to what you get with Eq. 2. It shares the same problem of
conditioning on sample size.

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