2. A priori violations
In the following cases, your data violates the normality and homoskedasticity assumption on a priori grounds: (1) count data  Poisson regression (2) binary data  logistic regression

3. A priori violations
In the following cases, your data violates the normality and homoskedasticity assumption on a priori grounds: (1) count data  Poisson regression (2) binary data  logistic regression

8. Output example
> summary(xglm) Call: glm(formula = error ~ alc, family = "binomial") Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) ** alc ***

9. The linear model Y ~ b0 + b1*X1 + b2*X2 so this is how we expand this…
do this on the blackboard …. make a list with X1 = gender, and X2 = focus or no focus …and then 0 times, 1 times etc.

10. p(Y) ~ logit-1(b0 + b1*X1 + b2*X2)
The logistic model
p(Y) ~ logit-1(b b1*X b2*X2)
linear predictor
so this is how we expand this…
do this on the blackboard …. make a list with X1 = gender, and X2 = focus or no focus …and then 0 times, 1 times etc.

11. Representative values
Probability
Odds
Log odds (= “logits”)
0.1
0.111
-2.197
0.2
0.25
-1.386
0.3
0.428
-0.847
0.4
0.667
-0.405
0.5 1
0.6
1.5
0.405
0.7
2.33
0.847
0.8 4
1.386
0.9 9
2.197
- So a probability of 80% of an event occurring means that the odds are “4 to 1” for it occurring
What happens if the odds are 50 to 50? -> ratio is 1
If the probability of non-occurrence is higher than occurrence, fractions
If the probability of occurrence is higher, positive numbers

12. Snijders & Bosker (1999: 212)

13. = inverse logit function
plogis()

14. Estimate Std. Error z value Pr(>|z|) (Intercept) -3. 643 1. 123 -3
Estimate Std. Error z value Pr(>|z|) (Intercept) ** alc ***
for probabilities: transform the entire LP with the logistic function
for odds: transform individual predictors with exp(x)
plogis()

15. General
Linear Model
Generalized
Linear Model

16. = “Generalizing” the General Linear Model to cases that don’t include continuous response variables (in particular categorical ones)
= Consists of two things: (1) an error distribution, (2) a link function
Generalized
Linear Model

17. = “Generalizing” the General Linear Model to cases that don’t include continuous response variables (in particular categorical ones)
= Consists of two things: (1) an error distribution, (2) a link function
Logistic regression: Binomial distribution
Poisson regression:
Poisson distribution
lm(response ~ predictor)
glm(response ~ predictor, family=”binomial”)
glm(response ~ predictor, family=”poisson”)
Logistic regression: Logit link function
Poisson regression:
Log link function

18. Simple linear regression & multiple regression
= generalized linear model with normal error structure and identity link function