Part5(April1-6) - 1 One Way ANOVA A factor is a variable...

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1 One Way ANOVA A factor is a variable that has different levels (treatment levels). Each treatment level (or treatment) corresponds to one treatment group (one study population). Denote the number of treatment groups as k . The one-way completely randomized design is an experiment where there are n i replicated observations of independent experimental subjects for each treatment i , and the observations are X i 1 , X i 2 , · · · , X in i for each treatment i , i = 1 , · · · , k . Denote the sample mean for each treatment group i as ¯ X i , where ¯ X i = X i 1 + X i 2 + · · · + X in i n i , i = 1 , · · · , k Denote the population mean for each treatment group i as μ i , i = 1 , · · · , k . Analysis of variance (ANOVA) is a method to use the data to compare mean responses μ i , i = 1 , · · · , k for k treatment groups. Usually, we are interested in testing whether the mean responses μ i , i = 1 , · · · , k are the same: H 0 : μ 1 = μ 2 = · · · = μ k and the alternative is H a : μ i 6 = μ j for at least a pair of ( i, j ), where 1 i, j k One-way ANOVA is used to analyze the effect of one factor (with different treatment levels), and Two-way ANOVA is used to analyze the effect of two factors. Example 1. Researchers are interested in testing whether a dieting program and an ex- ercise program have been effective on weight loss for participants. An SRS with sample size n 1 + n 2 + n 3 are recruited, and they are randomly assigned to the dieting group (sam- ple size n 1 ), the exercise group (sample size n 2 ) and a “control” group (for which there is no diet/exercise program) (sample size n 3 ). The amount of weight loss after 1 month of participation of these groups is measured for each subject, and the measurements are X i 1 , X i 2 , · · · , X in i , i = 1 , 2 , 3 . We are interested in testing whether the mean weight loss μ 1 , μ 2 , μ 3 for these three treatment groups are equal. In order to perform a one-way ANOVA test, there are basic assumptions to be fulfilled: 1
1. Normality - Each population (each treatment group) from which a sample is taken is assumed to be normal. 2. Independence of observations - All samples are randomly selected and independent. 3. Equality of variances, called homoscedasticity - The populations for different treat- ment groups are assumed to have equal standard deviations (or variances) . We are interested in comparing multiple mean response under different treatments. The statistical inferences usually have two steps: 1. An overall test to test whether there is any difference among all means under different treatments, H 0 : μ 1 = μ 2 = · · · = μ K . This test is the ANOVA F-test discussed below. 2. A follow-up analysis to carry out pair-wise comparison of means between different treatment groups. The test is Tukey’s test. 2 ANOVA F-Test Example 2. k = 4 and we have the following dataset from an experiment. Treatment 1 Treatment 2 Treatment 3 Treatment 4 21 32 22.5 28 19.5 30.5 26 27.5 22.5 25 28 31 21.5 27.5 27 29.5 20.5 28 26.5 30 21 28.6 25.2 29.2 Let X ij = an observation in the dataset: i = i th level of the treatment; j = j th observation in a treatment: subscripts i = 1 i = 2 i = 3 i = 4 Treatment 1 Treatment 2 Treatment 3 Treatment 4 j = 1 21 32 22.5 28 j = 2 19.5 30.5 26 27.5 j = 3 22.5 25 28 31 j = 4 21.5 27.5 27 29.5 j = 5 20.5 28 26.5 30 j = 6
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