Two-Way ANOVA: The F-Test Statistic•Factor A main effect:•Factor B main effect:•Interaction effect:
Two-Factor ANOVA - ExampleAirline company is concerned because many of its frequent flier program members have accumulated large quantities of free miles. They conducted an experiment in which each of three methods forredeeming frequent flier miles was offered to a sample of 16 customers divided in four agegroups. Factor A is the redemption offer type with three levels. Factor B is theage group of each customer with four levels.
Two-Factor ANOVA - Example Summary in Excel1. Open file.2. Select Data > Data Analysis.3. Select ANOVA: Two FactorWith Replication.4. Define data range(include factor A and B labels).5. Specify the numberof rows per sample: 46. Specify Alpha7. Indicate output range..8. Click OK.
Two-Factor ANOVA Table – Example Results
InteractionNo InteractionInteraction ispresent
Interaction•Test for interaction•If present, conduct a one-way ANOVA to test the levels of one of the other factors using only one level of the other factor•If NO interaction, test Factor A andFactor B
Section 13.1Goodness of Fit
Goodness of Fit Tests•In most of the analyses that we’ve done in this class and the Stats I class, we made assumptions about the shape and distribution of the data. •For instance, we assumed that the populations were normally distributed in all of our ANOVA calculations.•In these and other cases, we can use what is called a goodness of fit test to verify that the assumptions are being met.
Examples •Examples might include:•Is demand constant across each day of the week? (i.e. does it follow a uniform distribution?). Should we staff the same number of people on the weekends as the weekdays?•Is our manufacturing process normally distributed?•Do the data follow a binomial distribution?•1:The number of observations n is fixed.•2:Each observation is independent.•3:Each observation represents one of two outcomes ("success" or "failure").•4:The probability of "success" p is the same for each outcome.
Example from Our Book•Checker Cab Company is interested in matching the number of cabs in service with customer demand throughout the week. Currently, the company runs the same number of taxis Monday through Friday, with 25% reduced staffing on the weekends. The manager wants to find out if the assumed demand pattern still applies.•Data collected over 140 days •Use a .05 level of significance.DayCustomer CountSun4502Mon6623Tue8308Wed10420Thu11032Fri10754Sat4361
Problem--Visualized•Here’s a look at the observed vs. the expected demand (we’ll look at how to calculate expected demand in a minute). •Since the curves do not match, can we stop now and reject the null hypothesis?•No. We have to find out if the demand is STATISTICALLY different. They might just be different because drawing samples results in sampling error.