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Modelling strength of correlation
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Jay Weedon
Mar 16, 2013, 4:24:20 PM3/16/13
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Folks,
(Gosh it's a very long time since I last posted here!)
A medical colleague has longitudinal data concerning 2 biological
parameters (let's call them systolic [X] & diastolic [Y] blood
pressure): about 20 X,Y pairs for each of about 50 people.
He's made some X-Y scatterplots and has come to tentatively believe
that one group of people (let's call them men [M]) exhibits a stronger
correlation between X & Y than women. At this stage we're explicitly
disregarding any effect of time on X or Y - the time points are
relatively close together.
In terms of a formal analysis I've been tinkering with the idea of
fitting a random intercepts & slopes model. Let's assume that I can
find some transformation of X and/or Y that approximately linearizes
their relationship. Then if index i represents person and index j
time, and if M is an indicator variable, a model might be:
Yij = (beta0 + b0i + beta1.M) + (beta2 + b1i + beta3.M).Xij + eij,
such that eij are normally and independently distributed, uncorrelated
with b0i or b1i, which are binormally distributed with mean zero and
unstructured covariance. There are thus 4 fixed effect parameters and
4 covariance parameters.
The most obvious parameter of interest here is beta3; if this is
non-zero, then mean X-Y slope differs significantly between men &
women, right?
I have a couple of reservations about this idea:
1. Slope and correlation are not the same thing. In OLS regression
these quantitities are related by a function of the variances of X &
Y, which might be fairly constant across subjects and times. I can
investigate these questions empirically, but if the assumptions are
reasonable, then is it ok in the context of a mixed model to conclude
that (after accounting for sign), slope & correlation are
monotonically related across subjects?
2. From the point of view of assessing degree of correlation, the
decision of which variable is to be dependent is arbitrary - so do I
need to fit the model both ways? I should perhaps point out that in
the above example X & Y share the same metric; that is actually not
the case for the data under consideration.
Then further questions are:
A Is there any direct way to estimate subject-specific X-Y
correlations or covariances from the model above? I suppose I could
correlate Xs with the BLUPs of Y for each subject - is that more
helpful than Pearson correlations using raw data?
B. Is there a better way to approach the problem?
Insights much appreciated,
Jay Weedon
(Gosh it's a very long time since I last posted here!)
A medical colleague has longitudinal data concerning 2 biological
parameters (let's call them systolic [X] & diastolic [Y] blood
pressure): about 20 X,Y pairs for each of about 50 people.
He's made some X-Y scatterplots and has come to tentatively believe
that one group of people (let's call them men [M]) exhibits a stronger
correlation between X & Y than women. At this stage we're explicitly
disregarding any effect of time on X or Y - the time points are
relatively close together.
In terms of a formal analysis I've been tinkering with the idea of
fitting a random intercepts & slopes model. Let's assume that I can
find some transformation of X and/or Y that approximately linearizes
their relationship. Then if index i represents person and index j
time, and if M is an indicator variable, a model might be:
Yij = (beta0 + b0i + beta1.M) + (beta2 + b1i + beta3.M).Xij + eij,
such that eij are normally and independently distributed, uncorrelated
with b0i or b1i, which are binormally distributed with mean zero and
unstructured covariance. There are thus 4 fixed effect parameters and
4 covariance parameters.
The most obvious parameter of interest here is beta3; if this is
non-zero, then mean X-Y slope differs significantly between men &
women, right?
I have a couple of reservations about this idea:
1. Slope and correlation are not the same thing. In OLS regression
these quantitities are related by a function of the variances of X &
Y, which might be fairly constant across subjects and times. I can
investigate these questions empirically, but if the assumptions are
reasonable, then is it ok in the context of a mixed model to conclude
that (after accounting for sign), slope & correlation are
monotonically related across subjects?
2. From the point of view of assessing degree of correlation, the
decision of which variable is to be dependent is arbitrary - so do I
need to fit the model both ways? I should perhaps point out that in
the above example X & Y share the same metric; that is actually not
the case for the data under consideration.
Then further questions are:
A Is there any direct way to estimate subject-specific X-Y
correlations or covariances from the model above? I suppose I could
correlate Xs with the BLUPs of Y for each subject - is that more
helpful than Pearson correlations using raw data?
B. Is there a better way to approach the problem?
Insights much appreciated,
Jay Weedon
Ray Koopman
Mar 17, 2013, 6:18:40 PM3/17/13
to
so individual data sets -- means, SDs, correlations, regressions done
both ways, with scatterplots done on the same scale for everyone;
plus scatterplots of the pooled data for each group, and analyses
comparing the groups on the various statistics.
I agree that differences in r are probably epiphenomenal, that the
real differences are probably in the structural parameters. Do you
intend all the eij in your proposed model to come from the same
distribution? What about putting individual &/or group effects on the
error variance?
My biggest problem with all that is that there is no IV-DV relation
between X & Y. Unless your colleague can make a good argument for such
a relation, I would be inclined to think about a latent variable model
with both X & Y as DVs. The groups could differ on (some subset of)
the mean & SD of the latent variable, the slopes and intercepts of the
regressions of X & Y on the latent variable, and the SDs of the
regression errors. (Yes, identifiability would need some thinking
about.)
Rich Ulrich
Mar 18, 2013, 2:40:34 AM3/18/13
to
I like Ray's comments.
I will say much the same things, with my own emphasis.
On Sat, 16 Mar 2013 16:24:20 -0400, Jay Weedon <jwe...@earthlink.net>
wrote:
The reason that the test of raw regression is usually better than
the test of correlation is that the former is not distorted by
differences in variance. - If you want to talk about "correlation",
then you probably want to assure, first, that variances are
not unequal. In my own experience, the hypothesis was usually
better described in terms of regression coefficients. And, yes,
I have had the experience of telling a PI that, although two
correlations "differ significantly", the regression lines are not
different; one group had very little variance on the predictor (age),
and that explained the near-zero r.
With your setup, of multiple ratings on multiple persons, I doubt
that you really will have homogeneity... assuming that the two blood
pressure measures are a suitable analogy in this respect, for
whatever you are actually measuring.
>
>2. From the point of view of assessing degree of correlation, the
>decision of which variable is to be dependent is arbitrary - so do I
>need to fit the model both ways? I should perhaps point out that in
>the above example X & Y share the same metric; that is actually not
>the case for the data under consideration.
>
>Then further questions are:
>
>A Is there any direct way to estimate subject-specific X-Y
>correlations or covariances from the model above? I suppose I could
>correlate Xs with the BLUPs of Y for each subject - is that more
>helpful than Pearson correlations using raw data?
"Is there any direct way..."? I'm not sure what question you
are asking, since if you have many points on each of multiple people,
you might start this project by looking at the variances, covariances,
correlations and regressions for each subject. That might properly
be done *before* a larger model is attempted.
The questions about within-subject correlations might be different
from the questions about between-subject correlations. I think
you need to be clear about which you want to model.
>
>B. Is there a better way to approach the problem?
--
Rich Ulrich
I will say much the same things, with my own emphasis.
On Sat, 16 Mar 2013 16:24:20 -0400, Jay Weedon <jwe...@earthlink.net>
wrote:
the test of correlation is that the former is not distorted by
differences in variance. - If you want to talk about "correlation",
then you probably want to assure, first, that variances are
not unequal. In my own experience, the hypothesis was usually
better described in terms of regression coefficients. And, yes,
I have had the experience of telling a PI that, although two
correlations "differ significantly", the regression lines are not
different; one group had very little variance on the predictor (age),
and that explained the near-zero r.
With your setup, of multiple ratings on multiple persons, I doubt
that you really will have homogeneity... assuming that the two blood
pressure measures are a suitable analogy in this respect, for
whatever you are actually measuring.
>
>2. From the point of view of assessing degree of correlation, the
>decision of which variable is to be dependent is arbitrary - so do I
>need to fit the model both ways? I should perhaps point out that in
>the above example X & Y share the same metric; that is actually not
>the case for the data under consideration.
>
>Then further questions are:
>
>A Is there any direct way to estimate subject-specific X-Y
>correlations or covariances from the model above? I suppose I could
>correlate Xs with the BLUPs of Y for each subject - is that more
>helpful than Pearson correlations using raw data?
are asking, since if you have many points on each of multiple people,
you might start this project by looking at the variances, covariances,
correlations and regressions for each subject. That might properly
be done *before* a larger model is attempted.
The questions about within-subject correlations might be different
from the questions about between-subject correlations. I think
you need to be clear about which you want to model.
>
>B. Is there a better way to approach the problem?
Rich Ulrich
Ryan
Apr 26, 2013, 8:40:09 PM4/26/13
to
Jay,
I, too, have not been monitoring google groups for quite a while. In fact, I realize that this response is EXTREMELY delayed, and chances are you will not even see my reply. Even if you do, you probably have moved on from this problem. Still, knowing that you use SAS (since we have corresponded in the SAS google group in the distant past) and on the off-chance that you are still trying to figure out a way to calculate the correlation, I believe you will find utility by considering a type of repeated measures bivariate linear mixed model such as the one offered in this SAS-L post:
http://listserv.uga.edu/cgi-bin/wa?A2=ind0309B&L=sas-l&P=R10606
Best,
Ryan
I, too, have not been monitoring google groups for quite a while. In fact, I realize that this response is EXTREMELY delayed, and chances are you will not even see my reply. Even if you do, you probably have moved on from this problem. Still, knowing that you use SAS (since we have corresponded in the SAS google group in the distant past) and on the off-chance that you are still trying to figure out a way to calculate the correlation, I believe you will find utility by considering a type of repeated measures bivariate linear mixed model such as the one offered in this SAS-L post:
http://listserv.uga.edu/cgi-bin/wa?A2=ind0309B&L=sas-l&P=R10606
Best,
Ryan
Jay Weedon
May 3, 2013, 4:20:11 PM5/3/13
to
Many thanks for the contribution Ryan, I'll check it out.
Jay
Jay
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