Generalized Linear Models

Transcription

1 Generalized Linear Models Ordinal Logistic Regression Dr. Tackett / 26

2 Announcements HW 8 due Thursday, 11/29 Lab 10 due Sunday, 12/2 Exam II, Thursday 12/6 2 / 26

3 Packages library(knitr) library(broom) library(dplyr) library(tibble) library(ggplot2) library(readr) library(cowplot) library(arm) #binned residuals library(nnet) #multinomial regression models 3 / 26

4 Multinomial Logistic Regression 4 / 26

5 Multinomial Logistic Regression If we have an explanatory variable, then we want to be a function of Choose a baseline category. Let's choose. Then, 5 / 26

6 Multinomial Logistic Regression If we have an explanatory variable, then we want to be a function of Choose a baseline category. Let's choose. Then, We have a separate model for each category 5 / 26

7 Multinomial Logistic Regression Suppose we have a response variable that can take three possible outcomes that are coded as 1,2,3 Let 1 be the baseline category. Then 6 / 26

8 Calculating Probabilities For, we calculate the probabilities, as For the baseline category,, we calculate the probability as 7 / 26

9 Interpreting Coe cients For every one unit increase in, the log-odds of vs. increase by Con dence Interval: to For every one unit increase in, the odds of vs. multiply by a factor of Con dence Interval: to 8 / 26

10 Model Diagnostics For each category of the response, : Analyze a plot of the binned residuals vs. each continuous explanatory variable Look for any patterns in the residuals plots For each categorical explanatory variable: Examine the average residuals for each category of the response variable 9 / 26

11 Example: Sesame Street We will analyze data from an experiment to test the e ectiveness of the children's program Sesame Street. As part of the experiment, children were assigned to one of two groups: those who were encouraged to watch the program and those who were not We want to understand what e ect the encouragement had on the frequency of viewing after adjusting for other characteristics 10 / 26

12 Response Variable viewcat 1: rarely watched show 2: once or twice a week 3: three to ve times a week 4: watched show on average more than ve times a week 11 / 26

13 Explanatory Variables age: child's age in months prenumb: score on numbers pretest (0 to 54) prelet: score on letters pretest (0 to 58) viewenc: 1: encouraged to watch, 0: not encouraged site: 1: three to ve year old from disadvantaged inner city area 2: four year old from advantaged suburban area 3: from advantaged rural area 4: from disadvantaged rural area 5: from Spanish speaking home 12 / 26

14 Example: Sesame Street Full description of data Analysis 13 / 26

15 Example: Sesame Street We built a model that could be used to describe the relationship between the amount a child watched Sesame Street and whether or not they were encouraged to watch the show The response variable viewcat has a natural and meaningful ordering which gets lost in the multinomial logistic model We can build an ordinal logistic model that takes into account the natural ordering of the response variable 14 / 26

16 Models for Ordinal Responses 15 / 26

17 Ordinal Responses Suppose is a response variable that can take values is an ordinal response if there is a natural ordering for categories Examples: Movie Ratings: 1 to 5 stars Survey Questions: Strongly Disagree to Strongly Agree Levels of Viewership: Rarely, Weekly, Daily 16 / 26

18 Cumulative Probabilities Suppose is a variable that can take values Assume the ordering Let be the associated probabilities The cumulative probability of response level j is 17 / 26

19 Proportional Odds Model Let be an ordinal response variable that takes levels with associated probabilities The proportional odds model can be written as the following: 18 / 26

20 Understanding the Model Why "proportional"? The proportional odds model assumes that each explanatory variable has the same e ect on the log-odds of being at or below category, regardless of what is 19 / 26

21 Understanding the Model Why "proportional"? The proportional odds model assumes that each explanatory variable has the same e ect on the log-odds of being at or below category, regardless of what is Why? Form of the model calculated by the polr() function in R Leads to a more intuitive interpretation of the coe cients 19 / 26

22 Interpreting the Intercept, When all explanatory variables equal 0, the log-odds of falling at or below category are 20 / 26

23 Interpreting the Intercept, When all explanatory variables equal 0, the log-odds of falling at or below category are When all explanatory variables equal 0, the odds of falling at or below category are 20 / 26

24 Interpreting the Slope, When increases by one unit, the log-odds of falling at or below category decrease by 21 / 26

25 Interpreting the Slope, When increases by one unit, the log-odds of falling at or below category decrease by When increases by one unit, the odds of falling at or below category multiply by a factor of 21 / 26

26 Interpreting the Slope, When increases by one unit, the log-odds of falling at or below category decrease by When increases by one unit, the odds of falling at or below category multiply by a factor of Con dence Intervals: Log-Odds: Odds: to to 21 / 26

27 Sesame Street: Interpreting prenumbcent Analysis For every one additional point on the numbers pretest, the logodds of a child being in viewing category or below, are expected to decrease by about For every one additional point on the numbers pretest, the odds of a child being in viewing category or below are expected to multiply by about / 26

28 Predicted Probabilities Predicted Cumulative Probability: 23 / 26

29 Predicted Probabilities Predicted Cumulative Probability: Predicted Probability: 23 / 26

30 Model Diagnostics For each category of the response, : Analyze a plot of the binned residuals vs. each continuous explanatory variable Look for any patterns in the residuals plots For each categorical explanatory variable: Examine the average residuals for each category of the response variable 24 / 26

31 Change in Deviance Test Use the change in deviance test to compare models (similar to logistic regression) To test signi cance of predictor(s) Fit model with predictor(s) and without predictor(s) Use anova() function to perform change in deviance test (test="chisq") Null hypothesis: coe cient(s) of additional predictor(s) equal to 0 25 / 26

32 Proportional Odds Model in R Use the polr() function the MASS package #calculate the proportional odds model library(mass) my.model <- polr(y ~ X1 + X Xp, data=my.data) # calculate predicted probabilities of being in each cateogry predicted.probability <- predict(my.model,type="probs") # find the predicted category for each observation predicted.category <- predict(my.model,type="class") 26 / 26