## Saturday, April 13, 2013

### various choices of variance functions

• function in 2.2 to illustrate the the effect of the assumed variance function (using the identity link for all functions)
x <- 1:3
y <- c(1,2,9)

Call:  glm(formula = y ~ x, family = gaussian(link = identity))

Coefficients:
(Intercept)            x
-4            4

Degrees of Freedom: 2 Total (i.e. Null);  1 Residual
Null Deviance:      38
Residual Deviance: 6    AIC: 16.59

glm(y~x,family=poisson(link=identity))

Call:  glm(formula = y ~ x, family = poisson(link = identity))

Coefficients:
(Intercept)            x
-2.4          3.2

Degrees of Freedom: 2 Total (i.e. Null);  1 Residual
Null Deviance:      9.052
Residual Deviance: 1.69         AIC: 14.36

glm(y~x,family=Gamma(link=identity))

Call:  glm(formula = y ~ x, family = Gamma(link = identity))

Coefficients:
(Intercept)            x
-1.799        2.744

Degrees of Freedom: 2 Total (i.e. Null);  1 Residual
Null Deviance:      2.537
Residual Deviance: 0.4385       AIC: 14.6

glm(y~x,family=inverse.gaussian(link=identity))

Call:  glm(formula = y ~ x, family = inverse.gaussian(link = identity))

Coefficients:
(Intercept)            x
-1.370        2.353

Degrees of Freedom: 2 Total (i.e. Null);  1 Residual
Null Deviance:      0.8611
Residual Deviance: 0.1181       AIC: 13.48


### model on page 16 to illustrate the use of an offset

require(foreign)
m <- glm(los ~ hmo + white + type2 + type3, data=medpar, family=poisson(link="log"))
summary(m)

Warnmeldung:
Faktorbeschriftungen von Stata-5-Dateien können nicht gelesen werden

Call:
glm(formula = los ~ hmo + white + type2 + type3, family = poisson(link = "log"),
data = medpar)

Deviance Residuals:
Min       1Q   Median       3Q      Max
-5.3063  -1.8259  -0.6319   1.0081  15.3827

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)  2.33293    0.02721  85.744  < 2e-16 ***
hmo         -0.07155    0.02394  -2.988  0.00281 **
white       -0.15387    0.02741  -5.613 1.99e-08 ***
type2        0.22165    0.02105  10.529  < 2e-16 ***
type3        0.70948    0.02614  27.146  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1)

Null deviance: 8901.1  on 1494  degrees of freedom
Residual deviance: 8142.7  on 1490  degrees of freedom
AIC: 13868

Number of Fisher Scoring iterations: 5

• Wald test whether the coefficient on white is equal to -0.2
require(survey)
regTermTest(m,test.terms = "white",null=-0.2,method="Wald")

Wald test for white
in glm(formula = los ~ hmo + white + type2 + type3, family = poisson(link = "log"),
data = medpar)
F =  2.831664  on  1  and  1490  df: p= 0.092632

• define model with offset variable white=-0.2
• do the likelihood-ratio test
m1 <- glm(los ~ hmo + type2 + type3 + offset(-0.2*white), data=medpar, family=poisson(link="log"))
anova(m1,m,test="Chisq")

Analysis of Deviance Table

Model 1: los ~ hmo + type2 + type3 + offset(-0.2 * white)
Model 2: los ~ hmo + white + type2 + type3
Resid. Df Resid. Dev Df Deviance Pr(>Chi)
1      1491     8145.5
2      1490     8142.7  1   2.8657  0.09049 .
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1