Introduction

Recreational running raised as one of the most popular physical activities worldwide, gaining more adepts and increasing the number of running events [1]. In this sense, marathon (i.e., 42.195 km) running is a physically high-demanding challenge that become one of the most popular and beloved running distances in recent years [2, 3]. In fact, a great number of recreational runners systematically train year-by-year seeking their best marathon performance, in some cases trying to imitate world-class marathon runners. Scientific evidence has consistently demonstrated that the optimal manipulation of several training variables (e.g., accumulated training volume, training intensity distribution -TID-, training periodization and peaking strategies), as well as training experience, are key factors that affect running performance [4,5,6]. In this regard, it was previously suggested that an accurate and realistic prediction of the marathon race final time (MRT) is also crucial for optimizing running performance [7]. Moreover, planning optimal competition strategies (e.g., pacing behavior) increase the possibilities of a runner to successfully complete a marathon, or to reach a personal best MRT [8, 9]. Concerning this, it is important to consider that the age of peak performance in long-distance running races (i.e., from 5 km to marathon) is different between men and women [10]. More specifically, women seem to achieve their best half-marathon and marathon race time 1 year and 3 years earlier in life than men, respectively [10].

In recent decades, several mathematical models have been proposed to predict MRT in both male and female runners. These models are based on different endurance performance variables, including anthropometric (e.g., body mass, body mass index -BMI-, body fat percentage, calf circumference, trunk-to-leg proportion etc.) and physiological variables (e.g., maximum oxygen consumption -VO2max-, running economy -RE-, physiological thresholds etc.) [11, 12] as well as those related to training (e.g., weekly training distance or volume, training frequency, average weekly running speed or pace, TID etc.) [3, 13,14,15] and previous experience in the distance [16]. However, some of these variables are commonly assessed by qualified specialists through laboratory tests or by using specific and not easily available equipment for recreational runners and coaches. Although these models are valid and have reported relatively good accuracy [17], predictions based on race final time in shorter distances remain the best option for estimating MRT performance in recreational runners [18, 19]. Thus, participating in a shorter race before the main race, but not too close to cause fatigue during the marathon competition, might be an easy way to test and estimate endurance running performance of recreational runners. In this sense, several cities and organizations worldwide hold competitions over shorter distances (usually half-marathon, 21.1 km), weeks before the marathon race under the claim of “road to”. This is the case of Valencia Marathon, in Spain, one of the fastest, flattest and renamed marathons in the world. Through such preparatory races, athletes might estimate their future performance and pacing on a course profile that is partially similar to that of the main race (i.e., marathon race).

Undoubtedly, accurate prediction of MRT is the main factor in establishing rational and realistic competition pacing, and displaying optimal pacing behavior (PB) during the race is mandatory to follow the previous strategy. Pacing behavior is defined as the way in which effort is distributed along the race [9], and is a crucial skill to develop in order to increase marathon performance in both elite [20] and recreational runners [21]. In fact, Renfree and Casado (2018) suggested that choosing the optimal pacing strategy and PB over the duration of long-distance events will improve running performance. In this regard, there are several factors that affect PB, such as running performance [20], sex [22] and drafting strategies [23]. Concerning running performance, top runners tend to adopt a more even PB (i.e., the pace between the first half-marathon and the second one is almost the same) [22,23,24], while runners with lower performance level often perform a positive PB (i.e., the first half-marathon is faster than the second one) [24]. Other authors have also suggested that sex might influence runner’s PB, with positive PB being the most common in men, while women show fewer variations in PB during a marathon [25]. Nonetheless, in the last five decades, a negative PB (i.e., the second half-marathon is faster than the first one) has emerged as a good strategy to achieve the best MRT in elite marathon runners [20]. In addition, drafting strategies may also affect PB. Drafting enhances performance in distance running competitions, as demonstrated by previous studies [5]. Accordingly, it has been proposed that running behind an athlete during the second half of a marathon may result in a 5.9% decrease in the metabolic cost of running [26]. In other words, running behind another runner favors RE and therefore, running performance. Similarly, pack formation could help athletes draft one another during a race. Although the influence of pack formation on performance and PB has been investigated in global championships for half-marathons [27] and marathons [23], there are no consensus regarding the best strategy to follow another runner during the race. In this sense, Hanley (2016) suggested that packing strategies might be diverse, mostly when runners display an even PB.

To the best of our knowledge, no study has proposed a regression model based on previous time in half-marathon, age group, PB, and packing behavior during a marathon. Thus, the aims of the present study were: 1) to determine the influence of PB and packing behavior on marathon performance, and 2) to develop an accurate regression model, for men and for women, based on previous half-marathon time, PB, age group, and packing behavior. We hypothesize that the inclusion of these variables might explain MRT better than previous models, especially when runners’ performance level is considered.

Materials and methods

Ethical aspects

Informed consent and ethical approval from university Ethics Committee were not necessary because the data were public and freely available (https://www.valenciaciudaddelrunning.com/maraton/clasificaciones-maraton-2021/). This study was conducted in accordance with the Declaration of Helsinki (1964, amended in 2013) concerning human experimentation.

Study design

An observational approach was followed to conduct this study. Ten race lap splits were examined in order to categorize PB (0–5, 5–10, 10–15, 15–20, 20–21.1, 21.1–25, 25–30, 30–35, 35–40 and 40–42.2 km). Informed consent was not obtained from participants because the data were public and free. This study also met the standards of the Declaration of Helsinki regarding human experimentation.

Participants

Official electronic splits and finishing times were obtained from the official website of each race (Valencia Half-marathon and Marathon: https://www.valenciaciudaddelrunning.com/medio/clasificaciones-medio-maraton-2021/ and https://www.valenciaciudaddelrunning.com/maraton/clasificaciones-maraton-2021/) by means of web scraping technique. Subsequently, through matching technique we identified the names and surnames that took part in both races. Therefore, the inclusion criteria were: 1) to have taken part in the two races in 2021; 2) all official splits times in each race interval were properly recorded and available; 3) the absence of any atypical record (i.e., lack of a lap record or wrong measurement); and 4) to finish the marathon race between 2:06:08 and 5:33:46 [h:min:s] for men and from 2:26:00 to 5:20:03 [h:min:s] for women. This interval time ensures that the different athletes included in the present study run during the most part of the competition. After applying all the inclusion criteria, 1822 runners (1571 men and 251 female) were selected for further analysis.

Pacing behavior criteria

To define PB among runners regardless of their final performance, the average speed of each lap was normalized relative to the average individual race speed for each performance. The average race speed for each runner is represented by a value of 1.00. As a result, a number higher than 1.00 denotes a split pace faster than the average race speed, whereas a value lower than 1.00 shows a lap pace slower than the average race speed.

Pack formation

The split time gap with the nearest runner in each section must be less than one second to determine whether an athlete is running in a pack. This classification criterion, which separates six different forms of packing behavior throughout the race, was based on Hanley’s proposal (2016): ever-present pack, in which all athletes run together for at least eight of the nine splits, up to and including the 40th km; halfway pack, when all participants ran together until halfway but were then separated after at least 30th km; nomadic-pack, in which all racers ran with different rivals for at least seven of the nine segments; semi-nomadic pack, where all runners were in packs (not necessarily the same) for at least five of the seven splits until the 30th km, but then ran alone; regrouping-pack, where runners belonged to different packs for more than half of the nine splits and regrouped after having run alone for two splits; and short-lived packs, where athletes ran in a pack for fewer than five of the nine splits.

Race profile

Valencia Marathon altimetric race profile (i.e., course gradient) displays a minimum and maximum altitude over sea level of 1.8 and 18 m, respectively, which makes it one of the fastest and flattest marathons worldwide (Fig. 1).

Fig. 1
figure 1

Valencia Marathon altimetric race profile (recovered from https://www.valenciaciudaddelrunning.com/wp-content/uploads/2022/11/42k-recorrido-2022-con-avituallamiento.pdf)

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Marathon final time

Marathon range final time was established for ensuring runners included in our study to run during most time of the competition. Based on previous studies that established the preferred gait transition speed between walking and running at ⁓ 7.2–7.4 km/h [28,29,30], we set the average marathon speed threshold at 7.5 km/h for guarantying running the most part of the marathon.

Statistical analysis

The data homogeneity of variance test was performed using Levene’s test, and the Kolmogorov–Smirnov, Cramer-von Mises, and Anderson-Darling tests were used to analyze the normal distribution of all continuous variables. Thereafter, unpaired t-tests (between the first and second half-marathon times) were used to identify runners’ PB (i.e., negative, positive, or even) [31].

The χ2 Pearson test was performed to check possible dependencies between variables: PB × packing behavior, age group × packing behavior, and age group × PB. If the previous analysis showed a significant association, Cramérs’s V was set to establish the effect size (ES); thresholds for effects were: < 0.2 “small”, 0.2 ≤ 0.6 “medium”, and > 0.6 “large”. A subsequent correspondence analysis was carried out to determine the proximity relationship between the variables.

After these preliminary tests, we created two multiple regression models (one for men and another for women) with MRT as the dependent variable. The following independent variables were considered as possible predictors of MRT: age category, half-marathon time 6 weeks before marathon, PB and packing behavior displayed during marathon, the interactions between PB and packing behavior, age group and PB, and age group and packing behavior. The interaction is a combination of variables, making a new one that has a significantly larger effect on the dependent variable than the sum of individual independent variables alone.

The final regression model was selected using stepwise forward and backward method of the “caret” R package [32]. For internal validation, k-fold cross-validation (10 folds and five repetitions) was performed. Internal validation was performed to reduce possible overfitting of the model [33]. There was no sign of multicollinearity (small Variance Inflation Factor for all independent variables).

The R package dplyr was used to identify possible outliers and improve the fitting of the regression model [34]. The outlier data in the multiple regression model were identified and removed when the absolute value of the studentized residual (SRE) was ≥ 2. After this analysis, a final sample of 1416 men and 186 women was taken into consideration for the final regression models.

The Jarque-Bera test was used to check the normal distribution of the residuals in both regression models (men and women). In turn, homoscedasticity of the regression models was checked by Breusch-Pagan test.

To evaluate whether our proposed predictive models differed significantly from the actual time of the runners a concordance test was conducted.

Model performance was assessed using the root mean square error (RMSE) and R2. The root-mean-square error is the error of the model reported in the outcome units (i.e., min:s). All statistical analyses were two-sided, and the significance level was set at p <0.05. Statistical analysis was conducted using R software 4.2.2 (R Core Team, 2022) and RStudio version 2022.12.0.353 (Rstudio Team, 2022).

Results

The participants’ characteristics are presented in Tables 1 and 2 for both men and women, respectively. Descriptive data are structured regarding PB (i.e., even pace, positive pace or negative pace); age group according to the official categorization proposed by the World Athletics (i.e., Under 20, Under 23, Senior, 35, 40, 45, 50, 55, 60, 65 and 70 for both men “M” and women “W”); as well as marathon and half-marathon final time in hours, minutes and seconds including standard deviation. The main behaviors regarding PB and packing were even pace and nomadic in men, while in women were positive and nomadic.

Table 1 Men pacing behavior, age-group, pack formation, marathon, and half marathon performance distribution
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Table 2 Women pacing behavior, age-group, pack formation, marathon, and half marathon performance distribution
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A linear and positive association between Marathon time and Half-marathon time for men and women was observed (r = 0.89; 95% CI= 0.88–0.89; p <0.001 in men; r = 0.83; 95% CI= 0.77–0.87; p <0.001 in women).

The chi-square test showed a significant association between PB and packing behavior (χ2 (10) = 19.7, p = 0.032; Cramer’s V = 0.101) in men but not in women (p >0.05). Similarly, there was a significant dependency between age group and PB (χ2 (18) = 31.0, p = 0.029; Cramer’s V = 0.126) only for men. There was no association between age group and packing behavior of either men or women (p >0.05). Subsequent analyses did not reveal any clear associations between these factors. However, because of the low ES, these data did not allow us to infer the results obtained.

Regression models in men

Owing to the large heterogeneity of the sample, four groups of runners were established according to their performance in the half-marathon. The sample was divided into four quarters (Table 3).

Table 3 Regression models for men runners based on performance group in half marathon
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After stepwise regression, four different models for men were constructed considering the time in 21.1 km, PB, Packing, and age group as explanatory variables, but not in all models. There were no significant interactions between the independent variables (i.e., PB × packing behavior; PB × age group; packing behavior × age group). The final regression models for men are showed in Table 3 (R2= 0.88, 95%CI-0.86-0.9; RSE= 5:08 [min:s]; R2= 0.72, 95%CI-0.65–0.75; RSE= 4:03 [min:s]; R2= 0.87, 95%CI-0.85–0.88; RSE= 8:03 [min:s]; R2= 0.82, 95%CI-0.79–0.85; RSE= 8:31 [min:s], for 1st, 2nd, 3th, and 4th regression model, respectively). After applying concordance test, there were no significant differences between the real MRT and prediction time (p>0.05) (Fig. 2 A 95%CI-617.56–631.92, B 95%CI-476.42–478.45, and D 95%CI-982.6–1022.7). However, the regression model for the 3th performance group (Figure 2C) showed significant differences between the real and predicted times (p <0.01; t = 220, df = 609, 95%CI-931–1150). Although possible outliers were removed, analysis of the residuals from the final regression model showed that they did not follow a normal distribution. Other assumptions (linearity, multicollinearity, and homoscedasticity) were also checked. Thus, the final regression model was constructed using a generalized linear method.

Fig. 2
figure 2

Differences between the real marathon time and predicted time regarding performance group and sex. A: men performance group 1; half-marathon time (in h:min:s) between 1:00:06 and 1:27:41; B: men performance group 2; half-marathon time (in h:min:s) between 1:27:42 and 1:37:11; C: men performance group 3; half-marathon time (in h:min:s) between 1:37:12 and 1:46:10; D: men performance group 4; half-marathon time (in h:min:s) between 1:46:11 and 2:33:15; E: real vs predicted marathon time in women

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Regression model in women

Applying stepwise regression, the final 42.2 km model for women considered the time in 21.1 km and PB as explanatory variables. The resultant model was a multiple linear regression (R2= 0.95, 95%CI-0.94–0.96; RSE= 8:06 [min:s]) (Table 4). No meaningful interactions were observed between the independent variables (i.e., PB × packing behavior, PB × age group, packing behavior × age group). No significant differences were observed between real MRT and predicted times (p >0.05, Figure 2E). The residuals of the linear regression model were normally distributed and the, linearity, multicollinearity, and homoscedasticity were checked.

Table 4 Regression model using Marathon as the criterion for women
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For a better comprehension, Table 5 summarizes the five equation models (four for men and one for women).

Table 5 Regression equations summary
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Discussion

The main aim of this study was to propose different regression models for men and women based on the runner’s age group, time spent in the previous half marathon, PB, packing behavior, and performance group (only for men models). However, all predictor variables considered differed in relevance regarding performance groups in men and might partially explain the dependent variable (i.e., MRT), thus our hypothesis was in part validated. To the best of our knowledge, this is the first study that consider runners’ behavior during competitions for improving the accuracy of the regression models. Moreover, the participants of our study presented a great heterogeneity regarding marathon performance, which makes our predictive model implementation suitable in a wide performance range of runners, from recreational to high-level runners.

Analyzing our models for predicting marathon performance in male runners of different levels, it seems that, apart from the previous time in half-marathon, there is a tendency of PB being more relevant as runner’s performance level increase, while pack formation becomes more important in lower-level runners. Further, age group may also play a small role in predicting marathon performance in middle-level male runners (i.e., performance group 2). For instance, the variables explaining the marathon predicted performance in male runners under one and a half hour in half-marathon (i.e., performance group 1) are PB displayed and previous time in half-marathon, with a coefficient of determination (R2) of 0.88. On the other hand, marathon time prediction relies more on half-marathon time, pack formation (i.e., nomadic and regrouping), as well as positive, even, and negative PB (only performance group 3) pace in middle-to-low-level runners with a previous time in half-marathon between 1 h and 37 min and 2 h and 33 min (i.e., performance groups 3 and 4; R2 = 0.87 and 0.82, respectively).

The novelty of our model relies in the inclusion of PB and pack formation as explanatory variables. In both men’s and women’s regression models, only positive PB was significantly correlated with a reduction in runner’s performance (from 4:04 to 11:53 [m:s] for men and 11:40 [m:s] for women), while pack formation seems to be more relevant as runner’s level decrease. This tendency to decline the performance time during marathons by displaying a positive PB has been well documented in several studies of elite [20] and recreational athletes [21]. Although the present models did not show superior statistical significance of negative PB compared to even PB, it might be observed a small tendency (Tables 3 and 4) of improving performance following a negative PB for almost models. This superiority of negative and even PB compared to positive PB has been consistently reported in previous studies [22,23,24]. Therefore, our proposal allows athletes to evaluate their performance based on different PBs prior to marathon. Moreover, our models check how could PB affects runners’ MRT. Taking together, in almost all the proposed models, for both men and women, negative PB showed a tendency to perform better, although in most cases, this performance was not statistically different. Likewise, in the regression models (Groups 3 and 4) in which packing behavior was selected as an explanatory variable, nomadic packing was shown to be the best strategy for performance enhancement. In this sense, previous studies indicate that this packing behavior may be the most appropriate because it allows the athletes to develop a race pace according to their fitness [23, 35]. Further, adopting nomadic packing could prevent runners to follow those runners’ groups performing a too fast race pace they could not maintain during the whole competition.

On the other hand, age group was another major factor that slightly influenced the prediction models. However, only one regression model (men performance group 3) took into account age-group as explanatory variable. In this sense, based on the limitation of our data it is difficult to draw some conclusions. Thus, age might not be more determinant than the runner’s previous experience in marathon distance [18] or other predictive variables. Our results are in accordance with previous studies, since Nikolaidis and Knechtle (2017) reported only a trivial interaction between marathon performance and age group in the New York City Marathon.

As expected, the main prediction variable in our models was the time in the previous half-marathon for both men and women. In this sense, some popular approaches based on performance in shorter distances, close to the main sports event, might be right for establishing future marathon pacing and assessing a runner’s fitness status [18]. Prior studies have underlined the utility of using time in shorter races, such as 10 km, a mile [18, 19], and half-marathon to predict MRT [18], thus the use of these regression models allows the assessment of marathoners’ performance. On the other hand, other factors that can influence the prediction of time based on previous distances are weather conditions and race profile (i.e., relative humidity, temperature, course gradient etc.) [36]. In this regard, previous studies have highlighted the negative influence of increasing relative humidity and temperature on runners’ performance [3]. Taking this into account, our models have been based on practically the same circuit in the same city, with a flat profile (Valencia, Spain) and similar weather conditions (12 °C vs. 11 °C and relative humidity of 40% vs. 50% for half-marathon vs. marathon, respectively). Based on the relative humidity and temperature data for both distances, it can be assumed that these variables have not influenced the performance of a single race (21.1 or 42.2 km) and therefore, did not affect the predictive model.

Regarding the robustness of the different proposed regression models, it is also important to consider that the coefficients of determination of our models ranged from 0.72 to 0.88 for men, and 0.95 for women, which in all cases are superior than other studies that considered previous race time in shorter distances (10 and 21.1 km) [19]. Scientific literature reported other regression models for estimating MTR mainly based on training variables (i.e., average weekly training distance or volume, training frequency, mean weekly training speed or pace, maximum workout distance per week etc.).[13, 14, 37] In this regard, the equation proposed by Schmid and colleagues (2012) reported R2 values of 0.50 considering calf circumference and average running speed during training. Similarly, other works reported prediction equation R2 values of 0.44 based on body fat percentage and running speed during training,[38] while Nikkolaidis and coworkers (2021) reported higher R2 values (R2 = 0.61) when VO2max, weekly training distance and BMI are included in the prediction equation. Only two studies [14, 37] reported R2 values similar to those of our regression models (R2 = 0.72 for Tanda’s model and R2 = 0.81 for Tanda and Knechtle’s model vs R2 = 0.87-0.91 for ours). However, compared to our models, some of these proposals require monitoring of specific training and physiological variables that the majority of recreational runners do not conduct, making it more difficult to apply these regression models.

The present study has several limitations. The main one is based on the small sample size in the U-23, M-60, M-65, and M-70 age groups in males and in the U-23, W-60, and W-65 age groups in females. Regarding this, we encourage not to use this predictor (Table 3, group 3), which corresponds to each respective age group. This fact could limit the capacity of our models to accurately predict MRT in these specific age groups. A second limitation of our study is the estimation error (RMSE). In men, this prediction error is close to 08:30 [min:s] for lower-level runners, while for runners performing better this error is reduced in 4 min. For female runners, this estimation error rise to 8:06 [min:s]. These results might be interpreted considering the great heterogeneity of our sample when regarding marathon performance, since both high- and low-performance level runners were included in our work. However, the RMSE for our models is in accordance with previous regression models, with lower R2 values for men [14, 39, 40] and women [41]. More recent studies have been able to improve this estimation error based on previous race times [42] and several training variables (i.e., typical mileage, number of tempo and interval training sessions) [43, 44].

In summary, this is the first study to propose different regression models for both men and women based on a previous time in half-marathon, runners’ age, PB and pack formation, as well as performance level (only for men models). The high R2 values reported for all models and the heterogeneity of the sample included make them accurate for easily estimate MRT in runners of different performance levels.

Practical applications

This study has focused on the prediction of the marathon time based on the time achieved in a half marathon in the previous weeks. Bearing in mind that the vast majority of marathoners run a half marathon weeks before their main race, the MRT prediction capacity obtained in this study is very useful and easily applicable because it has no economic cost, has a great reliability in runners with different performance levels and it can be considered a pre-competition estimation tool that help runners to set realistic time and race pace goals for their marathons.

Conclusion

Participating in a half-marathon before the main competition (i.e., marathon) may be one of the easiest ways to accurately estimate MRT. In addition, selecting a competitive strategy that develops a negative or even PB during the marathon race will enhance marathon performance. Consequently, we encourage runners to plan in detail the competitive strategy to optimize his/her race pace during marathon according to their performance level and therefore, improve MRT. On the other hand, other factors such as packing formation and the runner’s age group can help to partially explain the final performance and should be taken into account for the prediction of the MTR.