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

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

3. 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

4. ANOVA We have the situation where we want to compare three populations
We want to compare the three means (μ1, μ2, and μ3)
The method of Analysis of Variance (ANOVA) to test whether the three means are equal
The same method also applies to four groups (with μ1 through μ4), for five groups (with μ1 through μ5), and so forth

5. Why ANOVA Why don’t we use 3 separate hypothesis tests? Two reasons
Test whether μ1 = μ2
Test whether μ1 = μ3
Test whether μ2 = μ3
Two reasons
If each has a Type I error of 0.05, then doing three tests in a row accumulates a higher Type I error (in fact, to about 0.14)
It can be time consuming to compute, particularly if we have more than three populations

6. ANOVA: Null Hypothesis
We begin by claiming that all the populations have the same mean
The null hypothesis (for three populations) is thus
H0: μ1 = μ2 = μ3
Examples are
Whether three different drugs have the same mean effects
Whether three different groups of students have the same mean test scores

7. ANOVA: Null Hypothesis
A diagram showing when the null hypothesis is true and when a possible alternative hypotheses is true
The null hypothesis The alternative hypothesis
(equal means) (different means)

8. ANOVA: Alternative Hypothesis
The alternative hypothesis would be the opposite of the null hypothesis
If it is not true that all three means are the same, then at least one of them is different from the other two
The three means do not have to all be different
The alternative hypothesis would thus be
Ha: At least one of the population means (one of μ1, μ2, or μ3) is different from the others

9. 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

10. 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

11. 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) (ni – 1)si2
MST = MSE =
k – l n – k Σ k Σ k
n = 1
n = 1

12. MSE
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

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

14. 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
Just like the table we saw in tests for homogeneity or independence:
Σ (ni – 1)si2
= MSE
n - k

15. ANOVA: Calculation & Interpretation
Calculation: We use an F-test. Our calculator and most software will calculate a p-value
Interpretation: Same as in all inference tests (Remember the three C’s!!)

16. Example 1
Researcher Jelodar Gholamali wanted to determine the effectiveness of various treatments on glucose level of diabetic rates. He randomly assigned diabetic albino rats into four treat groups. Group1 served as a control group; group 2 were served a dietary supplement, fenugreek; group 3 were served a dietary supplement, garlic; and group 4 were served a dietary supplement, onion. Ideas came from Persian folklore. After 15 days of treatment, the blood glucose was measured in milligrams per deciliter (mg/dL). The table summarizes the results.

17. Example 1 Data Animal Control Fenugreek Garlic Onion Rat 1 288.1 229.1
177.4
299.7
Rat 2
296.8
240.7
202.2
258.3
Rat 3
267.8
239.4
163.4
286.8
Rat 4
256.7
207.7
184.7
244.0
Rat 5
292.1
225.7
197.9
267.1
Rat 6
282.9
230.8
164.6
297.1
Rat 7
260.3
206.6
193.9
249.9
Rat 8
283.8
213.3
158.1
265.1

18. Verify the Requirements
SRS:
Independent:
Normal:
Same Variance:
Stated in the problem
Different rats in each group
Normality plots for each sample are reasonable
1Varstats for each sample shows smallest stdev=13.49 and twice it less than the largest stdev=21.18

19. 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

20. 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)

21. Example 2
An environmentalist wanted to determine if the mean activity of rain differed among Alaska, Florida, and Texas. He randomly selected six rain dates at each of the three locations and obtained the data in the following table:
Rain Dates
Alaska
Florida
Texas
Date 1
5.41
4.87
5.46
Date 2
5.39
5.18
6.29
Date 3
4.90
4.40
5.57
Date 4
5.14
5.12
5.15
Date 5
4.80
4.89
5.45
Date 6
5.24
5.06
5.30

22. Example 2 Data Run an appropriate Test Hypotheses:
H0: μ1 = μ2 = μ (mean acidity in rain is the same)
Ha: at least one mean is different (the mean acidity in rain is not the same in all 3 states)

23. Verify the Requirements
SRS:
Independent:
Normal:
Same Variance:
Random stated in the problem
Cities are independent (??)
Normality plots for each sample are ok as stated in problem
1Varstats for each sample shows that twice the smallest stdev, 0.252, is less than the largest stdev, 0.397

24. TI-83 Calculator Output One-way ANOVA F=5.81095 p=.013532
Factor
df=2
SS=1.1675
MS=
Error
df=15
SS=
MS=
Sxp=
Calculations:
F = MST / MSE = df = 2,15 p-value =
Fcritical = 3.68 (from F-table)
Interpretation:
Since p-value (0.014) < α = 0.05, we reject the null hypothesis and conclude that at least one cities’ mean acidity in rainfall is different than the others.

25. 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 ; 1-4, 6, 7, 11, 13, 14