Using non-linear decomposition to explain the discriminatory effects of male–female differentials in access to care: A cardiac rehabilitation case study

Abstract

This paper demonstrates the use of non-linear decomposition for identifying discrimination in referral to a cardiac rehabilitation (CR) program. The application is important because the methods are not commonly applied in this context. A secondary data analysis was conducted on a cohort of 2375 patients eligible for referral (as defined) to an Australian hospital outpatient CR program (1 July 1996 to 31 December 2000) on the basis of inpatient discharge diagnosis codes. Data from a population-based disease register were linked to hospital inpatient statistics and CR program records. Cohort selection was established in accordance with first register recorded hospital separations having specified cardiac inpatient diagnoses for which CR was recommended. Using the existing literature as a guide, multivariate logistic regression methods tested the strength of statistical association between independent variables (or ‘endowments’) and CR referral. Compared with males, females had 40% fewer odds of being referred. Non-linear decomposition was performed as a post-logistic regression technique to show the extent to which the sex-based inequality in referral (as defined here) was due to group characteristics (the relative distribution of endowments) compared with other influences not adjusted for in the model. The results showed that approximately 18% of the male–female inequality in referral was not explained by group characteristics, and on this basis was ‘discriminatory’. The extent to which individual endowments contributed to the explained part of the inequality was also of interest. The methods offer potentially useful tools for informing researchers, policy makers, clinicians and others about unfair discriminatory processes that influence access to health and social services.

Introduction

Demonstrations of regression-based decomposition methods date back to research into wage inequalities conducted by Blinder (1973) and Oaxaca (1973) in the United States (US). At that time the purpose was to estimate the average extent of discrimination, against female or black workers, and provide a quantitative assessment of the causes of the wage differentials. The approach, now known as Blinder–Oaxaca decomposition, statistically regressed independent variables or ‘endowments’ such as years of education, training and work experience, against national wage outcomes, separately for males versus females, or blacks versus whites. At the time, men had higher levels of education and training compared with women, and whites had higher levels of education and training compared with blacks. Wage differentials were therefore partly due to the fact that the higher wage group had relatively superior endowments. In order to identify the discrimination component of the wage inequality it was necessary to compare wages in the two groups after adjusting for the distribution of the endowments. If the wages paid to males were higher than the wages paid to equivalently endowed females, then the difference was understood to be the discriminatory part of the inequality. However if after allowing for the distribution of the endowments, there was no difference in wages, the inequality was attributed to differences in endowment distributions between the two groups. Blinder–Oaxaca decomposition separated the contribution to the wage inequality made by the distribution of the endowments from the ‘discriminatory’ contribution made by the regression coefficients.

Early applications of Blinder–Oaxaca decomposition were underpinned by the US Civil Rights Act requiring non-discriminatory wage structures (Blinder, 1973, Oaxaca, 1973). However when applied in other situations, the interpretation of ‘discrimination’ is not always obvious. In the health care sector for instance, there is agreement that inequalities and inequities exist, but less concordance on ways of measuring and evaluating them. In this paper ‘discrimination’ is defined statistically through the use of non-linear decomposition in an Australian hospital outpatient setting.

There has been some recent interest in using decomposition techniques to help explain inequalities in access to health and social services. Researchers in the US have applied decomposition methods to show the impacts of health insurance (Weinick et al., 2000, Zuvekas and Tallaferro, 2003, Zuvekas and Weinick, 1999) and racial differences (Kirby, Taliaferro, & Zuvekas, 2006) on access to health care. Jacobson, Robinson, and Bluthenthal (2007) used decomposition methods to estimate the role of program location and neighbourhood disadvantage on racial disparities in alcohol treatment completion. Non-linear techniques have been developed within the past few years and some applications are reported in the literature (Charasse-Pouélé and Fournier, 2006, Fairlie, 2003, Pylypchuk and Selden, 2008, Yun, 2004). Mortensen and Song (2008) examined differences in hospital emergency department utilisation between Medicaid enrollees and the uninsured, McGuire, Alegria, Cook, Wells, and Zaslavsky (2006) investigated disparities in US outpatient mental health care for whites compared with blacks and also Latinos, and Pylypchuk and Selden (2008) decomposed racial and ethnic differences in children's health insurance using the 2004–2005 Medical Expenditure Panel Survey in the US. Nevertheless given that health care access is a topic of considerable interest amongst public health and social researchers (Goddard and Smith, 2001, Gulliford et al., 2002, Oliver and Mossialos, 2004) it is somewhat surprising that decomposition methods have not been more widely discussed and applied in this field. The methods deserve attention because they offer tools for quantifying and statistically explaining factors that contribute to inequalities and inequities in gaining access to health and other services.

Linear methods were central to early applications and non-linear methods have been developed and applied more recently. Theoretical explanation of non-linear decomposition follows explanation of the Blinder–Oaxaca linear methods given in the next section.

Assume that policy makers seek to ascertain the extent to which higher wages paid to males in a particular industry discriminates against females. If females had the same endowments as males (e.g. levels of education, training, experience etc.) by how much would the wages paid to females need to be increased to achieve equal pay with males? The outcome is average income for males y^male and females y^female. Separate (male and female) linear regressions specify the relationships between independent variables and the wage outcome.

$$y^{\mathit{male}}=a^{\mathit{male}}+\sum {\beta }^{\mathit{male}}{x}^{\mathit{male}}$$

$$y^{\mathit{female}}=a^{\mathit{female}}+\sum {\beta }^{\mathit{female}}{x}^{\mathit{female}}$$

Based on Eqs. (1), (2) the difference in average incomes, between males and females, can be expressed in terms of the difference in predicted average income as shown in Eq. (3).

$$y^{\mathit{male}}-y^{\mathit{female}}=\left(a^{\mathit{male}}+\sum {\beta }^{\mathit{male}}{x}^{\mathit{male}}\right)-\left(a^{\mathit{female}}+\sum {\beta }^{\mathit{female}}{x}^{\mathit{female}}\right)$$

Re-arrangement and substitution of the terms in equation (3), results in the following equation (4).

$$y^{\mathit{male}}-y^{\mathit{female}}=\left(a^{\mathit{male}}-a^{\mathit{female}}\right)+\sum x^{\mathit{female}}\left({\beta }^{\mathit{male}}-{\beta }^{\mathit{female}}\right)+\sum {\beta }^{\mathit{female}}\left(x^{\mathit{male}}-x^{\mathit{female}}\right)+\sum \left({\beta }^{\mathit{male}}-{\beta }^{\mathit{female}}\right)\left(x^{\mathit{male}}-x^{\mathit{female}}\right)$$

The first term (a^male − a^female) is that part of the average income difference that is due to ‘group membership’. This refers to baseline differences between the two groups that are external and therefore not explained by the model.

The second term $\sum x^{\mathit{female}}\left({\beta }^{\mathit{male}}-{\beta }^{\mathit{female}}\right)$, represents that part of the inequality that is due to differences in the (regression) coefficients, valued here in terms of the endowments of the low earning group, which in this case is females. This is the discriminatory component resulting from what are called ‘group processes’. In this example, group processes refer to differences in the way females' endowments (e.g. education, training) are remunerated as wages, compared with how they would be if females were paid the same wages as males.

The third term $\sum {\beta }^{\mathit{female}}\left(x^{\mathit{male}}-x^{\mathit{female}}\right)$, measures that part of the outcomes' gap that is due to differences in the distribution of endowments or ‘group characteristics’. This component is non-discriminatory and measures how much women would earn if they worked as many hours as men reflecting the adjustment made for compositional differences between the two groups.

The fourth term$\sum \left({\beta }^{\mathit{male}}-{\beta }^{\mathit{female}}\right)\left(x^{\mathit{male}}-x^{\mathit{female}}\right)$, refers to the interaction between differences in the endowments and differences in the coefficients for males and females. This is an interaction term in the sense that it jointly depends upon the endowments and the coefficients (Jones & Kelley, 1984). The interaction has also been allocated proportionally across the decomposition (Cotton, 1988, Neumark, 1988, Oaxaca and Ransom, 1994, Reimers, 1983, World Bank, 2004).

In the example given here, males are the reference group, and the purpose of the decomposition is to quantify discrimination as ‘deprivation’ of females (Jones and Kelley, 1984, Oaxaca, 1973). On the other hand if females had been the reference group, the decomposition would have measured ‘privilege’ of males (Blinder, 1973, Jones and Kelley, 1984). While the inequality is the same, these two perspectives produce different estimates of discrimination or privilege in accordance with the choice of reference group. This is the familiar ‘index number problem’ which describes the index or reference point from which measurements are taken. For example when comparing the difference between 100 and 150, the disparity can be expressed in one of two ways, either 150 is 50% more than 100, or 100 is 67% of 150. In the first case the lower amount (100) is the reference point and in the second case the higher amount (150) is the reference point. Either approach is valid with presentation depending upon the purpose of the exercise. Because the traditional labour market Blinder–Oaxaca decompositions were intended to ascertain the extent of discrimination against females or blacks, the high-wage group (males or whites) was the reference position from which the inequality was evaluated.

When a binary variable is the subject of a linear regression, the model is formulated to compute the probability of one of the outcomes. However the linear expression is unlimited with increasing values of the independent variables, so a logistic (non-linear) transform is used to confine values of the probability to the conventional [0,1] range. Whilst linear decomposition applied to a non-linear problem may produce invalid results, the outcomes will have undetermined errors. Linear decomposition cannot be used directly “if the outcome is binary and the coefficients are from a logit or probit model” (Fairlie, 2003). Decomposition methods proposed by Robert Fairlie are used here.

In the non-linear model, estimation of the total inequality requires the calculation of two sets of predicted probabilities in order to measure the difference between the average predicted probabilities for the outcome of interest. The decomposition for a non-linear equation such as $Y=F\left(X\hat{\beta }\right)$ can be expressed as:

$${\overline{Y}}^{\mathit{male}}-{\overline{Y}}^{\mathit{female}}=\left[\sum _{i=1}^{N^{\mathit{male}}}\frac{F\left(X_i^{\mathit{male}}{\hat{\beta }}^{\mathit{male}}\right)}{N^{\mathit{male}}}-\sum _{i=1}^{N^{\mathit{female}}}\frac{F\left(X_i^{\mathit{female}}{\hat{\beta }}^{\mathit{male}}\right)}{N^{\mathit{female}}}\right]+\left[\sum _{i=1}^{N^{\mathit{female}}}\frac{F\left(X_i^{\mathit{female}}{\hat{\beta }}^{\mathit{male}}\right)}{N^{\mathit{female}}}-\sum _{i=1}^{N^{\mathit{female}}}\frac{F\left(X_i^{\mathit{female}}{\hat{\beta }}^{\mathit{female}}\right)}{N^{\mathit{female}}}\right]$$

where N^j is the sample size for j, which in this case is sex. The example in Fairlie's (2003) paper referred to race, but sex has been used here for consistency with the previous example. The above expression refers to males as the reference group which means that the discrimination is being measured towards the females. Male coefficient estimates ${\hat{\beta }}^{\mathit{male}}$ are used as weights in the first term and the female distributions of the independent variables X_i are used as weights in the second term. An equally valid expression is a reversal of this with the females being the reference group and privilege being directed towards men. The expression would then be:

$${\overline{Y}}^{\mathit{male}}-{\overline{Y}}^{\mathit{female}}=\left[\sum _{i=1}^{N^{\mathit{male}}}\frac{F\left(X_i^{\mathit{male}}{\hat{\beta }}^{\mathit{female}}\right)}{N^{\mathit{male}}}-\sum _{i=1}^{N^{\mathit{female}}}\frac{F\left(X_i^{\mathit{female}}{\hat{\beta }}^{\mathit{female}}\right)}{N^{\mathit{female}}}\right]+\left[\sum _{i=1}^{N^{male}}\frac{F\left(X_i^{\mathit{male}}{\hat{\beta }}^{\mathit{male}}\right)}{N^{\mathit{male}}}-\sum _{i=1}^{N^{\mathit{male}}}\frac{F\left(X_i^{\mathit{male}}{\hat{\beta }}^{\mathit{female}}\right)}{N^{\mathit{male}}}\right]$$

Irrespective of the reference group used, the non-linear decomposition includes two key components shown in the square brackets on each line of the preceding equation. The first term represents that part of the gap that is ‘non-discriminatory’ and due to group differences in distributions of the x's and the second term represents the part of the gap that is ‘discriminatory’ and due to ‘not explained’ differences in group processes that determine levels of y, the outcome.

The ‘non-discriminatory’ component is decomposed in terms of the relative contribution made by each of the observed explanatory variables by replacing the distribution of endowments in one group with the distribution in the other group, while holding all other variables constant. In the male female wage example, this means multiplying the male rates of pay by the female distribution, and vice versa. This approach shows how the variables contribute individually to the part of the inequality that refers to group characteristics and provides what Fairlie calls ‘decomposition estimates’. They sum to the total contribution that all variables in the model make towards that part of the inequality that results from differences in group characteristics. Decomposition estimates can be negative or positive; negative estimates indicate that the variable in question contributes to the inequality in the direction which runs counter to the overall inequality. Standard errors for the decomposition estimates were based on methods used by Oaxaca and Ransom (Fairlie, 2003, Oaxaca and Ransom, 1998).

Non-linear decomposition may be used, for example, to explain inequalities in access to health care between two racially different populations. Assume access is a binary measure of outcome. Decomposing the inequality will show the extent to which the difference in access can be attributed to group characteristics (such as age, sex and place of residence) compared with group processes. In this case group processes will reflect the fact that, even when two racial groups are equivalently endowed, they do not have the same level of access to health care. The component not explained by the distribution of group characteristics indicates discrimination (or privilege).

The aim of this paper is to show how non-linear decomposition methods can be used to explain the discriminatory effects of inequalities in access to care. The objectives are to:

• use a case study based on a hospital outpatient cardiac rehabilitation (CR) program, as a vehicle to show how non-linear decomposition methods can be used to explain male–female discrimination in access to care and

• elucidate issues in relation to the use of non-linear decomposition techniques in evaluating access to care.

The outpatient CR program at John Hunter Hospital (JHH) Newcastle, Australia, provided the setting for the analyses undertaken for this paper. John Hunter Hospital is the principal tertiary referral hospital for the Hunter New England Region of New South Wales (NSW) of which there are over 800,000 residents. The CR program at the JHH is a comprehensive best practice outpatient program which includes exercise and lifestyle education and counselling. Clinical practice guidelines endorse the effectiveness of CR recommending that those patients who have received a diagnosis of acute myocardial infarction (AMI), unstable angina pectoris or heart failure or have undergone coronary revascularisation, as being suitable for CR (Beswick et al., 2004, Leon et al., 2005).

This work focuses on explaining a measured sex-based inequality in ‘invitation to participate’ in a CR program. In this study ‘invitation’ was the end result of a hospital-based selection process in which patients were firstly identified by the system as being potential candidates for CR, and secondly assessed by staff for their suitability (e.g. in terms of whether they had illnesses that may be adversely impacted by CR attendance). Invitation coincided with patients being enrolled and having a booked commencement date. For the purposes of this paper ‘referral’ is used to denote ‘invitation’ or selection for CR. It is acknowledged however that these terms can have different meanings in other contexts.

Referral was a necessary prerequisite for access to this program. It was assumed that there were no competing CR programs, that the JHH program was sufficiently well resourced to accept all patients who were eligible by discharge diagnosis, and that patient and clinician influences, such as preferences, were negligible. The use of the term ‘inequity’ in this paper is consistent with a widely accepted definition of equity in health which states that “health inequities are health inequalities that are judged unfair, unjust and avoidable” (Whitehead, 1990).