1-factor analysis of variance (1-anova)• use to compare the means of 3 or more groups

in a pair-wise manner that differ by 1 factor• detects if there is a difference between the means• does not identify which pair(s) is / are different

motivation use of multiple pair-wise comparisons using multiple t-tests increases the error, while 1-anova does not have such an issue.

Basis:use F-test to compare 2 estimates of the variance• MS(Tr): based on the SEM• MSE: find the average of the SD2 from

each group

where

and

if the means are the same, then MS(Tr) would be a valid estimate of the variance, SD2; otherwise,the F-test should show that the 2 estimates of the variance are different.

MS(Tr): estimate of SD2 based on the SEM. To simplify the analysis, assume all groups have the same sample size; if invalid assumption, then analysis is more complicated.

recall,

where SEM = standard deviation of the means; thus

where k = # groups n = # samples in a group = mean of the ith group

.. = mean of all samples

MSE: estimate of SD2 by averaging the SD2 of each group. To simplify the analysis, assume all groups have the same sample size; if invalid assumption, then analysis is more complicated.

where k = # groups SDi

2 = variance of the ith group n = # samples in a group = mean of the ith group = jth data in the ith group

Tukey test • use only if preceding 1-anova detects

a difference among the means• multiple (pair-wise) comparison test,

which identifies the pair(s) of different means