I'm Geert Molenberghs and I'm very happy to continue
on the topic of surrogate markers in the second lecture.
In the first one, we dealt with everything from the definition by apprentice up until
the construction of the meta-analytic framework but that is still for Gaussian data.
So that means a Gaussian surrogate,
a Gaussian true endpoint,
and to get it very clear usually a binary treatment.
But we will now have to move in situations where
the outcomes become of a different type or are measured longitudinally,
and maybe we should also consider the question,
are there other frameworks?
If yes and the answer is yes,
how does our framework now relate to those other frameworks?
Good point and the first thing I will do is I will go from
a univariate surrogate true endpoint to multivariate versions. Like what?
Well, like in a psychiatric trial
where you will have the plans and the CGI that we just discussed,
measured repeatedly over time.
What we see is the true endpoint and the surrogate endpoint model
pretty much as before except they have accrued a third index.
We now have i standing for trial,
j is the patient within a trial,
and t is the time at which the measurement is taken.
I must emphasize that the models we propose here are examples.
Of course, in a given study,
people would have to think about what's
the most appropriate mixed model for example or generalized linear mixed model,
or whatever model they're willing to consider.
But just by example,
of course the situation becomes more complex even
with this relatively simple longitudinal model. Why is that?
Well, we have a correlation between the surrogate
and the true endpoint at each time point.
So that may evolve over time.
The true end points of the same patients but at different times will be correlated,
and so will be the surrogate endpoints.
So we have to come up with
a fairly complex correlation structure or
equivalently variance-covariance structure and has Sigma Ri.
The R matrix captures the repeated measures correlation,
and then the first part of the Sigma matrix
captures the correlation between the two endpoints.
At the second stage,
you would take the drought specific effects like treatment effects,
Alpha_i, Beta_i, and decompose them just as we did before.
But also there, depending on how you do it,
you may have a single treatment effect or you may have a treatment effect that
is made up of a series of
treatment effects really for everyone of the time points separately,
or the treatment effect may be increasing linearly or quadratically overtime etc.,
and then the question is, yeah,
you have a bunch of treatment effects on the surrogate,
a bunch of treatment effects on the true endpoint,
how do you decide on surrogacy?
How do you evaluate surrogacy?
Can we create measures?
What people have done over the years is they've recreated
several proposals for generalizing
the R-square that we considered up to now to
measure that can take repeated measures longitudinal data into account.
Briefly there is things called the variance reduction factor,
canonical correlation analysis, etc.,
and what they do is they borrow strength, and information,
and concepts predominantly from multivariate statistics,
normally based multivariate statistics,
where you have the so-called root statistics.
Wilk's Lambda, Pillai's trace, etc.
So the statistics we construct here look very similar to those measures.