Multiple Logistic Regression model with interactions. This model shows significant evidence that referees tend to even out calls (i.e., that the probability of a foul called on the home team increases by 10% as total home scores increase compared to total visiting team scores - that is, as score.diff increases) after accounting for foul differential, whether the home team has the lead, and time remaining. The extent of the effect of foul differential also appears to grow (in a negative direction) as the first half goes on, based on an interaction between time remaining and foul differential). The effect of foul differential increases by 10.7% if a foul is an offensive foul and by 6.2% if a foul is an personal foul rather than a shooting foul, after controlling for score differential, whether the home team has the lead, and time remaining.

 AIC =  6761.022 ;  BIC =  6832.649

Call:
glm(formula = foul.home ~ foul.diff + score.diff + lead.home + 
    time + offensive + personal + foul.diff:offensive + foul.diff:personal + 
    foul.diff:time + lead.home:time, family = binomial, data = refdata)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.7279  -1.1215  -0.8831   1.1818   1.7956  

Coefficients:
                     Estimate Std. Error z value Pr(>|z|)
(Intercept)         -0.119543   0.107758  -1.109  0.26727
foul.diff           -0.033293   0.031234  -1.066  0.28646
score.diff           0.020174   0.006665   3.027  0.00247
lead.home           -0.098019   0.160252  -0.612  0.54077
time                -0.013077   0.007957  -1.643  0.10029
offensive           -0.106409   0.106348  -1.001  0.31703
personal             0.054044   0.062151   0.870  0.38454
foul.diff:offensive -0.101786   0.051772  -1.966  0.04929
foul.diff:personal  -0.060534   0.030602  -1.978  0.04792
foul.diff:time      -0.007630   0.003023  -2.524  0.01161
lead.home:time       0.021790   0.011174   1.950  0.05116
                      
(Intercept)           
foul.diff             
score.diff          **
lead.home             
time                  
offensive             
personal              
foul.diff:offensive * 
foul.diff:personal  * 
foul.diff:time      * 
lead.home:time      . 
---
Signif. codes:  
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 6884.3  on 4971  degrees of freedom
Residual deviance: 6739.0  on 4961  degrees of freedom
AIC: 6761

Number of Fisher Scoring iterations: 4

Multilevel Regression model with Crossed Random Effects for examining referee bias taking into account the effect of foul differential, score differential, whether the home team has the lead, and time remaining, on the odds a foul is called on the home team, while accounting for three crossed random effects at Level Two (game, home team, and visiting team). For a plot of the estimated Random Effects by team, see next page.

Example interpretation of fixed effects: Intercept (-0.246506). The odds of a foul on the home team is 0.781 (exp((-0.246506))) at the end of the first half when the score is tied, the fouls are even, and the referee has just called a shooting foul. As the foul differential decreases by 1 (the visiting team accumulates another foul relative to the home team), the odds of a home foul increase by 18.8% (1/exp(-0.171473)). As the score differential increases by 1 (the home team accumulates another point relative to the visiting team), the odds of a home foul increase by 3.4% (1/exp(0.03352) = 1.034088), after controlling for foul differential, type of foul, whether or not the home team has the lead, and time remaining in the half. Referees are more likely to call fouls on the home team when this is leading, and vice versa. The effect of foul differential increases by 10.9% (1/exp(-0.103552)) if a foul is an offensive foul rather than a shooting foul, after controlling for score differential, whether the home team has the lead, and time remaining. Interpretation of random effects: 0.18463: the variance in log-odds intercepts from game-to-game after controlling for all other covariates in the model.

Generalized linear mixed model fit by maximum likelihood
  (Laplace Approximation) [glmerMod]
 Family: binomial  ( logit )
Formula: 
foul.home ~ foul.diff + score.diff + lead.home + time + offensive +  
    personal + foul.diff:offensive + foul.diff:personal + foul.diff:time +  
    lead.home:time + (1 | game) + (1 | hometeam) + (1 | visitor)
   Data: refdata

     AIC      BIC   logLik deviance df.resid 
  6731.0   6822.2  -3351.5   6703.0     4958 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.1268 -0.8958 -0.5831  0.9457  2.6829 

Random effects:
 Groups   Name        Variance Std.Dev.
 game     (Intercept) 0.18463  0.4297  
 hometeam (Intercept) 0.07831  0.2798  
 visitor  (Intercept) 0.04312  0.2077  
Number of obs: 4972, groups:  
game, 340; hometeam, 39; visitor, 39

Fixed effects:
                     Estimate Std. Error z value Pr(>|z|)
(Intercept)         -0.246506   0.133959  -1.840 0.065745
foul.diff           -0.171473   0.045363  -3.780 0.000157
score.diff           0.033520   0.008236   4.070  4.7e-05
lead.home           -0.150633   0.177211  -0.850 0.395314
time                -0.008746   0.008560  -1.022 0.306914
offensive           -0.080800   0.111233  -0.726 0.467593
personal             0.067218   0.065397   1.028 0.304026
foul.diff:offensive -0.103552   0.053869  -1.922 0.054567
foul.diff:personal  -0.055623   0.031948  -1.741 0.081671
foul.diff:time      -0.008690   0.003274  -2.654 0.007944
lead.home:time       0.026009   0.012172   2.137 0.032613
                       
(Intercept)         .  
foul.diff           ***
score.diff          ***
lead.home              
time                   
offensive              
personal               
foul.diff:offensive .  
foul.diff:personal  .  
foul.diff:time      ** 
lead.home:time      *  
---
Signif. codes:  
0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
            (Intr) fl.dff scr.df led.hm time   offnsv
foul.diff    0.208                                   
score.diff   0.269 -0.165                            
lead.home   -0.660  0.024 -0.659                     
time        -0.722 -0.132 -0.170  0.618              
offensive   -0.184 -0.060  0.007  0.007  0.030       
personal    -0.276 -0.063  0.013  0.001  0.010  0.334
fl.dff:ffns -0.016 -0.223 -0.007  0.003 -0.009  0.201
fl.dff:prsn -0.054 -0.435 -0.003  0.013 -0.005  0.057
foul.dff:tm -0.011 -0.398 -0.042 -0.008  0.049 -0.014
lead.hom:tm  0.564 -0.045  0.386 -0.828 -0.743 -0.011
            persnl fl.dff:f fl.dff:p fl.dff:t
foul.diff                                    
score.diff                                   
lead.home                                    
time                                         
offensive                                    
personal                                     
fl.dff:ffns  0.057                           
fl.dff:prsn  0.179  0.345                    
foul.dff:tm -0.015  0.012    0.013           
lead.hom:tm -0.020 -0.008   -0.005    0.024  
optimizer (Nelder_Mead) convergence code: 0 (OK)
Model failed to converge with max|grad| = 0.0827966 (tol = 0.002, component 1)

Estimated random effects and associated prediction intervals for 39 home teams and visiting teams in the Final Model. For instance, we see that the teams DePaul and Seton Hall have higher baseline odds of home fouls than those of Purdue or Syracuse. On the other hand, the teams ND and WV have higher baseline odds of visiting fouls than those of De Paul or IN.