The D1o and D1c models are mathematically equivalent and the two models display a simple mathematical relationship: the constant and the D1 variables are equal in magnitude, and the dummies are shifted away from zero by the value of the constant when it is replaced with the D1 variable. Using the D1c specification, two of the level 2 dummies are statistically nonsignificant.
Shifting all the dummy variable coefficients away from zero by the value of the constant makes them pass the test of statistical significance. Adding a regression variable that shifts other parameter estimates away from zero, thereby making them statistically significant, seems questionable, at least when the additional complexity of the model is not accompanied by other advantages. We are not generally opposed to including nonsignificant parameters in such regression models, but we must point out that the authors of the D1 tariff excluded the I22 variable because it was statistically nonsignificant. If the D1c specification is considered more appropriate, the I22 variable should be included or the nonsignificant dummy variables should be excluded. Removing the nonsignificant dummies would simplify the model considerably, while the resulting health state values would remain virtually unchanged.
The VIF scores for all D1o predictors were larger than those for their D1c counterparts. With suggested cut-offs of somewhere between five and 10, all the predictors display questionable levels of multicollinearity using the D1o model, while the I2 and I3 variables are questionable in both models. In addition to increasing the VIF of all other variables, the D1 variable itself had a VIF of more than 113, which is extreme in relation to the suggested cut-offs. While the reported SE estimates for the D1c coefficients aren't greatly reduced as compared to the D1o counterparts, the use of the D1o specification increases the SE of the estimated health state values, particularly for states with impairments on multiple dimensions. Consider when all five dimensions are impaired that the D1 variable is 4. This means that the SE of the D1 term is invoked four times, while the SE of the constant (in the D1c model) is invoked only once. Thus, using the D1c specification reduces the uncertainty around health state estimates.
With the exception of the very mildest health states, the D1c model required fewer calculation steps than the D1o model. Use of the D1 variable also implies more complex assumptions than suggested by the use of a constant; the use of a constant implies that the observed gap between perfect health and all other tariff values reflects a general tendency for substantial reductions in preference for all imperfect health states. The use of the D1 variable, on the other hand, implies that respondents attribute specific, additive losses in quality of life to specific impairments of health, but that they also include large "disutility discounts" to all health states where more than one dimension of health is impaired. Consider the calculations for health state 22222 (see Table 2); compared to the D1c calculations, the process of adding the constant five times (one time for each of the five dummy coefficients), then retracting it four times (the D1 variable) inflates the apparent magnitude of the dummy coefficients, while hampering interpretation. The D1o model has, as the authors of the US valuation study paper point out, a minor advantage in that it simplifies the calculation of marginal utility loss associated with single-dimension impairment. Unfortunately, the inflated dummy variable coefficients obscure two important properties of the tariff: that the utility loss of any movement away from perfect health is disproportionally large as compared to similar movements given other problems, and that the general public barely distinguishes between the five dimensions when they are at level 2 (moderate problems).
When considering the tariff algorithm, the relative magnitude of the different parameters may often be interpreted as a measure of their relative importance. Thus, the use of the D1o specification leads to the erroneous impression that level 2 problems are somewhere between one half and one quarter as serious as level 3 problems. However, the inflated dummy variable coefficients in the D1o model may systematically bias how people perceive the tariff when presented with the calculation algorithm. From the D1c model, it is apparent that adding level 2 problems to any reduced health state is associated with very little additional utility loss, except when invoking I2/I22. For "usual activities," the estimated utility loss of movement from level 1 to level 2 is zero. This has the interesting effect that, for instance, health states 33133 and 33233 have the same tariff values (see calculation examples in Table 2). This applies to all pairs, 15 in total, of states where "usual activities" moves from 1 to 2, while no other dimension is at level 2 (the 16th such pair, 11111 and 11211, is associated with a utility loss of .140, the value of the constant).
We contend that there are numerous problems with replacing the constant signifying any movement away from perfect health with the D1 variable. First, it constitutes a breach of the scientific guiding law of parsimony (often referred to as Ockham's razor), in that it introduces new premises that unnecessarily complicate the model without adding to its explanatory value. Second, the substitution complicates, and potentially biases, interpretation of the tariff. Third, it obscures the fact that the dummies representing moderate problems with mobility, usual activities, and anxiety/depression are not significantly different from zero, while the two remaining level 2 dummy coefficients are both close to zero. Finally, it obscures the disproportionate utility loss of any movement away from perfect health as compared to similar movements from reduced health to further reduced health.
In this paper, we have focused on the problems related to the way the authors of the US valuation study replaced the constant term from previous regression models with the D1 variable. However, this is not the only alteration they made to the regression model used in the UK valuation study, in which the authors used a model consisting of the 10-dimension dummy variables present in the D1o model, a constant term, and dummy N3, representing the presence of any level 3 problems. In the D1o model, the N3 was replaced by the I3, representing the number of dimensions at level 3, beyond the first. Substituting the N3 with the I3 has an effect similar to the introduction of the D1 variable, in that it inflates the values of the level 3 coefficients. Replacing the I3 variable in the D1o model with the N3 dummy variable results in identical predicted values for all EQ-5D health states, while further reducing the multicollinearity of the model. The VIF for N3 would be 3.01 when substituting the I3 in the D1c model, while the VIFs of the other variables would remain unchanged. Nevertheless, substituting the N3 with I3 is less problematic than replacing the constant term with the D1 variable, since this procedure does not alter the statistical significance of any of the coefficients.