The theory of coaching effectiveness

Research on the influence of coaches on athletes’ performance can be traced back to the intersection of sports psychology and management science, where theoretical frameworks have progressively shifted from qualitative descriptions to quantitative explorations. Early studies emphasized the “artistic” dimension of coaching, focusing on subjective and behavioral aspects that shape athlete outcomes. For instance, Jowett and Cockerill [14] explored the coach-athlete relationship, highlighting how interpersonal factors such as trust, mutual respect, and effective communication foster athlete motivation and team cohesion, based on qualitative interviews with elite athletes. Fransen et al. [15] investigated the impact of coaches’ motivational strategies on team performance, demonstrating through case studies that supportive behaviors, such as positive feedback and goal-setting, enhance collective efficacy in high school sports teams. Additionally, Bloom et al. [16] examined the role of coaches in fostering long-term athlete development, using qualitative case studies of elite youth coaches to underscore mentorship and individualized training as key drivers of expertise. González-García et al. [17] Analyze the athletes and coaches of various Chinese sports teams, finding that behaviors promoting team cohesion, assessed through athlete surveys, significantly influence short-term performance. Similarly, Moen and Federici [18] explored the role of coaches’ emotional intelligence in enhancing athlete motivation, reporting through descriptive analyses that empathetic leadership correlates with improved athlete engagement. These studies indicate that coaches play a crucial role in motivating teams, guiding training, and enhancing team cohesion, making them a key factor influencing the number of Olympic medals won. However, current methods are primarily based on qualitative or subjective factors, such as athletes’ personal perceptions or case-based observations, which limit their ability to capture the dynamic and causal nature of coaching effects.

Causal modeling in public policy and sports

In the field of public policy and sports, causal modeling primarily includes panel regression and fixed effects models, the difference-in-differences method (DID), time series change point detection, hierarchical and dynamic correction, and so on. Literature Valenti et al. [4] evaluated the impact of elite sports policies on national football team performance using panel data. However, this type of model assumes constant effects and ignores unobserved variables (such as fluctuations in the coach effect), which can lead to estimation errors. The traditional DID model, as discussed in Goodman et al. [19] for handling variation in treatment timing, has been adapted to assess the effect of coaching interventions in sports, though not directly applied in that study. For example, studies like Valenti et al. [4] used DID to assess coaching impacts, finding significant performance improvements in sports teams, though specific percentage gains vary by context. Barnett et al. [20] found that the average coach-athlete adaptation period is 1.2 Olympic cycles, resulting in traditional DID models overestimating the early effect by 63%. For time series change point detection, Wang et al. [9] was conducted using the STGCN-LSTM model, and the results obtained indicate the impact of coach mobility and strategic investment on medal prediction. De et al. [21] proposed the Bacon decomposition method, which solves the bias problem of traditional DID by stratifying heterogeneous intervention times, but has not been combined with attenuation models in the field of sports yet, and cannot capture the long-term dynamic of the effect.

Medal predictions and static evaluations

Traditional Olympic medal prediction and performance evaluation methods primarily rely on static indicators or correlation analysis. For instance, Schlembach et al. [12] developed a machine learning model to predict medal distributions using variables such as GDP, population, and sports investment. However, such models are essentially “correlation fits” and cannot distinguish between medal growth driven by economic investment versus coaching efficacy. Côté and Gilbert [22] proposed an evaluation method based on coaching leadership styles, analyzing their impact on athlete performance through qualitative approaches. However, this method lacks empirical quantitative verification and does not take into account other confounding factors. Its analytical ability for the contribution of coaches is also limited. In terms of static quantitative evaluation, methods relying on event results (such as recent world championship rankings) or expert ratings are susceptible to interference from incidental factors (e.g., athlete injuries) and cannot reflect coaches’ long-term contributions to “team building,” such as talent reserves fostered by youth training systems. For instance, Cheng et al. [23] demonstrated that static performance metrics, such as win-loss ratios, are heavily influenced by situational variables like team composition and external disruptions, limiting their ability to assess sustained coaching impact. Similarly, Truyens et al. [24] found that short-term indicators, such as international competition rankings, fail to account for long-term developmental strategies.

Building upon the limitations of the studies above, we propose a CPD-based model to address the following key issues: 1) Introduce the CUSUM algorithm to detect change points in medal outcomes for selecting viable data samples, and eliminate false detections of change points through the “Olympic cycle consistency test” and the “participation scale stability test”; 2) Incorporate Bacon decomposition to stratify the timing of coaching interventions, and combine it with a dynamic DID model to control for confounding factors such as home advantage and GDP; 3) Finally, quantify the persistence of different types of coaching interventions using an exponential decay function and a half-life model.

Data compliance statement

The collection methods of all the data in this article include the official data from the Olympic Games website, the competition questions released by the 2025 American Undergraduate Mathematical Modeling Contest, the downloaded data from the World Bank database, and the data from the Olympedia website. All data used in this study were collected from publicly available sources. The collection and analysis of these data complied with the terms and conditions of each data provider. Specifically, we adhered to the following: 1) Olympic data were used in accordance with the non-commercial research purposes stated on the official website. 2) Economic data from the World Bank are openly available for academic use under their Open Data License. 3) Coach and team performance data from Olympedia were used in compliance with their stated purpose of being an openly accessible statistical resource for Olympic history. 4) No ethical approval was required as the study used aggregated, publicly available data without individual identifiers.

Data cleaning and processing

• Missing value processing strategy: There are missing values in the continuous variables, such as the size of the competition in the original data, mainly due to incomplete records of early events. Since these variables have time series characteristics, KNN interpolation [25] can use the correlation of adjacent years to restore data integrity. For the missing eigenvalue in sample , its interpolation value can be calculated by the weighted average of the eigenvalue of the K adjacent values and the weight is determined according to the reciprocal of the distance. The formula is as follows:

(1)

where is the eigenvalue of the adjacent value , and the weight is calculated by the reciprocal of the distance.

• Data standardization: There is a significant magnitude difference between sports performance (e.g., medal count) and economic indicators (e.g., GDP) in the raw data: the medal count is usually in the range of 0–100 medals, while the GDP range is 104-fold (e.g., US GDP of 29 trillion in 2024, Saint Lucia’s GDP of 2.47 billion). The coach tenure data showed A right-skewed distribution (mean 4.2 years, median 3.5 years), and the normality was improved by Box-Cox transformation, with =0.3. Finally, Z-score standardization of multi-dimensional data is implemented.

(2)

• Data smoothing processing: In order to meet the requirements of the CUSUM algorithm for data, we first smooth the data. Sliding window smoothing: Using an 8-year (one Olympic cycle) sliding window to eliminate a single session of accidental fluctuations.

(3)

Classification standards

Project types.

The classification task of Olympic events in this study aims to quantify the heterogeneity of different event features and provide a basis for model parameter adjustment (such as dynamic weights for high-volatility events). A total of five categories are defined (see Table 2), and the classification criteria are as follows:

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Table 2. Parameter tuning for CPD-D³ model in olympic performance analysis.

https://doi.org/10.1371/journal.pone.0334635.t002

• High Fluctuation Events: In the past three consecutive Olympic Games, the gold medalists of at least two Games have come from different countries/regions, and the number of countries/regions involved in the medal distribution of this event in a single Games is ≥ 8 (such as shooting and equestrian).
• Talent Scarcity Events: The number of registered professional athletes worldwide is less than 5,000 (such as modern pentathlon), and the number of new national teams added in the last five years is <3.
• Emerging Events: The time since inclusion in the Olympics is 10 years (such as skateboarding and breakdancing), and the annual growth rate of participating countries/regions is >20%.
• Dominant Events: A single country/region has won >60% of the gold medals in the last three Olympic Games (such as Chinese diving and American basketball), and this country/region has held the top position in the world ranking for this event for five consecutive years.
• Unpopular Events: The global TV broadcast viewership is <5 million people per Olympic Games (such as softball and handball), and the number of international sponsors is <5.

Mutation point identification (CUSUM)

The CUSUM algorithm can effectively accumulate the historical monitoring data of each Olympic cycle, and realize the response to small changes through the accumulation and continuous increase. Therefore, it is necessary to first establish the accumulation and graph of each country on different items, and once the accumulation exceeds a certain threshold, it is considered that a mutation has occurred at this time.

Although some countries did not win in some events, the number of participants increased year by year, which increased the possibility of winning. For example, Brazil registered only 120 professional skateboarders in 2016, to 2023 has exceeded 1,800 people, an increase of about 15 times, the Tokyo Olympic Games skateboarding project entered the Olympic Games for the first time, and Brazil sent 6 players (accounting for 7.5% of the total number of participants), won 3 silver medals, and the Paris Olympic Games, The Brazilian skateboarding team increased the number of participants to 12 (13.6% of the total number of participants), and finally won 3 gold, 1 silver, and 1 bronze, becoming the biggest winner of the skateboarding project. This is because large events have a standard or threshold for participation, and the number of participants has increased year by year in the sense that the country’s comprehensive strength in sports has been increasing.

Traditional models do not capture this information, so the scoring method needs to be adjusted to take into account the variable “number of countries participating” and separate it out. Therefore, different weights are used to comprehensively define the score of a country in a certain event to reflect the difference in medal value. The equation is as follows:

(4)

In Eq (4), the total score of the project consists of two parts: Medal Score and Participation Score. The medal score is the number of gold/silver/bronze medals won in the event, expressed in turn by variable , while the participation score is defined as the total number of the country’s entries in the event. By referring to the research method of the Hybrid Weight Optimization Framework proposed by Schlembach et al. [12], parameters were optimized, and the weights of the four indicators were determined.

To prevent dimensional differences, the medal scores and participation scores need to be standardized and weighted separately:

(5)

where and and and represent the historical mean (8 – year sliding window) and standard deviation of medals and number of participants scores, respectively. is used to control the proportion of participation that contributes to the total score, which avoids the fact that some countries have a high number of participants but still have a low rate of award.

After obtaining the standardized score sequence , we only focus on the positive shift because we are more concerned about the positive mutation caused by the “Great Coach” Effect, that is, the performance improvement. In Eq (6), two parameters need to be defined: one is the reference value k, and the other is the dynamic threshold . The parameter k refers to the reference value of the allowable offset, which is used to measure whether the difference between the “current score” and the “historical baseline” is significant, and it determines whether the algorithm is more sensitive to “small offset” or “large offset”. The smaller k is, the more sensitive the algorithm is to a small offset. The larger the k is, the more attention is paid to large offsets. Parameter refers to the control threshold, which determines when to “trigger the alarm”. When the cumulative sum exceeds , an offset is considered to be detected. directly affects the false positive rate and detection delay of the algorithm. The larger is, the lower the false positive rate is, but the detection delay increases. Smaller means faster response but more false positives. The calculation equation is as follows:

(6)

where and represent the standard deviation and mean of the historical average number of participants, respectively. is defined as a cumulative sum, which accumulates and increases when the standardized score exceeds the historical fluctuation (). For high-participation projects ( is too large), the threshold should be appropriately raised to avoid frequent false positives due to data fluctuations. As mentioned earlier, if the cumulative sum of exceeds , then we consider this time to “trigger the alarm”, that is, this moment belongs to the mutation point. We retained these mutation points, generated mutation sequences, and plotted the CUSUM cumulative sum plot.

Double stability test

Since the presence of a flash point simply represents a “surge” in the country’s medal count during this Olympic cycle, the cause could be the “Great Coach” Effect, but it could also be a complex mix of other reasons, such as home-field advantage, rule changes, or structural changes. For example, before the 2016 Rio Olympic Games, the Thai women’s weightlifting team was not a traditionally strong team, and its all-time best result was 1 silver and 1 bronze at the 2004 Athens Olympic Games. However, the 2016 Rio Olympics suddenly won two gold and one silver, but then the doping problem was exposed, and the performance quickly collapsed, because of special circumstances (such as opponent mistakes, and individual athletes performing above average performance) caused by the performance of the increase, rather than the long-term influence of the coach. Therefore, it is necessary to ensure that the mutation point detected is not just a single event but a sustained increase.

Therefore, we also need to carry out the “Double Stability Test” for the mutation sequence; only after screening the “mutation suspect points” in the mutation sequence can the research be meaningful. The test can be divided into two steps; the first step is to consider the significance of the medal score improvement, and the second step is to test the stability of the number of participants.

First, check whether the average score of the two consecutive Olympic Games after the mutation point is significantly higher than the historical level. If the average score of the next two periods after the mutation point is significantly higher than the historical mean (more than c times the standard deviation), you can basically filter out the case for a single period of accidental improvement. In Eqs (7)–(8), is the window length, here we define it as 8 years (two consecutive Olympic Games), is defined as the historical average score before the mutation point, is defined as the historical standard deviation before the mutation point, and c is defined as the significance coefficient.

(7)

Secondly, check the stability of the number of participants to prevent a sharp decline in the number of participants resulting in an inflated score (the denominator is reduced to enlarge the score). The average number of participants in the following 8 years is the historical mean – d times the standard deviation. Like c, d is defined as a coefficient of significance necessary to eliminate interference from “great strategies” (such as sending only top players) and to ensure that performance gains are based on a stable talent base.

(8)

The two significant parameters in the equation are improved by using a grid search. Parameter c is used to determine whether the improvement or change in performance is statistically significant. Parameter d is to limit inflated or distorted scores due to fluctuations in the number of participants (denominator changes). Through the classification of project categories, most projects are determined to be 1.5 and 0.75 according to the experience value c and d(0). However, due to the high uncertainty or insufficient data volume of some projects, such as high-fluctuation projects or emerging projects Stefani et al. [26], we have made appropriate parameter adjustments according to the gradient boosting tree, as shown in Table 2. Only when and meet the conditions of PASS at the same time, we believe that it may be caused by the “Great Coach” Effect, so as to screen out other “mutation points”.

Dynamic effect evaluation (DID extension)

After selecting the appropriate “mutation point” in the mutation sequence, it is necessary to quantify the “Great Coach” Effect at this moment. That is, to isolate the net effect of “Great Coach” on the improvement of Olympic performance while controlling for other confounding factors (such as rule adjustment, host advantage, doping incidents, etc.). The traditional DID model [19] assumes that the intervention time of all individuals is the same, but in the actual Olympic cycle, there is heterogeneity in the appointment time of the coach (that is, the policy intervention time point) (for example, the Brazilian skateboarding coach took office in 2016, while the Japanese gymnastics coach intervened in 2021), which may lead to the estimation bias of the traditional DID. Therefore, in this study, the Bacon Decomposition Method [21] was used to stratify the heterogeneous intervention time to ensure that the model could capture dynamic effects.

First, it is assumed that in the absence of the intervention, the performance of the treatment group (the countries receiving the “Great Coach” intervention-project portfolio) and the control group (the countries receiving no intervention or the intervention time later than the treatment group – project portfolio) has the same trend over time, that is, the parallel trend hypothesis. The name effect can be captured by the difference between “performance improvement in the treatment group after the intervention” and “natural change in the control group during the same period.”

The sample is divided into multiple subgroups (layers) according to the intervention time of different country-project portfolios (i.e., time ), and each subgroup corresponds to a corresponding intervention time. Then, for each layer, the differential differences of all possible controls (later intervention or no intervention) were calculated, and finally, the effects of the layers were aggregated to quantify the “Great Coach” Effect.

Specifically, to ensure that the control group of each layer is comparable to the treatment group before intervention (i.e., the parallel trend hypothesis) is the control group of each layer, two basic conditions need to be met. First, the intervention time is later than the current layer (); Second, if a country has not been affected by the “Great Coach”, it needs to meet the condition . To ensure that the data for the treatment and control groups satisfy the parallel trends assumption, we need to conduct trend analysis on the data selected by CUSUM. To this end, we filter out data points that fall outside a predefined threshold range. If the trend deviations between the treatment and control group data are consistent (i.e., remain within the specified threshold range), the data are considered to satisfy the parallel trends assumption; otherwise, they are deemed unsatisfied, and the corresponding data are filtered out.

In view of the heterogeneity of coach tenure time , the Bacon decomposition method stratifies the mutation sequences according to . The treatment group and the control group were constructed in each layer, and the subscripts were defined as g and , respectively. However, the scoring situation in the treatment group is still complex, and in order to obtain a “clean” coaching effect, regression control is required for each intervention group, as shown in Eq (9):

(9)

where is used to control subjective factors such as host effect or rule adjustment; is used to control the improvement and change of the country’s comprehensive strength, such as the overall performance fluctuation of the Olympic cycle and the improvement of training level brought about by technological progress; is used to control the random error term, which contains unobserved perturbations. and represent the host country effect and GDP impact in treatment group g. The host country effect is explained by binary variables, and the GDP impact is calculated by the standard value Z obtained by Eq (2).

In Eq (10), and , in turn, represent the mean scores of the treatment group at layer g after intervention and before intervention, and are the mean values of the corresponding control group at layer , and the dynamic treatment effects at each layer are defined as and calculated by differential-difference. The effects of each layer were aggregated into the total effect weighted by the sample size, the sample weight reflected the contribution of different intervention queues and eliminated the bias of heterogeneous intervention time, and N was the sample size of the total treatment group.

(10)

After stratified weighted aggregation, the total effect can more fairly reflect the contribution of the intervention time layer and eliminate the influence of uneven sample distribution. The confidence interval CI was used to quantify the statistical uncertainty of the total effect, indicating that the total effect was significantly non-zero at 95% confidence level, that is, the coach intervention had a statistically significant effect.

Half-Life estimation

The “Great Coach” Effect may exist only in the current period, or it may last for a long time. Therefore, in order to capture the persistence and decay law of the “Great Coach” Effect, it is necessary to estimate the half-life of this effect.

Assuming that the Great Coach intervention effect decays exponentially over time, the dynamic effect can be expressed by the following equation, with defined as the time it takes for the Great Coach effect to decay to 50% of the initial value.

(11)

is defined as the initial effect of the intervention (i.e., the total effect value estimated by dynamic DID), λ is the decay rate (), the rate of decay of the counter-effect, and is the period after the intervention (in years). Here, λ represents the attenuation rate, which was obtained by fitting the effect sequence of 20 top coaches from 1980 to 2020 using the nonlinear least squares method. For the systematic training system (such as Zhou Jihong’s coaching of the Chinese diving team), , corresponding to a half-life of over 20 years; for the technology-driven intervention (such as the AI optimization of Japanese judo), , with a half-life of only 5.3 years, reflecting the difference in the long-term influence of system construction and technical tools. Then, based on the dynamic DID output year-by-year effect sequence , the nonlinear least square method of the optimization algorithm was used to fit the attenuation parameters λ and minimize the residual sum of squares between the observed effect and the model prediction, as shown in Eq (12).

(12)

Capture of performance mutations via the CUSUM algorithm

In order to elaborate on the accumulation and capture the fluctuation trend of medals in a country-event, the mutation accumulation and curve based on the output of the CUSUM algorithm are shown in Fig 2. In the United States Gymnastics CUSUM chart, for example, in the early 1980s, the blue line (performance fluctuation line) significantly exceeded the positive threshold for the first time, indicating that this is a “mutation suspect point”. It turned out that Bela Karolyi had moved to the United States in 1981 and led the U.S. gymnastics team to five medals (including one gold) at the 1984 Los Angeles Olympics, breaking the Eastern European monopoly. The reason for this is the strict training system and disciplined management it introduced, which became the turning point of the rise of American gymnastics.

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Fig 2. CUSUM chart for American women’s gymnastics.

https://doi.org/10.1371/journal.pone.0334635.g002

The blue line broke the threshold twice before and after 2000, and the slope increased further. Vela’s wife, Marta Karolyi, took over as national team coordinator in 2001 and introduced a system of “national training camps” to unify technical standards and selection processes. At the 2004 Athens Olympics, the U.S. women’s gymnastics team won nine medals (including two golds), ushering in an era of dominance. Around 2020 (2016–2020), the blue line remained high, indicating continued strengthening of the effect. Marta’s system produced superstars like Simone Biles, and the U.S. gymnastics team swept 12 medals (including four golds) at the 2016 Rio Olympics, continuing its dominance at the 2020 Tokyo Games.

Since the value of k was the one we previously tentatively set, it is now necessary to verify it.We tested alternative values of and , and redrew Fig 2 to observe the following:When changing k from to , the blue line (representing) breached the threshold earlier and more frequently, resulting in a steeper curve and an increased number of detected change points. The red line (representing) showed more pronounced negative accumulation, increasing negative change points. Overall, the plot exhibited greater fluctuations, with higher sensitivity but a potential increase in false-positive rates. When changing from to , the blue line breached the threshold later, with fewer detected change points and a smoother curve. The red line showed reduced negative accumulation, decreasing negative change points. The plot was more stable, reducing false positives but potentially missing smaller effects.It can be observed that is indeed the most appropriate value. By listing the mutation sequence points to screen out the “great coaching effect” detected by the “model cusum,” and then comparing it with the real data, the accuracy of the data can be obtained. After calculation, the accuracy rate of the mutation point reached 91.8% and remained basically stable at 92%.

The above analysis shows that CUSUM accumulation and graph can help us capture a certain number of “abrupt points”, but there are still some mistakes in judgment. Using 1000 bootstrap resamplings of historical data, the CUSUM algorithm achieved a change-point detection accuracy of 92% (SE = 0.9%, 95% CI: [90.1%, 93.7%]). A binomial test confirmed statistical significance compared to a 50% baseline (p < 0.001), indicating robust detection of coach-induced performance mutations, but there are also certain error situations. Therefore, it is necessary to carry out a “double robust test” and save all “mutation suspect points” to generate a mutation sequence.

Screening of effective mutation points and exclusion of pseudo-signals

By selecting COMAP data from some countries, mutation sequences were calculated separately according to Eqs (7)–(8), and the results are shown in Table 3. The four countries in the table, ROU, USA, AUS, and GBR, have passed the double robustness test and are highly likely to be affected by the “Great Coach”. The failure of THA, ITA, JPN, and other countries to test either the significance of the improvement in medal scores or the stability of the number of participants is considered to be outside the scope of the “Great Coach” Effect. And it turns out to be true.

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Table 3. Double test mechanism for detecting change points.

https://doi.org/10.1371/journal.pone.0334635.t003

The jump in performance in THA, ITA, JPN, and other countries in a given period is not driven systematically by the “Great Coach” Effect, but by external resource input, individual heroism, technical tool optimization, or other short-term influences. Specifically, the results of Thai women’s weightlifting (2004–2008) mainly relied on the “Chinese - style management” and high-intensity training system introduced by Chinese coach Zhang Xinmin, but the lack of localized youth training led to the subsequent talent gap (2012–2016 cycle Post 60%). The success of the star player Bhapavadi stems more from the breakthrough driven by personal will and economic pressure than from systematic training Chu et al. [27]. Although the Italian fencing team (2004–2008) achieved a short-term breakthrough with the star player Garrozzo by relying on the historical tactical tradition (such as tough close-in attack style), the insufficient investment in scientific research (only 5% of IJSPP papers) and the deficiencies in the stability of the team (Post 70%) exposed the limitations of relying on the inheritance of experience Zadorozhna et al. [28]. In Japan Judo (2008–2012), although the technology was enabled (40,000 video analysis and AI tactical optimization) to improve on-stage efficiency, over-reliance on technology tools led to a structural imbalance of the program (over-concentration of specific weight players), resulting in substandard athlete stability. The issue of sustainability was not addressed until the 2016 cycle when the coaching team was restructured and the youth academy was improved (Boguszewski et al. [6]).

These cases further demonstrate that external technology transfer, individual accidental breakthroughs, or instrumentalized improvements can bring short-term leaps in performance, but cannot replace the systemic innovation and talent supply synergy required by the “Great Coach” Effect, which requires the dual guarantee of scientific integration and echelon stability () like the British cycling team.

After passing the “Double Difference Test”, we need to specifically quantify the degree of improvement of this “Great Coach” Effect on the athletes’ ability to win awards and provide a scientific basis for the strategic decision-making of the later Olympic Games.

Quantification and stratified analysis of coaches’ net effects

Since the traditional difference model does not take into account the difference in intervention time, this study stratified heterogeneous intervention time through Bacon decomposition to ensure that the model can capture dynamic policy effects. Other factors were separated by Eqs (9)–(10), and the calculation results are shown in Table 4.

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Table 4. Allocation and impact analysis of great coaches in olympic performance.

https://doi.org/10.1371/journal.pone.0334635.t004

The following table provides a brief explanation of the data. From 1976 to 1984, Coach Béla Károlyi led the Romanian gymnastics team through intensive training, winning a lot of awards and fame. From 1991–2000, he coached the USA Gymnastics team and successfully elevated USA Gymnastics from a second-rate team to the top of the medal table by adapting to the North American market environment, namely commercial star formation (such as Kerri Strug), difficulty point rule games and family network continuation (Marta Karoly succession). In particular, the United States women’s team won the first team gold medal (+3 individual medals) at the 1996 Atlanta Olympics, and the total number of medals increased from 2 in 1992–7. Over several Olympic cycles, the U.S. Gymnastics team has generally won 4.3 medals, which is the A in the table, with a confidence interval of . In summary, it is basically possible to quantify the “Great Coach” effect brought by Coach Béla Károlyi.

Two aspects of information can be obtained from the data in Table 4. First, the intervention of top coaches has a significant impact on improving the performance of athletes, and the effect varies in different countries and periods. The other aspect is that the impact of a Great Coach will not only be the direct contribution of the individual coach but also the combined result of systemic intervention. Lang Ping’s case demonstrates the systematic nature of the effect: During her tenure as the head coach of the Chinese women’s volleyball team from 2013 to 2016, the Overall Effect was 4.1 (Standard Error = 0.3, p < 0.001). Specifically, China’s women’s volleyball team won 2 silver medals at the 2012 London Olympics, rose to 1 gold medal at the 2016 Rio Olympics, and the stability of the participation scale reached 92% (far exceeding the threshold), indicating that her systematic intervention of ‘team reconstruction + tactical innovation’ rather than relying on individual star athletes was effective. This is consistent with the statistical significance of the effect values in Table 4.

In other words, when a country imports one or more Great Coaches from other countries, the coaches not only act as technical mentors, but their role may also include the introduction of advanced training methods, team building, and so on. Variable reflects the overall improvement of a team or sport rather than the performance of individual athletes. For example, after the introduction of coach Dave Brailsford, France won medals in the men’s team sprint, men’s Kelsey, and men’s all-around at the 2020 Summer Olympics in Tokyo, Japan, winning a total of three more medals (including one gold) than the previous Olympics. This suggests that the DID effect size reflects the net effect of such a systematic intervention rather than the role of a single factor.

In view of the above analysis, we combined the analysis of existing data sets, focusing on the results of the Paris 2024 Olympic Games and the performance of previous Olympic Games. Table 5 shows three countries and the sports that should be considered for investment, as well as the possible impact analysis after the introduction of “Great Coach”. For the 2028 Los Angeles Olympic Games in the United States to make an improved forecast of the number of awards, the table in the gold/silver/bronze medals is in order with G/S/B abbreviations. Taking Brazilian swimming as an example, hiring Don Talbot as coach is expected to yield an intervention effect of 5.5 standard units (Overall-Effect = 5.5, 95% CI: 4.7–6.3). This result was derived using the CPD-D³ model and integrated with Don Talbot’s historical coaching experience. Reviewing Coach Don Talbot’s track record, the swimming teams he led consistently delivered outstanding performances at each Olympic Games. For instance, the Canadian team improved from 8 medals (0G + 2S + 6B) at the 1976 Montreal Olympics to 10 medals (4G + 3S + 3B) at the 1984 Los Angeles Olympics. Similarly, the Australian team progressed from 3 medals (1G + 1S + 1B) at the 1988 Seoul Olympics to 18 medals (5G + 9S + 4B) at the 2000 Sydney Olympics. Therefore, we conclude that if Brazil hires Don Talbot as the swimming team coach, it is expected to bring an improvement of 2 gold, 2 silver, and 1 bronze medal. It should be noted that this estimate is based on the Overall-Effect. After calculating from the Overall-Effect results, an improvement of 2G + 2S + 1B represents one possible outcome.

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Table 5. Allocation and impact analysis of great coaches on Olympic medal predictions.

https://doi.org/10.1371/journal.pone.0334635.t005

Taking Table 4 as an example: the Overall_Effect from Table 4 used as the effect size; the noise level determined by comparing the predicted medal counts with the actual medal counts; and the power analysis performed using a t-distribution, yielding the effect size.The final effect sizes of the power analysis are presented in Table 6. According to the results, the minimum power value is 80.6%, which is greater than 80%. Indicating that, with the sample size and the results calculated by the model, more than 80% confidence to demonstrate that the results are associated with the coach effect.

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Table 6. Power analysis of the final effect sizes.

https://doi.org/10.1371/journal.pone.0334635.t006

Half-life analysis and differential impacts of coach types

In the stage of capturing the persistence and decay law of the “Great Coach” effect, we conducted analysis through the Eq (10) half-life equation, as shown in Fig 3. The data in the figure shows that there is a huge gap in the half-lives of different coaches; for example, the half-life of coach Zhou Jihong from China is more than 20 years, while the half-life of a coach in high-tech driven programs is relatively short. The effect of Zhou Jihong’s systematic training system for the Chinese diving team can be directly demonstrated by the original data: Before she took over in 2000 (at the 1996 Atlanta Olympics), the Chinese diving team won 3 gold medals and 1 silver medals; after her appointment (at the 2000 Sydney Olympic), the number of medals rose to 5 gold and 5 silver, and at the 2004 Athens Olympics, it remained at 6 gold and 2 silver. At the 2008 Beijing Olympic, they still won 7 gold and 1 silver. This data shows that the effect is not a short-term fluctuation – the average number of medals during her tenure increased by 4, and in the simulation prediction after 10 years of her coaching (in 2016), the effect still maintained 100% of the initial value, with a half-life of over 20 years, confirming the long-term sustainability of the systematic system. The shaded part of the figure is the statistical uncertainty used to quantify the total effect, known as the confidence interval.

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Fig 3. Olympic coaches effect half-life.

https://doi.org/10.1371/journal.pone.0334635.g003

Finally, this study evaluated the accuracy of the detection of coaching effect mutations. A confusion matrix system was adopted, and the model was validated using the time-split cross-validation method. The construction of coaching replacement events was based on the cross-validation of the Olympipedia coaching database and the “International Sports Coaches” literature by Kim and Tak (2024). In the confusion matrix, the rows represent the classification we made based on the actual situation. The first row represents the real system-type coaching type, and the second row represents the real technology-driven coaching type. The columns represent the classification based on the calculated half-life according to the model. The first column represents the system-type coaching type with a coaching effect half-life of more than 10 years, and the second column represents the technology-driven coaching type with a coaching effect half-life of less than 10 years. The elements on the diagonal represent the number of correctly classified samples, while the off-diagonal elements reflect the model’s misjudgments.

(14)

where (FP) is the number of Technology-Driven Coach or non-coach effects incorrectly classified as Systematic Coach effects, and (TN) is the number of Technology-Driven Coach or non-coach effects correctly classified. The computed FPR of approximately 0.111 aligns with the model’s objective of maintaining a low false positive rate, reflecting the CPD model’s ability to effectively filter short-term accidental fluctuations through the dual robustness testing mechanism (significance testing with (c = 1.5) and stability test of participation size with (d = 0.75). To ensure the accuracy of the coach influence assessment (e.g., average effect size of 4.5 standardized units), the confusion matrix is shown in Fig 4.

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Fig 4. Confusion matrix for discriminating coaching effects.

https://doi.org/10.1371/journal.pone.0334635.g004

Based on the confusion matrix in Fig 4. Based on the confusion matrix in Fig 4, classification performance metrics are: Precision = 0.903 (SE = 0.038, 95% CI: [0.829, 0.977]), Recall = 0.789 (SE = 0.048, 95% CI: [0.695, 0.883]), F1 Score = 0.842 (SE = 0.032, 95% CI: [0.779, 0.905]), and Cohen’s Kappa = 0.70 (SE: Uncalculable, 95% CI: 0.020, 95% CI: [0.667, 0.745]). A binomial test on the overall classification accuracy (92%) confirms that the CPD model significantly outperforms a random classifier (p < 0.001), demonstrating robust discrimination of coaching effects. These metrics verify the CPD model’s ability to distinguish the effects of different types of coaches.

To prevent overfitting, we employ a time-based cross-validation approach to evaluate the fitted curves. Taking “China-fencing” (short-term training effects) and “Jamaica-athletics” (long-term training effects) as examples, we first train the model using data from the 20 years prior to the coach’s tenure, and then use the fitted curve to predict medal performance in subsequent Olympic cycles. The predicted curve is then compared with the actual performance trajectory. Subsequently, we extend the training window forward by one Olympic cycle and repeat the process. Finally, the predicted and actual trajectories are plotted together for comparison. The results are shown in Fig 5.

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Fig 5. Comparison of long-term and short-term coaching effects.

(a) China-Fencing. (b) Jamaica-Athletics.

https://doi.org/10.1371/journal.pone.0334635.g005

Refer to the images in Fig 5, specifically subfigures Fig 5a and Fig 5b respectively representing the predicted and actual fluctuation curves for China – Fencing (short-term coach) and Jamaica – Athletics (long-term coach).

Through the study of Fig 5a and Fig 5b, it can be concluded that the model has a slightly higher capability in predicting the impact generated by long-term coaches compared to short-term coaches. Furthermore, when trained with different training sets, the deviation between the predicted results and actual values remains within a small range, indicating that the model possesses strong resistance to overfitting.