1. Two-Factor Study with Random Effects In some experiments the levels of both factors A & B are chosen at random from a larger set of possible factor levels. This is a random effects model One of the advantages of this type of model is that the conclusions can be generalized to all factor levels. The form of the statistical model for the random effects case is the same as for the fixed effects case. However, assumptions about the parameters differ from the fixed effects case. We will assume that the a levels of factor A have been selected at random, and that the b levels of factor B have been selected at random. In addition, we assume that a total of n = abr experimental units are available for study and that r will be randomly allocated to each of the ab experimental conditions. 1STA305 week 6

2. Two-Factor Random Effects Model The model that we will use is: Y ijk = μ + α i + β j + γ ij + ε ijk The parameters have the same meaning as in the fixed effects case. The assumptions that we make are different, and they are as follows: 1. ε ijk are i.i.d. N(0, σ 2 ) 2.α i are i.i.d. 3.β j are i.i.d. 4.Since the levels of factors A and B are chosen at random, the interaction must be random as well. That is, we assume that γ ij are i.i.d. STA305 week 62

3. Component of Variance The variance of any single response is The variance due to each of the factors and to the interaction are called the components of variation. STA305 week 63

4. STA305 week 54 Sums of Squares The observed variation in the data is measure in the same manner as for the fixed effects case. In other words, the total variation in the data is measured by The sums of squares and the degrees of freedom for the other sources of variation are also the same as in the 2-factor fixed effects model. The difference between the fixed effects and the random effects cases are the expected mean squares.

5. STA305 week 55 Expected Mean Squares The expected mean squares can be found using same approach as for the fixed effect model. Exercise: verify that the following are true:

6. Hypothesis Testing Comparing the expected means squares to each other provides the rationale for the tests of the main effects and interaction effects. The tests for interaction and main effects are derived as follows… STA305 week 66

7. The ANOVA Table It is useful to add the expected mean squares to the table in order to remember which ratios to form for the F-tests. The ANOVA table is given below: STA305 week 67

8. Estimating the Model Parameters The variance of each of the factors as well as the interaction are often of interest to the researchers. Recall that an unbiased estimator of σ 2 is the MS E. So we can use this to get an unbiased estimate of Since it follows that an unbiased estimator is Similarly, we have unbiased estimates for factors A and B: STA305 week 68

9. Example The factors that influence breaking strength of a synthetic fiber are being studied. It is believed that the machine and that the machine operator both affect breaking strength. The study is designed by randomly selecting 4 machines from all those available and by randomly selecting 3 operators. Each operator uses each of the machines twice. The data are given in the following table: STA305 week 69