[R] Degrees of freedom in binomial glm

[Giovanni Petris]

Hello,

I am looking at the job satisfaction data below, from a problem in
Agresti's book, and I am not sure where the degrees of freedom come
from. The way I am fitting a binomial model, I have 168 observations,
so in my understanding that should also be the number of fitted
parameters in the saturated model. Since I have one intercept
parameter, I was thinking to get 167 df for the Null model, but 
R tells me it's 165. Where does this number come from?

Thanks in advance,
Giovanni


> ### Agresti, Problem 5.23
> race <- c("White", "Other")
> gender <- c("M", "F")
> age <- c("<35", "35-44", ">44")
> loc <- c("NE", "MidAtl", "S", "MidW", "NW", "SW", "Pac")
> sat <- factor(c("Yes", "No"), levels = c("No", "Yes"))
> Freq <- c(288,  60, 224, 35, 337,  70,   38, 19, 32, 22, 21, 15,
+           177,  57, 166, 19, 172,  30,   33, 35, 11, 20,  8, 10,
+            90,  19,  96, 12, 124,  17,   18, 13,  7,  0,  9,  1,
+            45,  12,  42,  5,  39,   2,    6,  7,  2,  3,  2,  1,
+           226,  88, 189, 44, 156,  70,   45, 47, 18, 13, 11,  9,
+           128,  57, 117, 34,  73,  25,   31, 35,  3,  7,  2,  2,
+           285, 110, 225, 53, 324,  60,   40, 66, 19, 25, 22, 11,
+           179,  93, 141, 24, 140,  47,   25, 56, 11, 19,  2, 12,
+           270, 176, 215, 80, 269, 110,   36, 25,  9, 11, 16,  4,
+           180, 151, 108, 40, 136,  40,   20, 16,  7,  5,  3,  5,
+           252,  97, 162, 47, 199,  62,   69, 45, 14,  8, 14,  2,
+           126,  61,  72, 27,  93,  24,   27, 36,  7,  4,  5,  0,
+           119,  62,  66, 20,  67,  25,   45, 22, 15, 10,  8,  6,
+            58,  33,  20, 10,  21,  10,   16, 15, 10,  8,  6,  2)
> satdata <- data.frame(Freq, expand.grid(gender=gender, age=age,
+                                         race=race, sat=sat, loc=loc))
> sat.glm0 <- glm(sat ~ gender + age + race + loc, weights = Freq,
+                 family = binomial, data = satdata)
> summary(sat.glm0)

Call:
glm(formula = sat ~ gender + age + race + loc, family = binomial, 
    data = satdata, weights = Freq)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-19.456   -6.839    0.000    6.309   17.635  

Coefficients:
             Estimate Std. Error z value Pr(>|z|)    
(Intercept)  0.334265   0.056491   5.917 3.28e-09 ***
genderF     -0.180480   0.047575  -3.794 0.000149 ***
age35-44     0.122422   0.051836   2.362 0.018191 *  
age>44       0.361610   0.051576   7.011 2.36e-12 ***
raceOther   -0.005883   0.061605  -0.095 0.923919    
locMidAtl    0.437342   0.103821   4.212 2.53e-05 ***
locS         0.178574   0.073033   2.445 0.014481 *  
locMidW      0.083189   0.066427   1.252 0.210449    
locNW        0.134337   0.067498   1.990 0.046563 *  
locSW        0.295874   0.073488   4.026 5.67e-05 ***
locPac       0.425480   0.096561   4.406 1.05e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 12987  on 165  degrees of freedom
Residual deviance: 12880  on 155  degrees of freedom
AIC: 12902

Number of Fisher Scoring iterations: 4

> str(satdata)
'data.frame':	168 obs. of  6 variables:
 $ Freq  : num  288 60 224 35 337 70 38 19 32 22 ...
 $ gender: Factor w/ 2 levels "M","F": 1 2 1 2 1 2 1 2 1 2 ...
 $ age   : Factor w/ 3 levels "<35","35-44",..: 1 1 2 2 3 3 1 1 2 2 ...
 $ race  : Factor w/ 2 levels "White","Other": 1 1 1 1 1 1 2 2 2 2 ...
 $ sat   : Factor w/ 2 levels "No","Yes": 2 2 2 2 2 2 2 2 2 2 ...
 $ loc   : Factor w/ 7 levels "NE","MidAtl",..: 1 1 1 1 1 1 1 1 1 1 ...
> sessionInfo()
R version 2.6.2 (2008-02-08) 
i686-pc-linux-gnu 

locale:
LC_CTYPE=en_US.UTF-8;LC_NUMERIC=C;LC_TIME=en_US.UTF-8;LC_COLLATE=en_US.UTF-8;LC_MONETARY=en_US.UTF-8;LC_MESSAGES=en_US.UTF-8;LC_PAPER=en_US.UTF-8;LC_NAME=C;LC_ADDRESS=C;LC_TELEPHONE=C;LC_MEASUREMENT=en_US.UTF-8;LC_IDENTIFICATION=C

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
[1] tools_2.6.2
> 

-- 

Giovanni Petris  <GPetris at uark.edu>
Associate Professor
Department of Mathematical Sciences
University of Arkansas - Fayetteville, AR 72701
Ph: (479) 575-6324, 575-8630 (fax)
http://definetti.uark.edu/~gpetris/