The ubiquity of the Simpson’s Paradox
 Alessandro Selvitella^{1}Email authorView ORCID ID profile
DOI: 10.1186/s4048801700565
© The Author(s) 2017
Received: 15 December 2016
Accepted: 14 March 2017
Published: 28 March 2017
Abstract
The Simpson’s Paradox is the phenomenon that appears in some datasets, where subgroups with a common trend (say, all negative trend) show the reverse trend when they are aggregated (say, positive trend). Even if this issue has an elementary mathematical explanation, it has a deep statistical significance. In this paper, we discuss basic examples in arithmetic, geometry, linear algebra, statistics, game theory, gender bias in university admission and election polls, where we describe the appearance or absence of the Simpson’s Paradox. In the final part, we present our results concerning the occurrence of the Simpson’s Paradox in Quantum Mechanics with focus on the Quantum Harmonic Oscillator and the Nonlinear Schrödinger Equation. We discuss how likely it is to incur in the Simpson’s Paradox and give some concrete numerical examples. We conclude with some final comments and possible future directions.
Keywords
Simpson’s Paradox Quantum mechanics Schrödinger Equation Prisoner’s DilemmaMathematics Subject Classification 2000
35Q55 34L40 62H20 62H17 62P35Introduction
Applicants  Admitted  Deny 

Female  1494  2827 
Male  3738  4704 
The Chisquare statistics for this test has one degrees of freedom with value χ ^{2}=111.25 and corresponding pvalue basically =0, while the Chisquare statistics with Yates continuity correction for this test has a value of χ ^{2}=110.849 and corresponding pvalue again approximately 0 (precision order 10^{−26}). A naïve conclusion would be that men were much more successful in admissions than women, which would clear be understood as a bad episode of gender bias. At that point, Prof. P.J.Bickel from the Department of Statistics of Berkeley, was asked to analyze the data.
Dpt  Male  Male  Female  Female 

applications  admissions  applications  admissions  
A  825  62%  108  82% 
B  560  63%  25  68% 
C  325  37%  593  34% 
D  417  33%  375  35% 
E  191  28%  393  24% 
F  191  28%  393  24% 
As Bickel, Hammel and O’Connell say in (Bickel et al. 1975), “The proportion of women applicants tends to be high in departments that are hard to get into and low in those that are easy to get into” and it is even more evident in departments with a large number of applicants. The examination of the aggregate data was showing a misleading pattern of bias against female applicants. However, if the data are properly pooled, and taking into consideration the tendency of women to apply to departments that are more competitive for either genders, there is a small but statistically significant bias in favour of women. The authors concluded that “Measuring bias is harder than is usually assumed, and the evidence is sometimes contrary to expectation” (Bickel et al. 1975). This episode is one of the most celebrated real examples of what is called Simpson’s Paradox: the trend of aggregated data might be reversed in the pooled data.
Note that the Simpson’s Paradox is not confined to the discrete case, but it can appear also in the continuous case. Even if less famous, we want to mention the following example which has been discussed on the New York Times recently (Norris 2013). Still today, the Simpson’s Paradox can be a source of confusion and misinterpretation of the data.
Group  Median change 

Total  +0.9% 
High School Dropouts  –7.9% 
High School Graduates, No college  –4.7% 
Some College  –7.6% 
Bachelor’s or Higher  –1.2% 
Here, the reason of the reversal is that the relative sizes of the groups changed greatly over the period considered. In particular, there were more welleducated and so higher wage people in 2013 than in 2000.
In both the cases described above (discrete and continuous, respectively), the variables involved in the paradox are confounded by the presence of another variable (department and level of education, respectively).
The problem of the occurrence of this paradox was considered already in the 19th century. The first author which treated this topic has been Pearson (1899), followed by the contributions of Yule (Yule 1903; Yule and Kendall 1937) and Simpson (1951).
In his paper (Simpson 1951), Simpson considered a 2×2×2 contingency table with attributes A, B, and C and illustrated the paradox using a heuristic example of clinic patients, divided into a Treatment Group and in a NoTreatment group. The data were examined by gender and showed that both males and females responded favorably to the treatment, with respect to who did not receive the treatment. On the other side, the aggregated data showed an opposite behaviour, since there seemed to not be anymore any association between the use of the treatment and the survival time (see (Goltz and Smith 2010) for more details).
The name “Simpson’s Paradox” was first used by Blyth (1972). Some authors prefer to not give full credit to Simpson, since he did not discover this phenomenon and to call it Amalgamation Paradox or YuleSimpson’s Effect instead.
In this paper, we outline that the Simpson’s Paradox is not confined to statistical problems, but it is ubiquitous in science. We give a series of formal definitions in Section 2. In Section 3, we show the ubiquity of the Simpson’s Paradox in several areas of technical and social sciences and we also give some examples of its occurrence. In Section 4, we outline our new result on the occurrence of the paradox in the context of Quantum Mechanics, with particular attention posed to the Quantum Harmonic Oscillator and to the Nonlinear Schrödinger Equation. We conclude with a brief discussion on how likely is the Simpson’s Paradox in Quantum Mechanics (Section 5), with a numerical example (Section 6) and some final comments (Section 7).
Very few papers in the literature treat the Simpson’s Paradox related to problems in Quantum Mechanics. At our knowledge, the only ones avaialable are the fast track communication by Paris (2012), an experimental result by Cialdi and Paris (2015), the preprint by Shi (2012) and a recent paper by the author (Selvitella 2017), which is the first paper that connects the Simpson’s Paradox to Partial Differential Equations and Infinite Dimensional Dynamical Systems.
Measures of amalgamation
In this section, we give the definition and some popular examples of Measures of Amalgamation. For more details, we refer to (Good and Mittal 1987).
2.1 Definitions
First, we define the Process of Amalgamation of contingency tables t _{ i },i=1,…,n.
Definition 1
Let t _{ i }=[a _{ i },b _{ i };c _{ i },d _{ i }], i=1,…,l be 2×2contingency tables corresponding to the ith of l mutually exclusive subpopulations, with a _{ i } b _{ i } c _{ i } d _{ i }≠0. Let N _{ i }=a _{ i }+b _{ i }+c _{ i }+d _{ i } denote the sample size for the ith subpopulation and let N=N _{1}+⋯+N _{ l } be the total sample size of the population. If the n tables are added together, the process is called Amalgamation. We obtain a table \(\mathbf {T}:= [A,B;C,D]:= \left [\Sigma _{i=1}^{l} a_{i},\Sigma _{i=1}^{l} b_{i}, \Sigma _{i=1}^{l} c_{i},\Sigma _{i=1}^{l}d_{i} \right ]\), where A+B+C+D=N.
After having amalgamated a group of contingency tables, we can define the Measure of Amalgamation.
Definition 2
A function \(\alpha : M_{p\times p} \rightarrow \mathbb {R}\) is called Measure of Amalgamation.
Given the definition of Measure of Amalgamation, we can formally define the Simpson’s Paradox.
Definition 3
with α defined on the set of contingency tables and real valued, as in Definition 2.
We fix some terminology that we are going to use in the list of examples below in the context of contingency tables (see (Good and Mittal 1987)). Sampling Procedure I, called also Tetranomial Sampling, is performed when we sample at random from a population. Sampling Procedure I I _{ R } (respectively I I _{ C }), called also ProductBinomial Sampling, is performed when the row totals (respectively columns) is fixed and we sample until this marginal totals are reached. Sampling Procedure III controls both row and column totals.
2.2 Examples
Consider the contingency table t=[a,b;c,d], given by

The Pierce’s measure:$$\pi_{Pearce}(\mathbf{t})=\frac{a}{a+b}\frac{c}{c+d}. $$Under Tetranomial Sampling and ProductBinomial Sampling with row fixed, this measure becomes$$\pi_{Pearce}=P(ST)P(S\bar{T}). $$
It compares the probability of an effect S under treatment and the probability of an effect S without any treatment (row categories are considered to be the “causes” of the column categories).

The Yule’s measure is given by the formula:It compares a/N with respect to its expected value under independence of rows and columns. In fact:$$\pi_{Yule}(\mathbf{t})=\frac{adbc}{N^{2}}. $$since N=a+b+c+d.$$\pi_{Yule}(\mathbf{t})=\frac{adbc}{N^{2}}=\frac{a}{N}\frac{(a+b)(a+c)}{N^{2}}=P(S \cap T)P(S)P(T), $$

The Odds Ratio is probably the most popular one:The Odds Ratio is the ratio between the probability of success and the probability of failure, given a treatment or a notreatment. In fact$$\pi_{Odds}(\mathbf{t})=\frac{ad}{bc}. $$$$\pi_{Odds}(\mathbf{t})=\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{\frac{a/(a+b)}{b/(a+b)}}{\frac{c/(c+d)}{d/(c+d)}}=\frac{P(ST)/P(\bar{S}T)}{P(S\bar{T})/P(\bar{S}\bar{T})}. $$

The Weight of Evidence is given by:Under Tetranomial Sampling or column fixed ProductBinomial Sampling, the Weight of Evidence represents the logarithm of the Bayes factor in favour of S, knowing that the treatment was T, namely:$$\pi_{{Weight}_{C}}(\mathbf{t})=\log \frac{a(b+d)}{b(a+c)}. $$$$\pi_{{Weight}_{C}}=\log \frac{P(TS)}{P(T\bar{S})}. $$

The Causal Propensity:under Tetranomial Sampling or ProductBinomial Sampling with row fixed, represents the propensity of T causing S rather than \(\bar {S}\):$$\pi_{Causal}(\mathbf{t})=\log \frac{d(a+b)}{b(c+d)}, $$$$\pi_{Causal}(\mathbf{t})=\log \frac{P(\bar{S}\bar{T})}{P(\bar{S}T)}. $$
The Simpson’s Paradox appears not just in statistics

Arithmetic: There exist quadruplets a _{1},b _{1},c _{1},d _{1}>0 and a _{2},b _{2},c _{2},d _{2}>0 such that a _{1}/b _{1}>c _{1}/d _{1} and a _{2}/b _{2}>c _{2}/d _{2} but (a _{1}+a _{2})/(b _{1}+b _{2})<(c _{1}+c _{2})/(d _{1}+d _{2}). Example: (a _{1},b _{1},c _{1},d _{1})=(2,8,1,5) and (a _{2},b _{2},c _{2},d _{2})=(4,5,6,8). In this case, the Measure of Amalgamation is given by:$$\pi(\mathbf{t})=\frac{a}{b}\frac{c}{d}=\frac{adbc}{bd}. $$If we consider the contingency tablesand$$\mathbf{t}_{1}=\,[\!a_{1},b_{1};c_{1},d_{1}] $$and the amalgamated one:$$\mathbf{t}_{2}=\,[\!a_{2},b_{2};c_{2},d_{2}] $$we have that:$$\mathbf{T}=\,\left[a_{1}+a_{2}, b_{1}+b_{2};c_{1}+c_{2}, d_{1}+d_{2}\right], $$and so we have the Simpson’s Paradox, accordingly to Definition 3.$$\max_{i=1,2} \pi(\mathbf{a_{i}})<0< \pi(\mathbf{T}) $$

Geometry: Even if a vector v _{1} has a smaller slope than another vector w _{1}, and a vector v _{2} has a smaller slope than a vector w _{2}, the sum of the two vectors v _{1}+v _{2} can have a larger slope than the sum of the two vectors w _{1}+w _{2}. Example: take w _{1}=(a _{1},b _{1}), v _{1}=(c _{1},d _{1}), w _{2}=(a _{2},b _{2}), v _{2}=(c _{2},d _{2}). The same Measure of Amalgamation of the previous example makes the game here as well.

Statistics: A positive/negative trend of two separate subgroups might reverse when the subgroups are combined in one single group. This happens in both the discrete and continuous case. We gave examples of this in the introduction, with the Berkeley Gender Bias (discrete) case and the “time vs US wage” case (continuous).

Linear algebra There exists A _{1},A _{2}∈M a t _{ n×n } such that$$det(A_{1})>0, \quad det(A_{2})>0, \quad {but} \quad det(A_{1}+A_{2})<0. $$
Consider for example A _{1}=t _{1} and A _{2}=t _{2}, as above.

Game theory: The Prisoner’s Dilemma shows why two players A and B might decide to not cooperate, even if it appears that, for both of them, it is more convenient to cooperate. If both A and B cooperate, they both receive a reward p _{1}. If B does not cooperate while A cooperates, then B receives p _{2}, while A receives p _{3}. Similarly, if viceversa. If both A and B do not cooperate, their payoffs are going to be p _{4}. To get the Simpson’s Paradox, the following must hold:$$p_{4}=a_{1}/b_{1}> p_{2}=c_{1}/d_{1}>p_{3}=a_{2}/b_{2} > p_{1}=c_{2}/d_{2}. $$Here p _{3}>p _{1} and p _{4}>p _{2}, and p _{4}>p _{3} and p _{2}>p _{1} imply that it is better to not cooperate for both A and B both given the fact that the other player does or does not cooperate (Nash Equilibrium). Note that, if we use these quadruplets for the table of rewards, we get for the rewards of player A:
Rewards for A
B cooperates
B does not
A cooperates
p _{1}
p _{3}
A does not
p _{2}
p _{4}
and for the rewards of player B:Rewards for B
B cooperates
B does not
A cooperates
p _{1}
p _{2}
A does not
p _{3}
p _{4}
Using the values in our examples, we get for the rewards of player A:Rewards for A
B cooperates
B does not
A cooperates
0.75
0.8
A does not
0.2
0.25
and for the rewards of player B:Rewards for B
B cooperates
B does not
A cooperates
0.75
0.2
A does not
0.8
0.25
Note that this implies that both players A and B are pushed, for personal convenience, to not cooperate, independently of what the other player does, but end up getting a worse reward than if they would have both cooperated. In fact, the amalgamated contingency table, gives:Rewards for A+B
B cooperates
B does not
A cooperates
1.5
1
A does not
1
0.5
that prizes the decision of cooperation. The Measure of Amalgamation considered here can be thought in the form of an Utility Function, such as:and$$U_{A}(a,b)=p_{1}ab+p_{3}a(1b)+p_{2}b(1a)+p_{4}(1a)(1b) $$$$U_{B}(a,b)=p_{1}ab+p_{2}a(1b)+p_{3}b(1a)+p_{4}(1a)(1b). $$Here a=1, means that A cooperates, while a=0 means that A does not. Similarly for B. Note that, under the conditions on p _{1}, p _{2}, p _{3} and p _{4} mentioned above, the Utility is bigger for the choice of not cooperation for both A and B, given any decision taken by the other player. In fact,and$$p_{1}=U_{A}(1,1)<U_{A}(0,1)=p_{2} $$and analogously for U _{ B }. However, when we combine the utilities, we get Utility Function$$p_{3}=U_{A}(1,0)<U_{A}(0,0)=p_{4} $$$$U_{A+B}(a,b)=2p_{1}ab+(p_{2}+p_{3})a(1b)+(p_{3}+p_{2})b(1a)+2p_{4}(1a)(1b). $$This utility is always bigger for cooperation, if we require 2p _{4}<p _{2}+p _{3}<2p _{1}, as we chose in our example. In fact:$$2p_{4}=U_{A+B}(0,0)<U_{A+B}(1,0)=p_{2}+p_{3}=U_{A+B}(0,1)<2p_{1}=U_{A+B}(1,1). $$In this way, we have restated the Prisoner’s Dilemma in the context of the Simpson’s Paradox.

Election polls: Suppose candidates T and C run for elections in two states S t a t e _{1} and S t a t e _{2}. Suppose that candidate T and C receive in S t a t e _{1} a percentage of votes:and that candidate T and C receive in S t a t e _{2} a percentage of votes:$$\% {votes for {T}}=\frac{a}{b}>1\frac{a}{b}=\% {votes for {C}} $$$$\% {votes for {T}}=\frac{2}{d}>1\frac{c}{d}=\% {votes for {C}}. $$Is it possible that overall candidate C receives a higher percentage of votes? Clearly, this is not possible because \(\frac {a}{b}>1\frac {a}{b}\) implies a>0.5b and \(\frac {c}{d}>1\frac {c}{d}\) implies c>0.5d and sowhich implies$$0.5b+0.5d<a+c, $$and so$$\frac{a+c}{b+d}>0.5 $$$$\frac{a+c}{b+d}>1\frac{a+c}{b+d}. $$
In this case, we do not have any paradox and this is related to the fact that there is an extra constraint on the construction of the contingency table. Note that since the set of real numbers for which these inequalities hold is an open set, the inclusion of a not strong third candidate will not change the situation. What happens if the third candidate is as strong as T and C?
The Simpson’s Paradox in quantum mechanics
In this section, we turn our attention to a novel result of us (Selvitella 2017) concerning the occurrence of the Simpson’s Paradox in Quantum Mechanics. In particular, we show how we can detect an unintuitive behaviour in the interaction between solitary wave solutions in the case of the Quantum Harmonic Oscillator and the Nonlinear Schrödinger Equation. We start with the Quantum Harmonic Oscillator.
4.1 The quantum harmonic oscillator

the profile u(x) for x∈R ^{ n } satisfies the equation$$\begin{array}{@{}rcl@{}}  \frac{\hslash^{2}}{2m} \Delta_{\mathbf{x}} u(\mathbf{x}) +\frac{1}{2}m \omega^{2}\mathbf{x}^{2} u(\mathbf{x})+\frac{\omega}{2} u(\mathbf{x})=0; \end{array} $$(3)

the position vector x(t) and the velocity vector v(t) satisfy the following system of ODEs:$$\begin{array}{@{}rcl@{}} \left\{ \begin{array}{l} {\dot{\mathbf{x}}}(t)=\frac{\hslash}{m}\mathbf{v}(t), \\ {\dot{\mathbf{v}}}(t)=\frac{m}{\hslash} \omega^{2}\mathbf{x}(t) ; \end{array} \right. \end{array} $$(4)

the complex phase γ(t) is such that$$\dot{\gamma}(t)=\frac{1}{\hslash}\mathcal{L}(\mathbf{x}(t),\dot{\mathbf{x}}(t); m, \omega), $$
where \(\mathcal {L}(\mathbf {x}(t),\dot {\mathbf {x}}(t); m, \omega):=\frac {1}{2}m\dot {\mathbf {x}}(t)^{2}\frac {1}{2}m\omega ^{2}\mathbf {x}(t)^{2}\) is the Lagrangian of the system of ODEs (4). For why this is true, we refer to (Berezin and Shubin 1991) and (Selvitella 2017).
4.2 The nonlinear Schrödinger Equation
4.3 The main theorems
In Quantum Mechanics and in the context of the Schrödinger Equation, there is a very natural Measure of Amalgamation, given by the L ^{2}(R ^{ n }) inner product.
Definition 4
Using the L ^{2}(R ^{ n }) inner product, we can show that, for the Quantum Harmonic Oscillator, there exist quadruplets of solitons, which exhibit the Simpson’s Paradox.
Theorem 1

In the stationary case, namely when v _{ i }(t)=0 and x _{ i }(t)=x _{ i } for every t; both when γ _{ i }=γ _{ j } for every 1≤i,j≤4 and when γ _{ i }≠γ _{ j }1≤i,j≤4, i≠j.

In the nonstationary case: if there exists t _{0}∈R such that the Simpson’s Paradox occurs at t _{0}, then the Simpson’s Paradox occurs at any t _{1} with t _{1}≠t _{0}.
Remark 1
As we can see from Theorem 1, the occurrence of the Simpson’s Paradox in the case of the Quantum Harmonic Oscillator is determined by the initial datum and so we can say that it is persistent under the flow of the Quantum Harmonic Oscillator.
Once we have proved the existence, we want to address the question of how robust this phenomenon is, namely if nearby a quadruplet of solitons, we can find plenty of quadruplets of solitons for which the paradox occurs. We have that the set of parameters for which the paradox occurs contains open sets.
Theorem 2
[Stability of the Simpson’s Paradox ] Suppose that there exists a set of parameters (x _{ i }(t),γ _{ i }(t),v _{ i }(t)) for i=1,…,4 such that the Simpson’s Paradox occurs in the stationary case. Then, there exists r>0 such that, for every \((\tilde {x}_{i}(t), \tilde {\gamma }_{i}(t), \tilde {v}_{i}(t))\) for i=1,…,4 inside B _{ r }((x _{ i }(t),γ _{ i }(t),v _{ i }(t)), i=1,…,4), the Simpson’s Paradox still occurs for initial data as above. Moreover, if the Simpson’s Paradox occurs for a ψ(t,x) at a certain time \(t=\tilde {t}\), then there exists an open ball in Σ:=L ^{2}(R ^{ n },d x)∩L ^{2}(R ^{ n },x^{2} d x) such that the Simpson’s Paradox still occurs for any \(\bar {\psi }(t,x)=\psi (t,x)+w(t,x)\) with w(t,·)∈Σ and the same time \(t=\tilde {t}\).
Now, we can discuss the nonlinear case.
Theorem 3
[Nonlinear case] Consider the nonlinear Schrödinger Equation in dimension n=1, with 1<p<5 (L ^{2} subcritical exponent). Then, there exist an initial datum ψ _{0}(x), in the form of a superposition of solitons (see (Martel and Merle 2006)), for which there exists \(t=\tilde {t}_{1}\gg 1\) where the Simpson’s Paradox occurs and \(t=\tilde {t}_{2}\gg 1\) where the Simpson’s Paradox does not occur.
Remark 2
In striking contrast with the Quantum harmonic Oscillator, for the Nonlinear Schrödinger Equation, the Simpson’s Paradox is not anymore persistent, but it is intermittent. In fact, we can detect it for large times but it appears and disappears indefinitely.
Proof
For the complete proofs of these theorems, we refer to (Selvitella 2017), while for a brief sketch of the proof of Theorem 1 in the stationary case, we refer to the upcoming Section 5. □
How likely is the Simpson’s Paradox in quantum mechanics?
An important question is: “How likely is the Simpson’s Paradox?”. It is in fact interesting to quantify, in some way, the chances that one has to run into the paradox.
In the case of 2×2×l contingency tables with l≥2, Pavlides and Perlman (2009) address the problem and, among the other things, they prove the following.
A similar question can be asked in the case of the Quantum Harmonic Oscillator and the Nonlinear Schrödinger Equation. In the constructions developed in (Selvitella 2017), we aimed just at finding one single choice of the parameters which gives the Simpson’s Paradox and we did it mainly with a perturbative method. But how large is (and in which sense it is large) the set of parameters which gives the Simpson’s Paradox?
To investigate a little bit further this issue, we briefly sketch the proof of Theorem 1, at least in the stationary case and deduce from it a preliminary result on the likelihood of occurrence of the Simpson’s Paradox.
for 1≤i≤j≤4 and with x(t), v(t) and γ(t) as in Subsection 4.1.
In the following discussion, we treat only the case γ _{ i }=γ _{ j }, for every i,j=1,…,4.
Remark 3
Note that the defining condition for the occurrence of the Simpson’s Paradox are all inequalities which is a hint of the fact that the Simpson’s Paradox occurs in a open set of the correct topology (see Theorem 2 and (Selvitella 2017)).
We have decided to test the inequality f(X,Y,Z)<0 over a grid of n×n×n values with n=1000 in the parallelepiped (X,Y,Z)∈ [ 0,1]× [ 0,1]× [ 0,4] and we discovered that about 1.2∗10^{−4} of the times (0.012%) the inequality is satisfied. Note that the choice of the uniform distribution on [ 0,1]×[ 0,1]×[ 0,4] has been made because for Z>4 the Simpson’s Paradox’s region is almost null (Fig. 2) and because already 0<X,Y<1. This result deserves further investigation. For reproducibility purposes, we give the Matlab Code that we used for the analysis:
Some numerical examples
For illustration purposes, we give some numerical examples of cases in which the Simpson’s Paradox occurs and on which it does not. We find interesting to give to each parameters their true physical value.
We choose \(L_{12}^{2}=1\epsilon _{1}^{2}\), \(L_{34}^{2}=1\epsilon _{2}^{2}\) and \(L_{23}^{2}=1+\delta ^{2}\) with ε _{1}≪1, ε _{2}≪1. The following R code produces and example of the paradox in our case:
Of course, there are cases in which the Simpson’s Paradox does not occur, like
Discussion
In this paper, we discussed the Simpson’s Paradox in several settings. In particular, we gave basic examples in arithmetic, geometry, linear algebra, statistics, game theory, gender bias in university admission and election polls, where we described the appearance or absence of the Simpson’s Paradox. Then, we moved to the presentation of our recent results on the occurrence of the Simpson’s Paradox in Quantum Mechanics with focus on the Quantum Harmonic Oscillator and the Nonlinear Schrödinger Equation (Selvitella 2017). We discussed the likelihood of the occurrence of the Simpson’s Paradox and we gave some numerical examples in which the Simpson’s Paradox occurs and some numerical examples in which the Simpson’s Paradox does not occur. This depends on in which parameter regions we are. Several problems remain to be addressed. An extended investigation of the question “How likely is the Simpson’s Paradox in Quantum Mechanics?” is appropriate. In particular, it would be interesting to construct and put a more suitable probability measure on the set of parameters and quantify the likelihood of the Simpson’s Paradox even further.
Declarations
Acknowledgements
I thank my family and Linda Forson for their constant support. I thank my supervisor Prof. Narayanaswamy Balakrishnan for his constant help and useful comments. Many thanks to the referees and the editors, whose suggestions improved the quality of the manuscript. This research has been supported by CANSSI and Central Michigan University in occasion of the ICOSDA 2016. The author has no competing interests in the manuscript. I dedicated this paper to the memory of Zia Vera and Zio Salvatore.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Bickel, PJ, Hammel, EA, O’Connell, JW: Sex Bias in Graduate Admissions: Data From Berkeley. Science. 187(4175), 398–404 (1975).View ArticleGoogle Scholar
 Berestycki, H, Lions, PL: Nonlinear Scalar field equations. Arch. Rational Mech. Anal. 82(3), 313–345 (1983).MathSciNetMATHGoogle Scholar
 Berestycki, H, Lions, PL, Peletier, LA: An ODE approach to the existence of positive solutions for semilinear problems in R ^{ N }. Indiana Univ. Math. J. 30(1), 141–157 (1981).MathSciNetView ArticleMATHGoogle Scholar
 Berezin, FA, Shubin, MA: The Schrödinger Equation. Translated from the 1983 Russian edition by Yu. Rajabov, DA Leı̆tes and NA Sakharova and revised by Shubin. With contributions by G. L. Litvinov and Leı̆tes. Mathematics and its Applications (Soviet Series), 66. Kluwer Academic Publishers Group, Dordrecht (1991). ISBN:079231218X 8101 (35J10 35P05 46N50 47F05 47N50).Google Scholar
 Blyth, CR: On Simpson’s Paradox and the SureThing Principle. J. Am. Stat. Assoc. 67(338), 364–366 (1972).MathSciNetView ArticleMATHGoogle Scholar
 Cialdi, S, Paris, MGA: The data aggregation problem in quantum hypothesis testing. Eur. Phys. J. D. 69, 7 (2015). doi:10.1140/epjd/e2014504257.View ArticleGoogle Scholar
 Good, IJ, Mittal, Y: The Amalgamation and Geometry of TwobyTwo Contingency Tables. Ann. Stat. 15(2), 694–711 (1987).MathSciNetView ArticleMATHGoogle Scholar
 Gidas, B, Ni, WM, Nirenberg, L: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209–243 (1979).MathSciNetView ArticleMATHGoogle Scholar
 Goltz, HH, Smith, ML: YuleSimpson’s Paradox in Research. Pract. Assess. Res. Eval. 15(15), 1–9 (2010).Google Scholar
 Kwong, MK: Uniqueness of positive solutions of Δ u−u+u ^{ p }=0 in R ^{ n }. Arch. Rational Mech. Anal. 105(3), 243–366 (1989).MathSciNetView ArticleMATHGoogle Scholar
 Martel, Y, Merle, F: Multi solitary waves for the nonlinear Schrödinger Equations. Ann. I. H. Poincaré. 23(6), 849–864 (2006).View ArticleMATHGoogle Scholar
 Norris, F: Can Every Group Be Worse Than Average? Yes (2013). https://economix.blogs.nytimes.com/2013/05/01/caneverygroupbeworsethanaverageyes/. Accessed 1 May 2013.
 Paris, MGA: Two quantum Simpson’s Paradoxes. J. Phys. A. 45, 132001 (2012).MathSciNetView ArticleGoogle Scholar
 Pavlides, MG, Perlman, MD: How likely is Simpson’s Paradox?Am. Stat. 63, 226–233 (2009).MathSciNetView ArticleGoogle Scholar
 Pearson, K, Lee, A, BramleyMoore, L: Genetic (reproductive) selection: Inheritance of fertility in man, and of fecundity in thoroughbred racehorses. Phil. Trans. R. Soc. A. 192, 257–330 (1899).View ArticleGoogle Scholar
 Selvitella, A: The Simpson’s Paradox in quantum mechanics. J. Math. Phys. 58(3), 37 (2017). 032101.MathSciNetView ArticleGoogle Scholar
 Shi, Y: Quantum Simpson’s Paradox and High Order BellTsileron Inequalities (2012). preprint available at arxiv.org/pdf/1203.2675.
 Simpson, EH: The Interpretation of Interaction in Contingency Tables. J. R. Stat. Soc. Ser. B. 13, 238–241 (1951).MathSciNetMATHGoogle Scholar
 Yule, GU: Notes on the Theory of Association of Attributes in Statistics. Biometrika. 2(2), 121–134 (1903).View ArticleGoogle Scholar
 Yule, GU, Kendall, MG: An Introduction to the Theory of Statistics. Griffin, London (1937).MATHGoogle Scholar