When we analyse ordered categorical data, with categories I<II<III,
should we expect an equivalent analysis if the ordering is changed to
I>II>III ? Not with a cloglog link (see example below). However, I
suspect the cloglog analysis on the reordered categories is equivalent
to a loglog analysis on the original ordering. Perhaps for consistency
polr should include a loglog link? Or warn that a different answer is
available, possibly more appropriate?

Cheese tasting example from McCullagh and Nelder:

> library(MASS)
> frqs <- c(0,0,1,7,8,8,19,8,1,
+ 6,9,12,11,7,6,1,0,0,
+ 1,1,6,8,23,7,5,1,0,
+ 0,0,0,1,3,7,14,16,11)
> Y1 <- ordered(rep(1:9,4))
> Y2 <- ordered(rep(9:1,4))
> trt <- factor(rep(c("A","B","C","D"),each=9))
> fit1 <- polr(Y1 ~ trt, method="clog",weight=frqs)
> fit2 <- polr(Y2 ~ trt, method="clog",weight=frqs)
> print(fit1)
Call:
polr(formula = Y1 ~ trt, weights = frqs, method = "clog")

Coefficients:
  trtB    trtC    trtD
-1.809487 -0.879997 0.987389

Intercepts:
    1|2      2|3      3|4      4|5      5|6      6|7
-2.50668784 -2.04055560 -1.42592187 -0.80166219 -0.01722569 0.55135471
    7|8      8|9
1.55688848 2.84302173

Residual Deviance: 725.2341
AIC: 747.2341

> print(fit2)
Call:
polr(formula = Y2 ~ trt, weights = frqs, method = "clog")

Coefficients:
   trtB     trtC     trtD
1.9513455 1.1033902 -0.9588354

Intercepts:
    1|2     2|3     3|4     4|5     5|6     6|7     7|8
-1.3832854 -0.5216350 0.4225419 0.9994969 1.8317538 2.5777980 3.4532818
    8|9
4.4026602

Residual Deviance: 718.0052
AIC: 740.0052


======================================
I.White
University of Edinburgh
Ashworth Laboratories, West Mains Road
Edinburgh EH9 3JT
Tel: 0131 650 5490 Fax: 0131 650 6564
E-mail: iwhite@(protected)

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