STAT 135: Anova Joan Bruna DepartmentofStatistics UC,Berkeley April 10, 2015 JoanBruna STAT135:Anova The color? The size? The brand? The material? Relevant groups should help me explain the variance. Motivation Say you want to buy a pair of shoes. You go to a store and observe a large sample of shoes with varying price. Where does the variance in the price come from? JoanBruna STAT135:Anova Relevant groups should help me explain the variance. Motivation Say you want to buy a pair of shoes. You go to a store and observe a large sample of shoes with varying price. Where does the variance in the price come from? The color? The size? The brand? The material? JoanBruna STAT135:Anova Motivation Say you want to buy a pair of shoes. You go to a store and observe a large sample of shoes with varying price. Where does the variance in the price come from? The color? The size? The brand? The material? Relevant groups should help me explain the variance. JoanBruna STAT135:Anova We will introduce the statistical framework for such problems. Motivation We want to compare the effect of several treatments or several groups. Also, analyze other factors simultaneously. Examples: Movie Recommendations. Testing Several Drugs. Include other factors, such as Age, Sex, etc. JoanBruna STAT135:Anova Motivation We want to compare the effect of several treatments or several groups. Also, analyze other factors simultaneously. Examples: Movie Recommendations. Testing Several Drugs. Include other factors, such as Age, Sex, etc. We will introduce the statistical framework for such problems. JoanBruna STAT135:Anova For each setting, two different approaches: Parametric: Methods based on the Normal distribution. Non-parametric: Mostly based on rank statistics. Different Settings One-way Layout: Independent Measurements under several Treatments. This generalizes the methods from previous Chapter. Two-way Layout: Analyze two factors simultaneously (ex: several Treatments + several age ranges). JoanBruna STAT135:Anova Different Settings One-way Layout: Independent Measurements under several Treatments. This generalizes the methods from previous Chapter. Two-way Layout: Analyze two factors simultaneously (ex: several Treatments + several age ranges). For each setting, two different approaches: Parametric: Methods based on the Normal distribution. Non-parametric: Mostly based on rank statistics. JoanBruna STAT135:Anova One-Way Layout Suppose we have I groups G ,...,G , with measurements 1 I G G G 1 2 I Y Y Y 1,1 2,1 I,1 Y Y Y 1,2 2,2 I,2 ... ... ... Y Y Y 1,J 2,J I,J We assume for now the same # of measurements in each group. Question: Are the means of each group the same? JoanBruna STAT135:Anova Main assumption: Errors (cid:15) are iid N(0,σ2). i,j I (cid:88) We define µ such that α = 0 . i i=1 Testing equality of means: Null hypothesis is H : ∀i , α = 0 . 0 i One-Way Layout Model Model observations as Y = µ+α +(cid:15) , where i,j i i,j µ: global mean across all groups. α: differential effect of i-th group. i (cid:15) : random error of each observation. i,j JoanBruna STAT135:Anova