- Word counts
Moving further
- Speech error counts
-Categorical
Metaphor countscount
data
- Active construction counts
Hissing Koreans
Winter & Grawunder (2012)
No. of Cases
Bentz & Winter (2013)
Alcohol Level
0
4
8
12
16
20
24
No. of Speech Errors
28
32
Alcohol Level
0
4
8
12
16
20
24
No. of Speech Errors
28
32
0
4
8
12
16
20
24
No. of Speech Errors
28
32
Poisson Model
Alcohol Level
The Poisson Distribution
few deaths
Army
Corps
with few
Horses
Siméon Poisson
Army
Corps
lots of
Horses
low
variability
many
deaths
high
variability
1898: Ladislaus Bortkiewicz
Poisson Regression
= generalized linear model
with Poisson error structure
and log link function
The Poisson Model
Y ~ log(b0 +
b1*X1
+
b2*X2)
In R:
lmer(my_counts ~ my_predictors +
(1|subject), mydataset,
family="poisson")
Poisson model output
ex
log
values
e
x
exponentiate
predicted
mean rate
0
4
8
12
16
20
24
No. of Speech Errors
28
32
Poisson Model
Alcohol Level
- Focus vs. no-focus
Moving further
- Yes vs. No
-Binary
Dative vs.categorical
genitive
- Correct vs. incorrect
data
Case yes vs. no ~ Percent L2 speakers
Bentz & Winter (2013)
1
0
Accuracy
5
10
Noise level (db)
15
20
1
0
Accuracy
5
10
Noise level (db)
15
20
1
0
Accuracy
5
10
Noise level (db)
15
20
1
0
Accuracy
5
10
Noise level (db)
15
20
Logistic Regression
= generalized linear model
with binomial error structure
and logistic link function
The Logistic Model
p(Y) ~ logit-1(b0 +
b1*X1
+
b2*X2)
In R:
lmer(binary_variable ~ my_predictors +
(1|subject), mydataset,
family="binomial")
Probabilities and Odds
p(x)
Probability of an
Event
p(x)
1- p(x)
Odds of an
Event
Intuition about Odds
What are the odds that I
pick a blue marble?
N = 12
Answer:
2/10
Log odds
æ p(x) ö
log ç
÷
è 1- p(x) ø
= logit function
Representative values
Probability
Odds
Log odds (= “logits”)
0.1
0.2
0.3
0.4
0.111
0.25
0.428
0.667
-2.197
-1.386
-0.847
-0.405
0.5
0.6
0.7
1
1.5
2.33
0
0.405
0.847
0.8
4
1.386
0.9
9
2.197
Snijders & Bosker (1999: 212)
Bentz & Winter (2013)
Case yes vs. no ~ Percent L2 speakers
Estimate Std.
(Intercept)
1.4576
Percent.L2
-6.5728
Error z value
0.6831
2.134
2.0335 -3.232
Log odds when Percent.L2 = 0
Pr(>|z|)
0.03286
0.00123
Bentz & Winter (2013)
Case yes vs. no ~ Percent L2 speakers
Estimate Std.
(Intercept)
1.4576
Percent.L2
-6.5728
Error z value
0.6831
2.134
2.0335 -3.232
For each increase in Percent.L2
by 1%, how much the log odds
decrease (= the slope)
Pr(>|z|)
0.03286
0.00123
Bentz & Winter (2013)
Case yes vs. no ~ Percent L2 speakers
Estimate Std.
(Intercept)
1.4576
Percent.L2
-6.5728
Error z value
0.6831
2.134
2.0335 -3.232
Pr(>|z|)
0.03286
0.00123
Exponentiate
Odds
Transform by
inverse logit
Probabilitie
Logits or
“log odds”
Case yes vs. no ~ Percent L2 speakers
Estimate Std.
(Intercept)
1.4576
Percent.L2
-6.5728
Error z value
0.6831
2.134
2.0335 -3.232
Pr(>|z|)
0.03286
0.00123
exp(-6.5728)
Odds
Transform by
inverse logit
Probabilitie
Logits or
“log odds”
Case yes vs. no ~ Percent L2 speakers
Estimate Std.
(Intercept)
1.4576
Percent.L2
-6.5728
Error z value
0.6831
2.134
2.0335 -3.232
Pr(>|z|)
0.03286
0.00123
exp(-6.5728)
0.001397
878
Transform by
inverse logit
Probabilitie
Logits or
“log odds”
Odds
p(x)
>1
1- p(x)
p(x)
<1
1- p(x)
Numerator
more likely
= event happens
more often than not
Denominator
more likely
= event is more
likely not to happen
Case yes vs. no ~ Percent L2 speakers
Estimate Std.
(Intercept)
1.4576
Percent.L2
-6.5728
Error z value
0.6831
2.134
2.0335 -3.232
Pr(>|z|)
0.03286
0.00123
exp(-6.5728)
0.001397
878
Transform by
inverse logit
Probabilitie
Logits or
“log odds”
Case yes vs. no ~ Percent L2 speakers
Estimate Std.
(Intercept)
1.4576
Percent.L2
-6.5728
Logits or
“log odds”
Error z value
0.6831
2.134
2.0335 -3.232
logit.inv(1.4576)
Pr(>|z|)
0.03286
0.00123
0.81
About
80%(makes
sense)
Bentz & Winter (2013)
Case yes vs. no ~ Percent L2 speakers
Estimate Std.
(Intercept)
1.4576
Percent.L2
-6.5728
Logits or
“log odds”
Error z value
0.6831
2.134
2.0335 -3.232
Pr(>|z|)
0.03286
0.00123
logit.inv(1.4576)
0.81
logit.inv(1.4576+
-6.5728*0.3)
0.37
Bentz & Winter (2013)
x
æ p(x) ö
log ç
÷
è 1- p(x) ø
e
x
1+ e
= logit
function
= inverse
logit function
x
This is the famous
“logistic function”
logit-1
e
x
1+ e
= inverse
logit function
Inverse logit function
logit.inv = function(x){exp(x)/(1+exp(x))}
(this defines the
function in R)
(transforms back to
probabilities)
General
Linear
Model
Generalized
Linear
Model
Generalized
Linear
Mixed
Model
General
Linear
Model
Generalized
Linear
Model
Generalized
Linear
Mixed
Model
General
Linear
Model
Generalized
Linear
Model
Generalized
Linear
Mixed
Model
= “Generalizing” the
General Linear Model to
cases that don’t include
continuous response
variables (in particular
categorical ones)
Generalized
Linear
Model
= Consists of two things:
(1) an error distribution,
(2) a link function
= “Generalizing” the
Logistic regression:
General
Linear
Model
to
Binomial distribution
cases that don’t include
Poisson regression:
continuous response
Poisson distribution
variables (in particular
categorical ones)
Logistic regression:
Logit link function
Poisson regression:
Log link function
= Consists of two things:
(1) an error distribution,
(2) a link function
=lm(response
“Generalizing”
the
~ predictor)
Logistic regression:
General
Linear
Model
to
Binomial distribution
cases
that don’t
glm(response
~ include
predictor,
Poisson regression: family="binomial")
continuous response
Poisson distribution
variables
(in particular
glm(response ~ predictor,
categorical
ones)
family="poisson")
Logistic regression:
Logit link function
Poisson regression:
Log link function
= Consists of two things:
(1) an error distribution,
(2) a link function
Categorical Data
Dichotomous/Binary
Count
Logistic
Regression
Poisson
Regression
General structure
Linear Model
continuous ~ any type of variable
Logistic Regression
dichotomous ~ any type of variable
Poisson Regression
count
~ any type of variable
For the generalized linear
mixed model…
… you only have to specify the family.
lmer(…)
lmer(…,family="poisson")
lmer(…,family="binomial")
That’s it
(for now)