Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Decomposition of whole-profile ratings into attribute part-worths in conjoint analysis

R_j = beta_0 + sum_(k=1)^K sum_(l=1)^(L_k) beta_kl * x_jkl + e_j

Maps a respondent's ratings of profiles, each described by levels of several attributes, to a part-worth for each level under an additive model, so that a profile's value is the intercept plus the part-worths of its levels. One level of each attribute is the reference, with a part-worth of zero. Relative importance and trade-offs between attributes follow from the part-worths. The notation follows the Conjoint Analysis article, whose outpatient clinic example is used throughout.

  • Part-worth of a two-level attribute from mean ratings in a balanced orthogonal design

    b_k = Rbar_1 - Rbar_0

    In a balanced orthogonal design with dummy coding, the least-squares part-worth of a two-level attribute equals the mean rating of the profiles showing the coded level minus the mean rating of those showing the reference level. Any design can be fitted by multiple regression on the dummy codes (HE-CF-CJA-001); the difference of means is a shortcut that holds only in a balanced orthogonal design, where every pair of levels of any two attributes appears equally often, as in a full factorial or an orthogonal fraction of it.

  • Predicted rating of a clinic profile under the additive part-worth model

    R_hat = b_0 + b_W * x_W + b_L * x_L + b_N * x_N

    Adds the part-worths of a profile's levels to the intercept, the predicted rating of the reference profile. With three two-level attributes it scores every one of the eight possible clinic designs, including ones not shown to respondents in a fractional design, as long as they combine levels used in the study.

  • Relative importance of a conjoint attribute as its share of the total part-worth range

    RI_1 = r_1 / (r_1 + r_2 + r_3)

    Takes the range of an attribute's part-worths, the difference between its most and least preferred levels over the levels in the study, and divides it by the sum of the ranges of all attributes. Written here for three attributes; with K attributes the denominator has K ranges. The share depends on the levels chosen, so it describes the study's design as much as the respondent.

  • Marginal rate of substitution between two conjoint attributes from part-worths

    dx_b = -b_a / b_b

    Gives the change in a continuous attribute b that offsets a one-unit change in attribute a, so that the profile's value is unchanged: minus the ratio of their part-worths. With waiting time as b it expresses a preference in weeks of waiting; with cost as b it is read as willingness to pay (HE-FM-WTP-001), which is a welfare measure only for choice data analysed under random utility theory.