Randomized Block and Two-Factor Notes
1. Randomized block: Read the material at
1.1 Experimental situation:
Two factors – one of the factors group observations (block). Observations are randomly
assigned within a block, one to each factor level
There are two sets of dummy variables, one set for the block and one for the factor.
Example page 542-543. We are blocking by combinations of age and weight. 25 groups
were formed. The four people in each group are randomly assigned one to each drug.
The model would have 24 dummy variables for the groups and 3 for the drugs. A total of
27 variables in the model.
What is of interest is whether the variation of the drug averages is more than would
expected if drugs are equal on average. This would be an F test.
The numerator is the variation of the drugs. It has 3 d.f.
The denominator is the variation in random data which has n-k-1 d.f. = 100-27-1=72
Ho 1 = 2 = 3 =4
H1 at least one differs from the others
Test Statistic F = ms(drugs)/mse = 4.12
Reject Ho if F > F 3, 72 = 2.76
We can say that the average reduction is not the same for all drugs after adjusting for the
age-weight conditions. Use the Tukey procedure to see which means differ and by how
Example of use in Information Systems (“Evaluating the Benefits of Augmented Reality …”
In this article a person is the block therefore the randomized block design is also called a
repeated measure design.):
2. Two factor randomized designs. (Section 14.5)
2.1 Data collection: Observations are randomly assigned to (experiment) or chosen from
(observational study) the combinations of the two factors.
There are three sets of dummy variables, one set for the factor 1, a second set for factor2, and
a set of product terms (first set times second set)
Example page 549-550: the first factor is the education level (less than high school, high
school, less than bachelors, at least one bachelors), second factor is the gender. The
response variable is the number of jobs held
Do you think that each education level will affect the average number of jobs-held in the
same way for males as for females? If it is possible that the effect of an education level
differs from males to females, the combinations of gender and education have to be
considered. If the effect of the education on average number of jobs held depends on the
gender, this is interaction.
Model contains three sets of dummy variables:
E1, E2, and E3 (education)
E1*g1 E2*g1, and E3*g1 and (education-gender)
Model degrees of freedom K=7 = 3 + 1 + 3. A total of 80 people, 10 are randomly chosen
from each combination. n-k-1 = 80-7-1 = 72
The degrees of freedom of the F tests we are considering are the degrees of freedom of
the effect and the degrees of freedom of random data
F-test for combinations is 3 and 72
F-test for gender is 1 and 72
F-test for education is 3 and 72
Ho : no combination effect
H1 : combination effect
Test: F = ms(interaction)/mse = 0.21
Reject Ho if F > F 3,72 = approximately F 3, 70 = 2.74
No significant combination effect was found. Effect of gender and education can be
2.3 Steps for analysis
Test interaction to see if there are combination effects. If so, see which combinations
are different from the others and by how much
If no interaction, test the factors separately. If a factor is important (large F), decide
which of its means are different and by how much
Use the Tukey’s approach: the margin of error for the difference in two sample means
is q (# of means, n-k-1) times the square root of (mse / number of observations in each mean)
Example: the sample means for education are 12.6, 11, 10,8, and 9. The MSE is
10.09. The Tukey table uses 4 and 72 degrees of freedom = 3.74 approximately. The
margin of error is then 3.74 times square root of (10.09/20) = 2.65. Therefore the
largest error you would expect in the difference in any two of the sample mean is ±
2.4 Design notes
If you have the same number of observations at each combination of factor levels, the
factors become uncorrelated (no multicollinearity)
If you only have one value at each combination and interaction is measured, the mean
squared error can not be calculated.
Example of use in Information Systems (“Psychological Factors Influencing World-Wide Web
Exercises: Check Blackboard for exercises to go with these two topics.