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- Word counts - Speech error counts - Metaphor counts - Active construction counts Moving further Categorical count data

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Hissing Koreans Winter & Grawunder (2012)

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No. of Cases Bentz & Winter (2013)

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Poisson Model

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Siméon Poisson 1898: Ladislaus Bortkiewicz Army Corps with few Horses Army Corps lots of Horses few deaths low variability many deaths high variability The Poisson Distribution

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Poisson Regression = generalized linear model with Poisson error structure and log link function

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The Poisson Model Y ~ log(b 0 + b 1 *X 1 + b 2 *X 2 )

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In R: lmer(my_counts ~ my_predictors + (1|subject), mydataset, family="poisson")

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Poisson model output log values predicted mean rate exponentiate

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Poisson Model

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- Focus vs. no-focus - Yes vs. No - Dative vs. genitive - Correct vs. incorrect Moving further Binary categorical data

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Bentz & Winter (2013) Case yes vs. no ~ Percent L2 speakers

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Logistic Regression = generalized linear model with binomial error structure and logistic link function

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The Logistic Model p(Y) ~ logit -1 (b 0 + b 1 *X 1 + b 2 *X 2 )

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In R: lmer(binary_variable ~ my_predictors + (1|subject), mydataset, family="binomial")

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Probabilities and Odds Probability of an Event Odds of an Event

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Intuition about Odds N = 12 What are the odds that I pick a blue marble? Answer: 2/10

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Log odds = logit function

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Representative values ProbabilityOddsLog odds (= “logits”) 0.10.111-2.197 0.20.25-1.386 0.30.428-0.847 0.40.667-0.405 0.510 0.61.50.405 0.72.330.847 0.841.386 0.992.197

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Snijders & Bosker (1999: 212)

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Bentz & Winter (2013)

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Estimate Std. Error z value Pr(>|z|) (Intercept) 1.4576 0.6831 2.134 0.03286 Percent.L2 -6.5728 2.0335 -3.232 0.00123 Case yes vs. no ~ Percent L2 speakers Log odds when Percent.L2 = 0

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Bentz & Winter (2013)

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Estimate Std. Error z value Pr(>|z|) (Intercept) 1.4576 0.6831 2.134 0.03286 Percent.L2 -6.5728 2.0335 -3.232 0.00123 Case yes vs. no ~ Percent L2 speakers For each increase in Percent.L2 by 1%, how much the log odds decrease (= the slope)

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Bentz & Winter (2013)

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Estimate Std. Error z value Pr(>|z|) (Intercept) 1.4576 0.6831 2.134 0.03286 Percent.L2 -6.5728 2.0335 -3.232 0.00123 Case yes vs. no ~ Percent L2 speakers Logits or “log odds” Exponentiate Transform by inverse logit Odds Proba- bilitie s

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Estimate Std. Error z value Pr(>|z|) (Intercept) 1.4576 0.6831 2.134 0.03286 Percent.L2 -6.5728 2.0335 -3.232 0.00123 Case yes vs. no ~ Percent L2 speakers Logits or “log odds” Transform by inverse logit Odds Proba- bilitie s exp(-6.5728)

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Estimate Std. Error z value Pr(>|z|) (Intercept) 1.4576 0.6831 2.134 0.03286 Percent.L2 -6.5728 2.0335 -3.232 0.00123 Case yes vs. no ~ Percent L2 speakers Logits or “log odds” exp(-6.5728) Transform by inverse logit 0.001397 878 Proba- bilitie s

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Odds > 1 < 1 Numerator more likely Denominator more likely = event happens more often than not = event is more likely not to happen

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Estimate Std. Error z value Pr(>|z|) (Intercept) 1.4576 0.6831 2.134 0.03286 Percent.L2 -6.5728 2.0335 -3.232 0.00123 Case yes vs. no ~ Percent L2 speakers Logits or “log odds” exp(-6.5728) Transform by inverse logit 0.001397 878 Proba- bilitie s

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Estimate Std. Error z value Pr(>|z|) (Intercept) 1.4576 0.6831 2.134 0.03286 Percent.L2 -6.5728 2.0335 -3.232 0.00123 Case yes vs. no ~ Percent L2 speakers Logits or “log odds” logit.inv(1.4576) 0.81

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Bentz & Winter (2013) About 80%(makes sense)

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Estimate Std. Error z value Pr(>|z|) (Intercept) 1.4576 0.6831 2.134 0.03286 Percent.L2 -6.5728 2.0335 -3.232 0.00123 Case yes vs. no ~ Percent L2 speakers Logits or “log odds” logit.inv(1.4576) 0.81 logit.inv(1.4576+ -6.5728*0.3) 0.37

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Bentz & Winter (2013)

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= logit function = inverse logit function

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This is the famous “logistic function” logit -1

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Inverse logit function (transforms back to probabilities) logit.inv = function(x){exp(x)/(1+exp(x))} (this defines the function in R)

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General Linear Model General Linear Model Generalized Linear Model Generalized Linear Model Generalized Linear Mixed Model

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General Linear Model General Linear Model Generalized Linear Model Generalized Linear Model Generalized Linear Mixed Model

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General Linear Model General Linear Model Generalized Linear Model Generalized Linear Model Generalized Linear Mixed Model

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Generalized Linear Model Generalized Linear Model = “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

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= “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 Logistic regression: Logit link function Poisson regression: Log link function

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= “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 Logistic regression: Logit link function Poisson regression: Log link function lm(response ~ predictor) glm(response ~ predictor, family="binomial") glm(response ~ predictor, family="poisson")

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Categorical Data Dichotomous/Binary Count Logistic Regression Poisson Regression

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General structure Linear Model continuous~any type of variable Logistic Regression dichotomous~any type of variable Poisson Regression count~any type of variable

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For the generalized linear mixed model… … you only have to specify the family. lmer(…) lmer(…,family="poisson") lmer(…,family="binomial")

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That’s it (for now)

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