[R] Can I use "mcnemar.test" for 3*3 tables (or is there a bug in the command?)

[Charles C. Berry]

On Sun, 19 Jul 2009, Peter Dalgaard wrote:

> Charles C. Berry wrote:
>
>>
>>  The test mcnemar.test() constructs is one of symmetry, which is equivalent
>>  to marginal homogenity in hierarchical log-linear models as I recall from
>>  Bishop, Fienberg, and Holland's 1975 opus on count data.
>
> Umm, er... Symmetry in the 3x3 table is a 3DF hypothesis, whereas marginal 
> homogeneity has 2DF, so unless I'm missing a fine point in the requirement of 
> "hierarchical log-linear", I'd say that one implies the other, but not the 
> other way around.


Right, symmetry equals marginal homogenity plus 'quasi-symmetry' - a 
condition on the odds-ratios of a two way table and here that condition 
uses one degree of freedom.

But, representing marginal homogenity in log-linear models gets sticky 
without that quasi-symmetry condition.

---

Taking m_{ij} to be the expected cell frequencies in a two way table, the 
log-linear model for the two way table is

log m_{ij} = \mu + \mu_{1(i)} + \mu_{2(j)} + \mu_{12(ij)}

with side conditions that any of the subscripted \mu terms sums to zero 
over any of its subscripts. In the notation here, \mu is an intercept, 
\mu_1 terms are row effects, \mu_2 terms are column effects, and \mu_{12} 
terms are interactions of the row and columns. The parenthical terms (i), 
(j), or (ij) index the row, column, or cell.

In the case of the 3 x 3 table, there are 1, 2, 2, and 4 degrees of 
freedom respectively for each of the sets of terms in the saturated 
log-linear model.

---

Marginal homogenity says m_{i+} = m_{+i}, all i, taking m_{ij} to be the 
expected cell frequencies and the {i+} notation to indicate summation over 
the missing subscript.

---

Trying to set up a log-linear model for marginal homogeneity would lead 
you to equate the row and column effects:

log m_{ij} = \mu + \mu_{1(i)} + \mu_{1(j)} + \mu_{12(ij)}

but this does not imply marginal homogenity given the side conditions 
unless the \mu_{12(ij)} obey additional constraints which also implies 
symmetry.

>
> E.g., you can easily check that the following two matrices have the same 
> homogeneous margins, but only one is symmetric
>
> 3 2 1
> 2 3 2
> 1 2 3
>
> 3 1 2
> 3 3 1
> 0 3 3
>

If you want to represent this last table as

 	m_{ij} = \exp(\mu + \mu_{1(i)} + \mu_{1(j)} + \mu_{12(ij)})

you cannot get there with the side conditions that are imposed on 
\mu_{12}. You need additional terms.

Chuck


> --
>   O__  ---- Peter Dalgaard             Øster Farimagsgade 5, Entr.B
>  c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
>  (*) \(*) -- University of Copenhagen   Denmark      Ph:  (+45) 35327918
> ~ ~ ~ ~ ~ ~ ~ ~ ~ ~  - (p.dalgaard at biostat.ku.dk)              FAX: (+45) 
> ~ ~ ~ ~ ~ ~ ~ ~ ~ ~  35327907
>
>

Charles C. Berry                            (858) 534-2098
                                             Dept of Family/Preventive Medicine
E mailto:cberry at tajo.ucsd.edu	            UC San Diego
http://famprevmed.ucsd.edu/faculty/cberry/  La Jolla, San Diego 92093-0901