CHAPTER 6
Two Factor Experiments with 𝑛 > 1 Observations per Cell
We now consider experiments where the response variable depends on two factors
with both factors of equal interest.
The statistical analysis is necessarily more complex because there are several
competing models that the scientist must consider before deciding which one best fits
the data.
We shall consider in detail the main effects model (no interactions), and the model
with interactions. A thorough analysis requires a combination of analytical and
graphical methods that are best understood in the context of a concrete example.
Statistical analysis of the two-factor experiment
1. The statistical model for the response variable We assume that we have
performed 𝑛 > 1 experiments at each combination of the factor-levels. We
denote the value of the 𝑘𝑡ℎ response at the (𝑖, 𝑗)𝑡ℎ combination of the factorlevels by 𝑦𝑖𝑗𝑘 , where
yijk     ijk
We assume the random variables  ~ N (o; 2 ) thus, their means and variances
are given by E ( yijk )  
and
Var ( y
ijk
) 2
2. Data format We denote the 𝑘𝑡ℎ observation in the (𝑖, 𝑗)𝑡ℎ cell by
Check the study guide!
We group these observations together by cell and arrange the ab cells in an 𝑎 × 𝑏
matrix consisting of 𝑎 rows and 𝑏 columns.
3. The dot subscript notation We denote the sample cell means, factor A (row)
means, factor B (column) means, and the grand mean by the dot subscript
notation; thus,
b
r
yij  
 yijk
k 1
r
yi 

j 1
r
 yijk
k 1
br
a
y j  

i 1
a
r
 yijk
k 1
ar
y 
b
r

i 1 j 1 k 1
abr
yijk
Question 1: Refer to the data set below. Compute the cell means, grand mean, and
the factor A and factor B means, and use the dot subscript notation to express your
results.
EXAMPLE:
This is a two-factor experiment with two levels of factor A (no fungicide, Captan) and
three levels of factor B (no pesticide, Dieldrin, and Diazinon). There are also 8
observations at each combination of the factor-levels; consequently in this experiment
we have: a = 2, b = 3, r = 8.
Solution:
Question 2: Use graphical method to check if there’s an interaction between factor
A and B.
4. The main effects and the interaction effects There are now two models to
consider depending on whether or not there are interactions between the two
factors.
The main effects model (also called the no interaction model) assumes that
𝑦𝑖𝑗 = 𝜇 + 𝛼𝑖 + 𝛽𝑗(𝑖 = 1, . . . , 𝑎; 𝑗 = 1, . . . , 𝑏).
5. The two factor model with interactions To allow for the possibility of
interactions we assume the mean response at the (𝑖, 𝑗) factor-level combination
is given by
𝜇𝑖𝑗 = 𝜇 + 𝛼𝑖 + 𝛽𝑗 + (𝛼𝛽)𝑖𝑗 .
The statistical model of the kth response at the (𝑖, 𝑗)𝑡ℎ factor level is given by:
𝑌𝑖𝑗𝑘 = 𝜇 + 𝛼𝑖 + 𝛽𝑗 + (𝛼𝛽)𝑖𝑗 + 𝑒𝑖𝑗𝑘,
The corresponding model statement format for the two-factor experiment with
interactions is Model (𝑌 = 𝑦|𝐴 = 𝑖, 𝐵 = 𝑗, 𝐴 ∗ 𝐵 = (𝑖, 𝑗)).
6. Least squares estimates of the model parameters for the two factor model
with interactions It can be shown that the least squares estimates of the
overall mean, the factor A and B, and interaction effects are:
7. How well does the model fit the data?
We analyze the model goodness of fit by first partitioning the total variability SST
into the sum of SSA (the variability explained by factor A), SSB (the variability
explained by factor B), SSAB (the variability explained by the interaction of factors
A and B), and SSE (the unexplained variability).
The coefficient of determination 𝑅 2 is the proportion of the total variability explained
by the model; it is given by:
R2 
SS A  SS B  SS AB
SSTO
The definitions of these sums of squares and their corresponding degrees of freedom
will now be given. We then arrange these quantities in the format of a two-way ANOVA
table. This will allow us to test hypotheses about the model parameters.
TASK 13: QUESTION 3
QUESTION 1
Table 1 records the result of an experiment to measure the performance of a computer
workstation, measured in million instructions per second (MIPS), as a function of
memory size (factor A) and cache size (factor B). Memory size has two levels: 4 Mbytes (low) and 16 M-bytes (high); and cache size has two levels: 1 K-bytes (low) and
2 Kbytes (high). This is a 22 experiment with three replicates at each factor-level
combination, resulting in a total sample size of 12.
Factor-Level Combination
Replicates
0,0
15
18
12
1,1
25
28
19
1,0
45
48
51
0,1
75
75
81
a) Find the mean of Main effects and interaction effects.
b) Evaluate the sums of squares for all factorial effects by the contrast method.
c) Evaluate the sums of squares for all factorial effects by the Yates method.
d) Construct an ANOVA Table and Test which effects are significantly different
from zero.
QUESTION 2
The following data come from a 23 factorial experiment with three replicates for each
treatment combination. Estimate the main effects and all multi-factor interactions.
Then analyze the data by constructing the ANOVA table and test for the relevant
hypothesis.
QUESTION 3
The data in the next table are the results of an experiment to investigate the sources
of variability in testing the strength of Portland cement. A sample of cement was
divided into small samples for testing. The cement was “gauged” (mixed with water)
by three different men called gaugers, and then it was cast into cubes.
Three men, called breakers, later tested the cubes for compressive strength. The
measurements are in pounds per square inch. Each gauger gauged 12 cubes, which
were then divided into three sets of four, and each breaker tested one set of four cubes
from each gauger. All the tests were carried out on the same machine. The purpose
of the experiment was to identify the source of and measure the variability among
gaugers and breakers.
(a) Compute the estimated cell means and display your results in the format of Table.
(b) Plot the estimated cell means of the breakers (factor B) against the three levels of
gaugers (factor A). Does the graph suggest the existence of an interaction?
(c) Test for the existence of main effects and interactions by constructing the ANOVA
table.
d) Perform pairwise comparison of the factors that are significant.