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Experimental Design and Choice modelling

Experimental Design and Choice modelling. 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:

malcolm-reynolds
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Experimental Design and Choice modelling

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  1. Experimental Design and Choice modelling

  2. 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 2nd order interaction terms P1*P2, P1*P3 and P2*P3, • And 3rd order interaction term P1*P2*P3

  3. 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) Motivating example

  4. 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(333); 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 63 9 So we may decide to go with n=18 scenarios

  5. 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

  6. 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:

  7. 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.

  8. 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.

  9. Let’s fit a model with main effects with 2nd 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

  10. 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

  11. Let’s fit a model with main effects with 2nd 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

  12. 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 2nd order (3rd 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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