[R-sig-ME] Calculations and interpretation of ordered logit model with clmm() and related functions

[Christian Brauner]

Hello all,

I fitted a proportional ordered logit model and I have some questions regarding
my interpretation of the R output and calculating certain values. I will
illustrate this with the wine dataset which can be found in the ordinal
package. The models won’t necessarily make sense but this will not have any
bearing on the case. I tried to formulate most questions as simple yes-no
questions to make it easier. I gathered that the r-sig-mixed-models help page
might be the right place to ask.

(A) Simple model:

library(ordinal)
mod1 <- clmm(rating
             ~ bottle
             + (1 | judge),
             data = wine,
             link = "logit")

summary(mod1)

> summary(mod1)
Cumulative Link Mixed Model fitted with the Laplace approximation

formula: rating ~ bottle + (1 | judge)
data:    wine

 link  threshold nobs logLik AIC    niter     max.grad cond.H 
 logit flexible  72   -80.26 184.52 607(1824) 2.50e-05 1.0e+02

Random effects:
 Groups Name        Variance Std.Dev.
 judge  (Intercept) 1.321    1.149   
Number of groups:  judge 9 

Coefficients:
        Estimate Std. Error z value Pr(>|z|)    
bottle2   1.1992     0.9652   1.242 0.214079    
bottle3   2.6115     1.0440   2.501 0.012367 *  
bottle4   2.2340     1.0174   2.196 0.028114 *  
bottle5   3.3366     1.0621   3.142 0.001680 ** 
bottle6   4.0071     1.0959   3.657 0.000256 ***
bottle7   5.9393     1.2201   4.868 1.13e-06 ***
bottle8   5.4545     1.1551   4.722 2.34e-06 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Threshold coefficients:
    Estimate Std. Error z value
1|2  -1.0696     0.8472  -1.263
2|3   2.1406     0.8945   2.393
3|4   4.9217     1.0580   4.652
4|5   6.8276     1.1852   5.761


 exp(mod1$beta)

> exp(mod$beta)
   bottle2    bottle3    bottle4    bottle5    bottle6    bottle7 
  3.317391  13.620129   9.337043  28.123588  54.987120 379.658280 
   bottle8 
233.819536

Questions for the simple model:

(1) Am I correct in assuming that the odds ratios which I get from
"exp(mod$beta)" for all the bottles are calculated with respect to
"bottle1"?

(2) If (1) is true is the reference level "bottle1" just the sum of all
the intercepts for every single bottle 1 to 8?

(3) If (1) and (2) are true are the following statements correct?:

(3.1) The odds ratio to be rated higher increases by 3.31739 if I choose
"bottle2" in comparison to "bottle1".

(3.2) The odds ratio to be rated higher increases by 13.620219 if I choose
"bottle7" in comparison to "bottle1".


(B) Additive model:

mod2 <- clmm(rating
             ~ contact
             + temp
             + (1 | judge),
             data = wine,
             link = "logit")

summary(mod2)

> summary(mod2)
Cumulative Link Mixed Model fitted with the Laplace approximation

formula: rating ~ contact + temp + (1 | judge)
data:    wine

 link  threshold nobs logLik AIC    niter    max.grad cond.H 
 logit flexible  72   -81.57 177.13 331(996) 1.04e-05 2.8e+01

 Random effects:
Groups Name        Variance Std.Dev.
  judge  (Intercept) 1.279    1.131   
Number of groups:  judge 9 

Coefficients:
           Estimate Std. Error z value Pr(>|z|)    
contactyes   1.8349     0.5125   3.580 0.000344 ***
tempwarm     3.0630     0.5954   5.145 2.68e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Threshold coefficients:
    Estimate Std. Error z value
1|2  -1.6237     0.6824  -2.379
2|3   1.5134     0.6038   2.507
3|4   4.2285     0.8090   5.227
4|5   6.0888     0.9725   6.261


exp(mod2$beta)

> exp(mod2$beta)
contactyes   tempwarm 
  6.264414  21.391566 

Questions for the additive model:

(1) Analogous to the non-additive model case.

(2) Is the reference level "tempcold-contactno" just the sum of all the
intercepts?

(3) Assuming I fit a simple linear model with dummy/treatment coding to the
wine dataset I would get a model matrix of the form:

        1 0 0 := tempcold + contactno
        1 1 0 := tempwarm + contactno
        1 0 1 := tempcold + contactyes
        1 1 1 := tempwarm + contactyes

How does this compare to the the clmm() model?

(3.1) How do I calculate the "x" in the following statement?: "The odds ratio
to be rated higher increases by "x" for "tempwarm-contactyes" in comparison to
"tempcold-contactno"?

(One last point: There is a related post here: 
http://stats.stackexchange.com/questions/89474/interpretation-of-ordinal-logistic-regression
While this post does address one point my inference algorithm is not sufficient to generalise this solution after having looked at this code for so long.)

Best,
Christian