2.3. Multinomial Logistic Regression

To understand whether or not EDs that provide high quality of care also produce the outputs in an efficient manner, we classified the EDs into four categories based on their performance levels. High efficiency refers to an efficient frontier ED (i.e., EDs with an efficiency score equal to one, or θ ^∗ = 1), whereas low efficiency refers to an inefficient ED (i.e., EDs with an efficiency score less than one, or θ ^∗ < 1). High quality refers to an ED whose quality measures (LWBT, PCI, and PAIN) were all better than the national mean (for each metric), whereas low quality refers to an ED with at least one of the three quality measures worse than its national mean. Hence, the EDs are classified into the four categories as follows (depicted in Table 2): high efficiency with high quality (category 1), high efficiency with low quality (category 2), low efficiency with high quality (category 3), and low efficiency with low quality (category 4).

Classification of EDs based on efficiency and quality.

We employed multinomial logistic regression to investigate the relationship between ED efficiency and quality performance and the factors affecting this relationship. EDs in categories 2 and 3 deviated from EDs in category 1, either for quality or for efficiency. However, EDs in category 4 significantly differed from those in category 1. To focus the analysis on identifying ED characteristics that contribute to compromised efficiency or quality, we included only the EDs in the first three categories in the multinomial logistic model. The response variable, ED performance or Y, can take on any of m = 1, 2, or 3 qualitative values (represented by categories 1, 2, and 3, resp.). Let π _ij denote the probability that the ith observation falls in the jth category of the response variable; that is, π _ij ≡ Pr(Y _i = j), for j = 1,   …  , m. We have k = 6 explanatory variables of interest, X ₁, … , X ₆, on which the π _ij depend. These six regressors (described in Table 3) represent operational and structural characteristics of the EDs that may be associated with the relationship between ED efficiency and quality performance.

Description of explanatory variables in econometric model.

This dependence between ED performance and the six explanatory variables is modeled using a multinomial logistic distribution:

In this multinomial logit model, there is one set of parameters, γ _0j, γ _1j,   …  , γ _kj, for each response category but the baseline. The use of a baseline category (category 1 in our model) is one way to avoid redundant parameters because of the restriction, reflected in (7), that the response category probability for each observations must sum to one. Upon some algebraic manipulation, we get the following model:

The regression coefficients in (8) represent effects on the log-odds of membership in category j versus the baseline category m. These regression coefficients are estimated using the method of maximum likelihood. Note that it is convenient to impose the restriction ∑_j=1 ^m π _ij = 1 by setting γ _m = 0 (making category m the baseline). This allows us to interpret γ _kj as the effect of X _k on the logit of category j relative to category 1 (baseline). In addition, we can form the log-odds of membership in any pair of category j and j′ (other than category m), where the regression coefficients for the logit between any pair of categories are the differences between corresponding coefficients for the two categories. Equation (9) allows us to interpret (γ _kj − γ _kj′) as follows: for a unit change in X_k, the logit of category j versus category j′ is expected to change by (γ _kj − γ _kj′) units, holding all other variables constant.