> pba.aov <- aov(score ~ oil*bowler*tourney)
> 
> summary(pba.aov)
                Df Sum Sq Mean Sq F value   Pr(>F)    
oil              4  337.1   84.27 157.858  < 2e-16 ***
bowler          36   84.4    2.35   4.394 9.55e-15 ***
tourney         10   97.7    9.77  18.293  < 2e-16 ***
oil:bowler     144  129.3    0.90   1.682 1.67e-05 ***
bowler:tourney 360  321.1    0.89   1.671 2.72e-08 ***
Residuals      555  296.3    0.53                     
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> 
> library(lmerTest)
> 
> pba.mod1 <- lmer(score ~ oil + (1|bowler) + (1|oil:bowler) + 
+    (1|oil:tourney) + (1|bowler:oil:tourney)) 
> 
> summary(pba.mod1)
Linear mixed model fit by REML t-tests use Satterthwaite approximations to
  degrees of freedom [lmerMod]
Formula: score ~ oil + (1 | bowler) + (1 | oil:bowler) + (1 | oil:tourney) +  
    (1 | bowler:oil:tourney)

REML criterion at convergence: 2811.3

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-2.97531 -0.60501  0.00816  0.64515  3.01851 

Random effects:
 Groups             Name        Variance Std.Dev.
 bowler:oil:tourney (Intercept) 0.178998 0.42308 
 oil:bowler         (Intercept) 0.001043 0.03229 
 bowler             (Intercept) 0.048254 0.21967 
 oil:tourney        (Intercept) 0.119908 0.34628 
 Residual                       0.533810 0.73062 
Number of obs: 1110, groups:  
bowler:oil:tourney, 555; oil:bowler, 185; bowler, 37; oil:tourney, 15

Fixed effects:
            Estimate Std. Error      df t value Pr(>|t|)    
(Intercept)  15.4356     0.2129 10.6020  72.508 1.33e-15 ***
oil2          0.4689     0.2967 10.0050   1.580  0.14510    
oil3          0.3714     0.2967 10.0050   1.252  0.23913    
oil4         -1.0830     0.2967 10.0050  -3.650  0.00446 ** 
oil5         -0.1148     0.2967 10.0050  -0.387  0.70697    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
     (Intr) oil2   oil3   oil4  
oil2 -0.697                     
oil3 -0.697  0.500              
oil4 -0.697  0.500  0.500       
oil5 -0.697  0.500  0.500  0.500
> anova(pba.mod1)
Analysis of Variance Table of type III  with  Satterthwaite 
approximation for degrees of freedom
    Sum Sq Mean Sq NumDF  DenDF F.value   Pr(>F)   
oil 18.414  4.6035     4 10.005  8.6239 0.002792 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> if(require(pbkrtest))
+ anova(pba.mod1, ddf = "Kenward-Roger")
Analysis of Variance Table of type III  with  Kenward-Roger 
approximation for degrees of freedom
    Sum Sq Mean Sq NumDF  DenDF F.value   Pr(>F)   
oil 18.414  4.6035     4 10.005  8.6239 0.002792 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> lsmeans(pba.mod1)
Least Squares Means table:
        oil Estimate Standard Error   DF t-value Lower CI Upper CI p-value    
oil  1  1.0   15.436          0.213 10.6  72.510     15.0     15.9  <2e-16 ***
oil  2  2.0   15.905          0.213 10.6  74.710     15.4     16.4  <2e-16 ***
oil  3  3.0   15.807          0.213 10.6  74.250     15.3     16.3  <2e-16 ***
oil  4  4.0   14.353          0.213 10.6  67.420     13.9     14.8  <2e-16 ***
oil  5  5.0   15.321          0.213 10.6  71.970     14.9     15.8  <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> difflsmeans(pba.mod1)
Differences of LSMEANS:
          Estimate Standard Error   DF t-value Lower CI Upper CI p-value    
oil 1 - 2     -0.5         0.2967 10.0   -1.58  -1.1299    0.192   0.145    
oil 1 - 3     -0.4         0.2967 10.0   -1.25  -1.0324    0.290   0.239    
oil 1 - 4      1.1         0.2967 10.0    3.65   0.4220    1.744   0.004 ** 
oil 1 - 5      0.1         0.2967 10.0    0.39  -0.5463    0.776   0.707    
oil 2 - 3      0.1         0.2967 10.0    0.33  -0.5636    0.758   0.749    
oil 2 - 4      1.6         0.2967 10.0    5.23   0.8909    2.213   4e-04 ***
oil 2 - 5      0.6         0.2967 10.0    1.97  -0.0774    1.245   0.077 .  
oil 3 - 4      1.5         0.2967 10.0    4.90   0.7934    2.115   6e-04 ***
oil 3 - 5      0.5         0.2967 10.0    1.64  -0.1749    1.147   0.132    
oil 4 - 5     -1.0         0.2967 10.0   -3.26  -1.6293   -0.307   0.008 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> 
>