Chapter 11

Analysis of Variance

11.1: The Completely Randomized Design: One-Way Analysis of

Variance

• vocabulary– completely randomized– groups– factors– levels– experimental units, – response

Analysis of Variance

• Total Variation: the sum of squared differences between each measurement and the overall average (or grand mean).

• You can divide Total Variation into parts. Today we use 2 parts: Among-Group and Within-Group.

• SST= SSA + SSW

ANOVA

• If a sum of squares is divided by the appropriate degree of freedom, it becomes a variance. Thus, ANOVA.

• We use these variances in the “F” test for Differences in More than 2 Means.

• The results are usually presented in an ANOVA summary table.

• The “F” test has the same parts as previous hypothesis tests.

Example: Problem 11.8

• What is the factor?

• What are the levels?

• What are the hypotheses?

• What is the decision rule?

• What is the test statistic?

• What is the “p” value?

• What can we conclude?

Post hoc Testing or Multiple Comparisons

• If at least 1 of the means is different, the next question is: which one(s) are significantly different?

• Use the Tukey-Kramer Procedure:– calculate the “critical range” (11.6)

– calculate the absolute differences for all pairs of means

– Any absolute difference bigger than the critical range signals a significant difference.

Post hoc testing on 11.8

• Critical range shows that experts differs from readers and that readers differs from darts.

• Note that more groups implies a lot more pairs.– text example: 4 goups: 6 pairs (page 404)

ANOVA Assumptions

• Randomness and Independence• Normality

– How do we test this?

• Homogeneity of Variance– Levine Test: uses medians.– Understand existence but not

mechanics.

11.2: The Randomized Block Design

• Do 11.1 again(!) except do it with blocks.

• What is a block?– Sets of experimental units that have been

matched.

– Experimental units that have repeated measurements performed on them.

– VERY SIMILAR to dependent means “t” test.

Theory

• Take SST and repartition it.• SSA, SSBL, and SSE.

– By doing this, you have increased calculated “F” because “F” is now SSA/SSE, and SSE is less than SSW.

– Thus, the experiment is more sensitive—it does a better job of discovering or revealing factor effects.

Practice

• Little has changed from the 1-way ANOVA:– The hypotheses are the same.– You are interested in the main factor, not the

blocking factor.– The ANOVA table has an additional row for

the Blocking source of variation.– Use Excel or Minitab.

Notes• You can test the block effects but most

statisticians do not report the results. The test will tell how useful blocking is. So will RE.

• The improvement in sensitivity is known as an improvement in Relative Efficiency (RE).

• We will not cover Relative Efficiency further.

• Post hoc testing is available and we will cover it.

Post hoc Testing

• Use the Tukey procedure.

• Equation 11.15.

11.3: The Factorial Design: Two-Way Analysis of Variance

• Similar to section 11-1 except uses 2 factors to partition variation.

• Think of the 2 factors as your 2 ways to account for the inherent variability in the data.

• Use software to do the math.

• The new wrinkle is interaction.

Theory

• This time, you have 2 independent factors.

• Take SST and (yet again) repartition it.• SST = SSA + SSB + SSAB + SSE• SSA and SSB result from imposing

your 2 factors (Factor A and Factor B) on the variability that lives in the data set.

In Practice with 2 Factors

Check for Interaction First!• What is interaction? Interaction occurs

when the level of Factor A affects or determines how the data react to Factor B (and vice versa).

• Look at the plot.

• H0: interaction equals 0 (no I/A).

• H1: interaction is not equal 0 (I/A exists).

When you find significant interaction

• If reject H0, don’t bother checking for significant A and B Factor effects.

• Conduct Post hoc testing for significantly different combinations of A and B.

• Post hoc testing for interaction involves more Tukey testing which is not presented in this text.

When there is no significant interaction:

• Examine the Main Effects.

• This examination is most often done by using the appropriate “F” test.

• It can also be done graphically--informally.

• If there is no interaction and there are 3 or more levels of the significant factor, post hoc testing is possible.