Negative log-log link function available in multi-level ordered logistic regression?

[Martijn Hogerbrugge]

Dear all,

I am currently analysing a cross-country dataset with a dependent variable that has the following answer categories:
1) Never
2) Less often
3) At least once every six months
4) At least once every three months
5) At least once a month
6) At least once a week

Although I could transform this variable into a count/frequency variable by recoding the answer categories into 'times per year' (i.e., 1=0; 2=1; 3=2; 4=4; 5=12; 6=52), I believe applying a (multi-level) ordered logistic regression on the original variable is preferable over analysing the (arbitrarily) recoded variable using either a linear regression, a Poisson regression, or a Negative binomial regression model.

Unfortunately, the dependent variable is not normally distributed. Instead, over half of the respondents chose the 'never' answer category, about 12.5% chose 'less often' and the remaining answer categories were each chosen by 4 to 6 per cent of respondents.

In SPSS, one is able to specify different link functions for ordinal regression models if the cumulative changes in the cumulative probabilities are not gradual. To quote the SPSS manual:
"The complementary log-log link may be a good model when the cumulative probabilities increase from 0 fairly slowly and then rapidly approach 1. If the opposite is true, namely that the cumulative probability for lower scores is high and the approach to 1 is slow, the negative log-log link may describe the data."
(see http://www.norusis.com/pdf/ASPC_v13.pdf, page 16)
Thus, for my dependent variable, I should use a negative log-log link instead of the (default) logit link function.

If I understand the answer provided on this page (http://www-01.ibm.com/support/docvie...id=swg21478495) correctly, the complementary log-log link in SPSS is called the log-log link in Stata, whereas the negative log-log link in SPSS is called the complementary log-log link in Stata. Please correct me if I am wrong in this.

I noticed that one is not able to specify a different link function in the ologit command, but one is able to choose different link functions in the glm command, including cloglog, loglog, and logc. Using the descriptions in the SPSS manual, and the Stata glm manual, I would specify the cloglog link function for my dependent variable (and loglog if the cumulative probabilities increase from 0 fairly slowly and then rapidly approach 1 in my dependent variable), right?

For my multi-level model, I have two questions:
- Am I correct that Stata's meologit command doesn't allow the user to choose a different link function (as is the case in the ologit command)?; and
- Does the ll (loglog) link function in gllamm correspond with the [negative] log-log link I need to use for my dependent variable?

I hope my description is clear enough and I look forward to your replies.

[Stephen Jenkins]

help meglm

is possibly the closest to what you're looking for in official Stata, but appears not to provide the link function you want. (I don't know the answer to your gllamm question.)
On a different issue: how many countries do you have? See ‘Regression analysis of cross-national differences using multi-level data: a cautionary tale’, European Sociological Review, 2015. doi: 10.1093/esr/jcv059 Open Access at http://esr.oxfordjournals.org/cgi/content/full/jcv059

[Martijn Hogerbrugge]

Yes, I am aware of the difficulties/issues when estimating multi-level models with a low number of second-level units. I have 29 countries in my analyses, so I should be OK (I hope), but I will follow the advice you provide in the conclusion of the cited paper.

Regarding the link functions in the glm command: the cloglog, loglog, and logc link functions are only available for models in which a binomial distribution is specified, and they are not available for ordered logistic models (which cannot be estimated with glm at all, if I am correct).

However, unlike glm, meglm allows one to specify the cloglog functions when an ordinal distribution is specified (the loglog, and logc link functions are not available in meglm).
(Note: as the Stata meglm help file explains, meologit is a derivative from meglm (to cite: Some combinations of families and links are so common that we implemented them as separate commands in terms of meglm), hence the lack of an option to change the link function, as I asked in my first post).

So I now have 2 options:

Ordinal logistic regression with a logit link function, which can be specified as:

meglm depvar x1 x2 || cntry:, family(ordinal) link(logit)
meologit depvar x1 x2 || cntry:
gllamm depvar x1 x2, i(cntry) family(ordinal) link(logit) 

or an ordinal logistic regression with a complimentary loglog link function, which can be specified as:

meglm depvar x1 x2 || cntry:, family(ordinal) link(cloglog)
gllamm depvar x1 x2, i(cntry) family(ordinal) link(ocll) 

Should a choice between the logit link function and the complimentary loglog link function be based purely on theory, or can I run both models and decide on either one after comparing the fit of both models using a loglikelihood difference test? (Of course, comparing the 2 models that were run with the same estimation command (either meglm, or gllamm).)

[Martijn Hogerbrugge]

I just found out that the cloglog and loglog link functions can be used when estimating (single-level) ordinal regression models, but one should use the user-written gologit2 command.
source: http://www.stata.com/statalist/archi.../msg00049.html

For multi-level models, only the complimentary loglog link function is available (either in meglm or gllamm; see my previous post).

If anyone can inform me whether a comparison of model fit is the right method when choosing between the different link functions; I would greatly appreciate that!