> tapply(y, trt, mean) # marginal mean for y
      pp       mp       cp      mps  control 
28.83333 27.83333 30.33333 28.66667 26.33333 
> tapply(x, trt, mean) # marginal mean for x
      pp       mp       cp      mps  control 
23.00000 25.16667 22.00000 25.00000 25.66667 
> 
> mean(x)
[1] 24.16667
> 
> # Fit Model with Interaction 
> trombone1.aov <- lm(y ~ x + pp + mp + cp + mps + ppx + mpx + cpx +mpsx)
> 
> summary(trombone1.aov)

Call:
lm(formula = y ~ x + pp + mp + cp + mps + ppx + mpx + cpx + mpsx)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.9160 -1.2527 -0.0376  1.0884  3.0191 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -4.1336     6.0508  -0.683   0.5023    
x             1.1870     0.2332   5.091 5.58e-05 ***
pp           17.6575     6.7271   2.625   0.0162 *  
mp           15.0516     9.8270   1.532   0.1413    
cp           13.2526    10.9310   1.212   0.2395    
mps          16.9400     7.3048   2.319   0.0311 *  
ppx          -0.5214     0.2631  -1.982   0.0614 .  
mpx          -0.5149     0.3844  -1.339   0.1955    
cpx          -0.2227     0.4733  -0.471   0.6430    
mpsx         -0.5526     0.2827  -1.955   0.0647 .  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 2.179 on 20 degrees of freedom
Multiple R-squared: 0.8225,     Adjusted R-squared: 0.7427 
F-statistic:  10.3 on 9 and 20 DF,  p-value: 9.164e-06 

> 
> # Fit Model without Interaction 
> trombone2.aov <- lm(y ~ x + pp + mp + cp + mps)
> 
> summary(trombone2.aov)

Call:
lm(formula = y ~ x + pp + mp + cp + mps)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.4156 -0.7855 -0.0574  1.1560  4.2078 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  7.39995    2.37356   3.118  0.00468 ** 
x            0.73766    0.08549   8.628 8.07e-09 ***
pp           4.46710    1.29984   3.437  0.00215 ** 
mp           1.86883    1.28041   1.460  0.15738    
cp           6.70477    1.31753   5.089 3.32e-05 ***
mps          2.82511    1.28096   2.205  0.03725 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 2.216 on 24 degrees of freedom
Multiple R-squared: 0.7797,     Adjusted R-squared: 0.7338 
F-statistic: 16.99 on 5 and 24 DF,  p-value: 3.371e-07 

> vcov(trombone2.aov)
            (Intercept)           x          pp          mp          cp
(Intercept)   5.6337865 -0.18759662 -1.31906414 -0.91260479 -1.50666077
x            -0.1875966  0.00730896  0.01949056  0.00365448  0.02679952
pp           -1.3190641  0.01949056  1.68958778  0.82855176  0.89027186
mp           -0.9126048  0.00365448  0.82855176  1.63944020  0.83220624
cp           -1.5066608  0.02679952  0.89027186  0.83220624  1.73587786
mps          -0.9438709  0.00487264  0.83180018  0.82124280  0.83667282
                    mps
(Intercept) -0.94387089
x            0.00487264
pp           0.83180018
mp           0.82124280
cp           0.83667282
mps          1.64086138
> 
> # Fit Model without treatment effects
> trombone3.aov <- lm(y ~ x)
> 
> summary(trombone3.aov)

Call:
lm(formula = y ~ x)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.2457 -1.7272  0.4491  2.0360  4.3123 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  13.6458     2.7901   4.891 3.74e-05 ***
x             0.6105     0.1131   5.397 9.37e-06 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 3.061 on 28 degrees of freedom
Multiple R-squared: 0.5099,     Adjusted R-squared: 0.4924 
F-statistic: 29.13 on 1 and 28 DF,  p-value: 9.368e-06 

>