Wow, I got such a useful response about that sampling question, I have another :)
I had another student come in with a question about modelling the influence of mean
outside temperature on temperatures in houses with various characteristics.
From the theory of his subject, there is a way to calculate mean external temperature
that takes into account the history of temperatures. For any parameter a in (0,1),
you can define a mean external temperature by
(1) T_a = (1-a)(T0 + aT1 + a^2 T2 + ....),
where T0 is the temperature at the current time period, T1 is the temperature during the previous
time period T2 is the temperature two periods ago, etc.
Then the theory proposes a simple linear model
(2) T_int = b + m T_a + e
for the influence of T_a on T_int.
He had two questions. the first was, is there a single value of a that should be used for all house
types, or should the value of a used depend on properties of the house?
The second was, after deciding on the appropriate value for a, can we determine how b and m depend
on properties of the house?
For the first question, he looked at what value of a in equation (1) gave the best fit regression equation (2)
for each house, and then we used a general linear model to look at the relationship between
recorded house characteristics and a. None of these turned out to be significant. Checking residuals,
they did seem to be normal and also homoscedastic.
We then fixed a as the mean value of a for all houses and used that build a new general linear model
(with repeated measures) for internal house temperature as a function of mean external temperature
and measured properties of the houses. In this case, there were significant effects on the coefficients.
However, I feel like this is fishy because we used the same data for the two analyses, and the second one
depends on the first one. I am also a bit leary of the way we did the first analysis, because of the way
we calculated the a. I have never created a derived variable in this sort of way before.
Any thoughts?
Eugenie
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