Modeling Linear Models Generalized Linear Models. Modeling in R. Peter Dalgaard. Department of Biostatistics University of Copenhagen
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1 Models in R Peter Dalgaard Department of Biostatistics University of Copenhagen Mixed Models in R, Copenhagen, January 2006
2 Outline Modeling Linear Models Generalized Linear Models
3 Modeling Tools: Overview Model formulas Model objects and summaries Comparing models Evaluating model fit Generalized linear models
4 Model Formulas Linear model, y = Xβ + ɛ In practice something like y = β 0 + β 1 height + β 2 1 (type=2) + β 3 1 (type=3) + ɛ Wilkinson-Rogers formulas: y = height + type (Interpretation depends on whether variables are categorical or continuous)
5 Model Formulas in R R representation y ~ height + type where type is a factor Interactions a:b, a*b = a + b + a:b Algebra (a:(b + c) = a:b + a:c etc.) Notice special interpretation of operators Special items: offset, -1 (no intercept)
6 Fitting Linear Models data(airquality) aq <- transform(airquality, Month=factor(Month)) fit.aq <- lm(log(ozone) ~ Solar.R + Wind + Temp + Month, data=aq) lm generates a fitted model object Extract information from model object Fit other models based on model object
7 Inspecting Model Objects Extract information about the fit summary(fit.aq) fitted(fit.aq), resid(fit.aq) anova(fit.aq, fit.aq2) plot(fit.aq) diagnostics predict(fit.aq, newdata)
8 Model Search anova(model) Type I sum of squares drop1 ( Type III ), add1 step (AIC/BIC) criteria update modifies a previous model
9 Demo 1 data(airquality) aq <- transform(airquality, Month=factor(Month)) fit.aq <- lm(log(ozone) ~ Solar.R + Wind + Temp + Month, data=aq) fit.aq2 <- update(fit.aq, ~. - Month) summary(fit.aq) plot(fit.aq) drop1(fit.aq, test="f") anova(fit.aq, fit.aq2)
10 Contrasts Categorical variables generally need a representation using one less coefficient than the number of categories Notice that the coefficients for Month are differences to the month of May. The R default is to set the first level of a factor to zero (treatment contrasts) for an unordered contrast However, other parametrizations are available: Sum contrasts, Helmert contrasts, successive differences etc. These can be set as attributes to individual factors, e.g. contrasts(fff) <- "contr.sum". The right hand side can be a name of a contrast generating function, or a matrix (n (n 1)) You can also set a specific contrast in a model formula, using terms like C(Month,"contr.sum")
11 Contrast Matrices Contrast matrices are typically created by calls like > contr.treatment(3) (some contrast types allow further parameters so that you can limit the degree of polynomial contrasts) Each row of the contrast matrix is inserted in the design matrix for observations in that group. Notice that the matrices map from contrasts to fitted values.
12 Generalized Linear Models Statistical distribution (exponential) family Link function transforming mean to linear scale Deviance Examples; Binomial, Poisson, Gaussian (σ known in principle) Canonical link functions: logit, log, identity Fit using glm in R
13 Demo 2 no.yes <- c("no","yes") smoking <- gl(2, 1, 8, no.yes) obesity <- gl(2, 2, 8, no.yes) snoring <- gl(2, 4, 8, no.yes) n.tot <- c(60,17,8,2,187,85,51,23) n.hyp <- c(5,2,1,0,35,13,15,8) data.frame(smoking,obesity,snoring,n.tot,n.hyp) hyp.tbl <- cbind(n.hyp,n.tot-n.hyp) glm.hyp <- glm(hyp.tbl~smoking+obesity+snoring, family=binomial("logit")) summary(glm.hyp) library(mass) confint(glm.hyp) confint.default(glm.hyp)
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