Sta s cal Models: Day 3

Transcription

1 Sta s cal Models: Day 3 Linear Mixed Effects Models and Generalized Linear Models Vincenzo Coia & David Kepplinger Applied Sta s cs and Data Science Group UBC Sta s cs February 9, 2017

2 Linear Mixed Effects Models

3 Mo va on for LME Example: esoph dataset in R Number of Cases Number of Controls Age Group Each age group has a different rela onship. Regression on each group? Not powerful. Linear Mixed Effects (LME): borrow strength across groups.

4 Defini on With one predictor X, model is: Y = (β 0 + b 0 ) + (β 1 + b 1 ) X + ε. β terms are constants. Called effects. b terms have normal dist n. Called effects. Are zero on average. Indicate group-to-group differences from average. Extensions: Can omit some random/fixed effects. Can add more predictors. Fit using lmer func on in lme4 package.

5 lmer Func on library(lme4) fit <- lmer(ncases ~ ncontrols + (ncontrols agegp), data=esoph) Write formula in this order: 1. Response variable. 2. ~ 3. Variables that get a effect, separated by +. Note: intercept included by default ( 5. Variables that get a effect, separated by +. Note: intercept included by default. 6. (read as grouped by ) 7. The grouping variables, separated by )

6 Exercise: lmer formula Using the sleepstudy dataset, fit an LME on Reaction against Days, grouped by Subject.

7 Challenge: lmer formula Using the Teams dataset from the Lahman package, fit a model on runs (R) from the variables walks (BB) and Hits (H), grouped by team (teamid). Hint: wrap the scale func on around each predictor variable.

8 lmer Output You can extract things out of the lmer model fit. Recall we called fit the output of lmer. summary(fit) prints to the screen a summary of the model fit. plot(fit) gives a residual plot. coef(fit) gives the slopes and intercepts for each regression line. The broom package can also be used to extract the es mates from the model: tidy(fit, fixed ) extracts the fixed effects. tidy(fit, ran_pars ) extracts the es mated covariance matrix for the random effects. tidy(fit, ran_modes ) extracts the es mated of the random effects.

9 lmer Output: Exercise Earlier, you fit a model in response to this exercise: sleepstudy Reaction Days Subject What is the intercept and slope of subject #310?

10 Generalized Linear Models

11 Background So far we covered only a response variable O en we have other responses, e.g. Count data (i.e., 0, 1, 2, 3, ) Binary data (e.g., no/yes, failure/success, 0/1) Models we discussed so far are not appropriate for these data

12 GLMs Similar idea as for linear regression models discussed so far: µ g(µ) = β 0 + β 1 x 1 + β 2 x β p x p The goal is to es mate the parameters β 0, β 1,..., β p such that we fit the observed data well.

13 What Distribu on to Use? The choice of the distribu on is guided by the type of the response data Binary data: distribu on Count data: or distribu on

14 Binomial Distribu on Corresponds the number of successes in m trials Parameterized by the probability of success in a single trial, p If m = 1 (e.g., tossing a coin once), Binomial distributed data can be either 0 or 1 A Binomial GLM is used to model the success probability p Natural link func on is the logit p g(µ) = log( 1 p )

15 Binomial Distribu on Link Func ons p Link cloglog logit probit β 0 + β 1 x β p x p

16 R Exercise HSAUR3

17 R Challenge womensrole HSAUR3?binomial glm

18 Poisson Distribu on Describes the number of events during a fixed period of me (or space, etc.) Characterized by the parameter, µ, the mean and the variance of the Poisson distribu on are equal to µ! A larger also implies a larger spread of the data (strong assump on!) Usual link func on is g(µ) = log(µ)

19 Poisson Distribu on 0.15 Rate Probability Number of Events

20 R Exercise number ~ treat + age polyps

21 Nega ve Binomial Distribu on (NB) Describes the number of successes before a specified number of failures occur Unlike the Poisson distribu on, the mean and variance can be different Similar to the Normal linear regression models, the variance is assumed to be constant in the model! We only want to model the mean µ of the distribu on Usual link func on is g(µ) = log(µ)

22 R Exercise Days ~ Sex * (Age + Eth * Lrn) quine