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

[James Pustejovsky]

Hi Luke,

Sure. I mean that the best-fit line is something like

mu-B = beta0 + beta1 mu-A

But if beta0 = 0, then mu-B = beta1 mu-A, or

beta1 = mu-B / mu-A,

so the two means are proportionally related, which is what the
response ratio metric describes.

On the other hand, if we had a non-zero beta0 but had beta-1 = 1, then
mu-B = beta0 + mu-A, or

beta0 = mu-B - mu-A,

so the two means differ by a constant, which is what the risk
difference metric (or difference-in-proportions) describes.

James

On Tue, Nov 2, 2021 at 10:19 AM Luke Martinez <martinezlukerm using gmail.com> wrote:
>
> Hi James,
>
> Thanks a lot for investing so much effort into my question! Let me ask
> a quick question regarding the second diagnostic in your post.
>
> In your post, you note that *"[Since] there is a strong linear
> relationship between the two [groups'] means, with a best-fit line
> that might go through the origin. . . the response ratio might be an
> appropriate metric."*
>
> Could you please elaborate on how this speaks to the appropriateness
> of LRR over SMD?
>
> Luke
>
> On Tue, Nov 2, 2021 at 8:12 AM James Pustejovsky <jepusto using gmail.com> wrote:
> >
> > HI Luke and listserv,
> >
> > I wrote up some thoughts on the question of using standardized mean
> > differences to analyze outcomes measured as proportions:
> > https://www.jepusto.com/mean-variance-relationships-and-smds/
> > Thoughts, comments, questions, and critiques welcome.
> >
> > James
> >
> > On Mon, Oct 25, 2021 at 9:07 PM James Pustejovsky <jepusto using gmail.com> wrote:
> > >
> > > All I mean is that a skewed distribution or one with large outliers
> > > does not necessarily *imply* that a mean-sd relationship exists. It
> > > could be the result of one, but skewness might be due to something
> > > else (such as selective reporting) instead.
> > >
> > > I would suggest that a well-behaved effect distribution is desirable
> > > and appropriate to the extent that it indicates empirical regularity
> > > of the phenomenon you're interested in. A less heterogeneous
> > > distribution means that effects are more predictable (at least in the
> > > corpus of studies that you're examining).
> > >
> > > On Mon, Oct 25, 2021 at 8:58 PM Luke Martinez <martinezlukerm using gmail.com> wrote:
> > > >
> > > > I thought the existence of outlying effect estimates under SMD and
> > > > lack of it under LRR could attest to the existence of
> > > > heterogeneity-generating artefacts like mean-sd relationships (and/or
> > > > variation in measurement error) across the studies.
> > > >
> > > > If not, then, would you mind commenting on why a more symmetric and
> > > > well-behaved effect distribution is equated with its appropriateness
> > > > for a set of summaries (e.g., means & sds) from studies?
> > > >
> > > > Luke
> > > >
> > > > On Mon, Oct 25, 2021 at 8:47 PM James Pustejovsky <jepusto using gmail.com> wrote:
> > > > >
> > > > > Responses below.
> > > > >
> > > > > On Mon, Oct 25, 2021 at 4:21 PM Luke Martinez <martinezlukerm using gmail.com> wrote:
> > > > > >
> > > > > > Sure, thanks. Along the same lines, if I see that the unconditional
> > > > > > distribution of the SMD estimates is multi-modal or right or left
> > > > > > skewed (perhaps due to extreme outliers), but the unconditional
> > > > > > distribution of the corresponding LRR estimates looks more symmetric
> > > > > > and well-behaved, does that also empirically suggest a mean-sd
> > > > > > relationship in one or more groups?
> > > > >
> > > > > I'm not sure that it implies a mean-sd relationship. But I think it
> > > > > does suggest that LRR might be a more appropriate metric.
> > > > >
> > > > > > PS. Is there a reason for exploring the mean-sd relationship
> > > > > > specifically in the control group?
> > > > >
> > > > > No, you could certainly examine the relationships in the treatment
> > > > > group(s) as well.