> # Tomato exp't
> y=scan("http://www.stat.ucla.edu/~hqxu/stat225/data/tomato.dat")  # scan or type data
> y <- c(16.1, 15.3, 17.5,   16.6, 19.2, 18.5,   20.8, 18.0, 21.0,   8.1, 8.6, 10.1,     12.7, 13.7, 11.5,   14.4, 15.4, 13.7)

> tom2 <- rep(c(0,1), c(9,9))
> denl.tom1 <- rep(c(-1,0,1,0,0,0), rep(3,6))/sqrt(2)
> denl.tom2 <- rep(c(0,0,0,-1,0,1), rep(3,6))/sqrt(2)
> denq.tom1 <- rep(c(1,-2,1,0,0,0), rep(3,6))/sqrt(6)
> denq.tom2 <- rep(c(0,0,0,1,-2,1), rep(3,6))/sqrt(6)
> 
> # nested model 
> g<- lm(y ~ tom2 + denl.tom1 + denl.tom2 + denq.tom1 + denq.tom2)
> summary(g)

Call:
lm(formula = y ~ tom2 + denl.tom1 + denl.tom2 + denq.tom1 + denq.tom2)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.93333 -0.82500 -0.01667  1.02500  1.20000 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 18.11111    0.40575  44.637 1.04e-14 ***
tom2        -6.08889    0.57381 -10.611 1.88e-07 ***
denl.tom1    2.56915    0.70277   3.656 0.003292 ** 
denl.tom2    3.93623    0.70277   5.601 0.000116 ***
denq.tom1    0.01361    0.70277   0.019 0.984869    
denq.tom2   -0.74846    0.70277  -1.065 0.307835    
---
Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 

Residual standard error: 1.217 on 12 degrees of freedom
Multiple R-Squared: 0.9296,    Adjusted R-squared: 0.9003 
F-statistic: 31.69 on 5 and 12 DF,  p-value: 1.625e-06 

> # drop the two quadratic terms
> g<- lm(y ~ tom2 + denl.tom1 + denl.tom2 )
> summary(g)

Call:
lm(formula = y ~ tom2 + denl.tom1 + denl.tom2)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.92778 -0.90556  0.09722  0.86944  1.67778 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  18.1111     0.3930  46.084  < 2e-16 ***
tom2         -6.0889     0.5558 -10.955 2.98e-08 ***
denl.tom1     2.5692     0.6807   3.774  0.00205 ** 
denl.tom2     3.9362     0.6807   5.783 4.75e-05 ***
---
Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 

Residual standard error: 1.179 on 14 degrees of freedom
Multiple R-Squared: 0.923,     Adjusted R-squared: 0.9064 
F-statistic:  55.9 on 3 and 14 DF,  p-value: 4.891e-08 

> 
> # final model, need a new variable for the linear effect
> denl <- rep(c(-1,0,1,-1,0,1), rep(3,6))/sqrt(2)
> g<- lm(y ~ tom2 + denl )
> summary(g)

Call:
lm(formula = y ~ tom2 + denl)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.4111 -0.5972  0.3333  0.6556  1.6889 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  18.1111     0.4061   44.60  < 2e-16 ***
tom2         -6.0889     0.5743  -10.60 2.30e-08 ***
denl          3.2527     0.4974    6.54 9.36e-06 ***
---
Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 

Residual standard error: 1.218 on 15 degrees of freedom
Multiple R-Squared: 0.9119,    Adjusted R-squared: 0.9001 
F-statistic: 77.58 on 2 and 15 DF,  p-value: 1.228e-08 

> 
> # an easy way
> tomato <- as.factor( rep( c(0,1), c(9,9)))
> density <- as.ordered( rep(rep( c(1,2,3), c(3,3,3)), 2))
> 
> g<-lm(y~tomato/density)
> summary(g)

Call:
lm(formula = y ~ tomato/density)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.93333 -0.82500 -0.01667  1.02500  1.20000 

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)    
(Intercept)       18.11111    0.40575  44.637 1.04e-14 ***
tomato1           -6.08889    0.57381 -10.611 1.88e-07 ***
tomato0:density.L  2.56915    0.70277   3.656 0.003292 ** 
tomato1:density.L  3.93623    0.70277   5.601 0.000116 ***
tomato0:density.Q  0.01361    0.70277   0.019 0.984869    
tomato1:density.Q -0.74846    0.70277  -1.065 0.307835    
---
Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 

Residual standard error: 1.217 on 12 degrees of freedom
Multiple R-Squared: 0.9296,    Adjusted R-squared: 0.9003 
F-statistic: 31.69 on 5 and 12 DF,  p-value: 1.625e-06 

> 
> g<-lm(y~tomato+density)
> summary(g)

Call:
lm(formula = y ~ tomato + density)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.2611 -0.7347  0.3028  0.6889  1.8389 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  18.1111     0.4126  43.891 2.22e-16 ***
tomato1      -6.0889     0.5836 -10.434 5.51e-08 ***
density.L     3.2527     0.5054   6.436 1.56e-05 ***
density.Q    -0.3674     0.5054  -0.727    0.479    
---
Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 

Residual standard error: 1.238 on 14 degrees of freedom
Multiple R-Squared: 0.9151,    Adjusted R-squared: 0.8969 
F-statistic: 50.27 on 3 and 14 DF,  p-value: 9.644e-08 

> 
> # compare with separate fitting, same estimates but different s.e. 
> g<-lm(y~density, subset=tomato==0)
> summary(g)

Call:
lm(formula = y ~ density, subset = tomato == 0)

Residuals:
   Min     1Q Median     3Q    Max 
-1.933 -1.000  0.400  1.067  1.200 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 18.11111    0.46600  38.865 1.94e-08 ***
density.L    2.56915    0.80714   3.183    0.019 *  
density.Q    0.01361    0.80714   0.017    0.987    
---
Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 

Residual standard error: 1.398 on 6 degrees of freedom
Multiple R-Squared: 0.6281,    Adjusted R-squared: 0.5041 
F-statistic: 5.066 on 2 and 6 DF,  p-value: 0.05145 

> g<-lm(y~density, subset=tomato==1)
> summary(g)

Call:
lm(formula = y ~ density, subset = tomato == 1)

Residuals:
   Min     1Q Median     3Q    Max 
-1.133 -0.800 -0.100  0.900  1.167 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  12.0222     0.3348  35.907 3.11e-08 ***
density.L     3.9362     0.5799   6.788   0.0005 ***
density.Q    -0.7485     0.5799  -1.291   0.2443    
---
Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 

Residual standard error: 1.004 on 6 degrees of freedom
Multiple R-Squared: 0.8883,    Adjusted R-squared: 0.8511 
F-statistic: 23.87 on 2 and 6 DF,  p-value: 0.001392