psychometric function fitting with lmer?

Hi, Mike

I think the terminology is confusing everybody.  Let me tell you how I
understand what you say, you can try again to tell us what you want.
Since I've just asked a similar kind of question comparing IRT scaling
with lmer, this is fresh in my mind.

Hi folks,

In some areas of psychology, we encounter binomial response data that,
when aggregated to proportions and plotted against a continuous
predictor variable, forms a sigmoid-like function. It is typical to
use OLS to fit a probit function to this data, yielding measures of
bias (mean of the Gaussian) and variability (SD of the Gaussian).

That kind of data is commonly called "grouped" data, as opposed to
individual level data.  In the olden days, that kind of grouped-data
regression you describe was sometimes called a "minimum chi-square
model".  I have no idea what you mean "bias" in this context.

I've never seen a probit function fitted with OLS, but I've seen a
logistic transformation of the proportions on the left hand side
leading to a regression like

ln( prop/(1-prop)) = X b + e

OLS is heteroskedastic, even in the olden days you'd have use WLS to
estimate these coefficients.  There is heteroskedasticity because 1)
there are different numbers of observations in each group and 2) the
variance of the error term is proportional to prop(1-prop).

This model is NOT perfectly equivalent to a probit (or logistic)
regression on individual level data,  the kind where

Pr(y=1 | x, b) = PHI(Xb)

where PHI is the cumulative distribution of a  Normal for probit or
Logistic for logit.

That individual model does not exactly coincide with the grouped
proportion model., partly because the "e" term in the OLS regression
has no direct counterpart in the individual level regression.  The
scaling of the parameters is arbitrary, they will be proportional to
one another.

These days, I'd suggest you fit the "proportion" model with a Beta
regression, for which there is a very excellent R package (betareg).
That is, if you want to analyze the grouped-level proportion data,
that is.

Between the group proportion and individual-level probit, the
estimates of the b's are, at least in theory, estimating the same
thing, except for some scaling effects.  But they never really are the
same.

This

fitting is typically done within each individual and condition of
interest separately, then the resulting parameters are submitted to 2
ANOVAs: one for bias, one for variability.

This one has me stumped.  Can you supply some citations?  "within each
individual" is puzzling to me.  Variability of what? bias in the sense
of a known mismatch between a "true" parameter value and its estimate?
  And the 2 separate ANOVA, well, I think you need to write it down.


 I wonder if this analysis

might be achieved more efficiently using a single mixed effects model,
but I'm having trouble figuring out how to approach coding this. Below
is an example of data similar to that collected in this sort of
research, where individuals fall into two groups (variable "group"),
and are tested under two conditions (variable "cue") across a set of
values from a continuous variable (variable "soa"), with each cue*soa
combination tested repeatedly within each individual. A model like

fit = lmer(
? ?formula = response ~ (1|id) + group*cue*soa
? ?, family = binomial( link='probit' )
? ?, data = a
)

This does not have random effects for group, cue, soa.  it gives fixed
estimates for group, cue, soa, and all interactions among them.  I
don't think  you mean that.

employs the probit link, but of course yields estimates for the slope
and intercept of a linear model on the probit scale, and I'm not sure
how (if it's even possible) to convert the conclusions drawn on this
scale to conclusions about the bias and variability parameters of
interest.

Thoughts?

I think I'm inclined to say that the "bias" and "variability"
parameters you mention are not sensible, because I've never seen a
publication that uses that approach you describe.  My guess is that
you are trying to replicate nonsense, which is, well, a time honored
tradition :)

But, in my last year of work in an interdisciplinary statistics
center, I've learned that all of the fields have their own nicknames
for things and so it is quite likely we have no idea what you are
asking because the nicknames you use are different than the nicknames
we use.  In particular, I bet your claim of estimating probit models
with OLS made some heads spin.

Mike

Paul E. Johnson
Professor, Political Science
1541 Lilac Lane, Room 504
University of Kansas