Introduction

Relative Age Effects (RAEs) occur when schools and sports group children into age-banded cohorts. Within each cohort, older children study and compete at an artificially conferred advantage over younger children [1–3]. Since the start-points and end-points for age-bandings rarely change, such advantages are conferred consistently throughout childhood. Initial success tends to breed longer-term success for two good reasons. First, there is a psychological advantage to being cohort-older. Because it is easy to mistake age-advantage for intrinsic talent, cohort-older children will tend to be nourished on a richer diet of positive regard from teachers, coaches, parents, and peers who see them as more competent than cohort-younger children–as indeed they are [4]. Affirmations from the world tend to become internalised by the child in the form of greater confidence [5] and willingness to engage in competitive activities [6]. Second, there is an organizational advantage, in that initial success can also institutionalise success. Specialised training centres, such as football academies, tend to select young ‒ when a year can make a big difference ‒ and so their squads grossly over-represent cohort-older children. They also often select once, meaning that an academy intake tends to resist future revision. Selecting young and once results in the same large RAE bias propagating from first- to last year in the academy [7–9]. In summary, RAEs are ubiquitous and long-lived into adulthood [1]; they operate by pervasively boosting self-esteem (the psychological advantage) and / or by a time-frozen selection bias onto elitist training programs (the organizational advantage).

The goal of this paper is to develop, explain and advocate a unified set of analyses that, in Boisot’s [10] terms, can help codify, and thus communicate, diffuse, accumulate and ultimately apply, knowledge. It is our contention that RAE researchers have not always extracted enough from their data. This limitation occurs both at the local level, in that analyses are often ended prematurely; and at the global level, in that analyses of a present set of results cannot easily be compared with past sets of results that were analysed differently. Perhaps this is to be expected, given that RAE research exists across many disciplines including sports science, education, psychology, health and economics, each with their own traditions of analysis. Although understandable, it is not to be encouraged.

A central place in our framework is occupied by the Poisson regression model (PRM) to analyse the frequency data typical of RAE research. Since many RAE researchers may not be familiar with Poisson regression, we offer a short tutorial introduction structured around substantive research questions raised in the recent RAE literature. Being an “advanced” technique, for an “advanced” audience, PRM is often presented in a way that is inaccessible to the majority—but for digestible overviews, see [11–13]. This is regrettable, as PRM is in fact a relatively straightforward analogue of the popular Ordinary Least Squares (OLS) regression method.

In particular, the PRM shape parameter (the counterpart of the OLS slope) acts within the PRM framework as the common format by which minimally-processed statistical analyses can be translated into more diagnostic but insightful indices and measures, thereby moving RAE research along the Data-Information-Knowledge-Wisdom trajectory [14]. The principal two are: the Index of Discrimination (I_D), which measures the relative odds of someone meeting a criterion given they are born at the start versus end of the cohort year; and two variants of the Index of Wastage (I_W1 and I_W2), which highlight the degree of talent lost because of RAE. The PRM model is also able to measure the change in RAE, consequent on policy changes to how the cohort year is defined [15].

The empirical analysis is based on an examination of all domestic players in the top two tiers of the “Big Five” European football leagues for the season 2016/17, involving nearly 4000 players. This is more than an illustrative example, but is a proper contribution to the RAE literature. Studies are beginning to emerge using Poisson regression. For instance, Doyle and Bottomley [16] searched, albeit unsuccessfully, for paragon clubs and countries within a European context from which others might learn how to minimise this form of age discrimination. Similarly Brustio et al. [17] documented a diminution of RAE among cohorts of Italian youth as they progressed through the Serie A academy system. Rada et al. [18] raised the interesting possibility that second-tier clubs may provide a second chance for cohort younger players to establish themselves. Sadly, the RAE was just as pronounced among professional players in second-tier clubs as in first-tier clubs in England, France, Germany, Italy, and Spain.

The rest of this paper proceeds as follows. We begin by discussing the basics of Poisson regression modelling and contrast this with the more conventional Ordinary Least Squares approach. Next, we provide a simple illustration with one predictor to the Big Five European football leagues, and show how the Indices of Discrimination and Wastage act as effective measures to summarise RAE bias. We then expand the model with the addition of a second predictor by examining the impact of population birth and death rates, and the challenges posed by multiple cohort year start and end-dates. Finally, we discuss the modelling of interaction terms as the search for paragon clubs, leagues and countries continues [16,18].

The basics of Poisson regression

Whereas simple bivariate Ordinary Least Squares (OLS) regression finds a best fit to paired (x, y) data of the form: (1) the Poisson regression model (PRM) finds a best fit of the form: (2)

Poisson is the default method of analysis when dealing with small expected counts, for instance, the mean number (λ) of unscheduled births a hospital might expect per hour. Students meet the Poisson in “Statistics 101”, where the problem is to build up a profile of the underlying discrete probability distribution (or density function). In this example, the distribution would be the probability of 0 births, 1 birth, 2 births, and so on per hour. Knowing this could help make informed decisions about staffing levels and bed requirements. If instead the window of analysis is one day rather than one hour, then we would expect λ to be 24 times larger as the event is assumed to occur at the same constant rate throughout the interval. Although the distribution is ever more positively skewed for smaller and smaller λ, the Poisson is still the appropriate method.

PRM extends the problem, from λ as a fixed average per unit of time or distance, to situations where λ might vary. The λ governing the expected number of punctures per 100 miles of cycling, for instance, might vary with terrain, speed, tyre pressure, depth of tread, and so on. But at any particular combination of these conditions, there will still be a λ that governs the distribution we might expect; and the Poisson distribution of punctures for that particular λ will look just as it did in Statistics 101. But as the λ for one combination of terrain, speed, and tyres, might be an order of magnitude greater than the λ for another combination, the scale and shape of each distribution may look very different, though in an entirely lawful and explicable way.

Returning to the algebraic formulation, y = e^{(a + bx)}, PRM starts from a given set of x and y and infers the a and b parameter estimates that best relate them. Just as the intercept, a, in OLS shifts the entire regression line up or down, the a in PRM does an analogous job by magnifying or shrinking the scale of the entire curve. To see this, notice y = e^{(a + bx)} = e^a.e^bx = Ae^bx, with A (= e^a) thereby being the multiplier. Likewise, whereas b measures the slope in OLS, the b in PRM governs the shape or degree of curvature of the (x, y) relationship. In both OLS and PRM, a positive b implies that y increases as x increases; and a negative b implies that y decreases as x increases. Statistical tests of the b coefficient are tests of the null hypothesis b = 0, which translates into a flat horizontal line in both PRM and OLS regression. Given RAE, we anticipate a decrease in frequency across the cohort year, thus b should be negative; and the more severe the RAE bias, the more negatively large b should be. In the less likely situation of reverse RAE, the b should be positive and significant for an underdog effect [19,20]. In what follows, we focus almost entirely on b, the Poisson shape parameter.

Just as there is multiple OLS regression, so too can more than one independent variable be included in the PRM estimating equation, and for all the same kinds of reasons. But, one unique feature of PRM not shared by OLS is that for a genuine Poisson process, the mean = variance (= λ), whatever level of λ is observed. The dispersion coefficient is an (aggregated) ratio of variance divided by mean. If dispersion exceeds 1 by a sufficient amount, the likelihood is either that variables are missing from the model which acts to cause more variation than expected, or that the assumptions of Poisson are not being met. The dispersion coefficient is sometimes called the Variance Inflation Factor (VIF) because standard errors of all regression parameter estimates should be scaled by the square root of the VIF to provide more accurate inference, a procedure known as "quasi-Poisson". One corollary is that if dispersion ≈ 1, the assumptions of Poisson are met and there are no missing variables that could improve the model. So the ultimate goal of Poisson regression is not to end up with no residual variance (i.e., R² = 1), as it is in OLS regression. This would result in suspicious levels of under-dispersion, implying that the randomness which underpins the Poisson process had been explained. Instead, the goal is to have residuals in about the “right amounts”, as suggested by a Poisson process. It follows that researchers must keep a careful eye on the dispersion coefficient.

Poisson application: Big Five European soccer leagues

The data comprise domestic players in the first-team squads of the 204 football clubs competing in the top two tiers of the “Big 5” European leagues (England, France, Germany, Italy, Spain), at the start of season 2016/17, compiled from club websites (see S1 Appendix for complete dataset). Domestic players are defined as those whose nationality matches the league they play in, (e.g., Germans in the Bundesliga). There are 882, 734, 609, 882, and 868 such players in England, France, Germany, Italy, and Spain respectively; in total 3975 of 6644 registered players. This restriction minimises ambiguity due to players learning their trade in a foreign academy, while ensuring that the correct cohort year definition is applied [20]. Players in France, Germany, Italy and Spain are analysed using 1^st Jan. to 31^st Dec. as the domestic competition (cohort) year, while in England it is defined 1^st Sep. to 31^st Aug.

How many players were born on each day of the year is the dependent variable (y). The independent variable (x) is time of birth, t_B, measured as a decimal fraction of the cohort year (0,1), such that players with earlier birthdays have smaller values of t_B and the mid-year point is t_B = 0.50. Specifically, t_B = (j ‒ 0.5) / 365, where (j—0.5) denotes that players are notionally born at midday on the j^th day of the cohort year, and these values are rescaled by 365 to lie in the one year interval (0,1).

Analysis begins with the overall model, pooled across both countries and tiers, before testing the legitimacy of these assumptions shortly. The best fitting Poisson equation is: (3)

And this model explains 40.8% of the variation (McFadden’s pseudo R²). The slope coefficient for t_B is negative and significantly different from zero (t (df = 363) = -15.81, p < .001). This shows that the frequency curve trends downwards over the cohort year, consistent with their being more early-born than later-born players, as per RAE. The Poisson a coefficient is of little interest. It merely scales the equation to account for the number of observations. Dispersion is 0.985, close to the ideal 1; but all reported significance tests still apply the quasi-Poisson correction. The best fitting model is also presented in Fig 1. From this, we illustrate next the derivation of the Indices of Discrimination and Wastage. These managerially useful, highly interpretable measures, quantify the size of the Relative Age Effect in a standardized way, thereby enabling comparison across future studies.

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Fig 1. Poisson regression curve with tercile frequencies shown as bars.

https://doi.org/10.1371/journal.pone.0213988.g001

Indices of Discrimination and Wastage: A graphical and algebraic understanding

The black curve in Fig 1 shows the best fitting Poisson model. Clearly, players born at the end of the cohort year (point B) have a much smaller chance of reaching the upper echelons of European soccer than players born at the start of the cohort year (point A). The odds of A being selected relative to B is shown by the relative length of the heavy vertical lines projecting to the x-axis from A and B. This is a graphical representation of the Index of Discrimination (I_D). In algebraic terms, given the Poisson equation y = e^{(a + bx)}, A occurs when x (= t_B) = 0; thus y = p(selected | t_B = 0) = e^a. Likewise, B occurs when x (= t_B) = 1; thus y = p(selected | t_B = 1) = e^{(a + b)}. It follows that: (4)

Returning to the overall model (Eq 3), and substituting in values for t_B = 0 and t_B = 1 for the selection cut-off dates gives expected player frequencies of 16.37 and 6.80 respectively. Taking the ratio of extremes, the Index of Discrimination (I_D) = 16.37 / 6.80 = 2.41. Or put more simply, I_D is e^−b = e^+0.8786. Thus players born right at the start of the cohort year are nearly two-and-a-half times more likely to reach the upper echelons of professional football than those born right at the end. This I_D accords quite closely with Doyle and Bottomley’s [16] 2.67, but their analysis involved 11 nations and was restricted to players in a "top 1000” worldwide listing, as judged by transfer value.

It is important to note that there are several alternative but less precise ways of defining I_D which, if ever they entered the RAE literature would immediately undermine the precision of using Poisson regression. In our definition, I_D is the ratio of two point estimates which lie at the extreme opposite ends of the cohort year. When seen in this way, forming the ratio of any other pair of points in the cohort year is clearly rather arbitrary (like measuring someone’s height from knee to nose, or from ankle to chin), and would necessarily reduce the size of I_D. Furthermore, it may be tempting to form a simplified I_D ratio, instead, from the numbers born in January versus December; or players in Quarter 1 versus Quarter 4. However, points are no longer being compared, but intervals. And given that such ratios ignore 10 / 12 (months) or 2 / 4 (quarters) of the data respectively, they are particularly vulnerable to sampling fluctuations. So, if researchers wish to make meaningful comparisons across different age groups, countries, sports, times, activities and so on, then it is imperative that they avoid arbitrary redefinitions of I_D in favour of a single definition of I_D.

The fact that A players per day born at the start of the cohort year reached a certain criterion suggests that in the absence of RAE, A players per day born anytime during the year should be able to reach that same criterion. If that were so, in Fig 2 the Poisson curve frequency distribution would run along the horizontal line going through A. The totality of players meeting this criterion is given by the area beneath that line. Importantly, this area is partitioned into two sub-areas by the Poisson regression curve. Below the curve are those players the model suggests are currently selected given existing RAE constraints (namely, area h + H + G). In contrast, above the curve are those might-have-beens, the anonymous but equally talented players that remain undiscovered (namely, area g + q). The ratio of undiscovered to discovered areas gives our second diagnostic statistic, the Index of Wastage (I_W1). It too can be derived algebraically (see S2 Appendix for mathematical details), as follows: (5)

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Fig 2.

Indices of Wastage: I_W1 and I_W2.

https://doi.org/10.1371/journal.pone.0213988.g002

Returning to the overall model (Eq 3) and substituting in b = -0.8786, we find I_W1 = 0.503. Thus, for every player selected, I_W1 is the estimated number of players who might have been selected, but have disappeared from the game because they passed through a selection process that rewarded greater maturity in youth. In this instance, for every two talented players who established themselves as professionals, there is one equally talented player that remains undiscovered. In a world without RAE, clubs from the top two tiers could expand their squads by 50%, without sacrificing overall quality. However, this is rather naïve because not only would the wage bill increase dramatically, but players in the now enlarged squads would struggle for game time. Indeed, an even more exaggerated form of squad enlargement would befall football academies, where typically I_W1 > 1, suggesting that their squads would at least double in size.

Consequently we have developed a second Index of Wastage I_W2, assuming clubs would not increase squad size, but keep to their same quota that they have in the current RAE-biased world. In Fig 2 maintaining quota amounts to finding a frequency α (instead of A), uniform across the cohort year that has the same area under it as the area under the exponential RAE curve. In other words, q + G + H = h + G + H. Immediately we see that q = h. To maintain quota, the number of newly selected cohort-younger players (q) must balance the number of cohort-older players de-selected or “let go”. The mathematical derivation of I_W2 is a little more involved (see S2 Appendix for details), but can be computed using the following equations: (6)

Rather more simply, α = N / 365, where N is the number of observations (here, N = 3975 players) in the analysis. For the Big Five leagues data α = 10.89, using either calculation method. Obviously from Fig 2, A > α > B, which it is: 16.37 > 10.89 > 6.80. Likewise T is the point in the cohort year where the horizontal no-RAE dashed line cuts the RAE curve. It marks the boundary between the unfairly advantaged early-borns and the unfairly disadvantaged, later-borns. For the Big Five leagues data, T = 0.4636, which corresponds to the boundary about two weeks (= (0.5–0.4636) * 365 days) before the cohort year mid-point. Finally, I_W2 = 0.122. Under the new, stricter criterion implied by α, the quota-preserving club would be able to reject h (the least able players born before t_B = T) in favour of q (those now retained who would before not have been). The new players (q) as a proportion of the total players selected (G + H + q) accounts for 11% of the players, which can also be calculated as:

While I_W2 is much smaller than I_W1 (0.122 versus 0.503) it is nonetheless economically important. Based on I_W2, out of domestic 3975 players, 437 (= 3975*0.11), around two players per first-team squad (437 / 204) actually have lower skills than another 437 players who are not included in this sample–either they play in lower leagues or gave up their dream of becoming professional footballers altogether–but would have had the potential to reach the top echelons and be included in this sample, had they faced a selection process that took into account their lower maturity in youth. In other words, because clubs rely on a selection process that rewards greater maturity in youth, two players per squad were wrongly selected and could have been replaced by better players to the benefit of owners of soccer clubs, soccer institutions and of course, soccer fans. (We thank our review team for these astute insights).

I_W2 concerns one-for-one player replacement, with no net loss or gain in player numbers overall. However, I_W1 concerns pure supplementation, necessarily leading to an increase in overall numbers (but with no players lost). It follows that the kinds of claims and language that can be made based on I_W1 will not be the same as those based on I_W2. In fact, I_W1 and I_W2 are really just special cases of a more general model in which players can be replaced and the overall numbers increased (or even decreased). See S2 Appendix.

To summarise, I_W2 measures how much more talent we could get out of the football system, assuming we put no more in beyond eliminating RAE. In contrast, I_W1 measures just how much more untapped talent is out there, with no provisos. The fact that I_W2 is much smaller than I_W1 may suggest one reason why clubs persistently ignore RAE. Although there are 2.4 times as many footballers born at the start of the cohort year than at its end (a stark inequality from the player perspective), the gains to the quota-preserving club of removing that inequality are less striking. Just slightly more than two players per squad (who are relatively older) would be replaced (by two relatively younger players).

Together, these indices emphasise the real bias inherent in RAE from different perspectives. This knowledge should also be useful for decision-making purposes where the consequences of RAE bias must be made explicit, and enable comparisons across clubs, tiers, countries, and ultimately time.

Having outlined the benefits of Poisson regression, in the interests of exposition, next we expand the estimating equation by adding a second predictor. Not only will this demonstrate the flexibility of this method compared to conventional chi-squared testing of contingency tables, but it enables us to address pertinent questions raised in the RAE literature, such as the benefits of including data on population births, and the implications of idiosyncratic competition years (e.g., England, Japan).

Discussion and conclusions

The Poisson regression model (PRM), despite being the standard statistical technique for analysing low frequency count data [11–13], remains under-used in Relative Age Effect (RAE) research [16–18]. This tutorial exposition hopes to demonstrate the application and benefits of this approach for this burgeoning research area which is attracting interest from not only those in sports science, but also economics, psychology, education and management. The standard practice is to use non-parametric Chi-squared tests [25] to show that observed quarterly or monthly player frequencies differ from the distribution of a reference population, typically peers that practice the same sport, who are born within the same cohort year. However, should such information be unavailable, for instance, when comparing across large numbers of countries, a uniform distribution may be substituted instead [26].

Yet RAE anticipates a decline in player frequency across the cohort year. Hence, the slope coefficient associated with the Frequency on Birth-time regression forms the basis of two useful and highly interpretable measures, quantifying the magnitude of the Relative Age Effect in a standardized way, namely the Indices of Discrimination (I_D) and Wastage (I_w). The I_D shows the relative odds of a player born at the very start (t_B = 0) of the competition year, compared to a player born on at the very end (t_B = 1) being selected. This should not be confused with the ratio of players born in Month 1 versus Month 12 or Quarter 1 versus Quarter 4 or whatever the granularity of the data maybe. Our results suggest that players born right at the start of the cohort year are nearly two-and-a-half times more likely (I_D = 2.407) to reach the upper echelons of European professional football than those born right at the end. This accords quite closely with Bottomley and Doyle’s [16] 2.67 based on a subset of players in the “top 1000” from 11 leading nations, and Rada et al.’s [18] 2.13 and 2.22 for comparable analyses of first-tier and second-tier players.

In addition, the Indices of Wastage (I_w1 and I_w2) capture the loss of talent because few players are fortunate to be born at the very start of the competition year, and thereby avail themselves of the greater opportunities it provides. I_w1 measures just how much more untapped talent is out there, with no provisos. Here, I_w1 = 0.503 which suggests that for every two players identified, there was one equally skilful player that remains undiscovered. In contrast, I_W2 measures how much more talent we could get out of the football system, assuming we put no more in beyond eliminating RAE. Here I_w2 = 0.122 which suggests just slightly more than one player per team, or two players per squad (3975 players / 204 teams) who were relatively older would have been replaced by one player who was relatively cohort-younger in an RAE “free world”. Consider only the English Premier League which comprises 20 teams. If one player per team was wrongly selected because they passed through a selection process that rewarded greater maturity in youth, in England there are at least 20 players (about one entire squad) who either play soccer at lower levels or gave up playing soccer altogether but would have had the potential to play in the Premier League and to be better than some current players. Not everybody has the persistence of Leicester City’s Jamie Vardy who was still playing non-league football aged 20. (Again, we thank our review team for these astute observations).

It is only by adopting standardised reporting procedures can we hope to facilitate comparison, build a coherent body of knowledge, and start addressing policy related issues. Unlike analyses based on chi-squared, Poisson regression can easily model more than one explanatory variable in the same analysis. For instance, we disentangled the effects of birthtime in the competition year (t_B) from background variations in the population birth and death rates across the year. We also investigated mixed cohort definitions, as when school year and sports year are different. While our findings remained unchanged, it nevertheless offers a more nuanced understanding of the RAE phenomena, and addresses concerns often raised as limitations in prior studies. With PMR it is also legitimate to perform more disaggregated analyses with smaller samples of data. Thus, we might anticipate witnessing an increase in more micro-analyses, such as club-by-club, academy-by-academy, or even year-group-by-year-group within academy analyses tracking the evolution of the RAE phenomena.

Finally, compared to OLS regression, the modelling of interaction terms is somewhat more complex. But such analyses exploring the conditional nature or contingencies linked to RAE perhaps offers the greatest value added and insights to policy-makers. Presently, there is no evidence to support the existence of paragon clubs, leagues or countries [16,18] from which other organisations may learn. Indeed, there have been few studies exploring the evolution of the RAE phenomena over time [17]. Clearly there is much work to do. Moving beyond another descriptive study documenting the existence of RAE, be it in a different country, competition, cohort, country or time-frame, is paramount. We believe that Poisson regression modelling provides scholars with the necessary tools to do this, and hope you agree!

Supporting information