Chapter 10

1. Chapter 10 Design of Experiments and Analysis of Variance

2. Elements of a Designed Experiment • Response variable • Also called the dependent variable • Factors (quantitative and qualitative) • Also called the independent variables • Factor Levels • Treatments • Experimental Unit

3. Elements of a Designed Experiment • Designed vs. Observational Experiment • In a Designed Experiment, the analyst determines the treatments, methods of assigning units to treatments. • In an Observational Experiment, the analyst observes treatments and responses, but does not determine treatments • Many experiments are a mix of designed and observational

4. Elements of a Designed Experiment Single-Factor Experiment Population of Interest Sample Independent Variable Dependent Variable

5. Elements of a Designed Experiment Two-factor Experiment

6. The Completely Randomized Design • Achieved when the samples of experimental units for each treatment are random and independent of each other • Design is used to compare the treatment means:

7. The Completely Randomized Design • The hypotheses are tested by comparing the differences between the treatment means to the amount of sampling variability present • Test statistic is calculated using measures of variability within treatment groups and measures of variability between treatment groups

8. The Completely Randomized Design • Sum of Squares for Treatments (SST) • Measure of the total variation between treatment means, with k treatments • Calculated by • Where

9. The Completely Randomized Design • Sum of Squares for Error (SSE) • Measure of the variability around treatment means attributable to sampling error • Calculated by • After substitution, SSE can be rewritten as

10. The Completely Randomized Design • Mean Square for Treatments (MST) • Measure of the variability among treatment means • Mean Square for Error (MSE) • Measure of sampling variability within treatments

11. The Completely Randomized Design • F-Statistic • Ratio of MST to MSE • Values of F close to 1 suggest that population means do not differ • Values further away from 1 suggest variation among means exceeds that within means, supports Ha

12. The Completely Randomized Design • Conditions Required for a Valid ANOVA F-Test: Completely Randomized Design • Independent, randomly selected samples. • All sampled populations have distributions that approximate normal distribution • The k population variances are equal

13. The Completely Randomized Design • A Format for an ANOVA summary table

14. The Completely Randomized Design • ANOVA summary table: an example from Minitab

15. The Completely Randomized Design • Conducting an ANOVA for a Completely Randomized Design • Assure randomness of design, and independence, randomness of samples • Check normality, equal variance assumptions • Create ANOVA summary table • Conduct multiple comparisons for pairs of means as necessary/desired • If H0 not rejected, consider possible explanations, keeping in mind the possibility of a Type II error

16. Multiple Comparisons of Means • A significant F-test in an ANOVA tells you that the treatment means as a group are statistically different. • Does not tell you which pairs of means differ statistically from each other • With k treatment means, there are c different pairs of means that can be compared, with c calculated as

17. Multiple Comparisons of Means • Three widely used techniques for making multiple comparisons of a set of treatment means • In each technique, confidence intervals are constructed around differences between means to facilitate comparison of pairs of means • Selection of technique is based on experimental design and comparisons of interest • Most statistical analysis packages provide the analyst with a choice of the procedures used by the three techniques for calculating confidence intervals for differences between treatment means

18. Multiple Comparisons of Means

19. The Randomized Block Design • Two-step procedure for the Randomized Block Design: • Form b blocks (matched sets of experimental units) of k units, where k is the number of treatments. • Randomly assign one unit from each block to each treatment. (Total responses, n=bk)

20. The Randomized Block Design Partitioning Sum of Squares

21. The Randomized Block Design Calculating Mean Squares Setting Hypotheses Hypothesis Testing Rejection region: F > F, F based on (k-1), (n-b-k+1) degrees of freedom

22. The Randomized Block Design • Conditions Required for a Valid ANOVA F-Test: Randomized Block Design • The b blocks are randomly selected, all k treatments are randomly applied to each block • Distributions of all bk combinations are approximately normal • The bk distributions have equal variances

23. The Randomized Block Design • Conducting an ANOVA for a Randomized Block Design • Ensure design consists of blocks, random assignment of treatments to units in block • Check normality, equal variance assumptions • Create ANOVA summary table • Conduct multiple comparisons for pairs of means as necessary/desired • If H0 not rejected, consider possible explanations, keeping in mind the possibility of a Type II error • If desired, conduct test of H0 that block means are equal

24. Factorial Experiments • Complete Factorial Experiment • Every factor-level combination is utilized

25. Factorial Experiments • Partitioning Total Sum of Squares • Usually done with statistical package

26. Factorial Experiments • Conducting an ANOVA for a Factorial Design • Partition Total Sum of Squares into Treatment and Error components • Test H0 that treatment means are equal. If H0 is rejected proceed to step 3 • Partition Treatment Sum of Squares into Main Effect and Interaction Sum of Squares • Test H0 that factors A and B do not interact. If H0 is rejected, go to step 6. If H0 is not rejected, go to step 5. • Test for main effects of Factor A and Factor B • Compare the treatment means

27. Factorial Experiments • SPSS ANOVA Output for a factorial experiment