"Comparing Multiple Samples of Numerical Data: Two Factors" 
Index to Module 4 Notes 
In Module Notes 4.4, we introduced ANOVA as a
powerful tool that allows us to compare samples of multiple groups
and make conclusions concerning the equality of the population mean
on one dimension or factor. The example we used was comparing mean
mile per gallon performance (the outcome measure) for automobiles
tested with different brands of gasoline (the factor). The factor,
brand of gas, has three levels, Brand A, B and C. Low Ratio High Ratio Long Peak 25 24 Long Peak 26 25 Long Peak 28 28 Long Peak 27 26 Short Peak 22 30 Short Peak 25 26 Short Peak 20 31 Short Peak 21 27
In this set of notes, we expand the ANOVA concept to the examination
of situations involving two factors. In the above example, perhaps we
want to test mean mile per gallon performance for automobiles tested
with different brands of gasoline (factor A) as well as different
driving conditions (factor B). The different driving conditions might
be city versus highway driving.
In this scenario, if there was a significant brand effect, then we
would know that mean mpg performances groups differ with respect to
at least two of the the three levels of the brand factor.
Likewise, if there was a significant driving condition, then we would
know the mean mpg performances differ with respect to the
two levels of the driving condition factor.
But wait a minute, you say! Back in multiple regression when we had
two independent variables, we also were concerned about
interaction. Same thing holds true in Two Factor ANOVA.
Interaction here would mean that we would have to be concerned about
mean mpg performance at six combination levels of the brand
and driving condition factors. That is, mean mpg for city driving
with Brand A, mean mpg for city driving with Brand B, mean mpg for
city driving with Brand C, mean mpg for highway driving with Brand A,
mean mpg for highway driving with Brand B, and mean mpg for highway
driving with Brand C.
That's the idea  I want to present two example situations, one with
interaction, and one without.
A Situation with
Interaction
This example is from a utility company
that was experimenting with variable pricing. Two factors are
involved. The first is the length of the peak period. At the long
peak situation (7 am  7 p.m.), customers would have a 12 hour
discount period between 7 p.m. and 7 am. At the short peak situation
(8 am  5 p.m.), customers would enjoy a 14 hour discount period
between 5 p.m. and 8 am. The other factor is ratio of the discount. A
low ratio is approximately a 2:1 discount on rates during offpeak
usage. A high ratio is approximately a 3:1 discount on rates during
offpeak usage.
The company prepared a satisfaction survey and measured satisfaction
on a 50 point scale (50 being high satisfaction, 0 being low). The
scores for a trial run of the survey are shown in Worksheet
4.5.1.
Worksheet 4.5.1
Worksheet 4.5.1 presents the data as
you would enter it in an Excel Worksheet in preparation for running a
TwoFactor ANOVA.
To run the ANOVA, select Tools from the Standard Toolbar, then
Data Analysis from the pulldown menu, then ANOVA:
TwoFactor with Replication. The dialog box is similar to those
you have seen before. I highlight all of the data in the three
columns and nine rows (including the labels) shown in Worksheet
4.5.1, and remembered to check Labels. Note an additional
question, "Rows per sample." This is asking how many observations are
in each of the two levels of the row factor, Long/Short Peak. Place
"4" in the adjacent dialog box.
The ANOVA output is shown in Worksheet 4.5.2.
Worksheet 4.5.2
Anova: TwoFactor With
Replication SUMMARY Low Ratio High Ratio Total Long Peak Count 4 4 8 Sum 106 103 209 Average 26.5 25.75 26.125 Variance 1.666667 2.916666667 2.125 Short Peak Count 4 4 8 Sum 88 114 202 Average 22 28.5 25.25 Variance 4.666667 5.666666667 16.5 Total Count 8 8 Sum 194 217 Average 24.25 27.125 Variance 8.5 5.839285714 ANOVA Source of Variation SS df MS F Pvalue F crit Sample 3.0625 1 3.0625 0.821229 0.382657 4.747221 Columns 33.0625 1 33.0625 8.865922 0.011538 4.747221 Interaction 52.5625 1 52.5625 14.09497 0.002749 4.747221 Within 44.75 12 3.729167 Total 133.4375 15
The first thing we get are the group means
and variances. Note that the average satisfaction scores are 26.5 for
the group long peak, low ratio; 25.75 for long peak, high ratio; 22
for short peak, low ratio; and 28 for short peak, high ratio.
Worksheet 4.5.3 shows a picture of these means.
Worksheet 4.5.3
This is interesting data and shows
interaction. Customer satisfaction depends upon the what
combination of ratio and peak the customer was considering. In
multiple regression, when interaction is present, we say the
relationship between customer satisfaction and low or high ratio
depends on length of the peak.
Statistically, to validate the presence of interaction, we examine
the ANOVA table in Worksheet 4.5.2. There are three rows of interest.
The Sample row pertains to variation attributed to the peak
factor; the Column row pertains to variation attributed to the
ratio factor; the Interaction row pertains to variation
attributed to the interaction of combinations of both peak and ratio
factors; and the Within row pertains to within group variation
(variation unexplained by peak period or ratio discount).
The first thing we test for is interaction in the TwoFactor ANOVA
model. The hypotheses are:
H_{0}: There is no interaction (Interaction is not important)
H_{a}: There is interaction (relationship between satisfaction and length of peak period depends on ratio discount; which is the same as stating the relationship between satisfaction and ratio discount depends on length of peak period).
Method 1 Method 2 Male 125 121 Male 117 119 Male 123 120 Female 106 102 Female 107 102 Female 100 103 Anova: TwoFactor With
Replication SUMMARY Method 1 Method 2 Total Male Count 3 3 6 Sum 365 360 725 Average 121.6667 120 120.8333 Variance 17.33333 1 8.166667 Female Count 3 3 6 Sum 313 307 620 Average 104.3333 102.3333333 103.3333 Variance 14.33333 0.333333333 7.066667 Total Count 6 6 Sum 678 667 Average 113 111.1666667 Variance 102.8 94.16666667 ANOVA Source of Variation SS df MS F Pvalue F crit Sample 918.75 1 918.75 111.3636 5.67E06 5.317645 Columns 10.08333 1 10.08333 1.222222 0.301061 5.317645 Interaction 0.083333 1 0.083333 0.010101 0.922417 5.317645 Within 66 8 8.25 Total 994.9167 11
Since the pvalue for Interaction (0.002749) is less than an alpha
value of 0.01 (I am using the lower value of alpha to recognize that
we are doing multiple tests with the same data), we reject the null
hypothesis and conclude there is interaction in the data. Thus, the
company needs to consider both the length of the peak period as well
as the discount ratio when setting peak/off peak price discounts.
A Situation without
Interaction
A production firm that assembles
surgical kits for hospital operating rooms noticed that female
employees seem to assemble kits faster than male employees (this
factor would be the gender factor). If there is a significant gender
effect, the company would need to somehow recognize and respond to
the different average assembly times. There is another factor that
must be considered: there are two methods of assembly.
In this situation, the ANOVA model is used to determine if we can
analyze average assembly time with respect to each factor independent
of the other, or if we have to address interaction. If there is no
interaction, the the average assembly time for males can be compared
to the average assembly time for females . Likewise, if there is no
interaction, then the average assembly time for Method 1 can be
compared to the average assembly time for Method 2. If there is
interaction, then averages involving combinations of the two factors
must be studied: average assembly time for males using Method 1; for
males using Method 2; for females using Method 1, and for females
using Method 2.
Worksheet 4.5.4 provides the data for
running the TwoFactor ANOVA model in Excel.
Worksheet 4.5.4
The TwoFactor ANOVA is run in Excel as
before. Select Tools, Data Analysis, ANOVA: TwoFactor with
Replication, and respond to the dialog box. This time, there
are three rows for the Sample or row factor, Gender. Worksheet 4.5.5
provides the result.
Worksheet 4.5.5
Also as before, with TwoFactor Anova, we first test interaction.
The hypotheses are:
H_{0}: There is no interaction (Interaction is not important)
H_{a}: There is interaction (relationship between assembly time and gender depends on assembly method; which is the same as stating the relationship between assembly time and assembly method depends on gender).
Since the pvalue (0.922417) is greater than
0.01, we do not reject the null hypothesis and conclude there is no
interaction.
Now we can independently test the factors. First, let's look at
gender.
H_{0}: Mean_{Male} = Mean_{Female }H_{a}: Mean_{Male} =/= Mean_{Female}
Since the pvalue for the Sample Factor (the
row or gender factor) (5.67E06) is less than 0.01, reject the null
hypothesis and conclude that the means are not equal. In this case,
we can use the top part of the ANOVA result to see which mean is
less. The Male mean is given in the total column for the Male data as
120.83 minutes per surgical kit, and for the Female data, the mean is
103.3. The females are significantly faster, on average, than the
males.
Finally, we can test the assembly method factor. The hypotheses
statements are:
H_{0}: Mean_{Method 1} = Mean_{Method 2}H_{a}: Mean_{Method 1} =/= Mean_{Method 2}
Since the pvalue (0.301061) for the Method
factor (the Column factor), is greater than 0.01, do not reject the
null hypothesis. These is no significant difference in the average
assembly time for Method 1 compared to Method 2. Estimator 1 Estimator 2 Estimator 3 Job 1 4.6 4.9 4.1 Job 2 6.2 6.3 5.6 Job 3 5 5.4 5.1 Job 4 6.6 6.8 6.0 Anova: TwoFactor Without
Replication SUMMARY Count Sum Average Variance Job 1 3 13.6 4.533333333 0.163333333 Job 2 3 18.1 6.033333333 0.143333333 Job 3 3 15.5 5.166666667 0.043333333 Job 4 3 19.4 6.466666667 0.173333333 Estim'r 1 4 22.4 5.6 0.906666667 Estim'r 2 4 23.4 5.85 0.736666667 Estim'r 3 4 20.8 5.2 0.673333333 ANOVA Source of Variation SS df MS F Pvalue F crit Rows 6.763333333 3 2.254444444 72.46428571 4.1954E05 4.757055194 Columns 0.86 2 0.43 13.82142857 0.005672508 5.143249382 Error 0.186666667 6 0.031111111 Total 7.81 11
The company can now study the gender factor without regard to method
of assembly. Here is a picture of this situation:
Worksheet 4.5.6
Note that the male averages are greater than the female averages, no
matter what the method of assembly  that is a picture of no
interaction.
TwoFactor Anova: Without
Replication
You may have noticed that you have two
choices in the Data Analysis selections for TwoFactor ANOVA. One is
with replication (multiple rows for the factors being analyzed as we
saw in both of the previous examples), and one without replication.
The "without replication" can be used for problems such as the
following.
A company employs three estimators in preparing bids for four
different types of construction jobs. The company is interested in
determining if the estimators are consistent in their bids. An
experiment is created to have each estimator independently prepare an
estimate for Job 1, Job 2, Job 3 and Job 4. So Estimator 1 prepares
an estimate for Job 1, Estimator 2 prepares an estimate for Job 1,
and Estimator 3 prepares an estimate for Job 1; this is repeated for
Job 2, then Job 3, then Job 4.
The results are shown in Worksheet 4.5.7. Estimates are in millions
of dollars.
Worksheet 4.5.7
To analyze this problem with TwoFactor ANOVA, we select Tools,
Data Analysis, Anova: TwoFactor without Replication,
fillin the entries on the dialog box, and obtain the following
output:
Worksheet 4.5.8
The hypothesis test of interest is as
follows:
H_{0}: Mean_{Estimator 1} = Mean_{Estimator 2} = Mean_{Estimator 3 }H_{a}: At least two means are not equal
The factor of interest is Estimator, the column
factor. Since the pvalue (0.005672508) is less than alpha of 0.05
(only one test is being done with the data), we do reject the null
hypothesis and conclude that there is a significant difference in the
mean estimates. This finding would have to be addressed by management
. Anova: Single Factor SUMMARY Groups Count Sum Average Variance Estim'r 1 4 22.4 5.6 0.906666667 Estim'r 2 4 23.4 5.85 0.736666667 Estim'r 3 4 20.8 5.2 0.673333333 ANOVA Source of Variation SS df MS F Pvalue F crit Between Groups 0.86 2 0.43 0.556834532 0.591564315 4.256492048 Within Groups 6.95 9 0.772222222 Total 7.81 11
This model takes into consideration the fact that the jobs themselves
had variability  Type 1 Jobs are smaller than Type 4 Jobs, for
example. The TwoFactor ANOVA model without replication accounts for
the variability of the row factor so that the column factor can be
more effectively studied.
If we would have assumed that there is no difference in the job
types, then we would assumed that jobs were independently and
randomly assigned to each estimator. This is the OneFactor ANOVA we
studied in Module Notes 4.4. Worksheet 4.5.9 shows the results of the
OneFactor ANOVA.
Worksheet 4.5.9
Note carefully that the pvalue for Between
Group variation (the Estimator factor) is now greater than 0.05, so
we would not reject the null hypothesis and conclude the average
estimates are the same. In this case, the wrong model would lead us
to an erroneous conclusion.
Summary
The two factor ANOVA models are very
powerful additions to the Single Factor ANOVA. The assumptions with
the twofactor models are that populations from which the samples
were drawn are normal and the variances are equal. The normality
assumption is not critical in the presence of large sample sizes, and
the variance assumption is not critical when there are equal sample
sizes for the factor level combinations. However, when we work with
small samples, samples with unequal sizes in the factor level
combinations, and/or when the sample indicates extreme values (skewed
data), we should revert to nonparametric techniques, as will be
described in Module Notes 4.6.
The twofactor ANOVA with replication assumes that observations are
randomly and independently assigned to each group of the two factors.
The twofactor ANOVA does not make that assumption and is used when
one wants to control for the variability of the row factor.
Readings:
Levine, D., Berenson, M. & Stephan,
D. (1999). Statistics for Managers Using Microsoft Excel (2nd.
ed.). Upper Saddle River, NJ: PrenticeHall, Chapter 10.
Mason, R., Lind, D. & Marchal, W. (1999).
Statistical Techniques in Business and Economics (10th. ed.).
Boston: Irwin McGraw Hill, Chapter
11.


