Prediction of marathon performance time on the basis of training indices

Abstract

The purpose of this study was to examine the relationship between marathon performance time (MPT) and some training factors recorded for a given number of weeks prior to a race. Twenty-two runners, age 28-54 years, participated as subjects in this investigation. They kept daily exercise records during their marathon training for an overall number of 46 races, whose marathon time ranged from 167 to 216 min. Among the several parameters investigated, MPT was found to be affected mainly by the mean distance run per week K, during the training period under observation, and by the mean training pace P. These two training parameters have been combined by a mathematical approach to give a correlation for the prediction of the mean marathon pace Pm (easily related to MPT), based on an 8-week training period, as follows: Pm (sec/km) = 17.1 + 140.0 exp[-0.0053 K(km/week)] + 0.55 P (sec/km). The above correlation is able to estimate the MPT with a SEE of about 4 min.

Full text

VOLUME 6 | ISSUE 3 | 2011 |
511
Prediction of marathon performance time on the
basis of training indices
GIOVANNI TANDA 1
DIMSET, University of Genova, Italy
ABSTRACT
Tanda G. Prediction of marathon performance time on the basis of training indices. J. Hum. Sport Exerc.
Vol. 6, No. 3, pp. 511-520, 2011. The purpose of this study was to examine the relationship between
marathon performance time (MPT) and some training factors recorded for a given number of weeks prior to
a race. Twenty-two runners, age 28-54 years, participated as subjects in this investigation. They kept daily
exercise records during their marathon training for an overall number of 46 races, whose marathon time
ranged from 167 to 216 min. Among the several parameters investigated, MPT was found to be affected
mainly by the mean distance run per week K, during the training period under observation, and by the
mean training pace P. These two training parameters have been combined by a mathematical approach to
give a correlation for the prediction of the mean marathon pace Pm (easily related to MPT), based on an 8-
week training period, as follows: Pm (sec/km) = 17.1 + 140.0 exp[-0.0053 K(km/week)] + 0.55 P (sec/km).
The above correlation is able to estimate the MPT with a SEE of about 4 min. Key words: EXERCISE,
ENDURANCE, TRAINING.
1 Corresponding author. DIMSET, University of Genova, via Montallegro 1, 16145 Genova, Italy
E-mail: giovanni.tanda@unige.it
Submitted for publication February 2011
Accepted for publication September 2011
JOURNAL OF HUMAN SPORT & EXERCISE ISSN 1988-5202
© Faculty of Education. University of Alicante
doi:10.4100/jhse.2011.63.05
Original Article

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INTRODUCTION
Efforts to correlate human endurance performance to physiological, physical and training factors have been
ongoing for about a century. For endurance sports like marathon running, the maximal oxygen uptake, the
lactate threshold and the energy cost of running appear to play key roles in endurance performance (see,
for instance, Sjödin & Svedenhag, 1985; Joyner & Coyle, 2008; Faude et al., 2009). Slovic (1977), Hagan
et al. (1981, 1987) and Bale et al. (1985) were among the first to point out that some training indices (such
as the training pace or the mean distance run per day) are also highly predictive of MPT. Florence & Weir
(1997) established a correlation between critical velocity CV (a velocity determined during tests involving a
series of fatiguing runs on a treadmill) and MPT; their data showed that marathon time correlated more
highly with CV than either peak oxygen consumption or ventilatory threshold. A study conducted by
Roecker et al. (1998), based on an incremental treadmill test, revealed that the individual anaerobic
threshold had the highest predictive value for long distance competitions among 16 other parameters. Billat
et al. (2001) showed that the discriminating factors for international top-class marathoners, when compared
with runners at a slightly lower level, are the maximal oxygen consumption V
̇O2max for males, and the
velocity on a 1000m run (after 10 km run at marathon velocity) for females. In a further investigation, Billat
et al. (2003) stated that the velocity at the V
̇O2max is the main factor predicting the 10-km performance in
elite male and female Kenyan runners; they also investigated the type of training and concluded that a
high-intensity training contributes to a higher V
̇O2max in men. More recently, Legaz et al. (2006) correlated
MPT for a top-level homogeneous group of males and females to some physiological measurements. Their
model for the male group used as independent variables the lactate value at 10km/h, left ventricular
telediastolic diameter (LVD) and lactate value at 22 km/h, whereas for the female group the independent
variables were the subscapular skinfold, serum ferritin and sum of six skinfolds.
Most of the above studies have been based on measurements of such variables as maximal oxygen
consumption, heart rates, lactic acid during submaximal and maximal treadmill running, performed on high-
level distance runners and involving expensive laboratory facilities. However, a method of predicting
performance based on training indices may be, for the increasing mass of non-competitive runners, an
attractive and inexpensive alternative to metabolic testing (typically reserved to elite athletes) in events like
the marathon.
The aim of this study is to correlate marathon performance time (MPT) only with training characteristics.
Correlations for MPT mainly or only based on training variables are already documented in the literature
(Slovic 1977; Hagan et al. 1981, 1987; Bale et al. 1985); moreover, a direct relationship between training
and physiological factors is likely to exist (Billat et al., 2002, 2003) and this could justify the assumption of a
correlation completely based on the training data. To comply with this purpose, a regression analysis has
been performed in order to identify those training factors, recorded for a given number of weeks prior to a
race, that contribute significantly to the prediction of final race time.
MATERIAL AND METHODS
Research participants
This study collected training data of twenty-two runners (twenty-one males, one female), age 28-54 years
for a five-year period, over which they run 46 marathons, with MPT ranging from 167 min to 216 min (2h
47min to 3h 36min). All runners gave informed consent to participate in this investigation. Table 1 gives the
anthropometric characteristics of the subjects as mean ± SD (standard deviation), minimum and maximum.

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The subjects provided their daily exercise records during their pre-marathon training (typically a three-
months period), including the distance and time run for each workout. To ensure the required level of
reliability of the study, the marathon races considered had the same level of difficulty (predominantly on flat
terrain), similar weather conditions and were run by the athletes at a regular pace throughout the race (with
a difference between first half and second half times less than 4 min) and at the highest intensity in line with
their training level.
Table 1. Anthropometric characteristics of the subjects.
Research design and training documentation
The purpose of the study was to develop a regression equation to predict the marathon performance time.
Unfortunately, it was not possible to record physiological measurements for the athletes’ sample; thus the
study was limited to training variables. The training data were accumulated over a twelve-week period
before the race. Since the workouts performed during the week immediately preceding the race are
typically of lesser importance, analysis was developed neglecting data collected in the final week.
The variables considered for the MPT prediction were: the number of previous marathons run Nm, the
number of training days per week Nd, the mean distance run for each workout Kday, the maximum workout
distance per week Kmax, the mean workout distance per week K, the mean training pace P. In order to
provide reliable data, subjects were used to train on track or in external flat environment with a GPS device
in order to correctly estimate their training distances. Moreover, each athlete included, in his/her training
programme, one or more long-distance workouts (typically a 30-35 km long steady run 3-4 weeks prior the
race).
Parameters based on the distance run in a given time lap (day, week) typically include the distance run also
in the warm-up and recovery; the same consideration applies to the calculation of the training pace, given
by the ratio between the time, employed to run the workout distance (including the warm-up and recovery),
and the distance itself.
No tests for a given distance (i.e. faster time for a 5km or 10km run) were considered in this study for the
following reasons: the difficulty in scheduling a running test under standard conditions (i.e., similar weather
and terrain, same time allocation prior to the marathon, etc.) for athletes living and training in different
cities, and the consideration that numerous subjects of the sample did not include a fast 5km or 10km
workout in their programme of marathon training.
Training and performance characteristics of the research participants are summarised in Table 2. Even
though the analysis has been conducted by averaging the collected training data over 11, 10 and 8
consecutive weeks prior to the race, the time-averaged parameters reported in Table 2 refer to the 8-week
period since no significant alterations of data were recorded by changing (from 8 to 11 weeks) the period of
mean
SD
min
max
Height (cm)
173
6.4
168*
185
Weight (kg)
65.2
6.5
57.5*
79
BMI (kg/m
2
)
21.7
1.3
19.2
24.7
Age (years)
42.8
6.7
28
54
*male sample group (for the sole female runner: height:155 cm, weight 53.5 kg)

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observation. The reference, 8-week, period started 9 weeks before the race and finished seven days prior
to the race.
Table 2. Performance and training of the subjects.
Statistical analysis
The identification of the parameters significantly affecting the MPT was performed by using a standard
software for the regression analysis of data according to different shapes (linear, polynomial, exponential,
power law, etc.) of the mathematical correlating function. The standard error of estimate SEE and the
correlation coefficient r were considered to evaluate the accuracy of a given regression curve. Too large
values of SEE or values of r relatively far from unity were considered as indices of poor quality of the
correlation.
RESULTS
Firstly, the effects of the selected training characteristics on the recorded marathon pace Pm (sec/km) were
investigated. It is useful to keep in mind that marathon performance time MPT, expressed in min, is related
to Pm, expressed in sec/km, by the equation: MPT = 42.195 Pm/60.
The relationship of marathon pace Pm to considered training indices is reported in Table 3, where the shape
of the best regression line, the correlation coefficient (r) and the standard error of estimate (SEE) are
indicated for each independent variable. Results showed that the mean distance for each workout Kday
(km/day) and the maximum distance per week Kmax (km/week) are not effective predictors for MPT.
Similarly, neither the number of workout days per week Nd, nor the number of previous marathons Nm were
found to be predictive of the marathon pace. Conversely, the mean distance per week K and the mean
training pace were found to be strongly correlated with MPT. The regression lines giving the marathon pace
Pm versus K and P are shown in Figures 1 and 2, respectively.
mean
SD
min
max
Finishing time MPT (min)
191
12
167
216
Race pace Pm (sec/km)
271.8
17.7
237.4
306.8
Previous marathons Nm
7.6
6.3
0
22
#No. of training days per week Nd
4.0
1.0
2.7
6.8
#Mean distance for each workout Kday (km/day)
15.6
1.4
11.4
19.1
Maximum workout distance per week Kmax (km/week)
87
15.2
59
132
#Mean workout distance per week K (km/week)
65.9
15.9
40.4
110.7
#Mean training pace P (sec/km)
284.6
18.1
253.3
330.6
# averaged over an 8-week training period prior to the race

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Table 3. Relationship of marathon pace to selected training indices.
K (km/week)
30 45 60 75 90 105 120
Pm (sec/km)
220
240
260
280
300
320
Pm
vs K
best regression line
Figure 1. Relationship between Pm (marathon pace) and K (mean distance run per week).
shape of best
regression
line
coefficient of
determination
r
SEE(sec/km)
Previous marathons Nm
linear
0.11
17.9
#No. of training days per week Nd
linear
0.72
12.8
#Mean distance for each workout Kday (km/day)
linear
0.36
17.2
Maximum workout distance per week Kmax (km/week)
linear
0.71
12.6
#Mean workout distance per week K (km/week)
exponential
decay
0.81 10.7
#Mean training pace P (sec/km)
linear
0.85
9.3
# averaged over an 8-week training period prior to the race.

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P (sec/km)
240 260 280 300 320 340
P
m
(sec/km)
220
240
260
280
300
320
P
m
vs P
best regression line
Figure 2. Relationship between Pm (marathon pace) and P (mean training pace).
Attention was then turned to the development, through a multiple non-linear regression analysis, of the
relationship giving the marathon pace Pm as a function of the best predictive factors, K and P.
According to the complete set of training data, the marathon pace can be predicted by the following
equation:
Pm (sec/km) = 17.1 + 140.0 exp[-0.0053 K(km/week)] + 0.55 P (sec/km)
where exp denotes the exponential function. The standard error of estimate SEE of the regression equation
is 5.77 sec/km, which corresponds to a SEE of only 4 min in the MPT prediction. Despite the reduced range
of variation of BMI for the sample group, it was found that, when the sample of runners with BMI<23 kg/m2
is considered (17 athletes out of 22, with 37 marathons run out of 46) the SEE of the regression equation
drops to only 5.10 sec/km (i.e. 3 min 35 sec for MPT).

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The predicted values of Pm (sec/km) are plotted in Figure 3 as functions of the recorded values. The solid
line represents the line of perfect agreement. The correlation predicts 69% of MPT values within 4 min and
50% within 2 min. These figures increase to 73% and 54%, respectively, when only subjects with BMI<23
kg/m2 are considered.
P
m, measured
(sec/km)
220 240 260 280 300 320
P
m, predicted
(sec/km)
220
240
260
280
300
320
Figure 3. Mean marathon pace: predicted values against measured values. Solid line: perfect agreement.
DISCUSSION
The present study converged to a correlation featured by a standard error of estimate typically lower than
that encountered in similar studies reported in the literature (Hagan et al., 1981, 1987; Florence & Weir,
1997), despite the heterogeneous physical and training characteristics of the subjects, and developed
utilizing only two independent variables, one related to the training volume and the other to the training
intensity. The reasons for such a successful result of the regression analysis are probably: (i) the good

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quality of the sample training data (each runner selected for this study performed his training seriously and
with great motivation) and of the achieved race performances (each marathon run by the athlete group has
been typically featured by a constant pace throughout the race), (ii) the correct identification of main
training indices, (iii) and the choice of a non-linear regression analysis. The main limitation of the regression
equation for MPT is its validity range, currently from 167 min (2h, 47min) to 216 min (3h, 36min), which is
the range of race finishing time of the sample group. However, it is worth noting that, for example, more
than 10 thousand Italian people ran, in 2010, at least one marathon with a finishing time included in the
above MPT range; therefore the results of this study are potentially addressed to a relatively large cohort of
marathoners.
The duration of time over which the training measurements were accumulated ranged, in similar studies
reported in the literature (Slovic, 1977; Foster 1975, 1983; Grant et al. 1984; Hagan et al. 1981, 1987; Billat
et al. 2001), from 8 to 12 weeks. Hagan et al. (1987) showed that regardless of the period of observation
(8, 9, or 12 weeks), their equation variables remained unchanged. The present investigation seems to
support this consideration; the processing of present data was not affected by the period (from 8 to 11
weeks) over which training data were averaged. The 8-week period was then selected for the regression
equation since it involves a lower amount of data to record and process.
The number of subjects tested for this study was 22, lower than that considered by Hagan et al. (1981,
1987), but comparable to the one used by Billat et al. (2001, 2003) and higher than the sample number of
Florence & Weir (1997). It should be emphasized that some of the 22 athletes provided training data for
more than a single marathon race (up to four) run typically at six months/one year of distance from each
other; therefore the number of overall training data (and relevant marathon races run by the 22 athletes)
was 46, satisfactory to develop an accurate regression analysis. The sample group is mostly formed by
male athletes (21 subjects out of 22), However, the training data (recorded for 3 marathons run) provided
by the sole woman of the sample group are accurately correlated by the present analysis. Of course, the
accumulation of further training data for a larger sample of women and of athletes, in general, including
elite marathoners, could lead in the near future to an even more accurate correlation within a larger field of
validity.
Of the several training indices investigated, the analysis suggests that some are poorly correlated to MPT,
like the mean distance for each workout, the maximum distance per week, the number of previous
marathons, and the number of workout days per week. This result is in agreement with the majority of
similar studies reported in the literature. Conversely, the mean distance run per week and the mean training
pace emerged as the most important training characteristics affecting the marathon performance. Hagan et
al. (1981) included (among the others) these two parameters in their correlation for MPT prediction based
on the analysis of male runners training data; in a following study (Hagan et al., 1987), conducted for
female distance runners, MPT was correlated to the mean training pace and to the daily (rather than the
weekly) workout. It is interesting to note that the maximal aerobic power was not related to MPT for the
female sample. A similar study (Bale et al., 1985) on female marathon runners indicated the number of
sessions per week, the ectomorphy, the distance run per week and the number of years of training as the
best predictors for MPT. Conversely, studies conducted for first-time marathoners (Grant et al., 1984) and
for middle-aged and older runners (Takeshima et al., 1995) found only a low or moderate level of
correlation between the weekly running distance and the endurance running performance.
A more recent investigation on elite runners (Billat et al., 2003) revealed that the velocity at the high-
intensity training is the main factor predicting the 10km-run performance; this finding is in agreement with

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the results of this study, showing that the mean training pace P, which is related to intensity of the training
period, is the best predictor for MPT (lowest SEE and highest r). However, incorporating also the mean
weekly distance K is found, in the present analysis, to enhance the predictive accuracy of the developed
correlations for MPT; this is probably ascribed to the characteristics of the sample, not including elite
runners but amateurs only.
The effect of the athletes’ physical characteristics on MPT was not investigated in this study, apart from the
body mass index (BMI) of each runner, recorded and included in the analysis. Hagan et al. (1981, 1987)
showed that BMI was a better predictor of MPT for experienced female runners than for male runners.
Campbell (1985) found that the most important predictors for running speed in half-marathon were the
distance per week and BMI. No statistical significance was reached for the relationship of BMI with finish
time for both men and women ultramarathon runners was recorded by Hoffman et al. (2010), whereas BMI
was shown to account for 10-11% of the variation in finish time in a previous study (Hoffman et al., 2008),
where a larger sample size was considered. The present regression analysis indicated that a BMI lower
than 23 kg/m2 (that is quite usual for an experienced long-distance runner) contributes to increasing the
accuracy of the regression equations for MPT. Since BMI values of the sample group varied in a too narrow
range, no attempt to relate it to MPT was make; however it is argued from results that BMI may (negatively)
affect MPT only beyond some critical values (say 23 kg/m2).
CONCLUSIONS
The results of this investigation, conducted by processing the marathon training data of 22 athletes,
showed a high correlation between marathon performance time and a couple of training indices that can be
easily calculated by processing training data accumulated over a given period of observation (8 weeks)
prior the marathon race. These indices (the mean weekly distance and the mean training pace) condensed
the volume and intensity of training respectively, regardless of the type of training programme followed by
each athlete. Currently, the validity range of the correlation for MPT is from 167 min to 216 min. The
standard error of estimate of the regression equation is 4 min but it goes down to about 3min 30sec when
only runners with BMI lower than 23 kg/m2 are considered.
ACKNOWLEDGEMENTS
The author is grateful to the athletes for their availability to be subjects for this investigation, and to Prof. S.
Lo Presti, for the technical assistance and suggestions.
DEDICATION
This paper is dedicated to the memory of Dr. Maurizio Magagnini, trainer and dear friend, who passed
away on July 7, 2010.
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