Principles behind the Median Paradox

Some methodological conclusions could be drawn immediately, but their relevance can be judged more concretely, if we understand some principles behind the paradox. This understanding may be supported by a closer analysis of the above example. Of course, the ogives for the RSO-process result from an application of Eq. (1) to the present situation with $\pi_i=\pi_j=0.5$, leading to

$$y_\pi(c,x)=0.5 \cdot{}f_i(c,x) + 0.5 \cdot{}f_j(c,x).$$ (3)

The medians for unit i under conditions a and bare $\mu{}^*_{ia} = 8.0$ resp. $\mu{}^*_{ib} = 10.0$, whereas they are $\mu{}^*_{ja} = 20.0$resp. $\mu{}^*_{jb} = 22.0$ for unit j, and $\mu{}^*_{\pi a} = 16.5$resp. $\mu{}^*_{\pi b} = 13.5$for the RSO-process. Finally, let x_i and x_j be the abscissas of the intersection points of the ogives characterizing unit i resp. j. Then $x_i \approx 11.501$, and $x_j \approx 18.499$. It can be left to the reader to choose between a graphical and an algebraic check of these assertions, the latter one being based on the detailed algebraic description of the underlying functions f_u(c,x) in the appendix.

A first approach to principles behind the median paradox is based on a geometrical interpretation of Eq. (3). If we consider only the three thin plots in Fig. 1 (i.e., the ogives for condition a) and draw a vertical line for an arbitrary abscissa x, then the intersection of this line with the ogive representing $y_\pi(a,x)$ is the midpoint between the intersections of the same line with the ogives for f_i(a,x) and f_j(a,x). Now f_i(a,x)is the first curve crossing the median line; but after this intersection it slows down considerably, and according to the above midpoint principle the same happens with $y_\pi(a,x)$, since f_j(a,x) is almost 0 for x<15. In other words, the ogive $y_\pi(a,x)$cannot cross the median line, before f_j(a,x)becomes greater than 0 to the same amount as f_i(a,x)is less than 1, and this happens only at x=16.5, since $f_j(a,16.5)=1 - f_i(a,16.5) \approx 0.094$. But beyond this point, f_j(a,x) increases rapidly and crosses the median line at an earlier point than f_j(b,x). The latter curve has contributed to a relatively early crossing of the ogive for $y_\pi(b,x)$with the median line at x=13.5. Certainly, this property is mainly due to the fast increase of f_i(b,x), but since the midpoint principle holds also for $y_\pi(b,x)$, and f_i(b,x) is decelerated as the cumulative probability approaches 1, the intersection of $y_\pi(b,x)$with the median line would occur at a later point, unless a function value considerably greater than zero (namely 0.094) would be taken by f_j(b,x) already at x=13.5. Beyond this point, f_j(b,x) increases only slowly and is overhauled by f_j(a,x) such that it crosses the median line at a later point.

Obviously, the intersections of the ogives characterizing unit i and j at x_i resp. x_jare crucial for the principle behind the median paradox. This fact can be used to understand this paradox in another approach, which is similar to the introductory examples of changing the median of a pooled group by manipulations not affecting the subgroup medians or viceversa. Translating these examples into a probabilistic framework, we can state the following property of the median: If probability mass is moved upward or downward such that the distribution has a well defined median¹⁶ before and after the move, then the median is changed, if and only if a part of the moved mass crosses the old median. This property may be used to perform moves affecting only one of the medians, although every move of probability mass in a distribution characterizing a unit under a condition c implies (due to Eq. (1)) a similar - although smaller - move in the distribution characterizing the RSO-process under the same condition. For instance, the crossing of the ogives characterizing unit j at $x_j \approx 18.499$can be considered to be the result of two moves of probability mass in opposite directions, these moves making up the individual effect¹⁷ for unit j of treatment b (vs. a) upon the distribution of the dependent variable: One part of the probability mass below x_j of the distribution for condition a has been shifted to the left such that the ogive for condition b is left of the one for condition a below x_j. Since the individual median isn't crossed by this shift, it is left unchanged; but a non-zero part of the same mass crosses the median for the RSO-process, contributing to the low median for this process under condition b. Conversely, a part of the probability mass above x_jis moved upwards. It doesn't cross the medians for the RSO-process, but for the individual distribution, leading to $\mu{}^*_{jb} > \mu{}^*_{ja}$.

Capitalizing systematically on this property of the median, we can reconstruct something like a cookbook recipe, which has served for the construction of the example in Fig. 1. Suppose that we have started with a situation with ogives $f^{\circ}$given by $f^{\circ}_i(a,x) = f^{\circ}_i(b,x) = f_i(b,x)$, and $f^{\circ}_j(a,x) = f^{\circ}_j(b,x) = f_j(a,x)$, the respective medians being denoted as $\mu{}^{\circ *}_{uc}$resp. $\mu{}^{\circ *}_{\pi c}$. In other words, under both conditions the cumulative distribution functions are identical with the steep ogive for the respective unit. Then it is easily verified that the medians of the ogives characterizing the RSO-process would be 15.0. Now the transition from this situation to the one represented in Fig. 1 can be considered as a result of the following steps (where $x_i \approx 11.501$ and $x_j \approx 18.499$ are the abscissas of intended intersection points of ogives characterizing the same unit):

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In a first step, we change $f^{\circ}_i(a,x)$by shifting a part of the probability mass from the interval ]x_i, 15.0[to the interval ]15.0, 30.0[. Obviously, this shift results in an increased value of $\mu{}^{\circ *}_{\pi a} > 15.0$, whereas $\mu{}^{\circ *}_{ia}$remains unaffected.

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A second step is again concerned with $f^{\circ}_i(a,x)$: We shift a part of the probability mass from $]\mu{}^{\circ *}_{ib}, x_i[$to $]0.0, \mu{}^{\circ *}_{ib}[$. Then the median $\mu{}^{\circ *}_{ia}$will fall below its hithertoo value of $\mu{}^{\circ *}_{ia} = 10.0 = \mu{}^{\circ *}_{ib}$ (leading to $\mu{}^{\circ *}_{ia} < \mu{}^{\circ *}_{ib}$), whereas $\mu{}^{\circ *}_{\pi a}$keeps its value resulting from the first step.

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In two further steps, we perform corresponding changes with $f^{\circ}_j(b,x)$: Shifting probability mass from ]15.0, x_j[to ]0.0,15.0[ leads to $\mu{}^{\circ *}_{\pi b} < 15.0 < \mu{}^{\circ *}_{\pi a},$and finally a shift from $]x_j, \mu{}^{\circ *}_{ja}[$to $]\mu{}^{\circ *}_{ja}, 30.0[$gives $\mu{}^{\circ *}_{jb} > 20.0 = \mu{}^{\circ *}_{ja}$.

The above steps have resulted in two units with intersecting ogives, and this is, indeed, the most effective way of obtaining a median paradox. Generalizing the results to RSO-processes with more than two units, we can conclude that the occurrence of a median paradox is favoured by a pattern consisting in an intersection of ogives at a point with cumulative probability greater than 0.5 for units with low medians (our unit i), and a cumulative probability less than 0.5 at the intersection points of units with high medians (our unit j). However, it should be noted that it isn't necessary for the occurrence of a median paradox to have intersections of both types. We could e.g. take the curves f_i(a,x), f_i(b,x)and f_j(a,x) as in the example and combine them with a curve $f^{\circ}_j(b,x)$resulting from a shift of the entire curve f_j(a,x)to the right, leading to $\mu{}^*_{ja} < \mu{}^{\circ *}_{jb}.$Then we will have again $\mu{}^*_{\pi a} = 16.5$, but $\mu{}^{\circ *}_{\pi b} > 15.0$. If the said shift of f_j(a,x) leading to $f^{\circ}_j(b,x)$isn't too big, we nevertheless get $\mu{}^{\circ *}_{\pi b} < 16.5 = \mu{}^*_{\pi a}$, and again the median paradox is complete. Similarly, a median paradox with an intersection of ogives only for unit jcould result from keeping the curves f_i(b,x), f_j(a,x) and f_j(b,x) of the example and obtaining $f^{\circ}_i(a,x)$by a suitable shift of f_i(b,x) to the left.

Understanding principles behind a paradox may be important to avoid erroneous conclusions from data. For instance, understanding that the Simpson paradox (see, e.g., Agresti, 1990, p. 155 [1]) is just a special case of confounding leads to the control of potential confounders as a method of avoiding misled conclusions. At first glance, the median paradox looks similar: The direction of a statistical association can change under aggregation. However, the median paradox is not based on confounding, but on the properties of the median which have beeen outlined above. Since the source of the paradox is inherent to the median by its definition, we will have to discuss in Section 3 whether avoiding the median as a parameter reflecting treatment effects isn't the adequate answer to the median paradox.