Statistical Analysis Overview

The predicted average score for individuals in the little training group, which is 60.13 points.

The coefficient (-1.96) indicates that the average season score for normal training is 1.96 points lower than the reference group's average.

The coefficient (-2.89) signifies that the average season score for heavy training is 2.89 points lower than the reference group's average.

A problem in causal analysis where the independent variable (X) is correlated with the error term (u). This means the relationship between X and Y isn't solely due to causality but also other hidden factors.

A crucial assumption for causal inference where the expected value of the error term (u) is zero given the value of the independent variable (X).

The strength and direction of the linear relationship between the hours of training and the season score. A -0.1121 correlation indicates a very weak negative linear relationship, meaning as hours trained increase, season score tends to decrease slightly.

A statistical method to model the relationship between hours trained and season score. The equation is season_score = constant + (coefficient * hours_trained).

For each hour increase in training time, the season score is predicted to decrease by approximately 0.357 points.

The assumption that there is no relationship between the independent variables and the dependent variable (meaning the coefficient would equal 0).

A variable that is divided into distinct categories (e.g., heavy training, normal training, low training).

An OLS regression where the independent variable is now a categorical variable (e.g., heavy, normal, or low training).

The probability of rejecting a true null hypothesis (false positive).

The predicted season score when the category is 'low training'.

A type of bias in statistical analysis when a relevant variable is missing from the model, leading to an inaccurate estimation of the relationship between the other variables.

When the direction of cause and effect is reversed in a statistical analysis, meaning the independent variable is actually influenced by the dependent variable.

γ represents the correlation between the omitted variable and the independent variable. Its magnitude and sign influence the direction and strength of the bias.

π represents the effect of the dependent variable on the independent variable. Its sign and magnitude influence the direction and strength of the bias.

The bias in reverse causality is calculated as the covariance between the independent variable and the error term divided by the variance of the independent variable.

Factors that could influence the dependent variable but are not included in the model. These include factors like training quality, player's initial skill level, nutrition, external stressors, and match-specific factors.

Integrating data on player physical state ('0' indicating injury or lack of stamina) can provide valuable insights and potentially reduce endogeneity by capturing another relevant variable.

Integrating physical state data may not completely solve endogeneity issues if other omitted variables or reverse causality still exist. It's important to consider multiple potential sources of endogeneity and address them appropriately.

The average treatment effect (ATE) quantifies the average causal effect of a treatment on a population, based on comparing treated and untreated groups.

The average treatment effect on the treated (ATET) measures how much a treatment, on average, affects individuals who actually received it.

The challenge of estimating the effect of a treatment when you can only observe one outcome (either with or without treatment) for each individual, but not both.

A method of randomly assigning individuals to treatment and control groups, ensuring that the groups are similar on average and any difference in outcomes is likely due to the treatment.

An intervention or change being studied, often represented as a binary variable (1 if treated, 0 otherwise).

The variable measured to see the effect of treatment. It can be anything like performance, health, or satisfaction.

Randomization ensures that the treated and untreated groups are comparable on average, allowing us to attribute any differences in outcomes to the treatment itself.

ATE is calculated by finding the difference in means between the treated and untreated groups under the assumption that randomization ensures comparability between the groups.

The first stage of the instrumental variable regression, where we estimate the impact of the instrument (Z) on the endogenous variable (X) using regression. This stage is essentially a regression of the exogenous variable (X) on the instrument (Z).

The predicted values of the endogenous variable (X) obtained from the first stage regression. These fitted values are then used in the second stage.

The second stage of instrumental variable regression where we regress the dependent variable (Y) on the fitted values (x̂) from the first stage. This stage helps us understand the causal relationship between the endogenous variable (X) and the dependent variable (Y).

A regression of the dependent variable (Y) on the instrument (Z) directly, without considering the endogenous variable (X). This is used to assess instrument relevance.

A crucial assumption for instrumental variable regression. The instrument (Z) should have a statistically significant impact on the endogenous variable (X).

The estimated coefficient in the second stage regression. This coefficient reflects the causal effect of the endogenous variable (X) on the dependent variable (Y), after controlling for the influence of the instrument.

A specific type of instrumental variable estimator that involves one binary instrument (Z) and one endogenous variable (X).

A common technique to estimate the causal effect of an endogenous variable on a dependent variable using an instrumental variable. It involves two stages: estimating the impact of the instrument and then regressing the dependent variable on the estimated values of the endogenous variable.

Part of the variation in an independent variable (X) that is correlated with the error term (u), meaning it's influenced by unobserved factors affecting the dependent variable (Y).

Part of the variation in an independent variable (X) that is not correlated with the error term (u), meaning it's only influenced by observable factors affecting the dependent variable (Y).

A variable (Z) that is correlated with the independent variable (X) but not directly correlated with the dependent variable (Y), allowing us to isolate the exogenous variation in X and estimate the true causal effect.

The instrument variable (Z) must be correlated with the independent variable (X), meaning it has a significant impact on X.

The instrument variable (Z) must not be directly correlated with the dependent variable (Y) or any other omitted variables that influence Y, except through its impact on X.

A regression model used to test the relevance of the instrument variable (Z) by examining its effect on the independent variable (X).

A regression model that directly estimates the effect of the instrument variable (Z) on the dependent variable (Y), capturing the total indirect effect through X.