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> options(show.signif.stars=FALSE)
> ### Several models fit to the Formaldehyde data
> lm1 <- lm(optden ~ 1 + carb, Formaldehyde)
> 
> ## This formula can also be written optden ~ carb because the
> ## intercept term is implicit.  Some people prefer to include it and
> ## some don't.
> 
> lm0 <- lm(optden ~ 0 + carb, Formaldehyde)
> ## To suppress the intercept term you must include 0 + or - 1, as in
> ## optden ~ carb - 1, in the formula
> 
> ## Residual sums of squares for the two models
> (RSS <- c(lm0 = deviance(lm0), lm1 = deviance(lm1)))
         lm0          lm1 
0.0003307264 0.0002992000 
> 
> ## Comparative anova
> anova(lm0, lm1)
Analysis of Variance Table

Model 1: optden ~ 0 + carb
Model 2: optden ~ 1 + carb
  Res.Df        RSS Df  Sum of Sq      F Pr(>F)
1      5 0.00033073                            
2      4 0.00029920  1 3.1526e-05 0.4215 0.5516
> 
> ## Corresponding t-statistic from coefficient table for lm1
> coef(summary(lm1))
               Estimate Std. Error    t value     Pr(>|t|)
(Intercept) 0.005085714 0.00783368  0.6492115 5.515953e-01
carb        0.876285714 0.01353454 64.7444207 3.409192e-07
> 
> ## Compare to F statistic in the comparative anova
> coef(summary(lm1))["(Intercept)", "t value"]^2
[1] 0.4214755
> 
> ### Testing for non-zero values of coefficients by using an offset
> 
> ## Suppose we wish to test
> ##   H_0: beta_0 = 0.001 versus H_a: beta_0 != 0.001
> ## within the full model lm1
> 
> ## We can fit
> lm2 <- lm(optden ~ 0 + carb + offset(rep(0.001, 6)), Formaldehyde)
> 
> ## and compare
> anova(lm2, lm1)
Analysis of Variance Table

Model 1: optden ~ 0 + carb + offset(rep(0.001, 6))
Model 2: optden ~ 1 + carb
  Res.Df        RSS Df  Sum of Sq     F Pr(>F)
1      5 0.00031955                           
2      4 0.00029920  1 2.0347e-05 0.272 0.6295
> 
> ## It may be more obvious to fit the full model as
> lm3 <- lm(optden ~ 1 + carb + offset(rep(0.001, 6)), Formaldehyde)
> 
> ## which is equivalent to model lm1
> all.equal(fitted(lm1), fitted(lm3))
[1] TRUE
> all.equal(coef(lm1), coef(lm3) + c(0.001, 0))
[1] TRUE
> 
> ## but more obvious about the comparison
> anova(lm2, lm3)
Analysis of Variance Table

Model 1: optden ~ 0 + carb + offset(rep(0.001, 6))
Model 2: optden ~ 1 + carb + offset(rep(0.001, 6))
  Res.Df        RSS Df  Sum of Sq     F Pr(>F)
1      5 0.00031955                           
2      4 0.00029920  1 2.0347e-05 0.272 0.6295
> ## and
> coef(summary(lm3))
               Estimate Std. Error    t value     Pr(>|t|)
(Intercept) 0.004085714 0.00783368  0.5215575 6.295245e-01
carb        0.876285714 0.01353454 64.7444207 3.409192e-07
> coef(summary(lm3))["(Intercept)", "t value"]^2
[1] 0.2720222
> 
> ## In general, the sequential sums of squares for different terms come
> ## from the "effects" vector
> effects(lm1)
 (Intercept)         carb                                                     
-1.121458054  0.559955028  0.005139411  0.009915668  0.010691926 -0.007755559 
attr(,"assign")
[1] 0 1
attr(,"class")
[1] "coef"
> effects(lm1)[2]^2                       # sum of squares due to carb
     carb 
0.3135496 
> sum(effects(lm1)[-(1:2)]^2)             # residual sum of squares
[1] 0.0002992
> anova(lm1)                              # arranged in an anova table
Analysis of Variance Table

Response: optden
          Df   Sum Sq  Mean Sq F value    Pr(>F)
carb       1 0.313550 0.313550  4191.8 3.409e-07
Residuals  4 0.000299 0.000075                  
> 
> proc.time()
   user  system elapsed 
  0.540   0.040   0.553