Re: Measurement error estimation

On Nov 2, 11:34 am, Ramzi <rwnah...@xxxxxxxxx> wrote:
It seems that there should be no way to partition total variance into
population variability and measurement error unless one has more than
one measurement per observation. Yet in SAS when I have a random
coefficient model (vis the RANDOM statement) + serial correlation (via
the REPEATED statement) with only one measurement per point in time,
and then I add in the LOCAL option to REPEATED, SAS does this very
partitioning.

I did the same thing in R with a simple simulation and got the same
results. ... It seems to me that the fitting algorithm
is guessing an answer somehow! Are these two variance components
really uniquely estimable with only one measurement per observation or
is the answer I am getting a particular solution to a non-unique pair
of quantities?

While trying to figure this out, I also asked the authors of a book
I'm reading, "Linear Mixed Models for Longitudinal Data" (Geert
Verbeke and Geert Molenberghs, 2000). The ensuing conversation is
copied below (with their permission):

I wrote:
....
I am considering a model very similar to your model in Section 9.4,
that is, random coefficients + serial correlation + measurement error.
I was attempting to fit this model in SAS using RANDOM, REPEATED, and
the LOCAL option in the REPEATED statement (as you do in Section 9.4).
My question is how can the measurement error component of the within-
subject variability be uniquely estimated unless there are repeated
measurements at the same point in time for an individual (along with
repeated observations over time for individuals)?
....
The algorithm somehow splits up the total variation into population
variability and measurement error components. But how? Are these
really separately estimable?
....
What am I missing?


REPLY #1:
This is a good point, that comes up every once in a while. It is true
that the three components of variability (random effects, serial
association, instantaneous replication or measurement error) are
somewhat harder to distinguish when there is no replication at a time
point, but it is still possible. This is largely due to the fact that
the observations still do not follow the "smooth" behavior that would
result if there only were random effects and serial association.
However, usually there is "raggedness". This raggedness is, as it
should, captures by the measurement error component. Distinguishing is
easier when the measurements are taken during a sufficiently lengthy
period of time, so that the differential behavior of REs and serial
process can be seen, and measurements are spaced sufficiently closely
so that the residual variability can be seen over and above the other
two. The ultimate situation is where some measurements are taken at
the same time, but this is not necessary.

Another way to see the same is when measurements are taken very close
to each other, but not exactly at the same time. Suppose that such two
measurements are quite different: this implies a large measurement
error component. In case they are typically very similar, the
measurement error component would be small. I hope this is of some
help. ...

Kind regards,
Geert (Molenberghs)


FOLLOW-UP QUESTION:
Dear Geert M.,

Thank you for your very helpful reply. I think I see what you are
saying for distinguishing between serial correlation and measurement
error. With just serial correlation, the model says that the
correlation approaches 1 as |ti-tj| -> 0 whereas with both, there is
an upper bound on the correlation. So if the correlations are really
getting close to 1 with decreasing time separation, then the algorithm
would estimate less measurement error, and vice versa.

But what about the case in my simulation? There I have population
variability and measurement error. The "serial" component is just
compound symmetry. With no replication of measurements for an
observation at a time point, how are the two sources of variability
distinguishable? Perhaps compound symmetry is an unrealistic
assumption in real life, but in this case I generated the data from
this model. Maybe your previous answer covers this case, as well. Am I
still missing something?

Sincerely,
Ramzi

REPLY #2:
Compound symmetry (CS) is an interesting special case, since it
actually corresponds to a random subject effect. As such, it is a
"degenerate" serial process, since the correlation between two
measurements does not decrease with increasing time lag.

CS without measurement error would imply that all correlations are
equal to 1. If the correlation is, say 0.73, it means that 73% of the
variability can be attributed to the subject effect, and the remaining
27% to measurement error.

A way to disentangle all three is the so-called variogram. It is
described in our book (Verbeke and Molenberghs 2000), as well as in
Diggle, Heagerty, Liang, and Zeger (2002).

Kind regards,
Geert (Molenberghs)

.

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