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Experimental Design and Choice modelling

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Motivating example Suppose we have three products which can be set at three price points Priced at $1, $2 and $3 (note equally spaced). These can be recoded as -1,0 1 respectively (-$2 i.e. –mean centred) We have is a 3x3x3 design. We can measure: – the main effects for price for each model, called P1, P2 and P3 (also P1^2, P2^2, P3^2 for quadratic effects) –The 2 nd order interaction terms P1*P2, P1*P3 and P2*P3, –And 3 rd order interaction term P1*P2*P3

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Motivating example What we wish to do is measure particular quantities of interest with the smallest number of scenarios (a.k.a. sets or runs) We want to have: – balance (equal sample sizes per combination) – and orthogonality (correlations between effects is zero)

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How may scenarios do we need? If we have a straight linear main effects we the following tells us how many runs we may need (in SAS): %mktruns(3 3 3); Some Reasonable Design Sizes Cannot Be (Saturated=7) Violations Divided By 9 0 18 0 12 3 9 15 3 9 7 6 3 9 8 6 3 9 10 6 3 9 11 6 3 9 13 6 3 9 14 6 3 9 So we may decide to go with n=18 scenarios

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Let’s fit a main effects only model %mktdes(factors=x1-x3=3,n=18) proc print; run; Prediction Design Standard Number D-Efficiency A-Efficiency G-Efficiency Error ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 1 100.0000 100.0000 100.0000 0.6236 2 100.0000 100.0000 100.0000 0.6236 3 98.4771 96.8616 84.3647 0.6336 4 98.4771 96.8616 84.3647 0.6336 5 98.4771 96.8616 84.3647 0.6336 Obs x1 x2 x3 1 3 3 2 2 3 3 1 3 3 2 3 4 3 2 2 5 3 1 3 6 3 1 1 7 2 3 2 8 2 3 1 9 2 2 3 10 2 2 2 11 2 1 3 12 2 1 1 13 1 3 3 14 1 3 3 15 1 2 1 16 1 2 1 17 1 1 2 18 1 1 2

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How does this work out? We have 100% efficiency for the effects we wish to measure (main effects) But if we look at the correlation matrix of effects we have the following:

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Is this good enough? We see that the main effects are all orthogonal, but we have some correlation between these and the higher order interaction terms. (eg: P3^2 and P1*P3.) Is this a problem? –Well yes and no. No, if these effects are not of interest (e.g. P1*P3) – i.e. we suspect they don’t exist in real life. Yes, if we suspect they might and/or we or not sure if they do or not.

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Is this good enough?… Well-known fact in almost cases involving real data (Louviere, Hensher, Swait, 2000) Main effects explain the largest amount of variance in respondent data, often 80% or more (70-90%); Two-way interactions account for the next largest proportion of variance, although this rarely exceeds 3%~6%; Three-way interactions account for even smaller proportions of variance, rarely more than 2%~3% (usually 0.5%~1%); Higher-order interactions account for minuscule proportions of variance.

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Let’s fit a model with main effects with 2 nd order interactions %mktdes(factors=x1-x3=3, interact = x1*x2 x1*x3 x2*x3 x1*x2*x3,n=18) proc print; run; Design Standard Number D-Efficiency A-Efficiency G-Efficiency Error ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 1 0.0000 0.0000 0.0000 0 2 0.0000 0.0000 0.0000 0 3 0.0000 0.0000 0.0000 0 4 0.0000 0.0000 0.0000 0 5 0.0000 0.0000 0.0000 0

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So how do we do now? This is an unmitigated disaster when we only have 18 scenarios. So let’s change the number of scenarios we investigate. We can increase this to 27 – as this is divisible by 3x3 = 9 –i.e. every possible combination for two 3 level factors

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Let’s fit a model with main effects with 2 nd order interactions (27 scenarios) %mktdes(factors=x1-x3=3, interact = x1*x2 x1*x3 x2*x3 x1*x2*x3,n=27) proc print; run; Design Standard Number D-Efficiency A-Efficiency G-Efficiency Error ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 1 100.0000 100.0000 100.0000 1.0000 2 100.0000 100.0000 100.0000 1.0000 3 100.0000 100.0000 100.0000 1.0000 4 100.0000 100.0000 100.0000 1.0000 5 100.0000 100.0000 100.0000 1.0000

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Conclusions Try to keep the number of scenarios (runs, sets) to less than 40 max. – otherwise you get respondent fatigue Only measure effects up to 2 nd order (3 rd order and above are difficult to explain and don’t account for much explanation If you have prior knowledge of which effects are more likely than others, then use this to establish which effects you want to measure.

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