Generalized Linear Models

Transcription

1 Generalized Linear Models

2 Assumptions of Linear Model Homoskedasticity Model variance No error in X variables Errors in variables No missing data Missing data model Normally distributed error GLM Error in Y variables is measurement error Observations are independent

3 Generalized Linear Models Retains linear function Allows for alternate PDFs to be used in likelihood However, with many non-normal PDFs the range of the model parameters does not allow a linear function to be used safely Pois(λ): λ > 0 Binom(n,θ) 0 < θ < 1 Typically a link function is used to relate linear model to PDF

4 Link Functions Canonical Link Functions Distribution Link Name Link Function Mean Function Xb = µ µ = Xb Normal Identity Exponential Inverse Xb = µ-1 µ = (Xb)-1 Gamma Xb = ln(µ) µ = exp(xb) Poisson Log exp Xb Binomial = Logit Xb=ln Multinomial 1 exp Xb 1 Can use most any function as a link function but may only be valid over a restricted range Most are technically nonlinear functions

5 Logistic Regression Common model for the analysis of boolean data (0/1, True/False, Present/Absent) Assumes a Bernoulli likelihood Bern(θ) = Binom(1,θ) Likelihood specification y~bern logit = X Data Model ~N B0, V B Parameter Model Process Model Bayesian

6 Logistic Regression y ~Binom 1, logit X 1 X Y Data Model β Process Model Β0,Vβ Parameter Model

7 Bayesian Logistic Regression model { ## priors for(i in 1:2) { beta[i] ~ dnorm(0,0.001)} } for(i in 1:n){ logit(mu[i]) <- beta[1]beta[2]*x[i] y[i] ~ dbern(mu[i]) }

8

9 Logisitic Exponential: Tree Mortality Logistic model of annual survival probability logit = X Bernoulli likelihood t Normal prior Y ~Bern ~N B 0, V 0 Metropolis MCMC 50K steps, 6-20K burn-in, thin 1/3 Dietze and Moorcroft 2001 Global Change Biology

10 Distribution of Eastern Forest

11 Mortality Covariates Climate Landscape/abiotic Precipitation Elevation Summer Max Temp Slope Winter Min Temp Radiation index Topographic Moisture Air Pollutants Stand scale biotic NO3 deposition SO4 deposition DBH Ozone Stand Basal Area Stand Age

12 Annual Mortality Probability Binned Data Mean Binned Data CI Model w/ average covariates Binned model w/ full covariates Annual Precipitation (meters)

13

14

15 Survival Covariate Effects Climate PFT Early HW N Mid HW S Mid HW Late HW N.Pine S. Pine Mid Con Late Con Evergreen Hydric int Pollution Precip Tmin Tmax no3 ns ns ns ns so4 Stand O3 age BA ns ns ns ns ns Topography slope rad elev TCI ns ns ns ns ns ns ns ns ns ns ns ns ns ns ns ns ns ns ns ns n 78, , ,437 78,635 18, ,476 10, ,397 7,891 22,352

16 Parameter Sensitivity Climate PFT Precip Tmin Early HW N Mid HW 1.82 S Mid HW Pollution Tmax Stand Topography no3 so4 NxS O3 age BA dia slope rad elev TCI MEAN Late HW N.Pine S. Pine Mid Con Late Con Evergreen Hydric MEAN i S i =1000 SD Y pred X

17 Logit-Normal Model Bernoulli likelihood only accounts for sampling error Can add additional extrabinomial variation to account for the fact that the covariates do not full account for the mean risk y~bern logit = X 2 ~N 0, Fairly sensitive to prior on sigma

18 Logit-Normal Regression model <- function(){ for(i in 1:2) { beta[i] ~ dnorm(0,0.01)} sigma ~ dgamma(1,0.1) for(i in 1:n){ Ey[i] <- beta[1]beta[2]*x[i] mu[i] ~ dnorm(ey[i],sigma) logit(theta[i]) <- mu[i] y[i] ~ dbern(theta[i]) } }

19

20 Poisson Regression Most common link is log(λ) y~poisson log = X Commonly used to model count data Especially for low counts were poor normal approx Easily generalized to Negative Binomial regression (not canned) More flexible in alternative link functions (λ > 0) Lab 4: cone counts

21 Poisson Regression y ~Pois exp X X Y Data Model β Process Model Β0,Vβ Parameter Model

22

23 Likelihood in R Option 1 glm(y ~ x,family=poisson(link= log )) Option 2 lnl = function(beta){ -sum(dpois(y,exp(beta[0] beta[1]*x,log=t))) }

24 AIC

25 Bayesian Poisson Regression y~poisson log = X ~N B0, V B model{ for(i in 1:2) { beta[i] ~ dnorm(0,0.001)} for(i in 1:n){ log(mu[i]) <- beta[1]beta[2]*x[i] y[i] ~ dpois(mu[i]) } }

26

27 Multinomial Regression Categorical variable as a function of covariates in a linear model Probability of falling within each group Categories can be ordered (ordinal) or unordered (nominal) Multivariate extension of the logistic regression to >2 catagories cumulative logit model End up with K-1 regression models for K classes

28 Logistic Regression n y~binom 1, = y y n y 1 logit =log =X 1 Multinomial Regression n y~multinom 1, 1,, K = y y 1 K th For the k class: 1 k log = X k 1 1 k K k=1 yk k

29 Inverting the link function for the FIRST class exp X 1 1 = 1 exp X 1 For the middle classes k 1 exp X k k = j 1 exp X k j=1 For the LAST class K 1 K =1 j j=1

30 Additional Restrictions Because the regression model for each of the K-1 categories is expressed in terms of the cumulative probability of all categories up to that one, these curves cannot cross each other Slopes and intercepts must be ordered β0,k < β0,k1 and β1,k < β1,k1 for all k Range restrictions can be accommodated in the prior using an indicator function I(A) I(A) = 1 if A is true and I(A) = 0 if A is false

31 Multinomial Regression Priors { { N b0,1, V b I 0,1 0,2 k=1 0, k ~ N b0, k, V b I 0, k 1 0, k 0, k 1 1 k K 1 N b0, K 1, V b I 0, K 2 0, K 1 k =K 1 N b1,1,v b I 1,1 1,2 k=1 1, k ~ N b1, k, V b I 1, k 1 1, k 1, k 1 1 k K 1 N b1, K 1, V b I 1, K 2 1, K 1 k=k 1

32

33 Multinomial BUGS code model{ beta[1, 1] ~ dnorm(0.0, 0.001) T(,beta[2, 1]) beta[2, 1] ~ dnorm(0.0, 0.001) beta[1, 2] ~ dnorm(0.0, 0.001) T(,beta[2, 2]) beta[2, 2] ~ dnorm(0.0, 0.001) for (i in 1:n) { logit(mu[i, 1]) <- beta[1, 1] beta[1, 2] * x[i] logit(cmu2[i]) <- beta[2, 1] beta[2, 2] * x[i] mu[i, 2] <- cmu2[i] - mu[i, 1] mu[i, 3] <- 1 - cmu2[i] y[i, ] ~ dmulti(mu[i, ], 1) } }

34

35

36 Assumptions of Linear Model Homoskedasticity Model variance No error in X variables Errors in variables No missing data Missing data model Normally distributed error GLM Error in Y variables is measurement error Observations are independent