R help - Mar 2007 - Mixed effects multinomial regression and meta-analysis

R Experts:

I am conducting a meta-analysis where the effect measures to be pooled
are simple proportions.  For example, consider this  data from
Fleiss/Levin/Paik's Statistical methods for rates and proportions (2003,
p189) on smokers:

Study	   N       Event P(Event)
 1       86       83    0.965
 2       93       90    0.968
 3       136     129    0.949
 4       82       70    0.854
Total    397     372    

A test of heterogeneity for a table like this could simply be Pearson'
chi-square test.  
------

smoke.data <- matrix(c(83,90,129,70,3,3,7,12), ncol=2, byrow=F)
chisq.test(smoke.data, correct=T)
> X-squared = 12.6004, df = 3, p-value = 0.005585
------

Now this test implies that the data is heterogenous and that pooling
might be inappropriate. This type of analysis could be considered a
fixed effects analysis because it assumes that the 4 studies are all
coming from one underlying population.  But what if I wanted to do a
mixed effects (fixed + random) analysis of data like this, possibly
adjusting for an important covariate or two (assuming I had more
studies, of course)...how would I go about doing it? One thought that I
had would be to use a mixed effects multinomial logistic regression
model, such as that reported by Hedeker (Stat Med 2003, 22: 1433),
though I don't know if (or where) it is implemented in R.  I am certain
there are also other ways...

So, my questions to the R experts are:

1) What method would you use to estimate or account for the between
study variance in a dataset like the one above that would also allow you
to adjust for a variable that might explain the heterogeneity?

2) Is it implemented in R?


Brant Inman
Mayo Clinic

R-Experts:

 

I just realized that the example I used in my previous posting today is
incorrect because it is a binary response, not a multilevel response
(small, medium, large) such as my real life problem has.  I apologize
for the confusion.  The example is incorrect, but the multinomial
problem is real.

 

Brant

 


	[[alternative HTML version deleted]]

Here is my suggestion. 

Let P_i denote the true proportion in the ith study and p_i the corresponding
observed proportion based on a sample of size n_i. Then we know that p_i is an
unbiased estimate of P_i and if n_i is sufficiently large, we know that p_i is
approximately normally distributed as long as P_i is not too close to 0 or 1.
Moreover, we can estimate the sampling variance of p_i with p_i(1-p_i)/n_i.
Alternatively, we can use the logit transformation, given by ln[p_i/(1-p_i)],
whose distribution is approximately normal and whose sampling variance is
closely approximated by 1/( n_i p_i (1-p_i) ).

So, let 

y_i = p_i with the corresponding sampling variance v_i = p_i(1-p_i)/n_i

or let

y_i = ln[p_i/(1-p_i)] with the corresponding sampling variance v_i = 1/( n_i p_i
(1-p_i) ).

With y_i and v_i, you can use standard meta-analytic methodology (if the
observed proportions are close to 0 or 1, I would use the logit transformed
proportions). You can fit the random-effects model, if you want to assume that
the variability among the P_i values is entirely random (and normally
distributed) and you are interested in making inferences about the expected
value of P_i. Or you can try to account for the heterogeneity among the P_i
values by examining the influence of moderators.


You might find a function that I have written useful for this purpose. See:

http://www.wvbauer.com/downloads.html

Alternatively, you could fit a logistic regression model with a random intercept
to these data (i.e., a generalized linear mixed-effects model). In other words,
knowing p_i and n_i for each study, you actually have access to the raw data
(consisting of 0's and 1's). This approach is essentially an
"individual patient data meta-analysis". Such a model may or may not
contain any moderators. You can find a discussion of this approach, for example,
in:

Whitehead (2002). Meta-analysis of controlled clinical trials. Wiley. 

Hope this helps,

-- 
Wolfgang Viechtbauer 
?Department of Methodology and Statistics 
?University of Maastricht, The Netherlands 
?http://www.wvbauer.com/ 



-----Original Message-----
From: r-help-bounces at stat.math.ethz.ch [mailto:r-help-bounces at
stat.math.ethz.ch] On Behalf Of Inman, Brant A. M.D.
Sent: Tuesday, March 06, 2007 00:56
To: r-help at stat.math.ethz.ch
Cc: Weigand, Stephen D.
Subject: [R] Mixed effects multinomial regression and meta-analysis



R Experts:

I am conducting a meta-analysis where the effect measures to be pooled are
simple proportions.  For example, consider this  data from
Fleiss/Levin/Paik's Statistical methods for rates and proportions (2003,
p189) on smokers:

Study	   N       Event P(Event)
 1       86       83    0.965
 2       93       90    0.968
 3       136     129    0.949
 4       82       70    0.854
Total    397     372    

A test of heterogeneity for a table like this could simply be Pearson'
chi-square test.
------

smoke.data <- matrix(c(83,90,129,70,3,3,7,12), ncol=2, byrow=F)
chisq.test(smoke.data, correct=T)
> X-squared = 12.6004, df = 3, p-value = 0.005585
------

Now this test implies that the data is heterogenous and that pooling might be
inappropriate. This type of analysis could be considered a fixed effects
analysis because it assumes that the 4 studies are all coming from one
underlying population.  But what if I wanted to do a mixed effects (fixed +
random) analysis of data like this, possibly adjusting for an important
covariate or two (assuming I had more studies, of course)...how would I go about
doing it? One thought that I had would be to use a mixed effects multinomial
logistic regression model, such as that reported by Hedeker (Stat Med 2003, 22:
1433), though I don't know if (or where) it is implemented in R.  I am
certain there are also other ways...

So, my questions to the R experts are:

1) What method would you use to estimate or account for the between study
variance in a dataset like the one above that would also allow you to adjust for
a variable that might explain the heterogeneity?

2) Is it implemented in R?


Brant Inman
Mayo Clinic

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> I just realized that the example I used in my previous posting today is
> incorrect because it is a binary response, not a multilevel response
> (small, medium, large) such as my real life problem has.  I apologize
> for the confusion.  The example is incorrect, but the multinomial
> problem is real.
Your data looks like it might be better considered as ordinal.
Whitehead and Whitehead discuss one proprtional odds random effects
approach (for a single binary covariate) in Statist Med 1991;
10:1665-1677, which is easy to implement.  The BUGS manual has an
example of random effects metaanalysis that you could expand.  You could
even partition out the studies using the party package (I believe it
does an ordinal logistic).

David Duffy.
-- 
| David Duffy (MBBS PhD)                                         ,-_|\
| email: davidD at qimr.edu.au  ph: INT+61+7+3362-0217 fax: -0101  /     *
| Epidemiology Unit, Queensland Institute of Medical Research   \_,-._/
| 300 Herston Rd, Brisbane, Queensland 4029, Australia  GPG 4D0B994A v