ANOVA

Sunday, 18 August 2019

9:23 PM

1 1 Analysis o' (ANOv'A) Analysis of Variance 11.2 sis ot Variarre o Comparing the between-group variability and the within-group variability is the purpose of the Analysis of Variance -+ often shortened to the acronym ANOVA Suppose that we have k different groups (k populations, or k sub-populations of a population) that we wish to compare. Often, each group is called a treatment or treatment level (general terms that can be traced back to the early applications of this methodology in the agricultural sciences) o The response for each of the k treatments is the random variable of interest, say X o Denote Xij the jth observation (j I , n,) taken under treatment i —+ we have k independent samples (one sample from each Of the treatments) MAT H20gg2859 (Statistics) Dr Jia

 

11.2 1 1 Analysis o' (ANOv'A) ANOVA model The ANOVA model is the following: Xij C i) where sis ot Variarre is the mean response for the ith treatment (i = 1 2 Cij is an individual random error component (j = 1,2 As usual for errors, we will assume that the random variables Cij are normally and independently distributed with mean 0 and variance 02: Cij N (O, a) for all Therefore, each treatment can be thought of as a normal population with mean and variance 02: XO NOG, a) Important: the variance "2 is common for all treatments MATH20gg2859 (Statistics) Dr Jia

 

11.2 1 1 Analysis o' (ANOv'A) sis ot Variarre Thus we have two potential estimators of 02: O MSEr, which always estimates 02 O MSTr, which estimates 02 only when Ho is true Actually, if Ho is not true, MS-rr tends to exceed 02, as we have E(MSTr) + 'true' variance between the groups + the idea of the ANOVA test now takes shape Suppose we have observed k samples Xil ,Xi2, . , in, x for 1,2, k, from which we can find through calculations the observed values ms-vr and msEr. Then: if MSTr MSEr, then Ho is probably reasonable if MSTr MSEr, then Ho should be rejected this will thus be a one-sided hypothesis test We need to determine what "MSTr MSE/' means so as to obtain a hypothesis test at given significance level o. MATH20gg2859 (Statistics) Dr Jia

 

 

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