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Re: st: interprating orthogonal polynomial regression

From   Steven Samuels <>
Subject   Re: st: interprating orthogonal polynomial regression
Date   Fri, 23 Jul 2010 14:24:45 -0400

I would add that cubic polynomials sometimes fit poorly at the extremes. Have you checked the fit of the data? Have you plotted the residuals vs time? I would do so before putting much faith in the interpretation of a cubic effect. I suggest instead fractional polynomials or cubic splines, e.g. -fracpoly- , -mvrs- ("findit mvrs"), or NJC's -rcspline- (ssc), among others. Also, if you are concerned about age-period-cohort effects, then try -apc- (from ssc).

On Jul 23, 2010, at 1:34 PM, wrote:

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.

Thanks a lot for any advice.


--- 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

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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