One-way ANOVA

• Example• Analysis of Variance Hypotheses• Model & Assumptions• Analysis of Variance• Multiple Comparisons• Checking Assumptions

Example:Days Absent by Job Type

11/26/2012 9:25.30 (4)

2 4 6 8 10 12 14 16A

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2 4 6 8 10 12 14 16B

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2 4 6 8 10 12 14 16C

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2 4 6 8 10 12 14 16D

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Analysis of Variance• Analysis of Variance is a widely used statistical technique that

partitions the total variability in our data into components of variability that are used to test hypotheses about equality of population means.

• In One-way ANOVA, we wish to test the hypothesis: H0 : 1 = 2 = = k

against:Ha : Not all population means are the same

Assumptions

• Each population being sampled is normally distributed and all populations are equally variable.

• Normality can be checked by skewness/kurtosis or normal probability plots. If any of the samples do not look like they come from a normal population the assumption is not met (unless the samples that do not look normal have a large sample size (n>30)

• Equal variability can be checked by comparing standard deviations. If no standard deviation is more than 2 times bigger than another equal variability can be assumed.

Example: Are the population mean days absent the same for all 4 job types?

ANOVA table Source SS df MS F p-value

Treatment 763.823 3 254.6076 69.48 5.72E-24Error 351.795 96 3.6645 Total 1,115.618 99

H0: μA = μB = μC = μD H1: Not all μ’s are equal

We can be almost 100% confident that population mean days absent differ in some way between the 4 job types.

Multiple Comparisons• A significant F-test tells us that at least two of the underlying

population means are different, but it does not tell us which ones differ from the others.

• We need extra tests to compare all the means, which we call Multiple Comparisons.

• We look at the difference between every pair of group population means, as well as the p-value for each difference.

• When we have k groups, there are:

possible pair-wise comparisons. For example 4 groups have 4*3/2 = 6 comparisons.

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Multiple Comparisons • If we estimate each comparison separately with 95% confidence, the overall confidence will be less than 95%.

• So, using ordinary pair-wise comparisons (i.e. lots of individual t-tests), we tend to find too many significant differences between our sample means.

• We need to modify our p-values so that we determine the true differences with 95% confidence across the entire set of comparisons.

• These methods are known as:multiple comparison procedures

Multiple Comparisons• We use Tukey simultaneous comparisons.• Tukey simultaneous comparisons overcome

the problems of the unadjusted pair-wise comparisons finding too many significant differences (i.e. p-values that are too small).

Tukey Pair-wise ComparisonsTukey simultaneous comparison t-values (d.f. = 96)

B D C A2.73 5.03 6.74 10.11

B 2.73 D 5.03 4.15 C 6.74 7.47 3.05 A 10.11 14.04 9.24 6.36

critical values for experimentwise error rate:0.05 2.620.01 3.21

We can be at least 99% confident that Job Type A has the highest population mean days absent. We can be also more than 99% confident that C and D have a larger population mean days absent than B. We can only be 95% confident that Job Type C has a higher population mean than D.