A8 Q1

For the recent General Social Survey data on X = gender (males, females) and Y = belief in an afterlife (no, yes), Table 7.12 shows results of fitting the independence loglinear model.

a. The deviance is 0.82 with df = 1. What does this suggest? Deviance is the test statistics for chi-square test of goodness of fit.

1-pchisq(0.82,1)

## [1] 0.3651802

0.365>0.5, we fail to reject the null hypothesis that the model is a good model of the goodness of fit test.

b. Report {\(\hatλ_j^y\) }. Interpret \(\hatλ_1^y\) − \(\hatλ_2^y\)

exp(1.4165)

## [1] 4.122666

The estimated odd of belief in afterlife is 4.12 for each level of gender

c. For the saturated model, software reports for \(\hatλ_{ij}^{xy}\)
Estimate the odds ratio.

exp(0.1368)

## [1] 1.146599

The estimated odd ratio is 1.147 for female who belief in haven to female who don’t b

d. The text website has a data file Postlife that cross-classifies belief in life after death with race (black, white, other). Fit the independence model and interpret {\(\hatλ_j^y\) }

fit<-read.table("http://users.stat.ufl.edu/~aa/cat/data/Postlife.dat",header=TRUE)
fit<-glm(formula = count ~ race + postlife, data = fit, family = poisson)
summary(fit)

## 
## Call:
## glm(formula = count ~ race + postlife, family = poisson, data = fit)
## 
## Deviance Residuals: 
##        1         2         3         4         5         6  
##  0.15781  -0.33688  -0.01717   0.03631  -0.20194   0.41917  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  4.05242    0.07309   55.44   <2e-16 ***
## raceother   -1.05209    0.11075   -9.50   <2e-16 ***
## racewhite    1.64927    0.06152   26.81   <2e-16 ***
## postlifeyes  1.49846    0.05697   26.30   <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: 2849.21758  on 5  degrees of freedom
## Residual deviance:    0.35649  on 2  degrees of freedom
## AIC: 49.437
## 
## Number of Fisher Scoring iterations: 3

1-pchisq(0.356,2)

## [1] 0.8369424

exp(1.49846)

## [1] 4.474793

Goodness of fit: deviance=0.356 and p -value is 0.8 which is >0.05. We accept the h0 that the model is a good fit.
The estimated odd of belief in afterlife is 4.47 for each level of gender.