One-Way and Two-Way ANOVA in Statistics: Model, Decomposition, and Testing Hypothesis, Study notes of Statistics

Download One-Way and Two-Way ANOVA in Statistics: Model, Decomposition, and Testing Hypothesis and more Study notes Statistics in PDF only on Docsity! STAT 324 Discussion #9 TA Jiale Xu Webpage www.stat.wisc.edu/∼xujiale/stat324 1 ANOVA 1.1 One Way ANOVA The model: yij = µ+ αi + eij, eij ∼ N(0, σ2) i = 1, 2, . . . , I, j = 1, 2, . . . , ni Decomposition: yij = y.. + (yi. − y..) + (yij − yi.) where y.. = ∑I i=1 ∑ni j=1 yij∑I i=1 ni , yi. = ∑ni j=1 yij ni Relation among Sum Squares: I∑ i=1 ni∑ j=1 (yij − y..)2 = I∑ i=1 ni∑ j=1 (yi. − y..)2 + I∑ i=1 ni∑ j=1 (yij − yi.)2 SST = I∑ i=1 ni∑ j=1 (yij − y..)2 SSB = I∑ i=1 ni∑ j=1 (yi. − y..)2 = I∑ i=1 ni(yi. − y..)2 SSW = SSE = I∑ i=1 ni∑ j=1 (yij − yi.)2 Test the Null Hypothese: The means in different groups are same. Mathematically α1 = α2 = . . . = αI = 0 ANOVA Table Source df SS MS F value p-value Group I-1 SSB MSb=SSB/(I-1) MSb/MSe P(F>MSb/MSe) Resid(error) n-I SSE MSe=SSE/(n-I) Total n-1 SST n = ∑I i=1 ni = Total number of observation Example: PlantGrowth data set 1