[R] comparing matched proportions using glm

Corry Gellatly corry.gellatly at newcastle.ac.uk
Thu Oct 4 21:35:47 CEST 2007

Dear R users,

Is it possible to use a generalized linear model to do a binomial
comparison of one list of proportions with a matched list of proportions
to test for a difference? 

So, for example: 

list 1  		list 2
a1  |  b1        	a2 |  b2

3   |  4          7  |  9
6   |  7          5  |  1
9   |  1          3  |  1               

I want to compare list 1 with list 2 and the samples are matched.
Obviously, I could add the columns and do a binomial test, i.e.
prop.test(c(18,15),c(30,26)), however, I have a large dataset so this
would reduce the power of my analysis. I could compare the ratios, i.e.
a1/(a1+b1) compared to a2/(a2+b2) for the samples in each list, however,
this does not account for the difference in sample sizes between samples
in each list.

I have tried a glm where I bind a2 and b2 as the y variable, i.e.
y<-cbind(a2,b2) and also bind a1 and b1 as the x variable, i.e.
y<-cbind(a1,b1) and run <-glm(y~x,binomial)

I get this type of output:

	glm(formula = y ~ x, family = binomial)

	Deviance Residuals: 
	     Min        1Q    Median        3Q       Max  
	-3.20426  -0.72686  -0.01822   0.68320   4.05035  

	             Estimate Std. Error z value Pr(>|z|)
	(Intercept)  0.178369   0.186421   0.957    0.339
	xa1     	 0.008109   0.017430   0.465    0.642
	xb1		-0.026666   0.018153  -1.469    0.142

	(Dispersion parameter for binomial family taken to be 1)

	    Null deviance: 565.14  on 467  degrees of freedom
	Residual deviance: 559.69  on 465  degrees of freedom
	AIC: 1883.3

	Number of Fisher Scoring iterations: 3

Is this output meaningful? It seems that y is not compared directly with
x, but rather compared with a1 and b1, which is not intended? 

I wonder if this is a suitable approach to the problem? I'll be very
grateful for any advice or suggestions.

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