One-Factor ANOVA Partitions of total variation Total Variation SST Variation Due to t Variation Due to Random Treatment SSA Sampling SSw Commonly referred to as Commonly referred to as: a Sum of Squares Among, or a Sum of Squares Within, or a Sum of Squares Between, or a Sum of Squares Error, or a Sum of Squares Model, or a Within Groups Variation d Among Groups Variation
One-Factor ANOVA Partitions of Total Variation Variation Due to Treatment SSA Variation Due to Random Sampling SSW Total Variation SST Commonly referred to as: Sum of Squares Within, or Sum of Squares Error, or Within Groups Variation Commonly referred to as: Sum of Squares Among, or Sum of Squares Between, or Sum of Squares Model, or Among Groups Variation = +
Total variation总变异 SST=∑∑(Xn-x)2 Xi s the ith observation in group j ni=the number of observations in group j n= the total number of observations in all groups c E the number of groups ∑∑xn =1i=1 the overall or grand mean n
Total Variation 总变异 2 1 1 SST ( X X ) c j n i ij j = − = = n X X c j n i ij j = = 1 = 1 the overall or grand mean Xij = the ith observation in group j nj = the number of observations in group j n = the total number of observations in all groups c = the number of groups
Among-Group Variation 组间变异 SSA sSA=∑n1(x1-x)2MsA c-1 ni- the number of observations in group j c= the number of groups Xi the sample mean of group j X the overall or grand mean Hr Hi Variation Due to Differences Among Groups
Among-Group Variation 组间变异 2 1 SSA n ( X X ) j c j = j − = nj = the number of observations in group j c = the number of groups the sample mean of group j the overall or grand mean j Variation Due to Differences Among Groups. − 1 = c SSA MSA Xj X _ __
Within-Group variation 组内变异 sSW=∑(xn-X1)2 MSW- SSW n-C Xi- the ith observation in group j the sample mean of group j Summing the variation within each group and then adding over all groups
Within-Group Variation 组内变异 2 1 1 SSW ( X X ) j c j n i ij j = − = = Xij = the ith observation in group j X j = the sample mean of group j j Summing the variation within each group and then adding over all groups. n c SSW MSW − =
Within-Group Variation MSW- SSW Forc=2 this is the pooled-variance in the n-C t- Test (n1-1)s2+(m2-1)s2+··+(n-1)S2 (n1-1)+(n2-1)+··+(nc-1) lf more than 2 groups, use F Test For 2 groups, use t-Test. F Test more limited
Within-Group Variation j ( n ) ( n ) ( n ) ( n )S ( n )S ( n )S n c SSW MSW c c c 1 1 1 1 1 1 1 2 2 2 2 2 2 1 1 − + − + • • • + − − + − + • • • + − = − = For c = 2, this is the pooled-variance in the t-Test. •If more than 2 groups, use F Test. •For 2 groups, use t-Test. F Test more limited