Using decision trees for measuring gender equity in the timing of angiography in patients with acute coronary syndrome: a novel approach to equity analysis

The Discharge Abstract Database (DAD) created, by the Canadian Institute for Health Information (CIHI) providing information on admission and discharge dates, diagnostic codes, hospital identifiers, age, sex, postal code, and discharge disposition was linked at the patient level to Statistics Canada’s Canadian Community Health Survey (CCHS) [18‐21]. The CCHS survey was started in 2001 and repeated every two years. It provides information on numerous demographic metrics including language, ethnicity, cultural group, age, sex, geographic region (urban versus rural), marital status, education, residence type, labor force participation, personal and household income, and a health utility index (HUI) developed at McMaster, measuring health status. Approximately 42,000 patients were surveyed from Ontario.

The primary outcome variable, delay to procedure was defined as the difference in days between date of admission to an acute care facility and date of angiography procedure. The importance of timing of angiography results from recent studies suggesting strong correlations between early invasive treatment (i.e. angiography with revascularization if indicated) and outcomes (death, myocardial infarction, stroke, or refractory ischemia) among patients with an ACS. For some types of ACS early angiography (i.e. within 24 h of admission) is recommended to optimize outcomes [22‐24], while for others a more conservative approach (i.e. three or more days) may be equally as effective [25‐27]. Thus rather then use a continuous measure for delay to angiography, we chose a discrete cutpoint. Subsequently the primary outcome was defined as a binary cutoff in excess of 1 day as a delay to angiography. Secondary outcomes were defined using 2 and 3 days to angiography from admission date.

Decision Trees, using CART (Classification And Regression Trees) software (Salford Systems, California) [28‐30] was used to examine the interactions of independent variables. The analysis begins with the complete population cohort. It then systematically chooses each available variable (continuous or categorical) and measures for each cut point its “impurity” or “disparity” in the defined distribution of the population according to some prescribed criteria, and for a given outcome measure. The criteria used to measure impurity include the Gini index. The algorithm searches amongst all the remaining “candidates” variables and selects that variable which provides the greatest degree of disparity (impurity) in the proportion of patients having a positive outcome (or negative). The form of the Gini impurity index is shown in Appendix 2 (equation 1 [29]). Once a split has occurred, the procedure is repeated in exactly the same fashion as before with the exception that the population is now defined by the population comprising the two “nodes” or subpopulations that were defined by the original split. Figure 1 illustrates the initial steps in model building. In this example with delay to angiography as an outcome (1 = No delay, 0 = delay), the most important clinical determinant of a delay occurs with a split on whether a patient age was below or above 65. Now the population is divided into two mutually exclusive subpopulations. Subsequently the next most important factor for the population aged 65 or less, is a health utility score > 0.80. Likewise the population aged above 65 has cognitive impairment as its next most important determinant with respect to delay to angiography. At this early stage of the model development we can begin to see a clear gradient of increasing delay rates going from left to right across the tree diagram. This process continues until the sample becomes sufficiently small as to render further splits unimportant. The final result is a graphical tree like structure describing a series of pathways or “branches” reflecting disparate distributions of the population with respect to the defined outcome measures. Finally, the different branches or strata are pruned back in order to further remove data artifacts and provide a more robust model (see Appendix 2, illustrating the pruning equation 2 [29]). However unlike regression modeling techniques, which simply indicate the significance levels and effect sizes, Decision Trees, or more generically recursive partition techniques, rank the overall importance of variables in their contribution towards disparities in outcomes. More practically this informs researchers and policymakers in developing strategies to target the gender equity gap, if it indeed exists.

Decision Trees allow the integration of a variety of factors reflecting clinical properties (such as co-morbid diagnoses in men and women), other social factors (such as income, education), and of the interaction of these two types of factors. In essence, each branch of the regression tree describes coefficients reflecting the rates of access formed by such factors as income or co-morbidities. The Decision Tree algorithm generates the vector of coefficients used in calculating the Gini coefficient, a cumulative measure of inequality, via a Lorenz curve [31]. If we rank order the groups corresponding to the coefficients so formed, and plot them, for example, against the cumulative rates of access to coronary angiography, the graph produced is the Lorenz curve. Furthermore, if we add a 45-degree line through the origin, representing equity, the departure of the Lorenz curve from this line characterizes the degree of inequity across groups [31]. The area between the Lorenz curve and the line of identity or equity, from here on in referred to as the departure from equity, is captured in the Gini coefficient (More precisely the Gini Coefficient is twice the difference between the Lorenz curve and the 45⁰ line of equity) whose formula can be expressed by equation 3, Appendix 2 (illustrating a more practical formulation of the Gini coefficient). The Gini coefficient can also be expressed in a more suggestive way using equation 4 [32, 33]: For purposes of inference it is much easier to work with equation 4, Appendix 2 for an alternative formulation of the Gini coefficient. The normative coefficient takes values in the range of 0 to 1, where 0 represents perfect equity and 1 represents complete inequity. By comparing the Gini coefficients (or Lorenz curves) developed around an outcome for men and women it is possible to describe the effect of gender on a health outcome. More generally, and for completeness we may generalize the Gini coefficient (Appendix 2, equation 5) by incorporating a parameter that captures the extent of aversion to gender inequitable differences [34, 35]. In the context of measuring gender equity, the algorithm is set out in the following sequence of steps:

Starting with the complete population (men and women):

The statistical significance of differences between Gini coefficients for men and women can be determined from the standard error of the Gini coefficient itself as denoted by equation 6 [38, 39] (Appendix 2) or more generally using re-sampling techniques such as the bootstrap. Having thus defined the asymptotic standard error for the Gini coefficient, testing for differences in Gini coefficients becomes a straight forward matter of examining coverage of confidence intervals using t critical values as shown in equation 7 [38] (Appendix 2) . Furthermore, by observing the particular points on the Lorenz curve (vertical line between corresponding point for men and women) where the differences are most pronounced, we can easily isolate the particular profiles that contribute greatest towards inequity. In using t-tests we can test for statistical significance

By extension, we can incorporate into the model variables to stratify the model along regional or area axis making it possible not only to determine gender-based inequities across different outcomes but also to determine gender-based inequities across regions such as provinces. A further extension of the model allows for analysis at the group level, lending itself to embedding a hierarchical structure within the Decision Tree framework.

By the nature of the design and the way Decision Trees work, potential issues or pitfalls of confounding and effect modification have been addressed. In layering social and clinical factors, we can control for confounding. In the process of building profiles through the interactions of variables and allowing a given variable to repeatedly enter the model, effect modification is dealt with naturally.

The study was submitted through the Sunnybrook Health Sciences Centre Research Ethics Board (REB) for approval and publication. Sunnybrook Health Sciences Centre is a fully affiliated teaching hospital of the University of Toronto in Ontario, Canada.

The study population consisted of 1349 patients of which 497 (36.8 %) were women and 852 (63.2 %) were men. Compared to men, women were older (68.3 vs. 64.3; P < 0.0001) had higher prevalence rates of hypertension, (57.7 % vs. 42.6 %; P < 0.0001), diabetes 27.2 % vs. 21.2 %; P = 0.01), arthritic and/or rheumatic conditions (58.2 % vs. 36.6 %; P < 0.0001) fair to poor self-reported health status (39.6 % vs. 34.6 %; P = 0.03), and lower health utility index scores [40] (0.75 vs. 0.82). Women were less likely to have a post-secondary/diploma (32.9 % vs. 44.8 %; P = 0.0003), married (48.3 % vs. 72.4 %; P < 0.0001), and have worked in the past 12 months (21.5 % vs. 47.0 %; P < 0.0001). Women had lower income ($20,281 vs. $38,428; P < 0.0001), and were more likely to report lower availability of health services (2.4 vs. 2.2; P = 0.02). A complete list of clinical/health and social determinants characteristics are shown in Table 1.

Using Decision Tree analysis our population cohort was originally partitioned into 18 distinct patient groups (tree not shown) delineated by varying rates of delay to angiography. Due to very small numbers of events dispersed among the 18 groups, potentially resulting in spurious conclusions, the decision tree was pruned, with 7 heterogeneous groups remaining (Fig. 1). The tree shown in Fig. 1 illustrates the clinical factors (including age) forced into the tree, followed by the socio-demographic factors in the bottom portion of the graph. These groups with delay rates separated according to sex are presented in Table 2, stratified by social and clinical determinants. Going from left to right across Table 2, delays rates exhibit an increasing gradient. For example, (group 1) patient’s aged 65 or more and otherwise fairly healthy (Health Utility Index (HUI) equal to or above 0.93), middle to upper class income, immigrated after 1962 and with a post secondary education had a 34.5 % delay rate to angiography. In contrast, patients aged 65 years or higher, with cognitive impairment and/or pre-existing heart disease, and limited in daily limited activities had an overall delay rate of 73.6 % (group 7).

The bottom row presents delay rates according to sex. The groups derived from the complete Decision Tree analysis tree generated the data points (delay rates for men and women) represented in Fig. 2. Figure 2 displays gender comparisons of socio-economic adjusted Lorenz curves of delays to angiography in excess of 1 day. Using the groups defined by Decision Tree analysis in the previous step, an equity measure, computed as Gini coefficients, resulted in no significant differences between men and women in delay to angiography post ACS (0.12 vs 0.14, P > 0.15). Geometrically, the derived value of 0.12 for men is simply the area between its Lorenz curve and the line of identity as shown in Fig. 2. Likewise, a value of 0.14 for women is similarly represented in by the Lorenz curve in Fig. 2. The difference between both curves reflect the degree of inequity between men and women. As a comparison, logistic regression models also showed gender had no significant impact on delay to angiography (OR = 0.81, p = 0.16) after adjusting for clinical and social determinants (Table 3).

When the definition of the outcome was defined as 2 or 3 days delay to angiography from hospital admission, little difference in inequity was observed. For 2 day delays the equity scores for men and women were 0.17 and 0.19 respectively (P > 0.10). Similarly, the difference in equity scores for a delay of 3 days was almost identical for men and women with scores of 0.23 and 0.22 respectively (P > 0.10). The Lorenz curves for outcomes of 2 and 3 days is shown in Figs. 3 and 4 respectively.