Chapter 10.1-10.3 - Math3200 Intermediate Probability and Statistics Prof Nan Lin Department of Mathematics Washington University Outline Simple linear

# Chapter 10.1-10.3 - Math3200 Intermediate Probability and...

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Math3200 Intermediate Probability and Statistics Prof. Nan Lin Department of Mathematics Washington University

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Outline Simple linear regression Model Ordinary least square Inference Nan Lin, Washington University 2
Regression analysis Response (outcome/dependent) variable ? Predictor (explanatory/independent) variables 𝒙 = (? 1 , ? 2 , … , ? 𝑘 ) Assume ? 𝑖 = ? 𝜃 (? 𝑖1 , ? 𝑖2 , … , ? 𝑖𝑘 ) + 𝜖 𝑖 Goal: infer the regression function ? 𝜃 (⋅) using the data ? 𝑖 , 𝒙 𝑖 , 𝑖 = 1, … , 𝑛 Simplest form of the regression function: linear ? ? = ? + ?? Nan Lin, Washington University 3

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What model to use? Response variable Explanatory variable discrete continuous categorical Contingency table analysis Logistic regression counts Poisson (log-linear) regression continuous ANOVA (two-sample t-test) Linear regression
Simple linear regression One response variable One explanatory variable Data: ? 𝑖 , ? 𝑖 , 𝑖 = 1, … , 𝑛 from 𝑛 independent subjects Model: ? 𝑖 = ? 0 + ? 1 ? 𝑖 + 𝜖 𝑖 , 𝑖 = 1, … , 𝑛, 𝜖 𝑖 ∼ ?(0, 𝜎 2 ) True regression line: ? ? 𝑖 = ? 0 + ? 1 ? 𝑖 ? 𝑖 ∼ ?(? 0 + ? 1 ? 𝑖 , 𝜎 2 ) Explanatory variable is regarded as non-random Goal: estimate (? 0 , ? 1 ) Nan Lin, Washington University 5

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Nan Lin, Washington University 6
Linear regression: Assumptions Assumption 1: linearity Assumption 2: errors have equal variances Assumption 3: errors are normally distributed Assumption 4: errors are independent

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Remarks Linear model refers to ‘linear’ in regression coefficients E.g. ? = ? + ??

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