Table 1 Algorithms used to predict ED-5D summary index utilities

prob(“no problems”) = ϑ₁ = 0.5 – tan^− 1 (−α₁ + β’X)/  π

prob.(“some problems”) = ϑ₂ = 0.5 – tan^− 1 (−α₂ + β’X)/  π- ϑ₁

prob.(“extreme problems”) = ϑ₃ = 1 – ϑ₁ – ϑ₂

Where α_i (i₌ 1; 2) were the constant terms in the linear

predictor for “no problems” and “some problems,”

respectively, and β’X in the linear predictor related the

functioning levels of an EQ-5D-3L dimension with the PDQ-39 subscale scores. Based on individual pattern X, the predicted functioning level can be obtained by assigning the category with the largest estimated probability (that is, the maximum of ϑ_1,ϑ_{2, and}ϑ₃)

Dimension i = β(Dimension, i)*x

Where i = 1,2,3. Dimension i corresponds to getting a response to an EQ-5D-3L question (i.e. Mobility 2 indicates the response “some problems” in the Mobility dimension).

β is the vector of the regression coefficients.

x is the matrix of the PDQ-39 scores of the subscales.

\( P\ \left( Dimension\ i\right)=\frac{e^{Dimension\ i}}{e^{Dimension\ 1}+{e}^{Domension\ 2}+{e}^{Dimension\ 3}} \)

Where i = 1,2,3, and Dimension 1 = 0 (The pivot outcome); e^x is the natural exponential function