# Robust analysis of second-leg home advantage in UEFA football through better nonparametric confidence intervals for binary regression functions

## TL;DR

This paper investigates the existence of second-leg home advantage in UEFA football using nonparametric methods, introducing new confidence intervals and bootstrap techniques to provide an objective analysis of game outcome data.

## Contribution

It develops novel nonparametric confidence intervals and bootstrap methods for binary regression, offering an unbiased approach to assess second-leg home advantage without model assumptions.

## Key findings

- A slight second-leg home advantage is observed.
- New nonparametric confidence intervals improve analysis accuracy.
- Bootstrap methods address bias in confidence interval estimation.

## Abstract

In international football (soccer), two-legged knockout ties, with each team playing at home in one leg and the final outcome decided on aggregate, are common. Many players, managers and followers seem to believe in the `second-leg home advantage', i.e. that it is beneficial to play at home on the second leg. A more complex effect than the usual and well-established home advantage, it is harder to identify, and previous statistical studies did not prove conclusive about its actuality. Yet, given the amount of money handled in international football competitions nowadays, the question of existence or otherwise of this effect is of real import. As opposed to previous research, this paper addresses it from a purely nonparametric perspective and brings a very objective answer, not based on any particular model specification which could orientate the analysis in one or the other direction. Along the way, the paper reviews the well-known shortcomings of the Wald confidence interval for a proportion, suggests new nonparametric confidence intervals for conditional probability functions, revisits the problem of the bias when building confidence intervals in nonparametric regression, and provides a novel bootstrap-based solution to it. Finally, the new intervals are used in a careful analysis of game outcome data for the UEFA Champions and Europa leagues from 2009/10 to 2014/15. A slight `second-leg home advantage' is evidenced.