1. DCAL Stats Workshop
Bodo Winter

2. What if your response is not continuous?
General
Linear Model
Generalized
Linear Model
Poisson regression
logistic regression

3. Background: logarithms
Logarithms transform a numbers into magnitudes Example: transforming a number by the log10 function log10(1) = 0 log10(10) = 1 log10(100) = 2 log10(1000) = 3

4. Background: logarithms
The exponential function is the inverse of the logarithmic function log10(1) = 0 log10(10) = 1 log10(100) = 2 log10(1000) = 3

5. Background: Logarithms of decimals
log10(0.1) = -1 log10(0.01) = -2 log10(0.001) = -3

6. Background: logarithms
The exponential function is the inverse of the logarithmic function log10(1) = = 0 log10(10) = = 1 log10(100) = = 100 log10(1000) = = 1000

7. Logarithms: a visual example

8. Logarithms: a visual example

9. In R:
log(x) exp(x)

10. What if your response is not continuous?
General
Linear Model
Generalized
Linear Model
Poisson regression
logistic regression

11. Generalized Linear Models: Three Ingredients
Assumed probability distribution of the response: normal (“Gaussian”), Poisson, binomial A linear predictor (LP) = “just a new name for your regression equation” A link function (identity, log, logit)

12. Three common GLMs (and their link functions)
𝑔 𝜇 =𝜇
Linear regression
𝑔 𝜇 =log⁡(𝜇)
Poisson regression
𝑔 𝜇 =log⁡( 𝜇 1−𝜇 )
Logistic regression

13. The Poisson distribution

14. Example where I used Poisson regression
Winter, B. & Ardell, D. (in prep.). Rethinking Zipf’s frequency-meaning relationship: The role of contextual diversity.

15. Another example: Bentz & Winter (2013)
Bentz, C., & Winter, B. (2013). Languages with more second language learners tend to lose nominal case. Language Dynamics & Change, 3:1, 1-27.

16. Example: modeling speech errors

17. Example: modeling speech errors, linear model

18. Example: modeling speech errors, Poisson model

19. Poisson model in R
xmdl <- glm(error ~ alcohol, data = xdata, family = ‘poisson’)
Estimate
(Intercept)
alcohol

20. Poisson model in R
xmdl <- glm(error ~ alcohol, data = xdata, family = ‘poisson’)
Estimate
(Intercept)
alcohol
log coefficients

21. Poisson model in R
xmdl <- glm(error ~ alcohol, data = xdata, family = ‘poisson’)
Estimate
(Intercept)
alcohol
log coefficients
example: What do we predict for alcohol = 2?
* 2 = 0.8 (logged value)

22. Poisson model in R
xmdl <- glm(error ~ alcohol, data = xdata, family = ‘poisson’)
Estimate
(Intercept)
alcohol
log coefficients
example: What do we predict for alcohol = 2?
* 2 = 0.8 (logged value)
exp(0.8) = 2.2

23. What if your response is not continuous?
General
Linear Model
Generalized
Linear Model
Poisson regression
logistic regression

24. Logistic regression with speech errors (yes / no)

25. Logistic regression with speech errors (yes / no)

26. Logistic regression with speech errors (yes / no)

27. Logistic regression with speech errors (yes / no)

28. Odds and log odds examples
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

29. Logistic function

30. for probabilities: transform the entire LP with the logistic function
Estimate Std. Error z value Pr(>|z|) (Intercept) ** alc ***
for probabilities: transform the entire LP with the logistic function
plogis()

31. Exercise I + II Poisson model: Nettle (1999)
Gesture data (height versus shape) from Hassemer & Winter (2016)

32. Hassemer & Winter (2016)
Shape
Height