Psychology 610 Handout #26, p. 1 Prof. Moore Options for Analyzing a One-Way Within-Subject Design with Replications Assume 5 levels of factor A, 3 participants who complete 4 replications of each of the 5 levels of factor A (20 observations per participant; 60 observations in the whole experiment). A. Average each participant=s responses within treatment condition. Now we have 5 observations per person, one for each level of factor A. It is now analyzed as a regular one-way within-subject design. Subjects is random; A is fixed. Source df E(MS) Mean 1 σe2 + aσS2 + anµ2 Subjects n - 1 = 2 σe2 + aσS2 A a - 1 = 4 σe2 + σSxA2 + nθA2 A x S 8 σe2 + σSxA2 Linear model: Yij = µ + αj + πi + απij + εij εij can’t be separated from απij B. Consider replications to be random and nested in Subject x A combinations. Subjects is also random. Source df E(MS) Mean 1 σe2 + arσS2 + anrµ2 Subjects n - 1 = 2 σe2 + arσS2 A a - 1 = 4 σe2 + rσAxS2 + rnθA2 A x S (a-1)(n-1) = 8 σe2 + rσAxS2 Reps/AS an(r - 1) = 45 σe2Handout #26, p. 2 B′. Replications is random and nested in S x A combinations, but subjects is fixed. Source df E(MS) Mean 1 σe2 + anrµ2 Subjects n - 1 σe2 + arθS2 A a - 1 σe2 + nrθA2 A x S (a-1)(n-1) σe2 + rθAxS2 R/AS an(r - 1) σe2 C. Consider replications to be a fixed factor crossed with other factors. Subjects is random. This analysis allows Reps to be tested. Source df E(MS) Mean 1 σe2 + arσS2 + anrµ2 Subjects n - 1 σe2 + arσS2 A a - 1 σe2 + rσAxS2 + arθA2 A x S (a-1)(n-1) σe2 + rσAxS2 Reps r - 1 σe2 + aσRxS2 + anθR2 R x S (r-1)(n-1) σe2 + aσRxS2 A x R (a-1)(r-1) σe2 + σAxRxS2 + nθAxR2 A x R x S (a-1)(r-1)(n-1) σe2 + σAxRxS2 D. Analyze the data of each subject in a separate ANOVA. Replications is a random factor in each analysis, nested in treatment. This allows the effect of factor A to be tested for each individual. Source df E(MS) Mean 1 σe2 + arµ2 A a - 1 σe2 + rθA2 Reps/A a(r - 1) σe2Handout #26, p. 3 D′. Individual subject ANOVAs; Replications is random and crossed with factor A (factor A is fixed). Source df E(MS) Mean 1 σe2 + aσR2 + arµ2 Reps r - 1 σe2 + aσR2 A a - 1 σe2 + σAxR2 + rθA2 A x R (a-1)(r-1) σe2 + σAxR2 Comments: A. In this approach, reps are not of interest. By averaging reps together for each subject, the scores to be analyzed will have more stability. Variance due to replications is averaged away. B. & B′. In these approaches, reps is also not of interest, but it becomes the error term for at least one test. The interaction of treatment (A) with Subjects can be tested. Subjects can also be tested. A question for BΝ is when it might be legitimate to consider subjects to be a fixed factor. C. In this approach, reps is a factor that the investigator is interested in testing, along with its interaction with the treatment effect. D. & D′. In these approaches the focus is on individual differences in the effect of factor A. Do some subjects show a treatment effect whereas others do not? There is a design issue in considering reps to be nested vs. crossed. Nested would be implied by a totally random order of the ar trials of the experiment. Crossed seems to be implied by a design in which all of the a trials of the first replication are completed before the second replication, etc. What are the implications of crossed versus nested for statistical power?Handout #26, p. 4 Four Alternatives for Analyzing Massaro and Anderson design (American Journal of Psychology, 1970) Data averaged over reps. Reps. nested within SOLA Reps. treated as factor Single subject analysis Source df Source df Source df Source df Mean 1 Mean 1 Mean 1 Mean 1 Subjects 15 Subjects 15 Subjects 15 Orientation 1 Orientations 1 Orientations 1 Orientations 1 Length 1 SxO 15 SxO 15 SxO 15 Angle 4 Length 1 Length 1 Length 1 OxL 1 SxL 15 SxL 15 SxL 15 OxA 4 Angle 4 Angle 4 Angle 4 LxA 4 SxA 60 SxA 60 SxA 60 OxLxA 4 OxL 1 OxL 1 OxL 1 Reps/OLA 80 SxOxL 15 SxOxL 15 SxOxL 15 100 OxA 4 OxA 4 OxA 4 SxOxA 60 SxOxA 60 SxOxA 60 Cells for systematic LxA 4 LxA 4 OxT 4 sources = 20 SxLxA 60 SxLxA 60 SxOxT 60 OxLxA 4 OxLxA 4 LxA 4 5 reps per cell = 100 obs. SxOxLxA 60 SxOxLxA 60 SxLxA 60 320 Reps/SOLA 1280 LxT 4 1600 SxLxT 60 Cells for systematic AxT 16 sources = 2 orientations Cells for systematic SxAxT 240 x 2 lengths x 5 angles = 20 sources = 20 OxLxA 4 20 cells x 5 trials = 100 per S SxOxLxA 60 20 scores per S x 16 S = 100 scores per S x 16 S = OxLxT 4 320 obs. 1600 obs. SxOxLxT 60 OxAxT 16 Reps is random SxOxAxT 240 S is random LxAxT 16 SxLxAxT 240 OxLxAxT 16 SxOxLxAxT 240 1400 Cells for systematic sources = 20 x 5 trials = 100 100 scores per S x 16 S = 1600