1. Design of Experiments and Analysis of Variance
Chapter 10
Design of Experiments and Analysis of Variance
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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