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

[David Winsemius]

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?

> 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