Randomized Block and Two-Factor Notes
1. Randomized block:
1.1Experimental 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.
1.2 Example:
A researcher is interested in testing for differences in the average reduction in blood pressure among four different drugs. Since the weight and age of people affect how a drug is absorbed, 25 groups of people are placed in groups of 4. In each group the people are similar in their combination of age and weight. The four people in each group are then 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 much.
Example of use in Information Systems(“Evaluating the Benefits of AugmentedReality …”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)
2.2 Example
A political scientist is interested in the effect that gender and educational has on the average number of jobs held by workers. Letthe first factor be the education leveldivided into four levels (less than high school, high school, less than bachelors, at least one bachelors) and let the second factor bethe gender. The response variable is the number of jobs held. Ten people were randomly selected from each of the 8 combinations of education and gender for a total of 80 observations.
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)
G (Gender)
E1*G E2*G, and E3*Gand (education-gender)
Model degrees of freedom,k, =7 = 3 + 1 + 3. A total of 80 people, 10 are randomly chosenfrom each combination. n-k-1 = 80-7-1 = 72
The degrees of freedom of the F tests we are consideringare 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 considered separately.
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 ± 2.65
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 Navigation”.):
Exercises: Check Blackboard for exercises to go with these two topics.