Here's a summary of some of the published studies used in "Referee Bias by Thomas Dohmen, University of Bonn, Maastricht University and Jan Sauermann of Stockholm University. Discussion Paper No.8857, dated February 2015:
Boyko et al. (2007): Premier League, England (1992/93–2006/07)
Goals 1.51 (1.10) goals for the home (visiting) team; significant effect of referee fixed-effect on goal differential and goals of home team;
Penalties .10 (.06) penalties for the home (visiting) team; significant effect of referee fixed-effect on penalty differential, penalties for home and visiting team
Red cards .06 (.09) red cards for the home (visiting) team; significant effect of referee fixed-effects for home and visiting team
Yellow cards 1.17 (1.62) yellow cards for the home (visiting) team; significant effect of referee fixed-effects for home and visiting team
Note: yellow card and penalty differential can partly be explained by referee-fixed effects; they note that results are not consistent over time; crowd size affects outcome significantly in most cases, crowd density does not
Johnston (2008): Premier League, England (2006/07)
Goals No effect of variation in attendance in explaining goal differential; no effect of referees on goal difference;
Page and Page (2010a): English competitions (1994–2007)
Goals Controlling for attendance and difference in team quality, their results suggest that there are still large difference in home bias between referees
Dohmen (2008): Bundesliga, Germany (1992/93–2003)
Goals 95.05% (95.99%) of home (visiting) team goals are correctly granted
Penalties 65.20% (72.57%) of home (visiting) team penalties are correctly awarded
Sutter and Kocher (2004): Bundesliga, Germany (2000/01)
Penalties 81% (51%) of home (visiting) team penalties found to legitimate
Dawson et al. (2007): English Premier League (1996–2003)
Yellow/red cards More disciplinary actions towards underdogs than to favorites; more sanctions in balanced games and games at the end of the season; evidence for variation of home team bias between referees
Buraimo, Forrest, and Simmons (2010): Premier League, England (2000/01–2005/06)
Yellow/red cards Underdog teams playing at home have a lower probability of receiving yellow and red cards
Buraimo, Forrest, and Simmons (2010): Bundesliga, Germany (2000/01–2005/06)
Yellow/red cards Underdog teams playing at home have a lower probability of receiving yellow and red cards; in games with tracks, home teams have an increased probability of being awarded with a yellow card
Dawson and Dobson (2010): UEFA a-tournaments, Europe (2002/03–2006/07)
Yellow/red cards More yellow/red cards awarded for visiting team; stadiums with track and referee’s nationality affects referee’s behavior
Buraimo et al. (2012): Primera Division, Spain (2003/04 and 2006/07)
Yellow/red cards Crowd effects larger in stadiums with running track; greater crowd size is related to stronger home bias
Buraimo et al. (2012): UEFA’s Champions League (2003/04 and 2006/07)
Yellow/red cards Crowd effects larger in stadiums with running track
Downward and Jones (2007): FA Cup, England (1996/97-2001/02)
Yellow cards Crowd size affects the probability of home team being awarded; effect is attenuated by games with largest crowds
Reilly and Witt (2013): English Premiership League (2003/04–2007/08)
Red Cards No evidence for social pressure effects
Goumas (2014): Champions League and Europa League (2009/10-2010/11)
Yellow cards Crowd density, not crowd size affects referee bias
Pettersson-Lidbom and Priks (2010): Serie A and B, Italy (2006/07)
Yellow cards .61–.68 fewer yellow cards for home teams in games with spectators
Red cards .07–.08 fewer red cards for home teams (hardly significant)
Fouls 4.36–4.56 fewer fouls for home teams
Note: Comparing games with and games without spectators
Nevill et al. (2002): laboratory study
Fouls 15.5% fewer fouls awarded to home team with crowd noise
Note: they analyse tackling scenes from the game between Liverpool (home) and Leicester City (visiting) (1998/99); referee experience matters for awarding fouls
Boeri and Severgnini (2011): Serie A, Italy (2004/05)
Game outcomes Use information on match rigging to detect rigging in previous seasons; rigging is also apparent in balanced matches
Torgler (2004): FIFA Football Worldcup (2002)
Game outcome Having a referee from the same region increases probability of winning; language does not have an effect
I assume we've all read it?
On Goals:
An alternative approach is to directly assess the correctness of decisions to award goals. Dohmen (2008) evaluates expert judgements on the correctness of referee decisions and finds that goals awarded to the home team were significantly less likely awarded correctly. Moreover, an analysis of all situations in which a goal was scored, reveals tentative evidence that referees grant the visiting team fewer disputable goals, and significant evidence that the home side is granted more illegitimate goals than the visiting side. However, there is no evidence that referees award the visitor fewer legitimate goals than the home team
Penalty Kicks:
Evidence of home biased refereeing with respect to penalty kick decisions is more clear cut. Dohmen (2008) finds that a larger fraction of awarded penalty kicks is either wrongly awarded or disputable when the home team is one goal behind. Evaluating all critical situations in the penalty kick area, he also finds that home teams are significantly more likely to be awarded a penalty kick in situations that are rated as disputable by experts who assess the correctness of these decisions after each match based on video recordings.
Similarly, Sutter and Kocher (2004), who use journalists’ reports on referees’ performance, find that during the 2000/01-Bundesliga season home teams were awarded 81 percent of all penalty kicks that should have been legitimately awarded, while visiting teams are awarded only 51 percent of all penalties that should have been awarded. This differential referee treatment is statistically significant. Corroborating suggestive evidence on biased penalty kick decisions is also provided by Boyko et al. (2007) who show that home teams in the English Premier League receive significantly more penalties than their model predicts.
Using the same approach that they used to judge referee bias in awarding goals, they find significant referee fixed-effects, which suggests that referees are biased in awarding penalty
kicks.
I will keep looking for the study that showed bias towards bigger clubs over smaller clubs.
The bottom line seems to be that referee bias is there but it's generally a result of being at home with a large boisterous home crowd. And I wouldn't have it any other way.
Other effects are that second-half extra time is typically a bit longer if the home team is a goal behind.