Comparing Three or More Means ANOVA (One-Way Analysis of Variance) Lesson 13 - 1 Comparing Three or More Means ANOVA (One-Way Analysis of Variance)

Objectives Verify the requirements to perform a one-way ANOVA Test a claim regarding three or more means using one way ANOVA

Vocabulary ANOVA – Analysis of Variance: inferential method that is used to test the equality of three or more population means Robust – small departures from the requirement of normality will not significantly affect the results Mean squares – is an average of the squared values (for example variance is a mean square) MST – mean square due to the treatment MSE – mean square due to error F-statistic – ration of two mean squares

One-way ANOVA Test Requirements There are k simple random samples from k populations The k samples are independent of each other; that is, the subjects in one group cannot be related in any way to subjects in a second group The populations are normally distributed The populations have the same variance; that is, each treatment group has a population variance σ2

ANOVA Requirements Verification ANOVA is robust, the accuracy of ANOVA is not affected if the populations are somewhat non- normal or do not quite have the same variances Particularly if the sample sizes are roughly equal Use normality plots Verifying equal population variances requirement: Largest sample standard deviation is no more than two times larger than the smallest

ANOVA – Analysis of Variance Computing the F-test Statistic 1. Compute the sample mean of the combined data set, x Find the sample mean of each treatment (sample), xi Find the sample variance of each treatment (sample), si2 Compute the mean square due to treatment, MST Compute the mean square due to error, MSE Compute the F-test statistic: mean square due to treatment MST F = ------------------------------------- = ---------- mean square due to error MSE ni(xi – x)2 (ni – 1)si2 MST = -------------- MSE = ------------- k – l n – k Σ k Σ k n = 1 n = 1

MSE and MST MSE - mean square due to error, measures how different the observations, within each sample, are from each other It compares only observations within the same sample Larger values correspond to more spread sample means This mean square is approximately the same as the population variance MST - mean square due to treatment, measures how different the samples are from each other It compares the different sample means Under the null hypothesis, this mean square is approximately the same as the population variance

ANOVA – Analysis of Variance Table Source of Variation Sum of Squares Degrees of Freedom Mean Squares F-test Statistic F Critical Value Treatment Σ ni(xi – x)2 k - 1 MST MST/MSE F α, k-1, n-k Error Σ (ni – 1)si2 n - k MSE Total SST + SSE n - 1

Excel ANOVA Output Classical Approach: P-value Approach: Test statistic > Critical value … reject the null hypothesis P-value Approach: P-value < α (0.05) … reject the null hypothesis

TI Instructions Enter each population’s or treatments raw data into a list Press STAT, highlight TESTS and select F: ANOVA( Enter list names for each sample or treatment after “ANOVA(“ separate by commas Close parenthesis and hit ENTER Example: ANOVA(L1,L2,L3)

Summary and Homework Summary Homework ANOVA is a method that tests whether three, or more, means are equal One-Way ANOVA is applicable when there is only one factor that differentiates the groups Not rejecting H0 means that there is not sufficient evidence to say that the group means are unequal Rejecting H0 means that there is sufficient evidence to say that group means are unequal Homework pg 685-691; 1-4, 6, 7, 11, 13, 14, 19

Problem 19 TI-83 Calculator Output One-way ANOVA F=5.81095 p=.013532 Factor df=2 SS=1.1675 MS=0.58375 Error df=15 SS=1.50686 MS=.100457 Sxp=0.31695