## Modelling the Effect of Age on the Speed of World Record Runners

In this article I develop a simple (linear) model of the effect of age on long distance running among world record holders. Extant models (WMA-Grubb-Jones, Fair, Vanderburgh, Dyer) are often sufficiently complex as to preclude any generalizable insights. However, the simplicity of the current approach will provide a robust basis for future new insights into the relationship between age and running performance.

The data used are from the Association of Road Racing Statisticians (arrs.net) single age world records for the road 5K, 10K, Half Marathon, and Marathon. The age range is from 30 to 90 years for all four male venues and for the female Marathon. The female 5K and 10K had an age range of 30 to 86 and the half Marathon had and age range of 30 to 85. The data were also adjusted for the effect of differences in the underlying population size associated with each single year of age; see Quantifying the Population Effect and also the Appendix to this article.

### Model

The model used has just three parameters (“λ”, “m”, and “b”):

Speedλ = m*Age + b

Or more compactly:

Sλ = mA + b

Whereas “m” and “b” are the slope and intercept of the linear regression, the power “λ” is taken from John Tukey’s linearizing “Ladder of Powers” (see OnLineStatBook, and Mosteller and Tukey). The parameters were determined independently for each of the eight venues so as to maximize the correlation between Sλ and Age; or equivalently to minimize the Fraction of Variance Unexplained (FVU) in Sλ.

Table 1 illustrates some examples of the Ladder of Transformations.

Note that models based on logs or speed are special cases of this ladder. However, also note they are suboptimal, as table 2 shows.

The linearizing effect of the Power Transform on road 5K records can be seen in Fig 1. For the 5K, the optimum λ was 2.78 and the correlation coefficient was 0.992.  Across all 8 venues, the optimized correlation coefficients ranged between 0.992 and 0.996, and the optimum value of λ ranged between 2.03 and 2.80.  However, near the optimum, the correlation was relatively insensitive to minor shifts in λ, suggesting a single value for all four venues within each gender.

### Concurrence

One possibly unexpected but very important result of using linearizing powers of speed to fit simple linear equations to population adjusted world records is the observation that, with equalized λ, the lines for all four venues are concurrent! That is, the fitted lines for the 5K, 10K, Half Marathon, and Marathon all intersect at the same point. The point of concurrence is different for males and females, with the concurrent point occurring at a later age for females. (Both points of concurrence are well beyond normal life spans).

Figure 2 illustrates the concurrent lines for males. Across all four venues the optimum λ, (power of speed) was 2.69. Using this power, the correlation coefficients were as follows: 5K: 0.992, 10K: 0.994, HalfM: 0.995, Marathon 0.994.

A future article will apply this linearizing power transformation to results for Local 5K Age Group Winners. The lines fitted for these local athletes have the same (male and female) points of concurrency as do the data for World Records Holders. As we will see later, this provides a method of age handicapping (age grading) that is both novel and elegant.

### Appendix

Based on the standard deviations obtained from fitting the linear models presented here, the Normal distribution can be used to estimate percentiles and hence adjust for the smaller population sizes observed at the older ages. [See Quantifying the Population Effect.] However, above 90 years, the population ratios become very extreme and the Normal approximation quickly loses accuracy.

For each of the 8 venues, the standard deviations of Sλ from the linear model were uniform across age. That is, the regressions of the absolute deviations on age were not significant, with p>0.05 in each case. And in only one case did the regression even approach significance, 0.10>p>0.05.

The details of the Normal approximation are as follows:

Let

• Φ-1 = the inverse Normal distribution
• Y = Sλ
• Yc = the population-corrected dependent variable
• σ = the standard deviation of Yc
• Pj = the population at age “j”
• rj = the ratio of population at age “j” in relation to the population at age 30; i.e. rj = Pj/P30.

Then Yc = Φ-1(0.5^rj)σ+Y.

For example, the relevant world male population at 80 is approximately 30% of the population of 30 year olds. Thus the adjusted world record for 80 year olds is Φ-1(0.5^0.3)σ+Y; i.e.

Yc = 0.886σ + Y.

Thus, if the population of 80 year olds were 3.33 times as large and thus equal to the population of 30 year olds, we anticipate that the median (transformed) age record for eighty year olds would be 0.886 standard deviations larger.