The IAU 24-Hour World Championships were held in Steenbergen, The Netherlands, in 2013. Over the 24-h period a total of 299 competitors ran on a course that had a 2.314 km lap. The athletes in the race were selected from their home country using qualification criteria set by their home ultrarunning association, thus most participants would be considered to be elite ultrarunners with experience in events that are challenging in distance and duration.

The weather over the two race days averaged 10°C which was 3°C lower than the mean temperature for that time of year in the region. Further, there was a considerable amount of precipitation, 8.8 mm of rain over the 2 days which included the race, and the winds were about 6 m/s with gusts as strong as 17 m/s. So, the race conditions were considered to be very difficult. These weather conditions increased the necessity to stop and change strategy during the race and contributed to the high drop-out rate due to injury, hypothermia, exhaustion and other factors during the 24 h.

A number of factors have been shown to influence the performance of a runner and their probability of completion in extreme ultrarunning events of the type being analyzed herein. Zingg et al. (2013) found that 40–44-year-old men and 35–39-year-old women had the fastest pace in 24-h races. Further, they found that female athletes have an average speed that is approximately 10% slower than male athletes. Lambert et al. (2004) studied the decrease in pace in 100 km ultra races and showed that the top runners were able to maintain their speed for 50 km but declined by 15% from their starting speed by the finish; however, in the IAU 24-h race data the runners tend to cover much greater distances than those studied therein. Further, Kao et al. (2008) found that runners in a 24-h race lose 5.05±2.28% of their body weight, so the nutritional and hydration demands of such events are crucial and thus the races can have high attrition throughout.

A significant amount of research has been completed on pacing in athletic events including running, cycling, swimming, speed skating and triathlon. Abbiss and Laursen (2008) review the strategies used and characterize them into: negative pacing (where speed increases throughout the event), all-out pacing (where there is an initial burst of speed followed by a slowly decreasing speed), positive pacing (where there is a gradually decreasing speed), even pacing (where the speed is constant), parabolic pacing (where the speed gradually decreases through an event but increases at a later stage in the event) and variable pacing (where the speed varies throughout an event, usually due to external factors like geography or environment). Within the parabolic pacing strategy, three shapes were characterized, a U-shaped (with a symmetric pacing profile), a J-shaped (where a small pacing decrease is followed by a steep rise in pace) and reverse J-shaped (where a strong pacing decrease is followed by a small rise in pace). Hanley (2015) studied pacing in the IAAF World Half Marathon (21.1 km) championships and found that the top runners maintained a constant speed for 15 km followed by a small decrease from 15 to 20 km and a strong increase in speed for the final 1.1 km whereas other runners had a gradual decrease in speed over the first 20 km followed by an increase for the final 1.1 km. Lima-Silva et al. (2010) did a similar analysis for a 10 km running race and observed similar pacing profiles where the pace was constant or slowly decreasing for the first 9600 m followed by a sudden increase for the final 400 m. Further, March et al. (2011) and Santos-Lozano et al. (2014) found similar pacing profiles in races over the marathon distance.

There have been fewer studies of pacing within ultra marathon events. However, Lambert et al. (2004) showed that in a race over 100 km, the pace of athletes tends to decrease over the duration of the event; they did not observe a strong increase at the end of the race, but this may because their data are aggregated over 10 km intervals.

Therefore, the performance in terms of speed, minimizing resting and avoiding dropping out are dictated by a number of factors for runners in ultra running events of long duration. Thus, it is of great interest to investigate the strategies used and to see how different clusters of runners utilize different strategies during such an event. This will also allow for comparison of the pacing strategies in 24-h ultra races with races of a shorter duration.

For each of the n = 299 runners we have a record of the lap time for each completed lap until the 24-h period was completed. Overall, 219 of the runners were still running at the end of the 24-h period and 80 runners finished running a significant time period before the end of the 24-h period.

In Table 1 we report some summary statistics for the runners: age at the start of the race, the number of completed laps, number of laps completed in a non-standard way (e.g. resting during the lap), speed per lap and average speed per athlete (km/h). In these summaries, we consider a lap to be completed in a non-standard way when the lap speed is below 4 km/h which indicates that the runner may have stopped during the lap or commenced walking during the lap. Further, 68% of the participants in the race were male and 32% were female. In order to properly read the table, note that the column “Speed” refers to single laps, whereas the column “Av. Speed” is referred to athletes and then it happens that the maximum average speed is lower than the maximum speed and the minimum average speed is higher than the minimum speed.

The only covariates available for the runners are age and gender. Whilst the performance of the runners may depend on many other factors including training, experience and nutrition, this information is not available for modeling purposes. However, the event is one for elite athletes selected by their home nation, so the athletes need to be experienced and well prepared to participate in the event.

The main variables of interest in this study are the runner’s speed per lap and a categorical variable that records the runner’s behavior (or status) during a lap (i.e. if the subject is running, resting, or leaves the race in a certain lap). The trajectory of the speed and of the categorical variable throughout the race are displayed in Figure 1.

Figure 1:

The top panel shows trajectories of speed for the individual runners (one every five) and the average speed per lap in red; the middle panel shows the proportion of subjects still running in a certain lap; the bottom panel shows the proportion of runners resting in a certain lap.

The overall appearance from Figure 1 is that the speed has a convex shaped behavior with respect to the lap number. Different explanations may be conjectured for this shape. A natural explanation is that runners tend to decrease the speed during the first part of the competition but they increase the speed when the end of the race is getting close. However, there is a drop-out effect due to the race being run for a fixed time (24 h) rather than a fixed distance. Thus, faster runners complete more laps, whereas slower runners complete fewer laps and they are not considered in the computation of the average speed when the lap number is large. Moreover, there is one further form of drop-out due to a runner leaving the race before the end of the competition and this may have a similar effect on the shape of the average lap speed.

We also note that the proportion of subjects still running dramatically decreases after the 50th lap and this is again due the two forms of drop-out mentioned above. On the other hand, we have a parabolic behavior for the proportion of subjects resting in a certain lap with the proportions being very low for low lap numbers and decreasing again for high lap numbers.

Thus, appropriate statistical modeling of the race data will need to account for the features found in this exploratory data analysis; the development of such a model is given in Section 3.

We adopt a latent trajectory model (Muthén and Shedden 1999; Roeder et al. 1999; Muthén 2004; Bollen and Curran 2006) that accounts for different possible strategies in running and at the same time facilitates clustering runners according to the adopted strategy by considering that for certain laps they may be running normally (B_il = 0), also they may have a rest (B_il = 1), they may finish before the end of the race (B_il = 2), or they finish because the 24-h time limit is reached. Essentially different lap performances are grouped into a finite number of possible states and different strategies are represented by specific probabilities for these states. Also the runners are clustered in finite number of latent classes according to their overall performance and the a priori probabilities to belong to each cluster are allowed to depend on individual covariates.

Let U_i denote a latent variable for the overall performance of runner i and let k denote the number of its possible values, labeled from 1 to k, with the corresponding mass probabilities indicated by π_iu = p(u_i = u) where π_iu may depend on runner-specific covariates. Each of the values (u) identifies a cluster of runners. The model is based on the following assumptions for every runner i given that he/she is in cluster u:

• on the first lap (l = 1), B_il has a generalized Bernoulli distribution with probabilities parametrized on the basis of multinomial logits, that is,

  $$\begin{array}{c}p\mathrm{(}{B}_{il}=0|U_i=u\mathrm{)}=\frac{1}{1+\text{exp}\mathrm{(}{x^{\prime }}_l{\gamma }_{1u}\mathrm{)}+\text{exp}\mathrm{(}{x^{\prime }}_l{\gamma }_{2u}\mathrm{)}}\mathrm{,}\\ p\mathrm{(}{B}_{il}=1|U_i=u\mathrm{)}=\frac{exp\mathrm{(}{x^{\prime }}_l{\gamma }_{1u}\mathrm{)}}{1+\text{exp}\mathrm{(}{x^{\prime }}_l{\gamma }_{1u}\mathrm{)}+\text{exp}\mathrm{(}{x^{\prime }}_l{\gamma }_{2u}\mathrm{)}}\mathrm{,}\\ p\mathrm{(}{B}_{il}=2|U_i=u\mathrm{)}=\frac{\text{exp}\mathrm{(}{x^{\prime }}_l{\gamma }_{2u}\mathrm{)}}{1+\text{exp}\mathrm{(}{x^{\prime }}_l{\gamma }_{1u}\mathrm{)}+\text{exp}\mathrm{(}{x^{\prime }}_l{\gamma }_{2u}\mathrm{)}}\mathrm{,}\end{array}\text{\hspace{1em}(1)}$$(1)

  where x_l is a function of l; in particular, every x_l is a column vector containing the terms of an orthogonal polynomial (Kennedy and Gentle 1980) of order r, which in our application is fixed equal to 3. If B_il = 0, then we assume the following model for the lap speed:

  $$Y_{il}\mathrm{|}{B}_{il}=\mathrm{0,}{U}_i=u\sim N\mathrm{(}{\mu }_{lu}\mathrm{,}{\sigma }^2\mathrm{}\mathrm{)}\mathrm{,}{\mu }_{lu}={x^{\prime }}_l{\beta }_u\mathrm{.}\text{\hspace{1em}(2)}$$(2)

  If B_il = 1 then the distribution of Y_il is left unspecified whereas if B_il = 2 then the process is stopped. The process is also stopped if the distance between the time of the lap (depending on the speed) is close to the end. Parameter vectors γ_1u, γ_2u, and β_u and cluster specific, whereas the variance σ² is common to all clusters.

• for the following laps (l > 1) and provided that the runner is still in the competition, B_il and Y_il are assumed to have the same distribution as above. Again, if the overall time is close to the end of the race or B_il = 2, then the precess is stopped as the runner leaves the competition.

Note that there are two forms of drop-out. The first is due to the overall time of the race which is non-informative as it deterministically depends on the previous values of response variables. The second, for the runner leaving the competition before the end, is informative and it is explicitly accounted in by the multinomial logistic regression model (1). Also note that, according to assumption (2), the lap speeds are conditionally independent given latent class and running normally. This assumption needs to be carefully checked on the basis of the corresponding residuals as we will show in Section 4.

Additionally, we allow for individual covariates to affect the distribution of the latent variables U_i. In particular, we adopt a parametrization based on multinomial logits of the following type:

$$\text{log}\frac{p\mathrm{(}{U}_i=u\mathrm{)}}{p\mathrm{(}{U}_i=1\mathrm{)}\mathrm{}}=\text{log}\frac{{\pi }_{iu}}{{\pi }_{i1}}={z^{\prime }}_i{\delta }_u\mathrm{,}u=\mathrm{2,}\dots \mathrm{,}k\mathrm{,}\text{\hspace{1em}(3)}$$(3)

where z_i is the vector of covariates (including a constant term for the intercept) for individual i, which are considered as fixed and given and δ_u is the corresponding vector of regression parameters for being in the u-th category instead of the first category. In the context of this race we have the age and gender of each runner available and these are included as covariates.

The labeling on the clusters is arbitrary and thus the model is only identifiable up to a permutation of the cluster labels. This problem is known as the label-switching problem in mixture models (Redner and Walker 1984). When studying the fitted models we label the clusters in terms of increasing race performance so that the results are presented in an intuitive manner.

In order to express the model likelihood, we have first to express the distribution of the response variables given the latent variables. In particular, for each subject i we observe the sequence b_i = (b_i1, …, b_ili)′; we also observe y_i,obs which corresponds to all or a part of the sequence y_i = (y_i1, …, y_ili)′. In particular, if all elements of b_i are equal to 0, then y_i,obs and y_i will coincide; if some elements of b_i are equal to 1 or 2, then y_i,obs will be a subvector of y_i.

Based on the assumptions formulated in the previous section, the distribution of interest has the following density function:

$$\begin{array}{c}f\mathrm{(}{b}_i\mathrm{,}{y}_{i\mathrm{,}obs}\mathrm{|}{U}_i=u\mathrm{)}=\left[{\displaystyle \prod _{l=1}^{l_i}}p\mathrm{(}{b}_{il}\mathrm{|}{U}_i=u\mathrm{)}\right]\left[{\displaystyle \prod _{l=1:b_{il}=0}^{l_i}}\varphi \mathrm{(}{y}_{il}\mathrm{|}{U}_i=u\mathrm{)}\right]\mathrm{,}\\ u=\mathrm{1,}\dots \mathrm{,}k\mathrm{,}\end{array}$$

where p(b_il|U_i = u) is defined in (1), the second product is extended to all observed elements of y_i, and ϕ(y_il|U_i = u) denotes the density of the normal distribution defined according to assumption (2). As in a standard finite mixture model, the manifest distribution has density that may be obtained as

$$f\mathrm{(}{b}_i\mathrm{,}{y}_{i\mathrm{,}obs}\mathrm{)}={\displaystyle \sum _{u=1}^k}{\pi }_{iu}f\mathrm{(}{b}_i\mathrm{,}{y}_{i\mathrm{,}obs}\mathrm{|}{U}_i=u\mathrm{}\mathrm{)}\mathrm{.}$$

This is the basis for the model log-likelihood, which has expression

$$\ell \mathrm{(}\theta \mathrm{)}={\displaystyle \sum _{i=1}^n}\text{log\hspace{0.17em}}f\mathrm{(}{b}_i\mathrm{,}{y}_{i\mathrm{,}obs}\mathrm{}\mathrm{)}\mathrm{,}$$

where θ is a vector containing all model parameters, that is, β_u, γ_1u, γ_2u, δ_u, for u = 1, …, k, and σ².

In order to maximize ℓ(θ) with respect to θ, we rely on the Expectation-Maximization (EM) algorithm (Dempster, Laird, and Rubin 1977). This algorithm has been used extensively for fitting mixture models (see McLachlan and Krishnan 1997; McLachlan and Peel 2000; Fraley and Raftery 2002) in the maximum likelihood framework.

The EM algorithm is based on alternating the following two steps until convergence in the target function:

• E-step: it consists of computing the conditional expected value, given the observed data and the current value of the parameters, of the complete data log-likelihood, which is defined as follows:

  $${\ell }^{\ast }\mathrm{(}\theta \mathrm{)}={\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{u=1}^k}{z}_{iu}\text{log}\mathrm{[}{\pi }_{iu}f\mathrm{(}{b}_i\mathrm{,}{y}_{i\mathrm{,}obs}\mathrm{|}{U}_i=u\mathrm{)}]\mathrm{.}$$

  In the above expression, z_iu is an indicator variable equal to 1 if subject i belongs to cluster u (i.e. U_i = u), and to 0 otherwise.

• M-step: the expected value resulting from the E-step is maximized with respect to θ and, in this way, this parameter vector is updated.

In practice, the E-step reduces to compute the (conditional) expected value of each indicator variable z_iu, denoted by $${\widehat{z}}_{iu},$$ by the following simple rule on the basis of the current value of the parameters:

$${\widehat{z}}_{iu}=\frac{{\pi }_{iu}f\mathrm{(}{b}_i\mathrm{,}{y}_{i\mathrm{,}obs}\mathrm{|}{U}_i=u\mathrm{)}}{f\mathrm{(}{b}_i\mathrm{,}{y}_{i\mathrm{,}obs}\mathrm{)}}\mathrm{.}$$

Regarding the M-step, we can use explicit solutions for the parameter vectors β_u and for the common variance σ²:

$$\begin{array}{c}{\beta }_u={\mathrm{(}{\displaystyle \sum _{i=1}^n}{\widehat{z}}_{iu}{\displaystyle \sum _{l=1:b_{il}=0}^{l_i}}{x}_l{x^{\prime }}_l\mathrm{)}}^{-1}{\displaystyle \sum _{i=1}^n}{\widehat{z}}_{iu}{\displaystyle \sum _{l=1:b_{il}=0}^{l_i}}{y}_{il}{x}_l\mathrm{,}u=\mathrm{1,}\dots \mathrm{,}k\mathrm{,}\\ {\sigma }^2=\frac{{\displaystyle {\sum }_{i=1}^n{\displaystyle {\sum }_{u=1}^k{\widehat{z}}_{iu}{\displaystyle {\sum }_{l=1:b_{il}=0}^{l_i}{\mathrm{(}{y}_{il}-{\mu }_{lu}\mathrm{)}}^2}}}}{{\displaystyle {\sum }_{i=1}^n{o}_i}}\mathrm{,}\end{array}$$

where o_i is the dimension of y_i,obs, that is, the number of regularly completed laps by runner i. On the other hand, updating the remaining parameters γ_1u and γ_2u in (1) requires an iterative algorithm of a Netwon-Raphson type. However, this is a simple algorithm since the objective function being maximized is of the same form as the objective function used when fitting a standard multinomial logit model with weights by maximum likelihood. The same Netwon-Raphson algorithm is also applied to update the parameters δ_u in (3) that affect the distribution of each latent variables U_i on the basis of the individual covariates. In the case where the π_iu probabilities are assumed to be equal for all subjects (i.e. π_iu = π_u), we have an explicit solution for the maximization of the expected complete-data log-likelihood with respect to the π_u probabilities:

$${\pi }_u=\frac{1}{n}{\displaystyle \sum _{i=1}^n}{\widehat{z}}_{iu}\mathrm{,}u=\mathrm{1,}\dots \mathrm{,}k\mathrm{.}$$

It is important, as for any other iterative algorithm, that the EM algorithm described above is suitably initialized; this amount to guessing starting values for the parameters in θ. We suggest to use both a simple rule providing sensible values for these parameters and a random rule which allows us to properly explore the parameter space. Just to clarify, we choose the starting values for the mass probabilities π_iu as 1/k for u = 1, …, k under the first rule, which is equivalent to fix the same size for all clusters. The corresponding random starting rule is instead based on first drawing each parameter π_iu from a uniform distribution between 0 and 1 and then normalizing these values.

We recall that trying different starting values for the EM algorithm is important to face the problem of multimodality of the likelihood function that may arise in finite mixture models and combining different initialization rules (deterministic and random) is an effective strategy in this regard.

Given that our application is focused on the clustering of individuals in separate groups, the main selection criterion we use for the number of these groups is the Normalized Entropy Criterion (NEC; Celeux and Soromenho 1996; Biernacki, Celeux, and Govaert 1999). This criterion is based on the following index:

$${\text{NEC}}_k=\frac{-{\displaystyle {\sum }_{i=1}^n{\displaystyle {\sum }_{u=1}^k{\widehat{z}}_{iu}\text{log\hspace{0.17em}}{\widehat{z}}_{iu}}}}{{\widehat{\ell }}_k-{\widehat{\ell }}_1}\mathrm{,}k\ge \mathrm{2,}$$

with NEC₁ = 1, where the numerator corresponds to the entropy and the denominator to the difference in maximum log-likelihood between the model with k classes and with 1 class. According to this approach, the value of k corresponding to the minimum of NEC_k has to be preferred, as it corresponds to the model being the best compromise between separation of the classes (as measured by the entropy) and goodness-of-fit (measured by the log-likelihood).

For completeness, we mention that another important criterion for selecting the number of components of a mixture model is the Bayesian Information Criterion (BIC; Schwartz 1978; Kass and Raftery 1995), which is based on the minimization of the index

$${\text{BIC}}_k=-2{\widehat{\ell }}_k+\text{log}\mathrm{(}n\mathrm{)}\mathrm{(}\#\text{par}\mathrm{}\mathrm{)}\mathrm{,}$$

where #par is the number of free parameters in the model; for an illustration see McLachlan and Peel (2000), Chapter 6. However, it is known that this criterion typically leads to a less parsimonious model than the model selected with NEC and, in particular, with classes not well separated. Therefore, given the target of our application, we prefer to rely on NEC.