Pace and Critical Gradient for Hill Runners: An Analysis of Race Records

Importance of descending skill for performance in fell races: a statistical analysis of race results

“…Our assumption that pace ratio varies only due to differences in descending skill means that these pace functions are expected to be the same for all runners at all uphill gradients, apart for a constant factor which could be removed by normalising pace relative to its value at zero gradient. However, the normalised pace function for an average runner (whose pace ratio would take the mean value), or indeed any other non-elite competitor, would diverge from that in Kay (2012b) as the downhill gradient increases, since the slopes of our regression lines for pace ratio on finish time become steeper for steeper races. Our regression equations could be used to derive the expected value of pace at some given downhill gradient for a competitor with some given overall performance level relative to race winners, but this derived pace would only apply to someone with a level of descending skill typical for runners at that level of overall performance.…”

“…Mean pace ratios for races may be compared with ratios calculated from the record (fastest ever) uphill and downhill paces on a course, as used by Kay (2012b); since pace ratio is negatively correlated with finish time, we would expect that the means are lower than the values for record-setters, and this is verified for 43 of the 44 races in our dataset (1 year's Scafell Pike race has an unusually high mean pace ratio, but this may be related to slight year-to-year variations in the route of the race). This has implications for the use of the functions derived by Kay (2012b) to model the variation of pace with gradient. Our assumption that pace ratio varies only due to differences in descending skill means that these pace functions are expected to be the same for all runners at all uphill gradients, apart for a constant factor which could be removed by normalising pace relative to its value at zero gradient.…”

“…Slieve Donard race (Northern Ireland): straight up and down the mountain, with a few hundred metres of flat road at the start and finish; a tourist path can be taken the whole way, but most runners opt for a more direct route over rough vegetation with scattered boulders (see Figure 2). (Kay, 2012b) 5. Ben Lomond race (Scotland): up and down a well made path with some short rocky sections; descent route differs from ascent on one short section.…”

“…An analysis of record (i.e., fastest ever) times for a selection of uphill and downhill foot races has previously been used to derive functions modelling runner's pace as a function of gradient (Kay, 2012b). These pace functions may be valid for the elite runners who achieve record times; for any other runner, we might expect the pace function to be a constant multiple of that for elite runners, but only if running performance at all gradients is purely a function of metabolism.…”

Waldo Tobler: An appreciation of the contributions made to geographic information science

“…This first appeared in Tobler (). It has been tested against fell (mountain) runner data from the UK in Kay (). Anthony Kay, who wrote this paper, is an academic mathematician and fell runner who, in initial correspondence with me, clearly had no idea of its originator’s status in geography and cartography.…”