1. 1 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

2. 2 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

3. 3  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 

4. 4 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 )

5. 5 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 )

6. 6 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”

7. 7 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).

8. 8 (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

9. 9 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

10. 10 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

11. 11 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

12. 12 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

13. 13 An issue to think about: We have:E ( MSI) =   + V int’n E (MSW) =   Since V int’n cannot be negative, and MSI =.0183 0. If this is true, E(MSI) =  , and we should combine MSI and MSW (i.e., “pool”) estimates. This gives: SSQdf MSSSQdfMS Int..11 6.0183 Error.4118.0228 Error.3012.025 to (Some stat packages suggest what you should do).

14. 14 FixedRandom Mixed MSB rows   + V R    + V I + V R   + V R MSB col   + V C   + V I + V C   + V I + V C MSB Int’n    + V I   + V I   + V I MSW error        Another issue: The table of 2 pages ago assumes what is called a “Fixed Model”. There is also what is called a “Random Model” (and a “Mixed Model”). MEAN SQUAREEXPECTATIONS col = fixed row= random col = fixed row= random Reference: Design and Analysis of Experiments by D.C. Montgomery, 4 th edition, Chapter 11.

15. 15 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

16. 16 “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].

17. 17 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.

18. 18 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:

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20. 20 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

21. 21 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

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25. 25 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

26. 26 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 )

27. 27 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

28. 28 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

29. 29 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

30. 30 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:

31. 31 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.

32. 32 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.

33. 33 Blocking We will add a factor even if it is not of interest so that the study of the prime factors is under more homogeneous conditions.This factor is called “block”. Most of time, the block does not interact with prime factors. Popular factors are “location”, “gender” and so on. A two-factor design with one block factor is called a “randomized block design”.

34. 34 For example, suppose that we are studying worker absenteeism as a function of the age of the worker, and have different levels of ages: 25-30, 40-55, and 55-60. However, a worker’s gender may also affect his/her amount of absenteeism. Even though we are not particularly concerned with the impact of gender, we want to ensure that the gender factor does not pollute our conclusions about the effect of age. Moreover, it seems unlikely that “gender” interacts with “ages”. We include “gender” as a block factor.