Study Guide for First-Year Interest Group Seminar | F A 1, Exams of Art

Download Study Guide for First-Year Interest Group Seminar | F A 1 and more Exams Art in PDF only on Docsity! 1 Summary from Last week: The complete two-way model: Yijt = µ + αi + βj + (αβ)ij + εijt To test for interaction: H0 AB: There is no interaction This can be restated as: H0 AB: [(αβ)ij - (αβ)iq] - [(αβ)sj - (αβ)sq] = 0 for all i ≠ s, j ≠ q or as H0 AB: [(αβ)i+1,j+1 - (αβ)i+1,j] - [(αβ)i,j+1 - (αβ)ij] = 0 for all i = 1, … , a-1, j = 1, … , b-1 If H0 AB is true, then the full model reduces to: Yijt = µ* + αi*+ βj *+ εijt where µ* = [µ - ( € αβ )••] αi* =[αi +( € αβ )i•] βj * = [βj + ( € αβ )•j ] This has the form of a main effects model, but the parameters are different from those in the complete model. We can test H0 AB using the F-statistic msAB/msE, where msAB = ssAB/(a-1)(b-1), ssAB = ss E0 AB - ssE. We obtained a formula for ssAB. This statistic has (a-1)(b-1) and n - ab degrees of freedom. Examples: 1. The battery experiment 2. The reaction time experiment (pp. 98, 148, 157 of textbook). The data are from a pilot experiment to compare the effects of auditory and visual cues on speed of response. The subject was presented with a "stimulus" by computer, and their reaction time to press a key was recorded. The subject was given either an auditory or a visual cue before the 2 stimulus. The experimenters were interested in the effects on the subjects' reaction time of the auditory and visual cues and also in the effect of different times between cue and stimulus. The factor "cue stimulus" had two levels, "auditory" and "visual" (coded as 1 and 2, respectively). The factor "cue time" (time between cue and stimulus) had three levels: 5, 10, and 15 seconds (coded as 1, 2, and 3, respectively). The response (reaction time) was measured in seconds. What next? This depends on whether or not interaction is significant and on what the original questions were in designing the experiment and on whether or not the analyzer wishes to engage in data-snooping and on the context of the experiment. We will spend a while discussing this. First, if the interaction is deemed not significant, then it is usually desirable to analyze the main effects. Note: The next section is a replacement of the section "Testing main effects with the complete model" from the notes posted for Friday, March 4. Testing the contribution of each factor in the complete model (equal sample sizes) Note: We are still assuming equal sample sizes. We wish to test whether or not the factor A is needed in the model. Since A is included in two ways, via the αi's and also via the interaction terms (αβ)ij, we can frame this question as a hypothesis test with null hypothesis H0 A+AB: Every αi and every (αβ)ij = 0 and alternate hypothesis Ha A + AB: At least one of the αi''s or (αβ)ij's is not zero. We will again use an F test comparing the full model with the reduced model where all H0 A+AB is true. If sample sizes are equal, it can be shown that the least squares estimate of E[Yijt] under this new reduced model (i.e, under H0 A+AB) is € yij ⋅ - € yi⋅⋅ + € y⋅⋅⋅ , giving sum of squares for the reduced model ssE0 A+AB = € tji ∑∑∑ (yijt - € yij ⋅ + € yi⋅⋅ - € y⋅⋅⋅ ) 2, which by appropriate algebraic manipulations becomes