Use of a nested bivariate binomial distribution in presence of covariates applied to a Brazilian soccer league data

Abstract

This study introduces a nested bivariate binomial (BVB) regression model under a Bayesian logistic regression framework to evaluate offensive efficiency in football, with application to the 2023 Brazilian Série A league. The BVB model characterizes offensive performance as a two-stage process: (i) the number of shots on target given total shot attempts, and (ii) the number of goals scored given shots on target. Financial covariates—namely, club investment during the 2023 season and estimated market value—are incorporated at both stages to assess their association with team efficiency. The analysis is based on data from 20 teams across the first 12 rounds of the season, mitigating mid-season variability and transfer effects. Results indicate that higher market value is positively associated with scoring efficiency, whereas recent financial investment does not exhibit a significant effect. These findings constitute a primary investigation into the influence of financial variables on offensive performance in football. By introducing the BVB model within a sports analytics framework, the study provides preliminary evidence of the explanatory and predictive utility of financial indicators—particularly market value—in modeling sporting outcomes.

Keywords

Association football financial investment goal kicks offensive efficiency sport analytics

Introduction

Football (also known as soccer) stands as the most widely followed sport globally, engaging billions of fans across diverse cultural and geographic contexts. Its global prominence is not only evident in its vast audience but also in the extensive economic systems that underpin the professional game. Modern football clubs operate as multifaceted enterprises, allocating substantial resources to player acquisitions, salaries, training infrastructure, and strategic planning. As a result, the sport increasingly reflects a complex interplay between financial investment and on-field performance, with club success often contingent upon effective financial management and resource allocation.

According to the McKinsey & Company report “The Value Pitch: The Importance of Team Value Management”, football clubs should regard their player rosters as portfolios of assets rather than mere cost centers. The report’s analysis of the 69 most valuable European clubs over five years demonstrates that activities such as developing contracted players, promoting youth talent, and optimizing player trading contributed approximately €13 billion of the total €17.2 billion increase in team market value. Importantly, this internally generated value frequently exceeded that derived from net investment in new acquisitions. For more than 70% of these clubs, including prominent teams like Liverpool and Real Madrid, effective asset management was the primary driver of value creation.

Building upon this strategic perspective, a substantial and growing body of academic research has examined the intricate relationships between financial inputs and sporting performance in football. For instance, Carmichael et al.¹ investigated the English Premier League and found that sustained competitiveness is strongly linked to prudent financial management, particularly the alignment between wage expenditure and revenue generation. In parallel, Dimitropoulos and Scafarto² analyzed the effects of UEFA’s Financial Fair Play regulations on Italian football, illustrating how budgetary discipline influences both club financial health and competitive outcomes.

Complementary to these studies,³ proposed a composite evaluation framework that integrates business, financial, and sporting dimensions to assess French football clubs’ overall performance, advocating for a multidimensional approach to understanding success. Similarly, Guzmán and Morrow⁴ employed data envelopment analysis to evaluate technical efficiency among English Premier League teams, revealing key factors that affect productivity and resource utilization.

Further studies have expanded the analysis to include governance and organizational factors. For example, Ruta et al.⁵ demonstrated that institutional governance structures critically influence club performance by shaping strategic decision-making processes. Meanwhile, Sakinc et al.⁶ examined the phenomenon of soft budget constraints in football, cautioning that expectations of external financial support may induce strategic mismanagement and undermine sustainability. Wicker et al.⁷ emphasized the complexity of linking monetary incentives to player effort and productivity, reinforcing the nuanced relationship between financial inputs and output quality. Moreover, Wilson et al.⁸ showed that ownership models in the English Premier League significantly impact sporting outcomes, underscoring the role of governance and stakeholder engagement in performance.

From an applied forecasting and betting market perspective,⁹ proposed a novel and profitable predictive model for the over/under betting market in professional football. The study focuses on the over/under 2.5 goals market, a popular form of total goals betting, and evaluates the profitability of statistical forecasts when applied to real betting odds. Utilizing historical match data from various European leagues, the author develops a logistic regression model that incorporates a range of covariates—including recent team performance, home advantage, and head-to-head statistics—to estimate the probability that a match will exceed 2.5 total goals.

Based on the context above, this study applies a nested bivariate binomial (BVB) regression model to data from the 2023 season of the Brazilian Série A league. Originally introduced by Crowder and Sweeting,¹⁰ the BVB model is well-suited for representing two hierarchically dependent binomial processes, thereby allowing the modeling of sequential outcomes commonly observed in sports analytics. In the context of offensive efficiency, we define a two-stage process: (i) the number of shots on target (i.e., goal attempts with accurate direction) given the total number of shots taken, and (ii) the number of goals scored given the number of shots on target.

This study also extends the BVB framework by incorporating financial covariates at both stages to examine whether club-level economic factors are associated with offensive efficiency. Specifically, we include two financial indicators: (1) total investment by the club during the 2023 season, measured in millions of Brazilian reais, and (2) the estimated market value of the club, measured in millions of euros. The total investment reflects short-term financial commitments related to player acquisitions and infrastructure, whereas market value captures longer-term assessments of player quality, squad stability, and perceived future potential. However, it is important to acknowledge that these financial covariates do not fully conform to the conventional definition of covariates as pre-treatment or exogenous variables observed prior to the outcomes of interest. In the present context, both financial indicators are measured contemporaneously with the performance data and may therefore be influenced by—or evolve in parallel with—the very outcomes they are intended to explain. Accordingly, the model should be interpreted primarily as an explanatory or retrospective framework, rather than one designed for predictive inference. The inclusion of these variables aims to assess whether cross-club financial disparities are systematically associated with variation in offensive efficiency during the early stages of the season.

The contributions of this study are threefold. First, it introduces the application of the Nested Bivariate Binomial (BVB) model to capture team-level nested proportions in a sports analytics context, specifically modeling sequential offensive actions in football. Second, the study extends this framework by incorporating financial covariates at both stages of the model to examine the association between club-level economic factors and offensive efficiency. Third, it provides a comparative evaluation of model performance with and without financial covariates, using Bayesian model selection criteria to assess the explanatory value added by economic variables. To the best of our knowledge, this is the first application of a nested BVB model incorporating covariates within soccer analytics, providing novel insights into the interplay between financial investment and on-field outcomes.

This paper is organized as follows: Section “Materials and Methods” presents the theoretical background of the BVB model, detailing its formulation, assumptions, and mathematical properties. Section “Inference Methods” details the Bayesian inferential framework employed for parameter estimation, including prior specification, computational algorithms, and model validation techniques used to assess goodness-of-fit. Section “Results” presents the application of the methodology to data from the Brazilian Série A league’s 2023 season and discusses the interpretation of the results. Finally, Section “Concluding Remarks” closes the manuscript.

Materials and methods

Study data

The analysis is based on data from the Brazilian Série A football league, comprising performance and financial indicators for 20 teams over the first 12 rounds of the 2023 season (Table 1). This early-season window was intentionally selected to ensure consistency in the number of matches played across all clubs, thereby minimizing potential biases arising from unequal scheduling or evolving team circumstances. By focusing on this initial phase, the study seeks to assess whether financial disparities among clubs are already associated with differences in offensive efficiency, prior to the full realization of cumulative performance outcomes and the potential endogeneity introduced by league standings and results-based adjustments. Furthermore, this restriction helps to mitigate the influence of mid-season confounders—such as player transfers, tactical reconfigurations, injuries, fixture congestion, and accumulated fatigue—that could obscure or distort the underlying relationships between financial structure and on-field performance over the course of a full season.

Table 1.

Dataset related to the 20 soccer clubs in Brazilian series A league (2023).

Club ID	Club Name	Goal	Total kicks	Goal kicks	Inv. (2023) (in million of Reais)	Club value (in million of Euros)
1	América (MG)	13	167	50	5.2	26.65
2	Atlético Paranaense	16	172	52	0.0	91.55
3	Atlético Mineiro	15	170	62	8.8	84.75
4	Bahia	12	163	49	33.0	54.70
5	Botafogo (RJ)	20	163	54	0.0	64.50
6	Bragantino	21	191	71	26.9	83.50
7	Corinthians	10	151	48	4.3	86.40
8	Coritiba	9	145	54	20.5	37.45
9	Cruzeiro	14	184	58	20.1	38.20
10	Cuiabá	8	156	46	7.6	17.28
11	Flamengo	24	167	51	123.0	171.65
12	Fluminense	19	157	59	16.8	63.40
13	Fortaleza	15	174	61	8.0	38.55
14	Goiás	10	150	45	0.0	22.25
15	Grómio	21	176	68	53.0	52.80
16	Internacional	13	140	45	25.1	71.35
17	Palmeiras	21	196	61	0.0	149.75
18	Santos	11	137	47	25.2	77.75
19	São Paulo	17	177	61	32.1	81.05
20	Vasco da Gama	11	155	43	89.0	62.30

Model formulation

For the data analysis of Brazilian Serie A soccer league, we employed a nested bivariate binomial (BVB) distribution introduced by Crowder and Sweeting.¹⁰ Since this model allows the modeling of two hierarchically dependent binomial processes, we focused on a two-step process in offensive efficiency:

The number of goal kicks (i.e., shots on target) out of the total number of kicks attempted denoted by $Y_{1}$ .

The number of goals scored out of the number of goal kicks denoted by $Y_{2}$ .

In this way, defining the bivariate random vector

Y = (Y_{1}, Y_{2}

) with

Y_{1} ⩾ Y_{2}

, the probability mass function of the BVB model is given by,

P (Y_{1} = y_{1}, Y_{2} = y_{2}) = (\begin{matrix} n \\ y_{1} \end{matrix}) p^{y_{1}} (1 - p)^{n - y_{1}} (\begin{matrix} y_{1} \\ y_{2} \end{matrix}) q^{y_{2}} (1 - q)^{y_{1} - y_{2}},

(1)

where

y_{1} = 0, 1, 2, \dots, n

;

y_{2} = 0, 1, 2, \dots, y_{1}

; and

0 < p < 1

0 < q < 1

. The parameters, in this case, are defined as follows:

–

$n \in N$ : a fixed, known positive integer representing the number of independent Bernoulli trials at the first level of the nesting structure, i.e., the total number of shots.

–

$p \in (0, 1)$ : the success probability associated with each of the $n$ Bernoulli trials that generate the first component $Y_{1} \sim Bin (n, p)$ , i.e., the probability of a shot being on target.

–

$q \in (0, 1)$ : the conditional success probability governing the second component $Y_{2}$ , such that $Y_{2} ∣ Y_{1} = y_{1} \sim Bin (y_{1}, q),$ i.e., the probability of a shot on target resulting in a goal.

The moment generating function (MGF) of

Y

following a BVB model with parameters

n, p

and

q

is given by,

M_{Y} (t_{1}, t_{2}) = [p q e^{t_{1} + t_{2}} + p e^{t_{1}} (1 - q) + 1 - p]^{n},

(2)

which implies that the first- and second-order moments of

Y_{1}

and

Y_{2}

are given, respectively, by:

\begin{aligned} E (Y_{1}) & = n p, \\ Var (Y_{1}) & = n p (1 - p), \end{aligned}

(3)

and

\begin{aligned} E (Y_{2}) & = n p q, \\ Var (Y_{2}) & = n p q (1 - p q) + n p (1 - p) q^{2} . \end{aligned}

(4)

The dependence between

Y_{1}

and

Y_{2}

is not arbitrary, however it arises through a shared binomial hierarchy:

Y_{2}

is binomial conditional on

Y_{1}

, and both are linked through the joint distribution in Equation (1). This hierarchical structure ensures:

–

Marginally, $Y_{1}$ follows a binomial distribution with parameters $n$ and $p$ .

–

Conditionally on $Y_{1} = y_{1}$ , $Y_{2}$ follows a binomial distribution with parameters $y_{1}$ and $q$ .

In this way, from Equations (3) and (4), the correlation coefficient for the BVB is given by:

ρ (Y_{1}, Y_{2}) = \frac{\sqrt{(1 - p) q}}{\sqrt{(1 - q) + (1 - p) q}} .

(5)

To incorporate covariate information into the analysis, we extend this formulation to a regression framework, incorporating a set of common covariates into both levels of the nested BVB model. Specifically, let $x_{i}$ denote the vector of covariates for unit $i$ (e.g., team in a given season). The success probabilities $p_{i}$ and $q_{i}$ are linked to $x_{i}$ via logit functions as follows:

logit (p_{i}) = β_{0} + x_{i}^{⊤} β, logit (q_{i}) = α_{0} + x_{i}^{⊤} α,

(6)

where

p_{i}

is the probability that a shot is on target for unit

i

;

q_{i}

is the conditional probability that a shot on target results in a goal;

β

and

α

are distinct vectors of regression coefficients governing the effects of covariates on

p_{i}

and

q_{i}

, respectively; and

x_{i}

is the vector of covariates (i.e., total investment in 2023 and club market value) used in both stages. The two-stage hierarchical data structure is preserved as:

Y_{1 i} \sim Bin (n_{i}, p_{i}), Y_{2 i} ∣ Y_{1 i} = y_{1 i} \sim Bin (y_{1 i}, q_{i}),

(7)

with

Y_{1 i}

denoting the number of shots on target and

Y_{2 i}

the number of goals, out of

n_{i}

total shot attempts.

Inference methods

In this section, we describe the Bayesian inferential procedure in two stages. First, we present the inference for the model without covariates, which establishes a baseline specification and allows assessment of the model structure under minimal assumptions. Next, we extend the model to incorporate financial covariates through logistic regression formulations. All models are estimated using a fully Bayesian framework implemented via Markov Chain Monte Carlo (MCMC) methods, and model comparison is conducted through the Deviance Information Criterion (DIC).

Likelihood function

Given a dataset consisting of $m$ independent and exchangeable observations $(y_{1 i}, y_{2 i}, n_{i})$ for $i = 1, \dots, m$ , where $y_{1 i}$ is the number of shots on target (goal kicks), $y_{2 i}$ the number of goals scored, and $n_{i}$ the total number of shots attempted by team $i$ , the joint likelihood under the BVB model is given by:

L (p, q) \propto \prod_{i = 1}^{m} p^{y_{1 i}} (1 - p)^{n_{i} - y_{1 i}} q^{y_{2 i}} (1 - q)^{y_{1 i} - y_{2 i}} .

(8)

Taking the logarithm and maximizing with respect to

p

and

q

leads to the maximum likelihood estimators (MLEs), which are:

\hat{p} = \frac{\sum_{i = 1}^{m} y_{1 i}}{\sum_{i = 1}^{m} n_{i}}, \hat{q} = \frac{\sum_{i = 1}^{m} y_{2 i}}{\sum_{i = 1}^{m} y_{1 i}} .

(9)

These estimators represent the overall empirical proportions across all observations. Under standard regularity conditions and for large $m$ , the joint distribution of $(\hat{p}, \hat{q})$ is approximately bivariate normal, that is, $(\hat{p}, \hat{q}) \overset{a p p r o x}{\sim} N_{2} ((p, q), I_{0}^{- 1} (p, q))$ , where $I_{0} (p, q)$ denotes the Fisher information matrix given by:

I_{0} (\hat{p}, \hat{q}) = [\begin{matrix} \sum_{i = 1}^{m} (\frac{y_{1 i}}{p^{2}} + \frac{n_{i} - y_{1 i}}{(1 - p)^{2}}) & 0 \\ 0 & \sum_{i = 1}^{m} (\frac{y_{2 i}}{q^{2}} + \frac{y_{1 i} - y_{2 i}}{(1 - q)^{2}}) \end{matrix}] .

(10)

Thus, the asymptotic variances and covariance can be consistently estimated using the empirical plug-in approach, leading to:

var (\hat{p}) \approx \frac{\hat{p} (1 - \hat{p})}{\sum n_{i}}, var (\hat{q}) \approx \frac{\hat{q} (1 - \hat{q})}{\sum y_{1 i}}, cov (\hat{p}, \hat{q}) \approx 0.

(11)

Bayesian inference

In this section, we consider a fully Bayesian framework to estimate the parameters $(p, q)$ of the BVB model under a data structure composed of $m$ independent and exchangeable observational units, where each unit corresponds to a distinct team. For each team $i = 1, \dots, m$ , we observe paired binomial outcomes $(Y_{1 i}, Y_{2 i})$ with sample size $n_{i}$ . In the case without covariates, we begin by presenting two distinct prior specifications for the parameters, uniform priors and Jeffreys’ prior, as the following:

(i) Uniform Priors. Assuming independent uniform priors $p \sim Uniform (0, 1)$ and $q \sim Uniform (0, 1)$ , the joint posterior distribution of $p$ and $q$ given the data $y = {(y_{1 i}, y_{2 i})}_{i = 1}^{m}$ is proportional to:

\begin{aligned} π (p, q ∣ y) \propto & p^{\sum_{i = 1}^{m} y_{1 i}} (1 - p)^{\sum_{i = 1}^{m} (n_{i} - y_{1 i})} \\ q^{\sum_{i = 1}^{m} y_{2 i}} (1 - q)^{\sum_{i = 1}^{m} (y_{1 i} - y_{2 i})} . \end{aligned}

(12)

which corresponds to two independent Beta posteriors:

\begin{aligned} p ∣ y \sim & Beta (\sum_{i = 1}^{m} y_{1 i} + 1, \sum_{i = 1}^{m} (n_{i} - y_{1 i}) + 1), \\ q ∣ y \sim & Beta (\sum_{i = 1}^{m} y_{2 i} + 1, \sum_{i = 1}^{m} (y_{1 i} - y_{2 i}) + 1) . \end{aligned}

(ii) Jeffreys’ Prior. Using Jeffreys’ prior derived from the Fisher information matrix, the joint prior is given by:

π (p, q) \propto {[p (1 - p) q (1 - q)]}^{- 1 / 2},

(13)

leading to the joint posterior:

\begin{aligned} π (p, q ∣ y) \propto & p^{\sum y_{1 i} - 0.5} (1 - p)^{\sum (n_{i} - y_{1 i}) - 0.5} \\ q^{\sum y_{2 i} - 0.5} (1 - q)^{\sum (y_{1 i} - y_{2 i}) - 0.5}, \end{aligned}

(14)

which corresponds to:

\begin{aligned} p ∣ y \sim & Beta (\sum_{i = 1}^{m} y_{1 i} + 0.5, \sum_{i = 1}^{m} (n_{i} - y_{1 i}) + 0.5), \\ q ∣ y \sim & Beta (\sum_{i = 1}^{m} y_{2 i} + 0.5, \sum_{i = 1}^{m} (y_{1 i} - y_{2 i}) + 0.5) . \end{aligned}

For both prior choices, posterior summaries such as means, medians, credible intervals, and highest posterior density (HPD) regions can be derived analytically from the respective Beta distributions. Now, to incorporate covariate information into the BVB model, we extend the parameterization of $p_{i}$ and $q_{i}$ for each observational unit (team) $i = 1, \dots, m$ using logistic regression structures as specified in Equation (6). Thus, we adopted a fully Bayesian framework by assigning Gaussian priors independently to all regression parameters:

β_{j} \sim N (0, τ_{j}^{2}), α_{j} \sim N (0, τ_{j}^{2}), j = 0, 1, 2,

(15)

where

0

and

τ_{j}^{2}

are hyperparameters chosen to represent weakly informative prior knowledge for each covariate. Thus, given the non-conjugacy of the posterior distribution, inference is performed using a Metropolis-within-Gibbs algorithm implemented via the R2jags package¹¹ within the R statistical environment.¹² The MwG procedure was executed with three independent Markov Chain Monte Carlo (MCMC) chains, each comprising 100,000 iterations. A burn-in phase corresponding to 5% of the iterations was applied to ensure convergence and reduce the impact of initial values. To mitigate autocorrelation and improve sample independence, thinning was performed by retaining every 95th iteration, resulting in a final posterior sample of 1,000 draws per chain. Posterior inference, including estimation of credible intervals and other summaries, was then based on the combined sample across all chains.

Model discrimination criterion

As goodness-of-fit criteria, we considered the Deviance Information Criterion (DIC), which is suitable for comparing models estimated via posterior simulation.¹³ The DIC is used here to evaluate competing Bayesian specifications—namely, models without covariates (using distinct prior choices) and models incorporating financial covariates through logistic regression. In this case, let $θ$ denote the full set of parameters, including either $(p, q)$ in the model without covariates, or the regression coefficients $(β_{j}, α_{j}, j = 0, 1, 2)$ in the model with covariates. The deviance function is defined as $D (θ) = - 2 \log L (θ)$ , where $L (θ)$ denotes the likelihood function specified in Equation (8). The DIC criterion is defined as

DIC = E_{θ} [D (θ)] + p_{D} = \bar{D} + p_{D},

where

\bar{D}

is the posterior mean deviance, and

p_{D}

is the effective number of parameters given by

p_{D} = \bar{D} - D (\hat{θ})

, with

\hat{θ}

denoting the posterior mean. Since the posterior expectation

E_{θ} [D (θ)]

cannot be computed analytically, we approximate it via Monte Carlo integration using posterior samples

{θ^{(i)}}_{i = 1}^{B}

obtained from the MCMC algorithm:

\hat{\bar{D}} = - \frac{2}{B} \sum_{i = 1}^{B} \log L (θ^{(i)}) .

Therefore, the estimated DIC is given by

\hat{DIC} = 2 \hat{\bar{D}} - D (\hat{θ})

Results

To illustrate the proposed methodology, we have adopted the Brazilian Serie A soccer league data introduced in Table 1. Figure 1 displays scatter plots of the performance metrics against the covariates: (i) the number of goal kicks (i.e., shots on target) out of the total number of kicks attempted by each team, and (ii) the number of goals scored out of the number of goal kicks. These quantities are examined in relation to the clubs’ investment in the 2023 season (measured in million Brazilian reais) and their estimated market value (measured in million euros).

Figure 1.

Scatterplots of the number of goal kicks (i.e., shots on target) relative to the total number of kicks attempted, and the number of goals scored relative to the number of goal kicks for each team, plotted against the covariates: investment in 2023 (in million Brazilian reais) and club market value (in million euros).

As illustrated in Figure 1, no discernible relationship is observed between either the 2023 financial investment or the club’s market value and the proportion of goal kicks–defined as shots on target–relative to the total number of attempts. On the other hand, the proportion of goals from successful goal kicks ( $Y_{2} / Y_{1}$ ) presents a positive relationship with the 2023 financial investment and with the club’s market value. However, despite this positive trend, the effect size is comparatively small, indicating that the early-season increases in scoring efficiency associated with greater financial input are present but modest. As outlined in Section “Introduction”, the inclusion of financial covariates serves an explanatory purpose, aiming to explore whether structural economic disparities across clubs are systematically associated with offensive efficiency during the early season. Nevertheless, within the limits of the current specification, the results do not provide empirical support for a definitive link between financial input and scoring outcomes.

Consistent with the descriptive observations in Figure 1, our initial analysis involved fitting the BVB model without incorporating the financial covariates investment in 2023 (measured in million Brazilian reais) and club market value (measured in million euros). Under this baseline specification, the random variable $Y_{1}$ represents the number of successful goal kicks (shots on target) out of the total number of kicks attempted by each team, with an associated success probability $p$ . The variable $Y_{2}$ denotes the number of goals scored from those successful goal kicks, governed by the success probability $q$ . The parameter estimates obtained from this model are summarized in Table 2, with the Bayesian model fit assessed via the DIC, which yielded values of 234.5 (Uniform priors) and 234.3 (Jeffrey’s priors). These results establish a reference point for subsequent models incorporating financial covariates.

Table 2.

Estimates of parameters (posterior means) $p$ and $q$ , under a Bayesian approach using uniform and jeffrey’s priors.

Prior Specification	Parameters	Estimates	Std. Dev.	95% Cred. Int.
Uniform	$p$	0.3298	0.0082	(0.3463, 0.6142)
	$q$	0.2769	0.0139	(0.2500, 0.3059)
Jeffreys	$p$	0.3295	0.0081	(0.3464, 0.6141)
	$q$	0.2767	0.0137	(0.2498, 0.3057)

The parameter $p$ represents the probability that a shot attempt by a team results in a successful goal kick, defined as a shot on target. Based on the results summarized in Table 2, under the Bayesian framework with uniform priors, the estimated value of $p$ is 32.98%, with a standard deviation of 0.82% and a 95% credible interval of (34.63%, 61.42%). The Jeffreys prior yields a nearly identical estimate of 32.95%, with a standard deviation of 0.81% and a 95% credible interval of (34.64%, 61.41%). This indicates that, on average, approximately one-third of shot attempts are successfully directed on target. The parameter $q$ corresponds to the probability that a shot on target is converted into a goal. The estimated value of $q$ under the uniform prior is 27.69%, with a standard deviation of 1.39% and a 95% credible interval of (25.00%, 30.59%), while the Jeffreys prior produces a similar estimate of 27.67%, with a standard deviation of 1.37% and a 95% credible interval of (24.98%, 30.57%). These findings suggest that slightly more than one in four shots on target result in goals. These parameters, in particular, provide a quantitative characterization of offensive efficiency, describing the transition from shot attempts to goals, thereby establishing a reference baseline for further exploration of the influence of financial covariates on team performance.

To further investigate the influence of financial factors on offensive efficiency within a model-based framework, we extended the baseline BVB model by incorporating the financial covariates—total investment in 2023 and club market value—at both stages, as detailed in Section “Materials and Methods”. As prior distributions, we adopted independent normal priors with hyperparameters specified as $τ_{j}^{2} = 100$ , for $j = 0, 1, 2$ , applied to each regression parameter $β_{j}$ and $α_{j}$ to ensure weakly informative priors. The results are presented in Table 3.

Table 3.

Posterior summaries for the BVB model in the presence of covariates $X_{1}$ and $X_{2}$ .

Parameters	Estimates	Std. Dev.	95% Cred. Int.
Goal Kicks
$β_{0}$ (Intercept)	$- 0.69771$	0.07551	$(- 0.8386, - 0.5439)$
$β_{1}$ (Investment)	$- 0.00480$	0.00130	$(- 0.0031, 0.0020)$
$β_{2}$ (Club Value)	$- 0.00002$	0.00101	$(- 0.0021, 0.0020)$
Goals Scored
$α_{0}$ (Intercept)	$- 1.40700$	0.14081	$(- 1.6890, - 1.1420)$
$α_{1}$ (Investment)	0.00210	0.00232	$(- 0.0022, 0.0065)$
$α_{2}$ (Club Value)	0.00540	0.00190	( 0.0020, 0.0092)

The posterior summaries presented in Table 3 indicate that neither total investment in 2023 nor club market value exert statistically significant effects on the probability of successful goal kicks (95% credible intervals for the regression parameters include zero). However, for the goal-scoring stage, the coefficient associated with club market value ( $α_{2}$ ) is positive and statistically significant (95% credible interval for the corresponding regression parameter does not include zero), indicating that each one-million-euro increase in market value corresponds to an increase of approximately 0.0054 in the log-odds of scoring from a goal kick, holding other variables constant. Although this effect achieves statistical significance, its practical impact on scoring efficiency is limited. This finding suggests that financial resources alone do not comprehensively account for differences in offensive outcomes across teams. Other determinants, including player skill levels, coaching methodologies, team cohesion, and contextual match factors, are likely to exert a more substantive influence on scoring performance. Moreover, model comparison using the Deviance Information Criterion (DIC) shows no meaningful improvement relative to the baseline model without covariates (DIC $=$ 236.1 versus 234.5), indicating that the addition of financial covariates offers minimal explanatory enhancement within this modeling framework.

Building on the previous findings, it is instructive to situate these results within the broader context of recent financial and organizational transformations among Brazilian Série A soccer clubs. A salient example is the case of Botafogo, which experienced a pronounced revival under the ownership of U.S. investor John Textor. Following a period marked by financial distress, Botafogo successfully restructured its operations, culminating in triumphs in both the Campeonato Brasileiro Série A and the Copa Libertadores in 2024.¹⁴ This resurgence is widely attributed to substantial capital injections and the implementation of a novel managerial framework, enabled by the 2021 legislative reform permitting Brazilian football clubs to transition into limited liability companies.¹⁵ The positive and statistically significant association identified in the BVB model between club market value and the likelihood of scoring from successful goal kicks ( $Y_{2}$ ) aligns with Botafogo’s recent performance trajectory, suggesting that the club’s increased market value—driven by strategic financial investments—is reflected in its improved offensive efficiency.

Concluding remarks

This study investigated the performance dynamics of Brazilian Serie A soccer clubs through the application of a nested bivariate binomial (BVB) model, focusing on the association between financial indicators—specifically club market value and investment. The model allowed for the examination of how these financial factors relate to two sequential outcomes: the number of successful goal kicks ( $Y_{1}$ ) and the number of goals scored from those successful kicks ( $Y_{2}$ ). The analysis revealed a statistically significant positive association between club market value and the probability of scoring goals from successful goal kicks ( $Y_{2}$ ). This finding aligns with empirical observations of certain clubs, notably Botafogo, which experienced a notable resurgence following organizational restructuring and increased financial backing.¹⁴ The elevated market value of such clubs, influenced by financial investment and changes in management, appears to be linked to enhanced offensive efficiency and improved on-field results.

In contrast, no statistically significant association was observed between financial investment and the probability of successful goal kicks ( $Y_{1}$ ), suggesting that financial resources alone may not directly influence all facets of team performance, especially those involving tactical execution or technical skill development. It is important to emphasize that the modeling approach employed in this study does not support causal inference; rather, it identifies statistical associations present within the observed data. The findings highlight associations between financial indicators—particularly club market value—and offensive performance metrics, yet these relationships are likely influenced by a multitude of additional factors not accounted for in the current model. Future research should incorporate a broader range of covariates, including tactical, technical, and organizational variables, and adopt longitudinal or quasi-experimental designs to more rigorously investigate the underlying mechanisms influencing team performance and the potential causal effects of financial resources.

In contrast, no statistically significant association was observed between financial investment and the probability of successful goal kicks ( $Y_{1}$ ), suggesting that financial resources alone may not directly influence all facets of team performance, particularly those involving tactical execution or technical skill development. It is worth noting that our analysis considered only the first 12 games of the season; therefore, some teams may have made substantial financial investments after this period, and any resulting improvements in performance may not yet be reflected in the observed data. The findings reveal a positive association between financial indicators—particularly club market value—and offensive performance metrics. However, these associations should not be interpreted as causal, as team performance is likely influenced by a multitude of additional factors not accounted for in the current model. Future research should incorporate a broader range of covariates, including tactical, technical, and organizational variables, and adopt longitudinal or quasi-experimental designs to more rigorously investigate the underlying mechanisms influencing team performance and the potential causal effects of financial resources.

Footnotes

Acknowledgments

The authors thank the editor and referees for their valuable suggestions that led to significant improvements to the manuscript.

ORCID iDs

Jorge Alberto Achcar

Emerson Barili

Ricardo Puziol de Oliveira

João Debastiani Neto

Ethical considerations

This article does not contain any studies with human or animal participants.

Consent to participate

There are no human participants in this article and informed consent is not required.

Consent for publication

Not applicable.

Declaration of conflicting interest

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding statement

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Data availability

https://fbref.com/pt/comps/24/shooting/Serie-A-Estatisticas

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