[R-meta] mean-variance relationships introduces additional heterogeneity, how?

[James Pustejovsky]

Hi Luke,

My original response to your question was not specifically about
mean-variance relationships within a study across multiple time points, but
rather was more generally about mean-variance relationships in summary
statistics. The simplest case here is where there's just one effect size
per study and where the studies vary in the mean level of the outcome in a
given group. I'll give you an example of the sort of data-generating
process that I had in mind. It's important to bear in mind, though, that
this is all entirely speculative---I'm *not* asserting that this model is
appropriate for your data specifically, only that the example is one
plausible situation where mean-variance relationships arise.

Suppose that we have k studies, each involving a two-group comparison, with
groups of equal size. In study i, the outcomes in group A follow a poisson
distribution with mean m_Ai, so that the variance of the outcomes in group
A is also m_Ai, for i = 1,...,k. The outcomes in group B follow a poisson
distribution with mean m_Bi, so the variance is also m_Bi. Now, suppose
that there is a fixed, proportional relationship between pi_Bi and pi_Ai,
so that pi_Bi = lambda pi_Ai for some lambda > 0. In other words, the
treatment contrast is *constant* on the scale of the response ratio.
However, the means in group A vary from study to study, according to some
distribution, say a gamma distribution with shape parameter alpha and rate
parameter beta.  What does this model imply about the distribution of
standardized mean differences across this set of studies?

The SMD parameter for study i (call it delta_i) is the ratio of the mean
difference to the square root of the pooled variance. So:

delta_i = (m_Bi - m_Ai) / sqrt[(m_Bi + m_Ai) / 2]
= (lambda - 1) m_Ai / sqrt[(1 + lambda) m_Ai / 2]
= sqrt(m_Ai) * (lambda - 1) * sqrt[2 * / (lambda + 1)]

The second and third term in the above expression are constants that only
depend on the size of the response ratio. The first term is random because
we have assumed that the group A means vary from study to study. It will
therefore create heterogeneity in the SMD parameters---the greater the
variance of the m_Ai's, the greater the heterogeneity in delta_i.

I've used poisson distributions for outcomes and the gamma distribution for
the m_Ai's only for sake of simplicity. Something very similar would hold
if we considered outcomes that were binomially distributed because the
binomial distribution has variance that is strongly related to its mean. If
the mean proportions vary from study to study, this implies that the
variances will also vary from study-to-study in a non-linear way, which
will induce heterogeneity in the SMD parameters.

James

On Sat, Oct 23, 2021 at 11:53 PM Luke Martinez <martinezlukerm using gmail.com>
wrote:

> Hello All,
>
> I wanted to follow up on an answer
> (
> https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2021-October/003354.html
> )
> on the list that, in a nutshell, says:
>
> Existence of a relationship between the Mean and SD for a given study
> group (e.g., Treatment), over a couple of time points, can potentially
> introduce additional heterogeneity in the SMD effect sizes across the
> studies.
>
> I was wondering how exactly this heterogeneity comes about?
>
> A part of me says that such a Mean-SD relationship for a given study
> group over time is indicative of what Hedges' (1981;
> https://doi.org/10.3102/10769986006002107) refers to as
> "subject-treatment interaction"?
>
> Another part of me says, no, "subject-treatment interaction" is
> controllable by adding a random effect for individuals, and thus it is
> different.
>
> I highly appreciate your insights regarding HOW Mean-SD relationship
> for study groups over time can potentially introduce additional
> heterogeneity in SMDs across studies?
>
> Luke
>
> PS:
> Here are the relation I see between mean and sd of a study group
> across 3 time points:
> group1_Ms = c(.39, .18, .13)
> group1_SDs = c(.25, .16, .13)
> plot(group1_Ms, group1_SDs, type="l")
>
> _______________________________________________
> R-sig-meta-analysis mailing list
> R-sig-meta-analysis using r-project.org
> https://stat.ethz.ch/mailman/listinfo/r-sig-meta-analysis
>

	[[alternative HTML version deleted]]