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Additive and Multiplicative Effects in a Fixed 2x2 Design using ANOVA

This article explores the difficulties in differentiating between additive and multiplicative effects in a fixed 2x2 design using ANOVA. It analyzes an example two-factor ANOVA and discusses the non-additive relation in the design.

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Additive and Multiplicative Effects in a Fixed 2x2 Design using ANOVA

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  1. Additive and multiplicative effects in a fixed 2 x 2 design using ANOVA can be difficult to differentiate J. A. Landsheer & G. van den Wittenboer

  2. Example two factor ANOVA • A, B: arbitrary coded independent variables {-1, 1} • Y: dependent variable

  3. Motivation * Ability interaction(Y = Performance) Means of the non-additive relation in the 2 x 2 design of Table 1 and 2 that are produced by a single product term. ANOVA analysis does not show a significant interaction effect (N = 96).

  4. Table 1: Exemplary ANOVA Results of 96 Cases. Demonstration of Enlarged Main Effects and no Interaction Effect when testing model Y = A + B + A*B. Partial eta Source SS df MS F p. squared Corrected Model 4125,954 3 1375,3 35,3 ,000 ,535 Intercept 19615,168 1 19615,1 503,0 ,000 ,845 A 2254,534 1 2254,5 57,8 ,000 ,386 B 1868,284 1 1868,2 47,9 ,000 ,342 A * B 3,136 1 3,1 ,1 ,777 ,001 Error 3587,566 92 38,9 Total 27328,688 96 Corrected Total 7713,520 95

  5. Four questions • What happened? Why did the ANOVA result not indicate an interaction? • Is this a real situation, which can be expected in real experimenting or is it just a simulation artefact? • Is there an analysis model that would be more appropriate for this kind of data? • How can the researcher decide that his model of analysis is not particularly fitting in this situation?

  6. Statistical interpretation: Rejection of the multiplicative model Simulated data: Y = A’ * B’ A’, B’: not directly observed treatment strengths A, B: {-1, +1} There are always linear relationships. In a simulation they are known. A’ = A + 3 B’ = B + 3 Y = (A + 3) (B + 3) + error What happened?

  7. What happened: diagnosis Suppose • real treatment values A’ (a1, a2) • real treatment values B’ (b1, b2) • Y = A’ * B’. with n observations in each cell. A -1 +1 B -1 O1 O2 +1 O3 O4

  8. Then: • Xmeantot = (O1 + O2 + O3 + O4) / 4. And • SSA = 4n [( (b1 + b2)(a1 – a2)) / 4] 2, • SSB = 4n [( (a1 + a2)(b1 – b2)) / 4] 2, • SSAB = 4n [(a1- a2)(b1 – b2) / 4] 2.

  9. Illustration of the dependency of the expected mean squares of main factor A on the treatment results b1 and b2 • Y = A + B + AB • E[MSA] = n {(b1 + b2 +2) (a1 – a2)}2/4 • E[MSAB] = n {(b1 - b2) (a1 – a2)}2/4 = 6 • Y = AB • E[MSA] = n {(b1 + b2)(a1 – a2)} 2/4 • E[MSAB] = n {(b1 - b2) (a1 – a2)}2/4 = 6 • Y = A + B • E[MSA] = n (2a1 – 2a2)2/4 = 6

  10. Seriousness of the problem ANOVA results of 21.000 simulations of a combinatory effect Performance = Ability * Motivation. The mean differences between the groups are kept constant (.5) for ability, while the mean manipulation difference of motivation is varied in 21 steps from 0 to 2. For each step the simulation is repeated 1000 times.

  11. Discussion • Can this situation arise in real experimenting? • Is centering helpful? • Sensitivity interactions / main effects • Wahlsten: sample 7 to 9 larger • McClelland & Judd: efficiency about 20% • Is Multiple Regression a better approach? • Measurement of the independent variables • Extended experimenting

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