1. Model Adequacy Checking in the ANOVA Text reference, Section 3-4, pg
Checking assumptions is important
Have we fit the right model?
Normality
Independence
Constant variance
Later we will talk about what to do if some of these assumptions are violated

2. Model Adequacy Checking in the ANOVA
Violations of the basic assumptions and model adequacy can be investigated by examination of residuals
Residuals (see text, Sec. 3-4, pg. 75)
Statistical software usually generates the residuals
Residual plots are very useful
Normal probability plot of residuals

3. Other Important Residual Plots

4. Other Important Residual Plots
A tendency of positive/negative residuals indicates correlation – violation of independence assumption
Special attention is needed on uneven spread on the two ends in residuals versus time plot
Peak discharge vs. fitted value (Ex. 3-5, page 81) – a violation of independence or constant variance assumptions
Figure 3-7 Plot of residuals versus fitted value (Ex. 3-5)

5. Outliers
When the residual is much larger/smaller than any of the others
First check if it is a human mistake in recording /calculation
If it cannot be rejected based on reasonable non-statistical grounds, it must be specially studied
At least two analyses (with and without the outlier) can be made
Procedure to detect outliers
Visual examination
Using standardized residuals
If eij: N(0,s2), then dij: N(0,1). Therefore,
68% of dij should fall within the limits ±1;
95% of dij should fall within the limits ±2;
all dij should fall within the limits ±3;
A residual bigger than 3 or 4 standard deviations from zero is a potential outlier.

6. Post-ANOVA Comparison of Means
The analysis of variance tests the hypothesis of equal treatment means
Assume that residual analysis is satisfactory
If that hypothesis is rejected, we don’t know which specific means are different
Determining which specific means differ following an ANOVA is called the multiple comparisons problem
There are lots of ways to do this…see text, Section 3-5, pg. 87
We will use pairwise t-tests on means…sometimes called Fisher’s Least Significant Difference (or Fisher’s LSD) Method

7. Contrasts Used for multiple comparisons
A contrast is a linear combination of parameters
with
The hypotheses are
Hypothesis testing can be done using a t-test:
The null hypothesis would be rejected if |to| > ta/2,N-a

8. Contrasts (cont.) Hypothesis testing can be done using an F-test:
The null hypothesis would be rejected if Fo > Fa,1,N-a
The 100(1-a) percent confidence interval on the contrast is

9. Comparing Pairs of Treatment Means
Comparing all pairs of a treatment means <=> testing
Ho: mi = mj for all i  j.
Tukey’s test
Two means are significantly different if the absolute value of their sample differences exceeds
for equal sample sizes
Overall significance level is a
for unequal sample sizes
Overall significance level is at most a

10. 100(1-a) percent confidence intervals (Tukey’s test)
for equal sample sizes
for unequal sample sizes

11. Example: etch rate experiment
a = f = 16 (degrees of freedom for error)
q0.05(4,16) = 4.05 (Table VII),

12. The Fisher Least Significant Difference (LSD) Method
Two means are significantly different if
where for equal sample sizes
for unequal sample sizes
For Example 3-1

13. Duncan’s Multiple Range Test (page 100)
The Newman-Keuls Test (page 102)
Are they the same?
Which one to use?
Opinions differ
LSD method and Duncan’s multiple range test are the most powerful ones
Tukey method controls the overall error rate

14. Design-Expert Output
Treatment Means (Adjusted, If Necessary) Estimated Standard Mean Error
Mean Standard t for H0 Treatment Difference DF Error Coeff=0 Prob > |t| 1 vs vs < vs < vs vs < vs <

15. For the Case of Quantitative Factors, a Regression Model is often Useful
Quantitative factors: ones whose levels can be associated with points on a numerical scale
Qualitative factors: Whose levels cannot be arranged in order of magnitude
The experimenter is often interested in developing an interpolation equation for the response variable in the experiment – empirical model
Approximations:
y = b0 + b1 x + b2 x2 + e (a quadratic model)
y = b0 + b1 x + b2 x2 + b3 x3 + e (a cubic model)
The method of least squares can be used to estimate the parameters.
Balance between “goodness-of-fit” and “generality”