Z-scores-based methods and their application to biological monitoring: An extended analysis of professional soccer players and cyclists athletes

The increase in the collection of biological data allows for the individual and longitudinal monitoring of hematological or urine biomarkers. However, identifying abnormal behavior in these biological sequences is not trivial. Moreover, the complexity of the biological data (correlation between biomarkers, seasonal effects, etc.) is also an issue. Z-score methods can help assess the abnormality in these longitudinal sequences while capturing some features of the biological complexity. This work details a statistical framework for handling biological sequences using three custom Z-score methods in the intra-individual variability scope. These methods can detect abnormal samples in the longitudinal sequences with respect to the seasonality, chronological time or correlation between biomarkers. One of these methods is an extension of one custom Z-score method to the Gaussian linear model, which allows for including additional variables in the model design. We illustrate the use of the framework on the longitudinal data of 3,936 professional soccer players (5 biomarkers) and 1,683 amateur or professional cyclists (10 biomarkers). The results show that a particular Z-score method, designed to detect a change in a series of consecutive observations, measured a high proportion of abnormal values (more than three times the false positive rate) in the ferritin and IGF1 biomarkers for both data sets. The proposed framework and methods could be applied in other contexts, such as the clinical patient follow-up in monitoring abnormal values of biological markers. The methods are flexible enough to include more complicated biological features, which can be directly incorporated into the model design.

💡 Research Summary

The paper presents a comprehensive statistical framework for monitoring longitudinal biological markers at the individual level, focusing on three custom Z‑score methods designed to capture intra‑individual variability. After describing two extensive datasets—3,936 professional French soccer players (five hematological biomarkers measured semi‑annually from 2006 to 2019) and 1,683 amateur/professional cyclists (ten biomarkers measured roughly every six days between 2003 and 2014)—the authors address the critical preprocessing step of achieving approximate normality. For each biomarker they evaluate a suite of deterministic transformations (identity, m‑th root, logarithm, Lambert W, and several Box‑Cox parameters), apply the Shapiro‑Wilk test to every individual series with at least four observations, and then use a Kolmogorov‑Smirnov test on the collection of p‑values to select the transformation that yields the most uniform distribution of p‑values. This pragmatic approach ensures that the subsequent Z‑score calculations rest on data that are as close to Gaussian as possible.

Four Z‑score statistics are defined. T(0) is the classic single‑observation Z‑score, comparing the newest measurement to the empirical mean and variance of all previous values; under the null hypothesis it follows a Student‑t distribution with n‑2 degrees of freedom. T(1) extends T(0) by taking the maximum standardized deviation over all observations in the series, thereby detecting any single outlier. T(2) further generalizes to detect abnormal subsequences: it scans all possible contiguous intervals I, computes the standardized difference between the mean of I and the mean of its complement, and selects the interval with the largest statistic. Because the statistic’s distribution depends on interval length, critical values are obtained via Monte‑Carlo simulation. T(3) is a multivariate extension that incorporates the correlation matrix of multiple biomarkers, using a Mahalanobis‑type distance for each observation after removing that observation from the sample. Finally, the DevianLM package embeds the T(1) concept within a Gaussian linear model, allowing additional covariates (season, age, training load) to be modeled through a design matrix M.

Applying these methods to the two athlete cohorts, the authors find that the subsequence detector T(2) identifies an unusually high proportion of abnormal values for ferritin and insulin‑like growth factor‑1 (IGF‑1). In both datasets the abnormal detection rate exceeds three times the expected false‑positive rate, suggesting that consecutive deviations in these markers are biologically meaningful—potentially reflecting iron supplementation cycles or hormonal fluctuations linked to training periods. The simpler T(0) and T(1) statistics yield more conservative detection rates, while the multivariate T(3) does not dramatically increase abnormal calls despite accounting for inter‑biomarker correlations.

The study highlights several strengths: interpretability of white‑box Z‑score methods, flexibility to incorporate seasonality or other covariates via the linear‑model extension, and applicability beyond sport to clinical monitoring of patients. Limitations include the requirement for at least four observations per individual, reliance on the iid normality assumption (which may be violated in real‑world data), and the potential for inflated false‑positives when the underlying distribution deviates from normality. The authors suggest future work could explore hierarchical Bayesian models or non‑Gaussian approaches to further improve robustness.

In summary, the paper delivers a well‑structured, statistically rigorous toolkit for longitudinal biomarker surveillance, demonstrates its practical utility on large athlete datasets, and opens avenues for broader adoption in personalized medicine and anti‑doping programs.