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RE: st: interprating orthogonal polynomial regression


From   "Nick Cox" <n.j.cox@durham.ac.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   RE: st: interprating orthogonal polynomial regression
Date   Fri, 23 Jul 2010 18:57:58 +0100

The short answer is that you are asking some very difficult questions.
Without seeing your data or knowing your research problem as well as you
do, good advice is difficult to produce. But

1. If you fit a bundle of terms, you need to see graphically what each
term implies and what collectively they imply. For example, even a cubic
term might be monotonic over the range of your data. It needn't have
turning points or inflections within that range. 

2. The converse of your statement is this: Someone confident that a
cubic term is not needed is confident that a second derivative never
changes. That is a pretty strong statement, especially with age as a
kind of time variable. 

3. Outside of Newtonian mechanics, it seems pretty rare to me that any
theory genuinely predicts any kind of polynomial behaviour. I'd welcome
examples to the contrary. Whether theorists, as well as empiricists,
find polynomials to be convenient descriptive tools is a different
matter. In several disciplines, including the sciences, theorists use
functional forms because they think that those have been shown to apply
empirically, and vice versa, to paraphrase Gabriel Lippmann's old quip
about the normal. 

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

Junqing

Thanks, Maarten. This is very helpful. I have additional question on the
age-period-cohort effects.

The study that i am working on interviewed the same individuals at five
waves(wave is my time variable) over 8 years. So the cohorts(subjects)
are
the same through the five interviews. I fitted a random intercept model
and i believe that the cohort effects(between-subject effects) should be
captured by the variance of the random intercept.

I included the three orthogonal polynomials of age(continuous variable)
and the three orthogonal polynomials of time as fixed effects in the
three
level random intercept model and found the six fixed coefficients are
all
significant. This means that age is not the only factor that contribute
to
the cubic trend of Y.

Then what i should do to possibly explain why y(mental health service
access) has a cubic trend. No much theory as I am aware of talks about
the
curvilinear trend of Y.

Maarten Buis replying to Junging 

>> I fitted a three level logistic regression of y on the
>> first, second, and third order of orthogonal polynomials
>> of time to examine the trend of y. Coefficients of the
>> three orthogonal polynomials are significant. The
>> signs of linear and cubic trend are negative and the
>> quadratic term is positive.
>>
>> I conclude that y has a cubic trend. The interpretation is
>> that as time increases, the probability of y first decrease.
>> With a further increase in time y appeared to increase. Then
>> at about 51 months(based on the graph of the sample mean of
>> y), y decreases again.
>>
>> What else should i interpret about the cubic trend? Do I
>> have to calculate the time points when the sings change?
>
> It is your argument, so you decide what you think is
> confincing or illuminating evidence and what is not. We
> can only make suggestions. Finding these points can be sorta
> nice, but they should not be taken too literaly, as they
> are to a large extend influenced by the functional form you
> assumed.
>
>> If so, i probably need to transform the coefficients of
>> orthogonal polynomials into coefficients for the original
>> time scale. I do not know how stata does this transform
>> after fitting a -mim:gllamm- model.
>
> *--------------- begin example ----------------
> sysuse auto, clear
> orthpoly weight, deg(3) generate(pw*)
> logit foreign mpg pw1-pw3 rep78
> orthpoly weight, deg(3) poly(P)
> matrix b = e(b)
> // extract the polynomials and the constant
> matrix b = b[1, "foreign:pw1".."foreign:pw3"], b[1,"foreign:_cons"]
> matrix b = b*P
> matlist b
>
> // check
> gen w1 = weight
> gen w2 = weight^2
> gen w3 = weight^3
>
> logit foreign mpg w1-w3 rep78
> *---------------- end example -----------------------
>
> Personaly, I like linear splines better, as they often provide
> a better balance between allowing for non-linear effects and
> giving directly interpretable coefficients. See -help mkspline-.
>
>> Then I need to think about why y has a cubic trend. One
>> possible explanation is age. With the increase in time, the
>> age of participants increase as well. The cubic trend may
>> because different age intervals have different trends.
>
> Assuming that participants aren't all born in the same year,
> you can add time and age, or time and year of birth, or age
> and year of birth, but not all three, as time - age = year of
> birth.
>
> There is a large literature on still trying to estimate
> these "age-period-cohort effects" which basically consists of
> proposing different constraints on one or more of these
> variables. Assuming that this constraint is true you can
> estimate all three, but you cannot test whether the constraint
> is true, so...
>
>> Does this mean i need to use age as the time variable
>> instead?
>
> There is only one person who can decide that, and that is you.

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