1. What If There Are More Than Two Factor Levels?
The t-test does not directly apply
There are lots of practical situations where there are either more than two levels of interest, or there are several factors of simultaneous interest
The analysis of variance (ANOVA) is the appropriate analysis “engine” for these types of experiments – Chapter 3, textbook
The ANOVA was developed by Fisher in the early 1920s, and initially applied to agricultural experiments
Used extensively today for industrial experiments

2. An Example (See pg. 61)
Consider an investigation into the relationship between the RF (radio-frequency) power setting and the etch rate for a wafer etching tool.
The response variable is etch rate.
The experimenter wants to determine the power setting that will give a desired target etch rate.
Other variables are fixed (gas, gap, etc.).
RF power levels: 160, 180, 200, and 220 W.
The experiment is replicated 5 times – runs made in random order

3. An Example (cont’ed)
The run order should be randomized. First, we need to assign numbers to the experimental run, e.g., as follows:
RF Power
Experimental Run Number
160 1 2 3 4 5
180 6 7 8 9 10
200 11 12 13 14 15
220 16 17 18 19 20

4. Test Sequence Run Number Power Level
An Example (cont’ed)
Choose a random number between 1 and 20, and assign it as the first to run/test, then the next. Until all 20 runs are assigned a test number. E.g.:
Test Sequence Run Number Power Level 1 14
200 2 17
220 3 19 4
160 5 … 20 15
What if the test was run in the original nonrandomized order?

5. An Example (cont’ed)

6. The Analysis of Variance (Sec. 3-2, pg. 63)
In general, there will be a levels of the factor, or a treatments, and n replicates of the experiment, run in random order…a completely randomized design (CRD)
N = an total runs
We consider the fixed effects case…the random effects case will be discussed later
Objective is to test hypotheses about the equality of the a treatment means

7. The Analysis of Variance
The name “analysis of variance” stems from a partitioning of the total variability in the response variable into components that are consistent with a model for the experiment
The basic single-factor ANOVA model is

8. Models for the Data
There are several ways to write a model for the data:

9. The Analysis of Variance
Total variability is measured by the total sum of squares:
The basic ANOVA partitioning is:

10. The Analysis of Variance
A large value of SSTreatments reflects large differences in treatment means
A small value of SSTreatments likely indicates no differences in treatment means
Formal statistical hypotheses are:

11. The Analysis of Variance
While sums of squares cannot be directly compared to test the hypothesis of equal means, mean squares can be compared.
A mean square is a sum of squares divided by its degrees of freedom:
If the treatment means are equal, the treatment and error mean squares will be (theoretically) equal.
If treatment means differ, the treatment mean square will be larger than the error mean square.

12. The Analysis of Variance is Summarized in a Table
Computing…see text, pp 70 – 74
The reference distribution for F0 is the Fa-1, a(n-1) distribution
Reject the null hypothesis (equal treatment means) if
F0 > Fa,a-1, a(n-1)
=> an upper tail, one tail critical region.

13. Calculation of Sum of Squares
The error sum of squares is
SSE = SST – SSTreatments
Usually these are done using a computer program.

14. Confidence Intervals in ANOVA
Can be established based on t-distribution
A 100(1- a)% confidence interval on the ith treatment mean mi is:
A 100(1- a)% confidence interval on the difference in any two treatments means is:

15. Simultaneous Confidence Intervals
Can be established based on one-at-a-time confidence intervals.
If there are r 100(1- a)% confidence intervals of interest, the probability that the r intervals will simultaneously be correct is at least 1-ra.
E.g. r = 5, a = 0.05, 1 – ra = 0.75
r = 10, a = 0.05, 1 – ra = 0.50
Bonferroni method:
Replacing a in the equations with a /r, then the simultaneous confidence intervals on treatment means/differences have a confidence level of at least 100(1- a)%.

16. ANOVA Computer Output (Design-Expert)
Response: etch rate ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares]
Sum of Mean F Source Squares DF Square Value Prob > F Model < A < Pure Error Cor Total
Std. Dev R-Squared Mean Adj R-Squared C.V Pred R-Squared
PRESS Adeq Precision