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

From: Prof Brian Ripley <ripley_at_stats.ox.ac.uk>
Date: Mon, 17 Mar 2008 06:30:43 +0000 (GMT)

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.

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.

>
>> with(bmi, table(BMI,Actual_Deaths))
> Actual_Deaths
> BMI 0 1 2 3 4 5 6 7 11 13
> 18.5-24.9 311 21 1 0 0 0 0 0 0 0
> 15.0-18.4 353 33 8 2 0 1 0 0 0 0
> 25.0-29.9 367 19 0 0 0 0 0 0 0 0
> 30.0-34.9 349 95 39 17 8 9 3 4 0 1
> 35.0-39.9 351 90 50 21 20 3 3 2 1 0
> 40.0-55.0 386 60 15 7 4 0 0 1 0 0
>
>> bmi.base <- with(bmi,glm(Actual_Deaths ~
> BMI + offset(log( MMI_VBT_Expected)), family="poisson"))
>> summary(bmi.base)
>
> Call:
> glm(formula = Actual_Deaths ~ BMI + offset(log(MMI_VBT_Expected)),
> family = "poisson")
>
> Deviance Residuals:
> Min 1Q Median 3Q Max
> -2.6385 -0.5245 -0.2722 -0.1041 3.4426
>
> Coefficients:
> Estimate Std. Error z value Pr(>|z|)
> (Intercept) 0.42920 0.20851 2.058 0.0395 *
> BMI15.0-18.4 0.31608 0.24524 1.289 0.1974
> BMI25.0-29.9 -0.22795 0.30999 -0.735 0.4621
> BMI30.0-34.9 -0.09669 0.21506 -0.450 0.6530
> BMI35.0-39.9 -0.04290 0.21455 -0.200 0.8415
> BMI40.0-55.0 0.19348 0.22569 0.857 0.3913
> ---
> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
>
> (Dispersion parameter for poisson family taken to be 1)
>
> Null deviance: 1485.0 on 2654 degrees of freedom
> Residual deviance: 1470.0 on 2649 degrees of freedom
> AIC: 2760.9
>
> Number of Fisher Scoring iterations: 6
> -----------------------------------------------------
>> bmi.no.int <- with(bmi,glm(Actual_Deaths ~
> BMI + offset(log(MMI_VBT_Expected)) -1 ,
> family="poisson"))
>> summary(bmi.no.int)
>
> Call:
> glm(formula = Actual_Deaths ~ BMI + offset(log(MMI_VBT_Expected)) -
> 1, family = "poisson")
>
> Deviance Residuals:
> Min 1Q Median 3Q Max
> -2.6385 -0.5245 -0.2722 -0.1041 3.4426
>
> Coefficients:
> Estimate Std. Error z value Pr(>|z|)
> BMI18.5-24.9 0.42920 0.20851 2.058 0.0395 *
> BMI15.0-18.4 0.74529 0.12910 5.773 7.79e-09 ***
> BMI25.0-29.9 0.20125 0.22939 0.877 0.3803
> BMI30.0-34.9 0.33251 0.05270 6.309 2.81e-10 ***
> BMI35.0-39.9 0.38631 0.05057 7.639 2.19e-14 ***
> BMI40.0-55.0 0.62268 0.08639 7.208 5.67e-13 ***
> ---
> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
>
> (Dispersion parameter for poisson family taken to be 1)
>
> Null deviance: 1630.7 on 2655 degrees of freedom
> Residual deviance: 1470.0 on 2649 degrees of freedom
> AIC: 2760.9
>
> 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?

>
> --
> David Winsemius

-- 
Brian D. Ripley,                  ripley_at_stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

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Received on Mon 17 Mar 2008 - 06:34:58 GMT

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