2Referred to as treatments Analysis of VarianceAllows us to compare ≥2 populations of interval data.ANOVA is:.a procedure which determines whether differences exist between population means.a procedure which analyzes sample variance.Referred to as treatmentsNot necessarily equal
3One Way ANOVA Example 14.1 (Stock Ownership) (1) New Terminology: Population classification criterion is called a factorEach population is a factor levelExample 14.1 (Stock Ownership) (1)The analyst was interested in determining whether the ownership of stocks varied by age. (Xm14-01)The age categories are:Young (Under 35)Early middle-age (35 to 49)Late middle-age (50 to 65)Senior (Over 65)
4Example 14.1 (Stock Ownership) (2) Question: Does the stock ownership between the four age groups differs?n=366; % of financial assets invested in equityFactor: Age Category“One Way” ANOVA since only 1 factor4 factor levels: (1)Young; (2)Early middle age; (3)Late middle age; & (4)SeniorHypothesis:H0:µ1 = µ2 = µ3 = µ4i.e. there are no differences between population meansH1: at least two means differ
8F >F ,(k–1),(n–k) = F5%, (4 –1), (366–4) = 2.61 Example 14.1 (Stock Ownership) (6)B-12F >F ,(k–1),(n–k) = F5%, (4 –1), (366–4) = 2.61H0:µ1 = µ2 = µ3 = µ4 H1: at least two means differ 2.79 > thus reject H0
9Example 14.1 (Stock Ownership) (7) Using Excel: Click Data, Data Analysis, Anova: Single Factor
10Example 14.1 (Stock Ownership) (8) p-value = < 0.05Reject the null hypothesis (H0:µ1 = µ2 = µ3 = µ4) in favor of the alternative hypothesis (H1: at least two population means differ).Conclusion: there is enough evidence to infer that the mean percentages of assets invested in the stock market differ between the four age categories.
11Multiple ComparisonsIf the conclusion is “at least two treatment means differ”i.e. reject the H0:We often need to know which treatment means are responsible for these differences3 statistical inference procedures highlights it:Fisher’s least significant difference (LSD) methodTukey’s multiple comparison method
12Fisher’s Least Significant Difference A better estimator of the pooled variances = MSESubstitute MSE in place of sp2Compares the difference between means to the Least Significant Difference LSD, given by:LSD will be the same for all pairs of means if all k sample sizes are equal.If some sample sizes differ, LSD must be calculated for each combination.Differ if absolute value of difference between means > LSD
13Example 14.2 (Car bumper Quality) (1) North American automobile manufacturers have become more concerned with quality because of foreign competition.One aspect of quality is the cost of repairing damage caused by accidents. A manufacturer is considering several new types of bumpers.To test how well they react to low-speed collisions, 10 bumpers of each of 4 different types were installed on mid-size cars, which were then driven into a wall at 5 miles per hour.
14Example 14.2 (Car bumper Quality) (2) The cost of repairing the damage in each case was assessed. Xm14-02Questions…Is there sufficient evidence to infer that the bumpers differ in their reactions to low-speed collisions?If differences exist, which bumpers differ?Objective: to compare 4 populationsData: intervalSamples: independentStatistical method: One-way ANOVA.
15Example 14.2 (Car bumper Quality) (3) F = 4.06, p-value =If 5% SL, then….Reject HoThere is enough evidence to infer that a difference exists between the 4 bumpersThe question now is…….which bumpers differ?
16Example 14.2 (Car bumper Quality) (4) The sample means are….We calculate the absolute value of the differences between means and compare them.
17Example 14.2 (Car bumper Quality) (5) Click Add-Ins > Data Analysis Plus > Multiple Comparisons
18Example 14.2 (Car bumper Quality) (6) Hence, µ1 and µ2, µ1 and µ3, µ2 and µ4, and µ3 and µ4 differ.The other two pairs µ1 and µ4, and µ2 and µ3 do not differ.
19Tukey’s Multiple Comparison Method As before, we are looking for a critical number to compare the differences of the sample means against. In this case:Note: is a lower case Omega, not a “w”Critical value of the Studentized rangewith n–k degrees of freedomTable 7 - Appendix Bharmonic mean of the sample sizesDifferent if the pair means diff >