[R] Std errors in glm models w/ and w/o intercept

[David Winsemius]

Prof Brian Ripley <ripley at stats.ox.ac.uk> wrote in
news:alpine.LFD.1.00.0803170624220.5706 at gannet.stats.ox.ac.uk: 

> On Mon, 17 Mar 2008, David Winsemius wrote:

>>
>> I am doing a reanalysis of results that have previously been
>> published. My hope was to demonstrate the value of adoption of more
>> modern regression methods in preference to the traditional approach
>> of univariate stratification. I have encountered a puzzle regarding
>> differences between I thought would be two equivalent analyses.
>> Using a single factor, I compare poisson models with and without
>> the intercept term. As expected, the estimated coefficient and std
>> error of the estimate are the same for the intercept and the base
>> level of the factor in the two models. The sum of the intercept
>> with each coefficient is equal to the individual factor
>> coefficients in the no- intercept model. The overall model fit
>> statistics are the same. However, the std errors for the other
>> factors are much smaller in the model without the intercept.
>>
>> The offset = log(expected) is based on person-years of follow-up
>> multiplied by the annual mortality experience of persons with known
>> age, gender, and smoking status in a much larger cohort. My logic
>> in removing the intercept was that the offset should be considered
>> the baseline, and that I should estimate each level compared with
>> that baseline. "18.5-24.9" was used as the reference level in the
>> model with intercept. Removing the intercept appears to be a
>> "successful" strategy. but have I committed any statistical sin?
> 
> No, but you have apparently not understood what the 'intercept'
> means here.  With a single factor and the default contr.treatment,
> it is the coefficient used to predict the first category of the
> factor, and the remaining coefficients are log ratios of mean rate
> for the named category to the first.  When you drop the intercept,
> the coefficients are no longer contrasts.

Thank you for your interest in my question, Prof Ripley. I did 
understand that the intercept coeff was the log(ratio) of the base 
group to the offset and that exp(coeff$intercept)) can be interpreted 
as a mortality ratio. Also, that the coefficients in the first model 
were log ratios of effect(BMI) to coefficient(BMI-reference), so that 
exp(coeff$level+coeff$intercept) would be a level's ratio relative to 
the "expected". My concern was with the markedly lower std errors 
around the "other" level coefficients when the intercept was removed. 
My preference would be to use the non-intercept model.
 
> When you drop the intercept, the coding (and hence the
> interpretation of the coefficients) of the first factor in the model
> changes. See MASS chapter 6.  So you are comparing apples with
> oranges. 

MASS.2ed.ch6, "Linear Statistical Models", says that the lm() models 
with and without intercepts have different contrast matrices and 
discusses interpretation of coefficients. If I to consult a later 
edition, will I find a discussion of the impact of those differences on 
the std errors of the coefficients?


>>
>>> with(bmi, table(BMI,Actual_Deaths))
>>           Actual_Deaths
>> BMI          0   1   2   3   4   5   6   7  11  13 SE.no-int SE.int
>>  18.5-24.9 311  21   1   0   0   0   0   0   0   0  0.20851  0.20851
>>  15.0-18.4 353  33   8   2   0   1   0   0   0   0  0.12910  0.24524
>>  25.0-29.9 367  19   0   0   0   0   0   0   0   0  0.22939  0.30999
>>  30.0-34.9 349  95  39  17   8   9   3   4   0   1  0.05270  0.30999
>>  35.0-39.9 351  90  50  21  20   3   3   2   1   0  0.05057  0.21455
>>  40.0-55.0 386  60  15   7   4   0   0   1   0   0  0.08639  0.22569

snipped model output ...appended SE(coeff)'s to factor counts

>> It does look statistically sensible that an estimate for BMI="40.0-
>> 55.0" with over 100 events should have a much narrower CI than
>> BMI="18.5-24.9" which only has 23 events. Is the model with an
>> intercept term somehow "spreading around uncertainty" that really
>> "belongs" to the reference category with its relatively low number
>> of events?

To my eye, the SE's in the no-intercept model make much more sense as 
far as their relationship to the sum of counts.  I also have a related 
concern that I may have in the past been using less efficient 
inferential methods when analyzing models with external standards by 
accepting the default intercept.

-- 
David Winsemius