I used R and the function polr (MASS) to perform an ordered logistic regression. The model is simple: there is only one dichotomous predictor (levels "normal" and "modified"). The question was if the frequency distribution of the response is shifted under the "modified" condition compared to the "normal" condition.
This is the script:
y = ordered(gl(5,1,labels=c("cal++","cal","even","hk","hk++")))
yy = rep(y,2)
condition = gl(2,5,labels=c("normal","modified"))
f.norm = c(2,0,4,13,5)
f.mod = c(0,0,1,11,12)
ff = c(f.norm, f.mod)
m = polr(yy~condition,weights=ff)
summary(m)
This is the result:
Coefficients:
Value Std. Error t value
conditionmodified 1.513 0.6034 2.508
Intercepts:
Value Std. Error t value
cal++|cal -2.6137 0.7456 -3.5053
cal|even -2.6135 0.7456 -3.5053
even|hk -1.1836 0.4607 -2.5691
hk|hk++ 1.4620 0.4864 3.0059
Residual Deviance: 97.08686
AIC: 107.0869
I also attached a plot with the predicted probabilities and empirical frequencies.
My question is: what is the practical meaning of the coefficient 1.513? From book I know that exp(-1.513) = 0.22 this is the ratio of odds for lower to higher outcome per unit increase in the predictor (here from "normal" to "modified"). So I read it as the odds for a higher outcome in "modified" is 0.22-times as high as in "normal". But then this would mean that the odds for higher outcomes should be lower in the "modified" group, from what I would expect that the "modified"-frequencies peak at lower response levels. But in fact I see that they peak at higher levels ("normal" peaks at hk, "modified" at hk++). Where am I wrong? How can I understand this?
Thanks for any help!
Dear Jochen,
the proportional odds model relates to cumulative probabilities of an ordered categorical response.
The analysis suggests that for every category of predictor (in this case - for "modified" ), the odds of being below given level of response are multiplied by exp (-1.513)=0.22
So, the odds of being below given level of response decrease by 78% for "modified" condition.
In R authors of function polr() wrote in help "Note that it is quite common for other software to use the opposite sign for...."
May be exp (1.513) ?
Moreover, there is the package epicalc with fucntion ordinal.or.display () -
Display modelling results in a medically understandable format. They use exp(beta), not exp(-beta).
If I'm understanding this question -- don't forget an OR of .22 means that a variable (or group) is associated with a 78% DECREASE in risk. . .
Dear Jochen,
the proportional odds model relates to cumulative probabilities of an ordered categorical response.
The analysis suggests that for every category of predictor (in this case - for "modified" ), the odds of being below given level of response are multiplied by exp (-1.513)=0.22
So, the odds of being below given level of response decrease by 78% for "modified" condition.
In R authors of function polr() wrote in help "Note that it is quite common for other software to use the opposite sign for...."
May be exp (1.513) ?
Moreover, there is the package epicalc with fucntion ordinal.or.display () -
Display modelling results in a medically understandable format. They use exp(beta), not exp(-beta).
Thank you, Olga. I missed this sentence in the R help file! With the opposite sign the interpretations makes more sense to me.
When you use ordered logistic regression, the dependent variable (here is "group"?) should be ordinal variable in nature. For instance, if the levels of group include cal++","cal","even","hk","hk++", to treat it as ordinal outcome means there is a natural order of al++" > "cal" > "even" > "hk" >"hk++" or reverse. By using ordinal logistic regression, we assume the proportional odds ratio, The OR of 0.22 means the odds of being in hk++ vs. in all other four combined for normal are 0.22 times the odds for the modified, which is sames as being in "hk++" plus "hk" vs. all other three, or other four combined vs. "al==". If the proportional odds model is not justified, multinormial logistic regression should be used. You can see one example here: http://www.ats.ucla.edu/stat/stata/whatstat/whatstat.htm#orderedlogistic
I don't know if this could help you, but I found it close to your question:
http://www.ats.ucla.edu/stat/stata/dae/ologit.htm
Look here:
listcoef, help
ologit (N=400): Factor Change in Odds
Odds of: >m vs |z| e^b e^bStdX SDofX
-------------+--------------------------------------------------------
pared | 1.04766 3.942 0.000 2.8510 1.4654 0.3647
public | -0.05868 -0.197 0.844 0.9430 0.9797 0.3500
gpa | 0.61575 2.363 0.018 1.8510 1.2777 0.3979
----------------------------------------------------------------------
b = raw coefficient
z = z-score for test of b=0
P>|z| = p-value for z-test
e^b = exp(b) = factor change in odds for unit increase in X
e^bStdX = exp(b*SD of X) = change in odds for SD increase in X
SDofX = standard deviation of X
**************************************************************************************
The probably exact answer is:
e^b = exp(b) = factor change in odds for unit increase in X
I see that they also use exp(+b) and not exp(-b).
Maybe that was the problem with R.
Olga is correct. More generally the ordinal logistic regression model can be parameterized in (at least) two different ways (altering the interpretation of coefficients) and the choice of reference category (first or last) can vary (also altering the interpretation of coefficients) .
It is wise to both check the documentation for your software and look at the predicted probabilities for the categories to double check the interpretation.
Your empirical results look really messy. I think you may not use the right model. I suggest you to read G. S. Maddala (1983) "Limited-Dependent and Qualitative Variables in Econometrics". You need figure out you want to conduct Discriminant analysis (Chapter 4) or Censored and truncated regression models (Chapter 6).
You should take a look at Scott Long:
http://www.amazon.de/Regression-Categorical-Dependent-Quantitative-Techniques/dp/0803973748
He goes into lengthy detail about the interpretation of the most important noncontinuous models
Intuitively, I think your research should be designed by censored or truncated regression model. Fair (1977) suggested an iteration method in handling empirical data with truncated normal distribution. I just check chapter 6 of Maddala (1983) now. It requires using FORTRAN function to conduct the iteration converge. Nowadays probably very little scholars know how to use FORTRAN. So, just forget about my opinions.
Lee-in, honestly, I don't understand your advice. For instance, I don't see where truncation or censonring plays a role here. I am not an expert in this topic. This is why I am asking. I am thankful for advices and hints to learn. Fortran, btw, would not be a problem - I know this language. It is also likely that in the meanwhile there are R packages available to tackle the calculations.
Jochen, I'm sorry, maybe following question should be placed apart in a new topic, but it concerns Your data.
One category ("cal") has no observations .
The R function polr() takes this category in consideration.
I try to reproduce that data and analysis in the statistical tool LogXAct. The results is different. LogXAct doesn't use this category ("cal"). Another words only four categories of response are in analysis.
When Your data are transformed to raw data, category "cal"does not exist (I try analyze the data in STATISTICA).
This fact not significantly changes the result of regression, but not only "... is quite common for other software to use the opposite sign ...", they change design?
What somebody thinks about that? And how do handle that data another stat tools?
Olga, yes, thanks. I also used "clm" from the package "ordinal" what also drops the unused level. I have not yet found out what the pros and cons of these different behaviours are.
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