IntroductionLinear Models and Generalized Linear ModelsThe Big PictureVariablesModelsDataOverview of Statistics 572Bret LargetDepartments of Botany and of StatisticsUniversity of Wisconsin—MadisonJanuary 22, 2008Statistics 572 (Spring 2008) Overview January 21, 2008 1 / 11IntroductionWelcome to Statistics 572!IntroductionBret LargetComment on syllabus.Textbook (better written than last year’s choice)Web for notes and grades (print notes before lecture)ObjectivesComputing (go R!)Assignments (late policy)Exams (save dates)GradingAcademic honestyDiscussion sections (attend the one you want)Statistics 572 (Spring 2008) Overview January 21, 2008 2 / 11IntroductionSome Changes from Last YearThe textbook is better written, so I can count on you to learn throughreading more than in the past. I will not try to cover every topicin lecture.Instead of attempting to blend some new ideas into the structure ofwhat other instructors have done with 572 in the past, I am (mostly)going to cover what I want to and will not cover some topics thatused to be in the course. I will stay pretty close to the textbook.I will attempt to incorporate some more active learning opportunitiesin class.I want you to become adept in R so I will spend more time on it inclass and will ask you to do more with it on homework.Statistics 572 (Spring 2008) Overview January 21, 2008 3 / 11Linear Models and Generalized Linear Models The Big PictureThe Big PictureA statistical approach to data analysis can lend insight to biologicalunderstanding of a wide variety of problems.In a statistical approach, measurable variables are treated asrealizations from a model that relates biological meaningfulparameters and stochastic sources of variation.No model accounts for all aspects of the underlying biology, but anappropriately selected model can be very useful.Many data analysis problems arising from the biological sciences areappropriate for linear and generalized linear models, a rich family ofpossible models.Statistics 572 (Spring 2008) Overview January 21, 2008 4 / 11Linear Models and Generalized Linear Models VariablesVariablesTypically, one variable of interest is modeled as a response variablewhich is related to one or more explanatory variables.Variables can be categorized as quantitative or categorical.Quantitative variables are typically either measured on a continuousscale or are discrete, variables that are counts.The appropriate choice of model is determined in part by the types ofthe response and explanatory variables.A linear combination of the variables X1, . . . , Xktakes the formβ1X1+ β2X2+ · · · + βkXkLinear and generalized linear models include linear combinations ofexplanatory variables.Statistics 572 (Spring 2008) Overview January 21, 2008 5 / 11Linear Models and Generalized Linear Models ModelsExamples of Linear ModelsSimple Linear Regression.—response variable: continuous quantitative variableexplanatory variable: one quantitative variableerror structure: normal distributionmodel: yi= β0+ β1xi+ ei, ei∼ N(0, σ2)example: response variable is phosphorous concentration in planttissue, explanatory variable is phosphorous concentrationin the soil.Multiple Linear Regression.—response variable: continuous quantitative variableexplanatory variables: more than one quantitative variableserror structure: normal distributionmodel: yi= β0+ β1x1i+ · · · + βkxki+ ei, ei∼ N(0, σ2)example: response variable is soybean yield, explanatory variablesare hours of daylight and amount of nitrogen.Statistics 572 (Spring 2008) Overview January 21, 2008 6 / 11Linear Models and Generalized Linear Models ModelsExamples of Linear Models (cont.)One-way ANOVA.—response variable: continuous quantitative variableexplanatory variable: one categorical variableerror structure: normal distributionmodel: yij= αi+ eij, eij∼ N(0, σ2)example: response variable is milk yield explanatory variable isdiet (four treatments)Multi-way ANOVA.—response variable: continuous quantitative variableexplanatory variables: more than one categorical variableserror structure: normal distributionmodel: yijk= αi+ βj+ (αβ)ij+ eijk, eijk∼ N(0, σ2)example: response variable nitrogen level in manure, explanatoryvariables are diet treatment, period, and interaction.Statistics 572 (Spring 2008) Overview January 21, 2008 7 / 11Linear Models and Generalized Linear Models ModelsExamples of Linear Models (cont.)Linear models with both types.—response variable: continuous quantitative variableexplanatory variables: both quantitative and categoricalerror structure: normal distributionmodel: yij= β0+ β1xij+ αi+ eij, eij∼ N(0, σ2)example: response variable is milk yield, explanatory variables arediet (four treatments) and days in milk.Polynomial regression.—response variable: continuous quantitative variableexplanatory variables: single quantitative explanatory variableerror structure: normal distributionmodel: yi= β0+ β1xi+ β2x2i+ β3x3i+ ei, ei∼ N(0, σ2)example: response variable is disease area, explanatory variable isageStatistics 572 (Spring 2008) Overview January 21, 2008 8 / 11Linear Models and Generalized Linear Models ModelsExamples of Linear Models (cont.)Mixed models.—response variable: continuous quantitative variableexplanatory variables: variables of both fixed and random effect.error structure: normal distributionmodel: yij= β0+ β1xij+ ai+ eij, eij∼ N(0, σ2), ai∼ N(0, σ2a)example: response variable is percentage cover of vegetation, siteis modeled as a random effect, quantitative variablesinclude soil moisture.Repeated measures.—response variable: continuous quantitative variableexplanatory variables: one or more including random effect forindividualerror structure: normal distributionexample: response variable is hormone concentration, explanatoryvariables include individual and day.Statistics 572 (Spring 2008) Overview January 21, 2008 9 / 11Linear Models and Generalized Linear Models ModelsExamples of Generalized Linear ModelsLogistic Regression.—response variable: categorical variable with two levelsexplanatory variables: one or moreerror structure: binomialmodel: P {yi= 1} is a function of β0+ β1x1i+ β2x2i.example: response variable is seed germination, explanatoryvariables include temperature and treatment.Poisson regression.—response variable: non-negative integer-valued variableexplanatory variables: one or moreerror structure: Poissonmodel: P {yi= k} is a function of β0+ β1x1i+