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Background

While dealing with data, I very often find my data to be "halfway between discrete and continuous".

For example, let's say my variable of interest is age. In principle, age should be continuous, say $67.34$ years old. But one will just write $67$ on a questionnaire, essentially discretizing the value. Furthermore, my questionnaire usually has a target audience, say the senior citizen group. As a result, the collected ages will be something like $60, 61, 78, \ldots, 96$. That's right: now my supposedly continuous data now becomes a series of integer $\in[60, 96]$.

Strictly speaking, under this kind of scenario, I should not use the generalized linear model, since its $y$ models continuous values. Instead, I should use multinomial logistic regression. Then, my headache is, I have so many ($96-60+1 = 37$) categories!

Question

My main purpose of fitting the model is to do some linear hypothesis testing, e.g., testing if $\beta_1=\beta_2$. Under this consideration, doing multinomial logistic regression causes more trouble, since sometimes the $\beta$'s are not comparable across models. On the contrary, linear hypothesis testing is very straightforward under a general linear model setting.

So what is the price I have to pay if I just go for the general linear model?


A Simpler Version

$y$ is trinomial, $0$ or $1$ or $2$. What happens if I fit a general linear model instead of a trinomial logistic regression, given my purpose of fitting is to compare $\beta$'s?

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    $\begingroup$ If you think age to be discrete, then it remains ordered and at worst you should consider ordinal logit (not multinomial) for age as a response or outcome (itself an unlikely scenario; age in my experience is usually a predictor). But your worry is over-fastidious. Arguably every measurement in principle continuous is subject to some rounding convention: e.g. in meteorology rainfall is reported as multiples of 0.1 mm, temperature as multiples of 0.1$^\circ$C; human heights are rarely reported other than in cm or half-inches., etc. Treatment of age as if continuous is fine. $\endgroup$ – Nick Cox Jun 15 '15 at 12:49
  • $\begingroup$ @NickCox Thanks! So viewing age as continuous is fine. What about the "simpler version" above, where I only have three possible $y$? Is fitting general linear model acceptable? $\endgroup$ – Sibbs Gambling Jun 15 '15 at 14:07
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    $\begingroup$ Depends entirely on how 0,1,2 arise and which GLM you are considering. $\endgroup$ – Nick Cox Jun 15 '15 at 14:51
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    $\begingroup$ Could be a binomial model. Could be an ordinal logit. My genetics is totally inadequate to advise further. $\endgroup$ – Nick Cox Jun 15 '15 at 16:33
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    $\begingroup$ Just adding a bounty without editing the question implies that you think that the question is clear, but the existing answer and comments are not what you seek. I can readily sense that the answer didn't address your main concern, but otherwise the comments have at least one clear implication: you can treat age as continuous. But, your question seems really to be two questions, (a) about age and (b) about a trinomial response, and the link between them seems obscure. I think that's why this got little response, and just adding a bounty doesn't undo that obscurity. $\endgroup$ – Nick Cox Jun 23 '15 at 15:03
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Sibbs Gambling...step back from the question of which model to use as being premature. You need to answer some really nitty gritty questions first. What is the data structure really like? What's the unit of analysis? Perhaps more importantly, what are you trying to predict, i.e., what's the dependent variable? You don't make explicit the distinction between the dependent variable, what you are trying to predict, and the predictors, the information that is to be used to understand the DV. Are these groupings, e.g., for age, your dependent variable? How can that be? You kind of imply that they are since your question is concerned with the functional form the model should take in dealing with data of this type. To me, this suggests potential confusion about the fundamental structure of regression models before anything else. Why not do some homework and read about how to set up a regression model first? And then worry about whether it's a multinomial logistic regression or not...

Regardless, you still have to deal with some very funky data: DVs, predictors, whatever. You've indicated the possibility that some values are at a unique, individual level, e.g., an individual's age, while other units represent aggregations of individuals up to a bigger unit, e.g., senior center. Based on the limited information provided, individual level data should present few problems.

The aggregated information is another story. With data of this type, you have stepped out of any traditional framework. The question is, how do you deal with it in a model? How can you decompose it to develop a meaningful analysis? Whatever you do to decompose this information, you definitely do not want to model all 37 possible categories for age (again, as an example).

So, what additional information do you have about these "centers?" For instance, and just to take a single example, have you been given frequencies of the people across different ages? Probably not...but only you know that for sure. If you do have some frequencies, then one option is to create proportions of the totals that fall into discrete age buckets (e.g., 60-64, 65-69, etc.). You can create a "coefficient of variation" for the age values which is the std dev divided by the mean (times 100). The CV would be a measure of the diversity across age (again, an example) for a grouping. Another option is to create a single field that contains the age range, min to max. If you have a zip code assigned to your data units, integrate geodemo information such as zip code population size, percentage by ethnicity, median home values, etc.

In my opinion whatever you do, you've got a ways to go before you even get to the question of multinomial logistic regression...

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  • $\begingroup$ Thanks, but I have my reason putting age as a dependent variable instead of a predictor. Sorry but I don't really get it which part of question actually leads you to think I don't even know "how to set up a regression model first". I think the question is a valid and practical question. Maybe I am not getting your logic in the answer. I will read it up again. Thanks a lot for helping! :-) $\endgroup$ – Sibbs Gambling Jun 15 '15 at 14:13
  • $\begingroup$ @SibbsGambling Whatever...please note that I said "potential." For all I know, you're a first year grad student. Also note that you failed to address any of my questions. $\endgroup$ – Mike Hunter Jun 15 '15 at 15:14
  • $\begingroup$ hmmmm... So may I ask your opinion on this particular question? Thanks. $\endgroup$ – Sibbs Gambling Jun 15 '15 at 15:18
  • $\begingroup$ @SibbsGambling I made every effort to ask clear and purposeful questions about your data. I am unable to make any further suggestions in the absence of answers to those questions. Please give a little thought to explaining what is going on. $\endgroup$ – Mike Hunter Jun 15 '15 at 15:22

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