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

Young et al. Kent et al. 
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 – ϑ1ϑ2
Where αi (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 ϑ1,ϑ2, andϑ3)
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); ex is the natural exponential function 