# Multinomial logit model and predicting estimated uptake of an alternative

Dear all,

We have conducted a Choice based conjoint analysis to estimate public preferences for different cancer screening methods. The CBC consisted of 5 attributes with 4 levels each. A dual response none was included. We estimated the utilities of the levels with a multinomial logit model using Sawtooth software. We estimated the utility for a screening alternative using the following formula:

U = V + E = β0 + Sum (β1Xi) + E

Whereby:
βi are the part-worths for all levels of the attributes (estimated from the MNL model)
β0 is a constant term, indicating the relative weight on average placed by individuals on screening programmes compared to no screening.

We are struggling with filling in the formula and with predicting the expected uptake of various screening test. We know that in Sawtooth effect coding is used. Hence, the reference level is defined as the negative sum of the estimated coefficients.

Example:
We have 3 questions regarding this example.

CBC with two attributes with 3 levels each:
Pain
Heavy pain                    -0.658
Mild pain                     0.079
No pain (reference) 0.579
Accuracy of the test
100%                             0.496
75%                             0.265
50% (reference)    -0.761
None                                      -0.519

1.    We are wondering how we can estimate the β0 alternative specific constant. Is this the utility value of the none (-0.519), is this the negative of the none (0.519), or is this another value?

2.    Furthermore, we do not know exactly how to fill in the formula.
We heard that Xi has a value of 1 if the associated level of a certain attribute is present in the screening alternative, -1 if the associated level is present and belongs to the reference level of the attribute and zero otherwise, when effects coding is used.

Suppose we have a screening test with heavy pain and 100% accuracy. Is it accurate to predict the total utility of the test, as follows: U = β0 + 1x-0.658 + 1x 0.496 ?
And a test with heavy pain and 50% accuracy, as follows: U = β0 + 1x-0.658 + -1x -0.761 ?

However, then the test with heavy pain and 50% accuracy has a higher utility than the test with 100% accuracy. What do we wrong?
Or can we just add up the path-worth utilities for the different attributes corresponding with the different levels?

3.    Can we predict the expected uptake of a screening alternative using the following formula?

P (accepting alternative test i) = 1 / (1+e^-V)

And how is the value of the none-utility incorporated in this formula?

We hope you can provide assistance with this problem.