1 Experimental Statistics Spring week 6 Chapter 15: Factorial Models (15.5)

2 STIMULUS EXAMPLE: Personal computer presents stimulus, and person responds. Study of how RESPONSE TIME is effected by a WARNING given prior to the stimulus: 2-factors of interest: Warning Type --- auditory or visual Time between warning and stimulus -- 5 sec, 10 sec, or 15 sec.

Auditory Visual 5 sec 10 sec 15 sec Warning Time Note: “Sort of like RCB” -- what is the difference? Question: How would you randomize? - 18 subjects - 1 subject

4 Observed data Level of Factor A Level of Factor B Replication (warning type) (time) (response time) Stimulus Data

5 Factor A Factor B 2-Factor ANOVA Data

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7 A Possible Model for STIMULUS Data Note: so according to this model Note: The model assumes that the difference between types is the same for all times i = type, j = time

8 Auditory Visual Hypothetical Cell Means

9 Similarly i.e. the model says the difference between times j and j' is the same for all types We may not want to make these assumptions!!

10 Auditory Visual Hypothetical Cell Means Auditory Visual

11 Model for 2-factor Design where

12 Sum-of-Squares Breakdown (2-factor ANOVA) SSA SSB SSAB SSE

13 2-Factor ANOVA Table (2-Factor Completely Randomized Design) Source SS df MS F Main Effects A SSA a  1 B SSB b  1 Interaction AB SSAB ( a  1)(b  1) Error SSE ab(n  1) Total TSS abn  See page 900

14 Hypotheses: Main Effects: Interactions:

15 data stimulus; input type$ time response; datalines; A A A A A A A A A V V V V V V V V V ; PROC GLM; CLASSES type time; MODEL response=type time type*time; means type/lsd; means time/lsd; TITLE ‘Stimulus Data'; run; Stimulus Data -- SAS

16 The GLM Procedure Dependent Variable: response Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Corrected Total R-Square Coeff Var Root MSE response Mean Source DF Type I SS Mean Square F Value Pr > F type <.0001 time type*time GLM Output

17 Testing Procedure 2 factor CRD Design Step 1. Test for interaction. Step 2. (a) IF there IS NOT a significant interaction - test the main effects (b) IF there IS a significant interaction - compare cell means

18 Stimulus Example Test for Interaction: Therefore we DO NOT reject the null hypothesis of no interaction.

19 Stimulus Data

20 Stimulus Example Test for Interaction: Therefore we DO NOT reject the null hypothesis of no interaction. Thus - based on the testing procedure, we next test for main effects.

Testing Main Effects: For each main effect (i.e. A and B) Note: I’ll use LSD from this point on unless otherwise noted. In General: where N denotes the # of observations involved in the computation of a marginal mean.

22 Auditory Visual 5 sec 10 sec 15 sec Warning Time

23 Stimulus Example Test for Main Effects: Thus, there is a significant effect due to type but not time A (type): B (time): - i.e. we can use LSD to compare marginal means for type - we will do this here for illustration although MC not needed when there are only 2 groups

24 The GLM Procedure t Tests (LSD) for response NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square Critical Value of t Least Significant Difference Means with the same letter are not significantly different. t Grouping Mean N type A V B A GLM Output -- Comparing “Types”

25 The GLM Procedure t Tests (LSD) for response NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square Critical Value of t Least Significant Difference Means with the same letter are not significantly different. t Grouping Mean N time A A A A A GLM Output -- Comparing “Times”

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28 Pilot Plant Data Variable = Chemical Yield Factors:A – Temperature (160, 180) B – Catalyst (C1, C2) 160 C C C C C C C C C C C C C C C C2 81

29 Pilot Plant Data Variable = Chemical Yield Factors:A – Temperature (160, 180) B – Catalyst (C1, C2) Catalyst Temperature

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32 Pilot Plant -- Probability Plot of Residuals

33 DATA one; INPUT temp catalyst$ yield; datalines; 160 C C C C2 81 ; PROC GLM; class temp catalyst; MODEL yield=temp catalyst temp*catalyst; Title 'Pilot Plant Example -- 2-way ANOVA'; MEANS temp catalyst/LSD; RUN; PROC SORT;BY temp catalyst; PROC MEANS; BY temp catalyst; OUTPUT OUT=cells MEAN=yield; RUN;

34 Pilot Plant Example -- 2-way ANOVA General Linear Models Procedure Dependent Variable: YIELD Sum of Mean Source DF Squares Square F Value Pr > F Model Error Corrected Total R-Square C.V. Root MSE YIELD Mean Source DF Type I SS Mean Square F Value Pr > F TEMP CATALYST TEMP*CATALYST Pilot Plant -- GLM Output

35 RECALL: Testing Procedure 2 factor CRD Design Step 1. Test for interaction. Step 2. (a) IF there IS NOT a significant interaction - test the main effects (b) IF there IS a significant interaction - compare cell means

36 Pilot Plant Example Test for Interaction: Therefore we reject the null hypothesis of no interaction - and conclude that there is an interaction between temperature and catalyst. Thus, we DO NOT test main effects

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39 Since there is a significant interaction, we do not test for main effects! - instead compare “Cell Means” - NOTE: interaction plot is a plot of the cell means

40 Pilot Plant Data Variable = Chemical Yield Factors:A – Temperature (160, 180) B – Catalyst (C1, C2) Catalyst Temperature

41 Pilot Plant Data -- cell means Catalyst Temperature

Comparing Cell Means: If there is significant interaction, then we compare the a x b cell means using the criteria below. Procedure similar to that for comparing marginal means: where N denotes the # of observations involved in the computation of a cell mean.

43 The GLM Procedure t Tests (LSD) for yield NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square 14.5 Critical Value of t Least Significant Difference Means with the same letter are not significantly different. t Grouping Mean N temp A B GLM Output -- Comparing “Temps” - disregard

44 The GLM Procedure t Tests (LSD) for yield NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Alpha 0.05 Error Degrees of Freedom 12 Error Mean Square 14.5 Critical Value of t Least Significant Difference Means with the same letter are not significantly different. t Grouping Mean N catalyst A C2 A A C1 GLM Output -- Comparing “Catalysts” - disregard

45 Note: - SAS does not provide a comparison of cell means

46 Pilot Plant Data -- cell means CatalystCatalyst Temperature LSD: MSE = N = LSD = C2/160 C1/160 C1/180 C2/

47 Testing Procedure Revisted 2 factor CRD Design Step 1. Test for interaction. Step 2. (a) IF there IS NOT a significant interaction - test the main effects (b) IF there IS a significant interaction - compare a x b cell means (by hand) Main Idea: We are trying to determine whether the factors effect the response either individually or collectively.