Ben Baumer Instructor

Transcription

1 MULTIPLE AND LOGISTIC REGRESSION What is logistic regression? Ben Baumer Instructor

2 A categorical response variable ggplot(data = hearttr, aes(x = age, y = survived)) + geom_jitter(width = 0, height = 0.05, alpha = 0.5)

3 Making a binary variable hearttr <- hearttr %>% mutate(is_alive = ifelse(survived == "alive", 1, 0))

4 Visualizing a binary response data_space <- ggplot(data = hearttr, aes(x = age, y = is_alive)) + geom_jitter(width = 0, height = 0.05, alpha = 0.5)

5 Regression with a binary response data_space + geom_smooth(method = "lm", se = 0)

6 Limitations of regression Could make non-sensical predictions Binary response problematic

7 Generalized linear models generalization of multiple regression model non-normal responses special case: logistic regression models binary response uses logit link function logit(p) = log ( p 1 p ) 0 1 = β + β x

8 Fitting a GLM glm(is_alive ~ age, data = hearttr, family = binomial) binomial() ## Family: binomial ## Link function: logit

9 MULTIPLE AND LOGISTIC REGRESSION Let's practice!

10 MULTIPLE AND LOGISTIC REGRESSION Visualizing logistic regression Ben Baumer Instructor

11 The data space data_space

12 Regression data_space + geom_smooth(method = "lm", se = 0)

13 Using geom_smooth() data_space + geom_smooth(method = "lm", se = 0) + geom_smooth(method = "glm", se = 0, color = "red", method.args = list(family = "binomial"))

14 Using bins data_binned_space

15 Adding the model to the binned plot data_binned_space + geom_line(data = augment(mod, type.predict = "response"), aes(y =.fitted), color = "blue")

16 MULTIPLE AND LOGISTIC REGRESSION Let's practice!

17 MULTIPLE AND LOGISTIC REGRESSION Three scales approach to interpretation Ben Baumer Instructor

18 Probability scale y^ = exp ( β^0 + β^1 x) 1 + exp( β^0 + β^1 x) hearttr_plus <- mod %>% augment(type.predict = "response") %>% mutate(y_hat =.fitted)

19 Probability scale plot ggplot(hearttr_plus, aes(x = age, y = y_hat)) + geom_point() + geom_line() + scale_y_continuous("probability of being alive", limits = c(0, 1))

20 Odds scale y^ odds( y^ ) = = exp ( β^0 + β^1 x) 1 y^ hearttr_plus <- hearttr_plus %>% mutate(odds_hat = y_hat / (1 - y_hat))

21 Odds scale plot ggplot(hearttr_plus, aes(x = age, y = odds_hat)) + geom_point() + geom_line() + scale_y_continuous("odds of being alive")

22 Log-odds scale y^ logit( y^ ) = log [ ] = β^0 + β^1 x 1 y^ hearttr_plus <- hearttr_plus %>% mutate(log_odds_hat = log(odds_hat))

23 Log-odds plot ggplot(hearttr_plus, aes(x = age, y = log_odds_hat)) + geom_point() + geom_line() + scale_y_continuous("log(odds) of being alive")

24 Comparison Probability scale scale: intuitive, easy to interpret function: non-linear, hard to interpret Odds scale scale: harder to interpret function: exponential, harder to interpret Log-odds scale scale: impossible to interpret function: linear, easy to interpret

25 Odds ratios odds( y^ x + 1) exp ( β^0 + β^1 (x + 1)) OR = = = exp β odds( y^ x) exp ( β^0 + β^1 x) 1 exp(coef(mod)) (Intercept) age

26 MULTIPLE AND LOGISTIC REGRESSION Let's practice!

27 MULTIPLE AND LOGISTIC REGRESSION Using a logistic model Ben Baumer Instructor

28 Learning from a model mod <- glm(is_alive ~ age + transplant, data = hearttr, family = binomial) exp(coef(mod)) ## (Intercept) age transplanttreatment ##

29 Using augment() # log-odds scale augment(mod) ## is_alive age transplant.fitted.se.fit.resid.hat ## control ## control ## control ## control ## control ## control ## control ## treatment ## control ## control

30 Making probabilistic predictions # probability scale augment(mod, type.predict = "response") ## is_alive age transplant.fitted.se.fit.resid.hat ## control ## control ## control ## control ## control ## control ## control ## treatment ## control ## control

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32 Out-of-sample predictions cheney <- data.frame(age = 71, transplant = "treatment") augment(mod, newdata = cheney, type.predict = "response") ## age transplant.fitted.se.fit ## 1 71 treatment

33 Making binary predictions mod_plus <- augment(mod, type.predict = "response") %>% mutate(alive_hat = round(.fitted)) mod_plus %>% select(is_alive, age, transplant,.fitted, alive_hat) ## is_alive age transplant.fitted alive_hat ## control ## control ## control ## control ## control ## control ## control ## treatment ## control ## control

34 Confusion matrix mod_plus %>% select(is_alive, alive_hat) %>% table() ## alive_hat ## is_alive 0 1 ## ##

35 MULTIPLE AND LOGISTIC REGRESSION Let's practice!