1. 1 Love does not come by demanding from others, but it is a self initiation.

2. 2 Two Factor Designs Consider studying the impact of two factors on the yield (response): Here we have R = 3 rows (levels of the Row factor), C = 4 (levels of the column factor), and n = 2 replicates per cell [n ij for (i,j) th cell if not all equal] NOTE: The “1”, “2”,etc... mean Level 1, Level 2, etc..., NOT metric values NOTE: The “1”, “2”,etc... mean Level 1, Level 2, etc..., NOT metric values 1234 17.9, 18.1 17.8, 17.8 18.1, 18.2 17.8, 17.9 18.0, 18.2 18.0, 18.3 18.4, 18.1 18.1, 18.5 18.0, 17.8 17.8, 18.0 18.1, 18.3 18.1, 17.9 BRAND 123123 DEVICE

3. 3 MODEL: i = 1,..., R j = 1,..., C k= 1,..., n In general, n observations per cell, R C cells. Y ijk =  i  j  ij  ijk

4. 4  the grand mean  i  the difference between the ith row mean and the grand mean  j  the difference between the jth column mean and the grand mean  ij  the interaction associated with the i-th row and the j-th column  ij  i  j 

5. 5 Where Y = Grand mean Y i = Mean of row i Y j = Mean of column j Y ij = Mean of cell (i,j) [ All the terms are somewhat “intuitive”, except for (Y ij -Y i - Y j + Y ) ] Y ijk = Y + (Y i - Y ) + (Y j - Y ) + (Y ij - Y i - Y j + Y ) + (Y ijk - Y ij )

6. 6 The term (Y ij -Y i - Y j + Y ) is more intuitively written as: how a cell mean differs from grand mean adjustment for “row membership” adjustment for “column membership” We can, without loss of generality, assume (for a moment) that there is no error (random part); why then might the above be non-zero? (Y ij - Y )(Y i - Y )(Y j - Y )

7. 7 ANSWER: Two basic ways to look at interaction: B L B H A L 58 A H 10? If A H B H = 13, no interaction If A H B H > 13, + interaction If A H B H < 13, - interaction -When B goes from B L  B H, yield goes up by 3 (5  8). - When A goes from A L  A H, yield goes up by 5 (5  10). -When both changes of level occur, does yield go up by the sum, 3 + 5 = 8? Interaction = degree of difference from sum of separate effects 1) “INTERACTION”

8. 8 2) -Holding B L, what happens as A goes from A L  A H ?+5 -Holding B H, what happens as A goes from A L  A H ?+9 If the effect of one factor (i.e., the impact of changing its level) is DIFFERENT for different levels of another factor, then INTERACTION exists between the two factors. B L B H A L 58 A H 1017 NOTE: - Holding A L, B L B H has impact + 3 - Holding A H, B L B H has impact + 7 (AB) = (BA) or (9-5) = (7-3).

9. 9 (Y ijk - Y ) = (Y i - Y ) + (Y j - Y ) + [(Y ij - Y i ) - (Y j - Y )] + (Y ijk - Y ij ) Going back to the (model) equation on page 4, and bringing Y... to the other side of the equation, we get If we then square both sides, triple sum both sides over i, j, and k, we get, (after noting that all cross-product terms cancel): Effect of column j at row i. Effect of column j

10. 10 TSS = SSB Rows + SSB Cols + SSI R,C + SSW Error and, in terms of degrees of freedom, R. C. n-1 = (R-1) + (C-1) + (R-1)(C-1) + R. C. (n-1); DF of Interaction = (RC-1)-(R-1)-(C-1) = (R-1)(C-1). OR,  (Y ijk - Y )   n. C.  Y i - Y   ijk i + n. R.  Y j - Y ) 2 + n.  Y ij - Y i - Y j +Y   ij  (Y ijk - Y ij   ijk j

11. 11 17.9, 18.1 17.8, 17.8 18.1, 18.2 17.8,17.9 18.117.818.1517.85 18.2, 18.0 18.0, 18.3 18.4, 18.1 18.1, 18.5 18.118.1518.25 18.3 18.0, 17.8 17.8, 18.0 18.1, 18.3 18.1, 17.9 17.917.9 18.2 18.0 12341234 18.00 17.95 18.20 18.05 1 2 3 In our example: D E V I C E 17.95 18.20 18.00 18.05 BRAND

12. 12 SSB rows =2 4[(17.95-18.05) 2 + (18.20-18.05) 2 + (18.0-18.05) 2 ] =8 (.01 +.0225 +.0025) =.28 SSB col =23[(18-18.05) 2 +(17.95-18.05) 2 +(18.2-18.05) 2 +( 18.05-18.05) 2 ] = 6 (.0025 +.001 +.0225 + 0) =.21 SSI R,C =2 (18-17.95-18+18.05) 2 + (17.8-17.95-17.95+18.05) 2....… + (18-18-18.05+18.05) 2 [ ] = 2 [.055] =.11 SSW = (17.9-18.0) 2 + (18.1-18.0) 2 + (17.8-17.8) 2 + (17.8-17.8) 2 + …....... (18.1-18.0) 2 + (17.9-18.0) 2 =.30 TSS =.28 +.21+.11 +.30 =.90

13. 13 F TV (2, 12) = 3.89 Reject H o F TV (3, 12) = 3.49 Accept H o F TV (6, 12) = 3.00 Accept H o 1)H o : All Row Means Equal H 1 : Not all Row Means Equal 2)H o : All Col. Means Equal H 1 : Not All Col. Means Equal 3)H o : No Int’n between factors H 1 : There is int’n between factors ANOVA .05 SOURCESSQdfM.S.Fcalc Rows.28 2.14 5.6 COL.21 3.07 2.8 Int’n.11 6.0183.73 Error.3012.025

14. 14

15. 15 Minitab: Stat >> Anova >> General Linear Model General Linear Model: time versus brand, device Factor Type Levels Values brand fixed 4 1, 2, 3, 4 device fixed 3 1, 2, 3 Analysis of Variance for time, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P brand 3 0.21000 0.21000 0.07000 2.80 0.085 device 2 0.28000 0.28000 0.14000 5.60 0.019 brand*device 6 0.11000 0.11000 0.01833 0.73 0.633 Error 12 0.30000 0.30000 0.02500 Total 23 0.90000 S = 0.158114 R-Sq = 66.67% R-Sq(adj) = 36.11%

16. 16

17. Assumption:   ijk follows N(0,  2 ) for all i, j, k, and they are independent. 17

18. 18 Test for Normality 1.Restore residuals when doing Anova. 2.Stat >> Basic Statistics >> Normality Test Mean-4.44089E-16 StDev0.1142 N24 AD0.815 P-Value0.030 Not really normal but not too far from normal.

19. 19 Test for Equal Variances Minitab: Stat >> Anova >> Test for Equal Variances Test for Equal Variances: time versus device, brand 95% Bonferroni confidence intervals for standard deviations device brand N Lower StDev Upper 1 1 2 0.0463363 0.141421 49.6486 1 2 2 * 0.000000 * … 3 4 2 0.0463363 0.141421 49.6486 Bartlett's Test (normal distribution) Test statistic = 2.33, p-value = 0.993 * NOTE * Levene's test cannot be computed for these data.

20. Fixed Effect Model 20

21. Random Effect Model Additional assumptions:  i follows N(0,  2  ) for all i, and they are independent.  j follows N(0,  2  ) for all j, and they are independent.  ij follows N(0,  2  ) for all I, j, and they are independent. All these random components  i  j,  ij  ijk are (mutually) independent. 21

22. Mixed Effect Model (fixed rows and random columns) 22

23. 23 FixedRandom Mixed MSR ows    cn         n       cn        n       cn   MSC ol      Rn         n       Rn       n       Rn    MSRC      n        n        n    MSE rror            Another issue: Table 17.17 (O/L 6 th ed., p. 1057) MEAN SQUAREEXPECTATIONS col = random row= fixed col = random row= fixed Reference: Design and Analysis of Experiments by D.C. Montgomery, 4 th edition, Chapter 11.

24. 24 Fixed:Specific levels chosen by the experimenter Random:Levels chosen randomly from a large number of possibilities Fixed: All Levels about which inferences are to be made are included in the experiment Random:Levels are some of a large number possible Fixed:A definite number of qualitatively distinguishable levels, and we plan to study them all, or a continuous set of quantitative settings, but we choose a suitable, definite subset in a limited region and confine inferences to that subset Random:Levels are a random sample from an infinite ( or large) population

25. 25 “In a great number of cases the investigator may argue either way, depending on his mood and his handling of the subject matter. In other words, it is more a matter of assumption than of reality.” Some authors say that if in doubt, assume fixed model. Others say things like “I think in most experimental situations the random model is applicable.” [The latter quote is from a person whose experiments are in the field of biology].

26. 26 My own feeling is that in most areas of management, a majority of experiments involve the fixed model [e.g., specific promotional campaigns, two specific ways of handling an issue on an income statement, etc.]. Many cases involve neither a “pure” fixed nor a “pure” random situation [e.g., selecting 3 prices from 6 “practical” possibilities]. Note that the issue sometimes becomes irrelevant in a practical sense when (certain) interactions are not present. Also note that each assumption may yield you the same “answer” in terms of practical application, in which case the distinction may not be an important one.

27. How to Fit these Models in Minitab 27 “Balanced ANOVA” can fit restricted and unrestricted version. By default, it shows unrestricted model. “General Linear Model” can only fit unrestricted model. There are no difference between restricted or unrestricted versions for fixed effect and random effect model. It only matters for the mixed effect model.

28. More on Minitab 28 The notation in EMS output under restricted model matches with ours but it under unrestricted model is different to ours. General suggestion: use “General Linear Model” to fit the models BUT use “Balanced ANOVA, option of restricted model” to find the EMS for fixed and random effect models.

29. 29 Two-Way ANOVA in Minitab Stat>>Anova>>General Linear Model: Model device brand device*brand Random factors Results Factor plots Graphs device Tick “Display expected mean squares and variance components” Main effects plots & Interactions plots Use standardized residuals for plots

30. 30 General Linear Model: time versus device, brand Factor Type Levels Values device random 3 1 2 3 brand fixed 4 1 2 3 4 Analysis of Variance for time, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P device 2 0.28000 0.28000 0.14000 7.64 0.022 brand 3 0.21000 0.21000 0.07000 3.82 0.076 device*brand 6 0.11000 0.11000 0.01833 0.73 0.633 Error 12 0.30000 0.30000 0.02500 Total 23 0.90000

31. 31 Exercise: Lifetime of a Special-purpose Battery It is important in battery testing to consider different temperatures and modes of use; a battery that is superior at one temperature and mode of use is not necessarily superior at other treatment combination. The batteries were being tested at 4 different temperatures for three modes of use (I for intermittent, C for continuous, S for sporadic). Analyze the data.

32. 32 Battery Lifetime (2 replicates) Mode of use 1234 I12, 1615, 1931, 3953, 55 C15, 1917, 1730, 3451, 49 S11, 1724, 2233, 3761, 67 Temperature

33. 33 MFMF Interesting Example:* Frontiersman April 50 people per cell Mean Scores “Frontiersman” “April” “Frontiersman” “April” Dependentmalesmalesfemalesfemales Variables(n=50)(n=50)(n=50) (n=50) Intent-to- purchase4.443.502.044.52 (*) Decision Sciences”, Vol. 9, p. 470, 1978 Brand Name Appeal for Men & Women:

34. 34

35. 35 ANOVA Results DependentSource d.f. MS F Variable Intent-to-Sex (A)123.805.61* purchaseBrand name (B)129.646.99** (7 pt. scale)A x B1146.2134.48*** Error1964.24 *p<.05 **p<.01 ***p<.001

36. 36 Two Factors with No Replication, When there’s no replication, there is no “pure” way to estimate ERROR. Error is measured by considering more than one observation (i.e., replication) at the same “treatment combination” (i.e., experimental conditions). 123 1734 21068 3625 4957 A B

37. 37 Our model for analysis is “technically”: Y ij =  i  j + I ij i = 1,..., R j = 1,..., C We can write: Y ij = Y + (Y i - Y ) + (Y j - Y ) + (Y ij - Y i - Y j + Y )

38. 38 After bringing Y to the other side of the equation, squaring both sides, and double summing over i and j, We Find:  Y ij - Y ) 2 = C  Y i -Y ) 2 + R  Y j - Y ) 2 +  (Y ij - Y i - Y j + Y ) 2 R i = 1 C j=1 R i=1 C j=1 R i=1 C j=1

39. 39 TSS = SSB ROWS + SSB Col + SSI R, C RC - 1 = (R - 1) + (C - 1) + (R - 1) (C - 1) Degrees of Freedom : We Know, E(MS Int. ) =    V Int. If we assume V Int. = 0, E(MS Int. ) =  2, and we can call SSI R,C SSW MS Int MSW

40. 40 And, our model may be rewritten: Y ij =  +  i +  j +  ij, and the “labels” would become: TSS = SSB ROWS + SSB Col + SSW Error In our problem:SSB rows = 28.67 SSB col = 32 SSW = 1.33

41. 41 Source SSQ df MSQ F calc rows col Error 28.67 32.00 1.33 9.55 16.00 00.22 326326 43 72 TSS = 62 11 at  =.01, F TV (3,6) = 9.78 F TV (2,6) = 10.93 ANOVA and:

42. 42 What if we’re wrong about there being no interaction? If we “think” our ratio is, in Expectation,  2 + V ROWS, (Say, for ROWS) 22 and it really is (because there’s interaction)  2 + V ROWS,  2 + V int’n being wrong can lead only to giving us an underestimated F calc.

43. 43 Thus, if we’ve REJECTED Ho, we can feel confident of our conclusion, even if there’s interaction If we’ve ACCEPTED Ho, only then could the no interaction assumption be CRITICAL.