1. Logistic Regression

2. Example: Birth defects
We want to know if the probability of a certain birth defect is higher among women of a certain age
Outcome (y) = presence/absence of birth defect
Explanatory (x) = maternal age a birth
> bdlog<-glm(bd$casegrp~bd$MAGE,family=binomial("logit"))
Since “logit” is the default, you can actually use:
> bdlog<-glm(bd$casegrp~bd$MAGE,binomial)
> summary(bdlog)

3. Example in R
Call: glm(formula = bd$casegrp ~ bd$MAGE, family = binomial("logit")) Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) * bd$MAGE <2e-16 *** --- Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Null deviance: on degrees of freedom Residual deviance: on degrees of freedom AIC:

4. Plot
> plot(MAGE~ fitted(glm(casegrp~ MAGE,binomial)), xlab=“Maternal Age”, ylab=“P(Birth Defect)”, pch=15)

5. Example: Categorical x
Sometimes, it’s easier to interpret logistic regression output if the x variables are categorical
Suppose we categorize maternal age into 3 categories:
> bd$magecat3 <- ifelse(bd$MAGE>25, c(1),c(0))
> bd$magecat2 <- ifelse(bd$MAGE>=20 & bd$MAGE<=25, c(1),c(0))
> bd$magecat1 <- ifelse(bd$MAGE<20, c(1),c(0))
Maternal Age
Birth Defect
< 20 years
20-24 years
> 24 years
Yes
101
105 36 No
1385
3755
6511

6. Example in R
> bdlog2<-glm(casegrp~magecat1+magecat2,binomial)
> summary(bdlog2)
Remember, with a set of dummy variables, you always put in one less variable than category

7. Example in R
Call: glm(formula = casegrp ~ magecat1 + magecat2, family = binomial) Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) <2e-16 *** magecat <2e-16 *** magecat <2e-16 *** --- Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: on degrees of freedom Residual deviance: on degrees of freedom AIC:

8. Interpretation: odds ratios
> exp(cbind(OR=coef(bdlog2),confint(bdlog2)))
Waiting for profiling to be done...
OR % %
(Intercept)
magecat
magecat
This tells us that:
women <20 years have a 13 times greater odds of a birth defect than women >24 years
women years have a 5 times greater odds of a birth defect than women >24 years

9. What variables can we consider dropping?
> anova(bd.log,test="Chisq") Analysis of Deviance Table Model: binomial, link: logit Response: casegrp Terms added sequentially (first to last) Df Deviance Resid. Df Resid. Dev P(>|Chi|) NULL magecat < 2.2e-16 *** magecat < 2.2e-16 *** bthparity *** smoke e-05 ***
Small p-values indicate that all variables are needed to explain the variation in y

10. Goodness of fit statistics
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) < 2e-16 ***
magecat < 2e-16 ***
magecat e-14 ***
bthparity2parous e-05 ***
smokesmoker e-05 ***
---
Signif. codes: 0 ‘***’ ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Null deviance: on degrees of freedom
Residual deviance: on degrees of freedom
AIC:
-2 log L
AIC

11. Binned residual plot
> x<-predict(bd.log)
> y<-resid(bd.log)
> binnedplot(x,y)
Plots the average residual and the average fitted (predicted) value for each bin, or category
Category is based on the fitted values
95% of all values should fall within the dotted line

12. Poisson Regression
Using count data

13. What is a Poisson distribution?

14. Example: children ever born
The dataset has 70 rows representing group-level data on the number of children ever born to women in Fiji:
Number of children ever born
Number of women in the group
Duration of marriage
1=0-4, 2=5-9, 3=10-14, 4=15-19, 5=20-24, 6=25-29
Residence
1=Suva (capital city), 2=Urban, 3=Rural
Education
1=none, 2=lower primary, 3=upper primary, 4=secondary+

16. Poisson regression in R
> ceb1<-glm(y ~ educ + res, offset=log(n), family = "poisson", data = ceb) Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) <2e-16 *** educnone <2e-16 *** educsec <2e-16 *** educupper <2e-16 *** resSuva * resurban * --- Null deviance: on 69 degrees of freedom Residual deviance: on 64 degrees of freedom AIC: Inf
Need to account for different population sizes in each area/group unless data are from same-size populations

17. Assessing model fit 1. Examine AIC score – smaller is better
2. Examine the deviance as an approximate goodness of fit test
Expect the residual deviance/degrees of freedom to be approximately 1
> ceb2$deviance/ceb2$df.residual
[1]
3. Compare residual deviance to a 2 distribution
> pchisq(2646.5, 64, lower=F)
[1] 0

18. Model fitting: analysis of deviance
Similar to logistic regression, we want to compare the differences in the size of residuals between models
> ceb1<-glm(y~educ, family=“poisson", offset=log(n), data= ceb)
> ceb2<-glm(y~educ+res, family=“poisson", offset=log(n), data= ceb)
> 1-pchisq(deviance(ceb1)-deviance(ceb2), df.residual(ceb1)-df.residual(ceb2))
[1]
Since the p-value is small, there is evidence that the addition of res explains a significant amount (more) of the deviance

19. Overdispersion in Poission models
A characteristic of the Poisson distribution is that its mean is equal to its variance
Sometimes the observed variance is greater than the mean
Known as overdispersion
Another common problem with Poisson regression is excess zeros
Are more zeros than a Poisson regression would predict

20. Overdispersion
Use family=“quasipoisson” instead of “poisson” to estimate the dispersion parameter
Doesn't change the estimates for the coefficients, but may change their standard errors
> ceb2<-glm(y~educ+res, family="quasipoisson", offset=log(n), data=ceb)

21. Poisson vs. quasipoisson
Family = “poisson”
Family = “quasipoisson”
Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) <2e-16 *** educnone <2e-16 *** educsec <2e-16 *** educupper <2e-16 *** resSuva * resurban * --- (Dispersion parameter for poisson family taken to be 1) Null deviance: on 69 degrees of freedom Residual deviance: on 64 degrees of freedom
Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) < 2e-16 *** educnone educsec ** educupper * resSuva resurban (Dispersion parameter for quasipoisson taken to be ) Null deviance: on 69 degrees of freedom Residual deviance: on 64 degrees of freedom

22. Models for overdispersion
When overdispersion is a problem, use a negative binomial model
Will adjust  estimates and standard errors
> install.packages(“MASS”)
> library(MASS)
> ceb.nb <- glm.nb(y~educ+res+offset(log(n)), data= ceb) OR
> ceb.nb<-glm.nb(ceb2)
> summary(ceb.nb)

23. NB model in R > ceb.nb$deviance/ceb.nb$df.residual [1] 1.124297
glm.nb(formula = ceb2, x = T, init.theta = , link = log) Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) < 2e-16 *** educnone educsec ** educupper resSuva resurban (Dispersion parameter for Negative Binomial(3.3872) family taken to be 1) Null deviance: on 69 degrees of freedom Residual deviance: on 64 degrees of freedom AIC: Theta: Std. Err.: x log-likelihood:
> ceb.nb$deviance/ceb.nb$df.residual
[1]

24. What if your data looked like…

25. Zero-inflated Poisson model (ZIP)
If you have a large number of 0 counts…
> install.packages(“pscl”)
> library(pscl)
> ceb.zip <- zeroinfl(y~educ+res, offset=log(n), data= ceb)