Overview of Statistics 572 Bret Larget Departments of Botany and of Statistics University of Wisconsin Madison January 22 2008 Statistics 572 Spring 2008 Overview January 21 2008 1 11 January 21 2008 2 11 Introduction Welcome to Statistics 572 Introduction Bret Larget Comment on syllabus Textbook better written than last year s choice Web for notes and grades print notes before lecture Objectives Computing go R Assignments late policy Exams save dates Grading Academic honesty Discussion sections attend the one you want Statistics 572 Spring 2008 Overview Introduction Some Changes from Last Year The textbook is better written so I can count on you to learn through reading more than in the past I will not try to cover every topic in lecture Instead of attempting to blend some new ideas into the structure of what 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 that used to be in the course I will stay pretty close to the textbook I will attempt to incorporate some more active learning opportunities in class I want you to become adept in R so I will spend more time on it in class and will ask you to do more with it on homework Statistics 572 Spring 2008 Overview Linear Models and Generalized Linear Models January 21 2008 3 11 The Big Picture The Big Picture A statistical approach to data analysis can lend insight to biological understanding of a wide variety of problems In a statistical approach measurable variables are treated as realizations from a model that relates biological meaningful parameters and stochastic sources of variation No model accounts for all aspects of the underlying biology but an appropriately selected model can be very useful Many data analysis problems arising from the biological sciences are appropriate for linear and generalized linear models a rich family of possible models Statistics 572 Spring 2008 Overview January 21 2008 4 11 Linear Models and Generalized Linear Models Variables Variables Typically one variable of interest is modeled as a response variable which is related to one or more explanatory variables Variables can be categorized as quantitative or categorical Quantitative variables are typically either measured on a continuous scale or are discrete variables that are counts The appropriate choice of model is determined in part by the types of the response and explanatory variables A linear combination of the variables X1 Xk takes the form 1 X1 2 X2 k Xk Linear and generalized linear models include linear combinations of explanatory variables Statistics 572 Spring 2008 Overview Linear Models and Generalized Linear Models January 21 2008 5 11 Models Examples of Linear Models Simple Linear Regression response variable continuous quantitative variable explanatory variable one quantitative variable error structure normal distribution model yi 0 1 xi ei ei N 0 2 example response variable is phosphorous concentration in plant tissue explanatory variable is phosphorous concentration in the soil Multiple Linear Regression response variable continuous quantitative variable explanatory variables more than one quantitative variables error structure normal distribution model yi 0 1 x1i k xki ei ei N 0 2 example response variable is soybean yield explanatory variables are hours of daylight and amount of nitrogen Statistics 572 Spring 2008 Overview January 21 2008 6 11 Linear Models and Generalized Linear Models Models Examples of Linear Models cont One way ANOVA response variable continuous quantitative variable explanatory variable one categorical variable error structure normal distribution model yij i eij eij N 0 2 example response variable is milk yield explanatory variable is diet four treatments Multi way ANOVA response variable continuous quantitative variable explanatory variables more than one categorical variables error structure normal distribution model yijk i j ij eijk eijk N 0 2 example response variable nitrogen level in manure explanatory variables are diet treatment period and interaction Statistics 572 Spring 2008 Overview Linear Models and Generalized Linear Models January 21 2008 7 11 Models Examples of Linear Models cont Linear models with both types response variable continuous quantitative variable explanatory variables both quantitative and categorical error structure normal distribution model yij 0 1 xij i eij eij N 0 2 example response variable is milk yield explanatory variables are diet four treatments and days in milk Polynomial regression response variable continuous quantitative variable explanatory variables single quantitative explanatory variable error structure normal distribution model yi 0 1 xi 2 xi2 3 xi3 ei ei N 0 2 example response variable is disease area explanatory variable is age Statistics 572 Spring 2008 Overview January 21 2008 8 11 Linear Models and Generalized Linear Models Models Examples of Linear Models cont Mixed models response variable continuous quantitative variable explanatory variables variables of both fixed and random effect error structure normal distribution model yij 0 1 xij ai eij eij N 0 2 ai N 0 a2 example response variable is percentage cover of vegetation site is modeled as a random effect quantitative variables include soil moisture Repeated measures response variable continuous quantitative variable explanatory variables one or more including random effect for individual error structure normal distribution example response variable is hormone concentration explanatory variables include individual and day Statistics 572 Spring 2008 Overview Linear Models and Generalized Linear Models January 21 2008 9 11 Models Examples of Generalized Linear Models Logistic Regression response variable categorical variable with two levels explanatory variables one or more error structure binomial model P yi 1 is a function of 0 1 x1i 2 x2i example response variable is seed germination explanatory variables include temperature and treatment Poisson regression response variable non negative integer valued variable explanatory variables one or more error structure Poisson model P yi k is a function of 0 1 x1i 2 x2i example response variable is number of seeds produced explanatory variables include treatment and light intensity Statistics 572 Spring 2008 Overview January 21 2008 10 11 Data Data request I will present each type of model with an example and data These case studies will be more interesting if they are related to genuine