It seems you don't want an example of multiple logistic regression, but of polytomous/ordinal logistic regression.

In multiple logistic regression, your outcome variable (y) still is binary, but you predict the log-odds through multiple x-variables (essentially the same as in multiple linear regression).

When the outcome variable has more than two options, the model is called polytomous logistic regression. With this term, you can find many examples through Google. Some nice slides with an example are available at http://sites.stat.psu.edu/~jls/stat544/lectures/lec19.pdf and http://www.columbia.edu/~so33/SusDev/Lecture_10.pdf .

It seems you don't want an example of multiple logistic regression, but of polytomous/ordinal logistic regression.

In multiple logistic regression, your outcome variable (y) still is binary, but you predict the log-odds through multiple x-variables (essentially the same as in multiple linear regression).

When the outcome variable has more than two options, the model is called polytomous logistic regression. With this term, you can find many examples through Google. Some nice slides with an example are available at http://sites.stat.psu.edu/~jls/stat544/lectures/lec19.pdf and http://www.columbia.edu/~so33/SusDev/Lecture_10.pdf .

Thanks, and apologies for the poor choice of words. What I was meaning is a LR where the dependent variable has more than two levels, and the independent variables are more than one.

I made some searches in Google but nothing like what I am after showed up.

I will look at the link you provided.

Best

Gm

Logistic regression can beextended to handle responses that are polytomous, i.e.taking r > 2 categories. When analyzing a multi-levels response,it’s important to note whether the response is ordinal(consisting of ordered categories) or nominal(consisting of unordered categories). Some types ofmodels are appropriate only for ordinal responses;other models may be used whether the response isordinal or nominal. If the response is ordinal, we donot necessarily have to take the ordering into account,but it often helps if we do. Using the natural orderingcan

lead to a simpler, more parsimonious model andincrease power to detect relationships with other variables.

For such data we can use log-linear model or logistic model, but loglinear model is more complex. If you are analyzing a set of categorical variables, and one of them is clearly a “response” while the others are predictors, It usually uses logistic rather than loglinear models.

The response is multinomial. That is, the “response” for row i, yi = (yi1,yi2,...,yir)T, is assumed to have a multinomial distribution with index ni =sum  of yij  for all j, and the parameter πi = (πi1,πi2,...,πir)T.   (here T is traspose, superscript)

If the data are grouped, then ni is the total number of “trials” in the ith row of the dataset, and yij is the number of trials in which outcome j occurred. If the data are ungrouped, then yi has a 1 in the position corresponding to the outcome that occurred and 0’s elsewhere, and ni = 1. Note, however, that if the data are ungrouped, we do not have to actually create a dataset with columns of 0’s and 1’s; a single column containing the response level 1, 2,...,r is sufficient.

Because the multinomial distribution can be factored into a sequence of conditional binomials, we can fit these three logistic models separately. The overall likelihood function factors into three independent likelihoods. This approach is attractive when the response can be naturally arranged as a sequence of binary choices. Hence you should have no problem to do so.  But in situations where arranging such a sequence is unnatural, we should probably fit a single multinomial model to the entire response.

For the multi-level, if there are r levels, then we need define r-1 response functions. There are many different ways to define the response functions (dependent variable).

Baseline-category logit model.  Log (πi/ πr), i=1,2,…,r-1. The regression model is

Log(πij/ πir)=XiTβj, j ǂ r.

We have r-1 interceptions and r-1 groups of regression coefficients.

Baseline is the common used for the response functions in multinominal logistic regression.

For the model

The kth element of βj can be interpreted as: the increase in log-odds of falling into category j versus category r resulting from a one-unit increase in the kth covariate, holding the other covariates constant.

Removing the kth covariate (predictor) from the model is equivalent to simultaneously setting r-1coefficients to zero.

Any of the categories can be chosen to be the baseline, not necessary to be the r category. The model will fit equally well, achieving the same likelihood and producing the same fitted values. Only the values and interpretation of the coefficients will change.

You can add any interaction among the p predictors.

The other response function for multinominal logistic regression such as for the ordinal response: we may use the proportional odds assumption . The model is

Log((p(yj) )=XiTβj,  j=1,2,…r-1.

There are more response functions.

Hope it help!

In credit scoring for example.

I wander if Artak asks an example:

Here it is: You may see attachment, which is more easy to read the SAS output.

/*The data set contains the results of a hypothetical taste test of three brands of ice cream.The three brands are rated for taste on a five-point scale from very good (vg) to very bad (vb). An analysis is performed to assess the differences in the ratings of the three brands. The variable taste contains the ratings, and the variable brand contains the brands tested. The variable count contains the number of testers rating each brand in each category. */

data Icecream;

input count brand$ taste$ @@;

datalines;

70 ice1 vg 71 ice1 g 151 ice1 m 30 ice1 b 46 ice1 vb

20 ice2 vg 36 ice2 g 130 ice2 m 74 ice2 b 70 ice2 vb

50 ice3 vg 55 ice3 g 140 ice3 m 52 ice3 b 50 ice3 vb

;

proc

logisticdata=Icecream rorder=data;

freq count;

class brand /param=glm;

model taste = brand /link=glogit;

See Attachment for the paramater estimation and odds ratio

How to read the results:

The baseline or reference group for taste = vb, hence there are four ligits

ln(p(vg)/p(vb)), ln(p(g)/p(vb)), ln(p(m)/p(vb)), ln(p(b)/p(vb))

for predictor, the reference group is ice3

1.       The first intercept is for ln(p(vg)/p(vb))

Intercept vg 1 -109E-13 0.2000

This is log odds for ice3, it is zero, reading the raw data: for ice3, vg=vb=50

2.       Similar to 1, all intercepts are the log odds for ICE3

3.       The first β in the row 5 is

brand ice1 vg 1 0.4199 0.2757 2.3187 0.1278

Which gives the difference of log odds between ICE1 and ICE3 for the first response function ln(p(vg)/p(vb)),

Hence exp(0.4199)=1.522 is the odds ratio between ICE1 and ICE3 for comparing the very good taste vs very bad taste. The percentage of the tasting ICE1 to be very good VS very bad, is roughly higher about 52% than ICE3, however the difference is not significant.

4.       The  explanation is similar for others, say for ICE2, the taste vg VS vb is significant lower than ICE3, the odds ratio is 0.286. It means ICE2 is worse than ICE3 for comparing vg and vb

 If we only use taste=vg and taste=vg, it is a binary outcome, use the logistic we get the same results for the multinomial logistic regression for the response function log(p(vg)/(1-p(vg))=log(p(vg)/p(vb)).

data ice2; set icecream;

if taste in ('vg','vb');

proclogisticdata=Ice2 rorder=data;

   freq count;

   class brand/param=glm;

   model taste = brand ;

  run;

See Attachment for the paramater estimation and odds ratio

very bad (vb) as reference group.

The same results are get by either way.

Note we use PARAM=GLM as the option, which tell SAS to go 1 0 codding. Without PARAM, SAS/logistic use 1 and -1 coding. the parameter estimation for refrence group is not zero. I will not discuss it here.