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Re: st: interprating orthogonal polynomial regression

From   Maarten buis <>
Subject   Re: st: interprating orthogonal polynomial regression
Date   Fri, 23 Jul 2010 07:16:31 +0000 (GMT)

--- On Thu, 22/7/10, wrote:
> 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

> 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

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.

Hope this helps,

Maarten L. Buis
Institut fuer Soziologie
Universitaet Tuebingen
Wilhelmstrasse 36
72074 Tuebingen


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