Simpson’s paradox in GDP and per capita GDP growths

Abstract

Simpson’s paradox occurs frequently in economic data analysis, wherein aggregation is a common practice. Yet, this paradox is not well known among researchers in economy. In this article, we present several real-world examples of Simpson’s paradox in economic statistics, including gross domestic product (GDP) growth and per capita GDP growth aggregations across developing and developed countries. These manifestations of Simpson’s paradox highlight some important issues in developing economies and have implications on social and economic policies. We also present Simpson’s paradox for continuous variables, and its relationship with ecological correlation using empiric economic data. We show that failure to recognize Simpson’s paradox and ecological correlation can cause inaccurate interpretations of economic data. Furthermore, even when one recognizes Simpson’s paradox in the data, one may still make wrong interpretations, wrong policy decisions, or business decisions. We recommend causal analysis to discern the confounding variable(s) and to apply sound statistical modeling in the face of such a paradox.

Appendices

Appendix 1: Mathematical formulation

Given three random events, $A, B,$ and $C$, and their complements, ${A}^{\mathrm{c}},\,{B}^{\mathrm{c}},\, {C}^{\mathrm{c}}$, Simpson’s paradox occurs when

$$ \begin{aligned} P({A|B \& C})&> P(A|B^{\mathrm{c}} \& C)\nonumber \\ P(A|B \& C^{\mathrm{c}})&> P(A|B^{\mathrm{c}} \& C^{\mathrm{c}}) \end{aligned}$$

(4)

Yet also

$$\begin{aligned} P({A|B}) \le P(A|B^{\mathrm{c}}) \end{aligned}$$

(5)

This happens when the random events B and C are dependent. If B and C are independent, it is not possible that the above inequalities, (4) and (5), are true at the same time. Thus, the paradox results from the dependence or interaction between B and C (Simpson 1951; Blyth 1972). Random variable C can be interpreted as a confounding variable that alters the conditional relationships when the conditional cells are collapsed into marginal cells. Notice that random variable C can represent one physical variable or the composite effect of several variables that cause the confounding. The converse is obviously true by symmetry; that is, the paradox can also occur in such a way that the inequality signs in (4) and (5) are flipped.

Several boundary conditions exist. When inequality (5) is simply equality, the aggregate of two larger (or smaller) quantities is equal to the aggregate of two smaller (or larger) quantities. Another boundary condition happens when two inequalities in (4) are equalities, but the inequality (5) remains an inequality of either larger or smaller sign.

The main reason that people are perplexed with the reversal is that they are intuitively confused between aggregation and arithmetic addition. In fact, another simpler way of formulating the paradox is the following inequalities. Given that

$$\begin{aligned} {a}/{b}<{A}/{B} \quad \quad \hbox {and} \quad \quad {c}/{d}<{C}/{D} \end{aligned}$$

(6)

Simpson’s paradox occurs when

$$\begin{aligned} ({a}+{c})/({b}+{d}) \ge ({A}+{C})/({B}+{D}) \end{aligned}$$

(7)

The reader can find other mathematical formulations of the paradox and related boundary conditions in Pavlides and Perlman (2009).

Appendix 2: Birth weight paradox

The birth weight paradox, as an example of Simpson’s paradox, appears in comparing infant mortalities of two or more different groups of mothers, such as smokers and nonsmokers, different races, different social status (Wilcox 2002, 2006). It is generally agreed that lower birth weight is strongly correlated to the risk of infant mortality. The Wilcox-Russell hypothesis describes the relationship between birth weight and infant mortality (see Fig. 3).

Fig. 3

a Mortality rates for the two groups (solid line is Group A in Table 7), with illustration of the third variable’s double effects (reducing the weight and increasing mortality risk), and b frequency distribution of birth weight for two different groups (adapted from Wilcox 2002, 2006). This figure is the so called “the Wilcox-Russell hypothesis”

Consider Group A as the maternal nonsmoking population and Group B as the maternal smoking population. Mortality of the infants born to smokers is generally lower than that of the infants born to the nonsmoking mothers for the same birth weight, as shown in Table 7. If the conditional associations are genuine, maternal smoking reduces the risk of infant mortality. The conclusion would be that the maternal smoking is a good prenatal care for infant survivorship.

Table 7 Illustration of birth weight paradox as a manifestation of Simpson’s paradox

In fact, maternal smoking reduces the infants’ birth weights and increases their mortality risk, but the effect of birth weight reduction obscures the effect of mortality (see the two split effects in Fig. 3). For a given birth weight, the infants of smokers have a lower mortality because the reduction of the birth weights of those infants masks the increased mortality. Had smoking not decreased the birth weight of the infants, and the within-class comparisons conditioning to birth weight would show higher mortality rates for infants of smokers.

The overall mortality of the infants from the maternal smoking group is higher than that of the infants from the nonsmoking mothers, and inference using the aggregated data is not bad, but the within-class comparisons are false. It is possible to adjust the confounding variable so that the within-class comparisons are meaningful. The adjustment, however, is not always straightforward and could distort the results (Hernandez-Diaz et al. 2006; Wilcox 2006).

According to a study by Wilcox (2002), high altitude can lead to lower birth weight as well. Colorado-born infants, for instance, have a lower average birth weight than the US national average. For each bracket of the birth weights, infants born in Colorado have a lower mortality, but the overall mortality of Colorado’s infants is the same as the US national average. In other words, the third variable (high altitude) causes a reduction of the birth weight, but not an increased or decreased mortality. The conditional associations are spurious, but the marginal association is more meaningful.