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RE: st: Dependent continuous variable with bounded range


From   "Nick Cox" <n.j.cox@durham.ac.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   RE: st: Dependent continuous variable with bounded range
Date   Tue, 15 Apr 2008 17:49:29 +0100

The numeric result for skewness doesn't quite match the fact that the mean 
is nearer the maximum than the minimum, not that that need that be the case. 

You possibly have a bit of a tail of fairly lousy firms, but otherwise this distribution 
looks quite healthy to me. How about 

gen repute = reputation / 10 
xtgee repute ..., link(logit) family(<continuous>) 

P.S. for "likert" read "Likert" (Rensis Likert, fl.C20)

Pavlos C. Symeou

Dear Nick, Anders, and Jay,

thank you very much for your responses. First, I need to agree with you 
that this problem can not be handled with interval regression. But as 
Anders recommends, I need to give further information about my variable 
in question, "reputation". The variable is a construct consisted of 
three other variables. These three variables take discrete values that 
range from 0-10 on a 11-point likert scale, namely 0,1,2,3...,10. These 
are responses to a survey questionnaire. These three variables' values 
are first added together and divided by 3 to yield my focal variable 
"reputation" (unfortunately I only have access to the final variable and 
not its components). Values for "reputation" closer to 0 suggest lower 
firm reputation. Values closer to 10 suggest higher firm reputation. A 
histogram shows a relatively normal distribution as you can see from its 
Skewness below. I also provide you with statistics on max and min values 
to respond to Anders' inquiry.

Nick advised that I transform my response so that is unbounded using a link function e.g. logit on response/10. (I am not sure how I can do this) Yet, Jay's suggestion is that this might not be that beneficial. Can you please advise on how to proceed?

Min                  2.95      
Max                 8.32
Mean              6.267978
Variance       .5091705
Skewness    -.057921
Kurtosis        3.642905

Yours truly,

Pavlos




I find both Nick Cox's and Jay Verkuilen's comments very reasonable.
But Pavlos does not mention how the variable "reputation" is created.
How would reputation get a value of, say, "9.6"? Where does the
boundary 0-10 come from, e.g., from the sample's population or from a
questionnaire
? Is reputation a scored variable or does it represent
original data?

Anders Alexandersson


Nick Cox outlined several strategies, to which I will add just a bit:

(1) How close to the boundaries are your observations? If the distribution looks reasonably symmetric, you probably won't gain much from using a specialized model that "knows" about the boundaries. If you have some skew due to the ceiling or floor but not a truly L- or J-shaped distribution, the logit transformation will probably normalize your errors enough to do ordinary panel regression models. If the distribution is truly L- or J-shaped, no transformation will fix things up. 

(2) -betafit- (by Nick, Maarten, et al) with clustered robust SEs is a viable alternative that uses the linking strategy. This uses the beta distribution as an error model. It won't adjust for the autoregressive nature of panel data, though, but maybe that'll work well enough. 

(3) If you want to use the GLM approach and are willing to move to different software (SAS or winBUGS), I can give you examples to do random effects and AR-adjusted beta regression. (One of these days, I want to port a random effects and GEE betareg over to Stata; no time right now.)

Jay



This does not sound like an interval regression problem to me. Interval
regression 
is for when at least some individual values are known as intervals not
points. 
Here values are known but must lie in a prescribed interval. 

Divide by 10 and the problem has exactly the same form as that often met
for
proportions. 

There are correspondingly various options: 

1. Go ahead regardless. The advantage is simplicity. The disadvantage is
that there is no 
guarantee that predicted values will lie in [0,10] certainly for some
values of the predictors
and possibly for your observed data. Arguably that is quite wrong in
principle and it may be a real
nuisance for your project. There are implications all the way downstream
in terms of assessing fit. 

2. Transform your response so that is unbounded. Model in terms of that
and then back-transform. 

3. A variant of 2, and in many ways preferable, to use a link function
e.g. logit on response/10. 

4. No doubt others. 

Nick
n.j.cox@durham.ac.uk 

Anders Alexandersson

See -help xtintreg-. It does random-effects interval data regression
models.

Pavlos C. Symeou <p.symeou@jbs.cam.ac.uk> wrote:

>  I have panel data for 100 firms for five years and I want to examine
the
>  effects of various variables on "reputation". My variable
"reputation"
>  takes  continuous values in the range of 0-10. Namely, it can take
>  values of 1,2,3 but also of 2.5, 9.6, etc. The values that the
variable
>  "reputation" takes in my sample range between 2.6 - 8.3. Can you
please
>  advise if I can still use panel OLS estimation for panel data or
should
>  I use a different model? In essence, my main concern is the
limitations
>  of the bounded range of my variable.


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