MATH 376 -- Mathematical Statistics
Example for Lab Class on 5/4

> x <- c(0,1,2,3,4,5,6)
> y <- c(30.6,44.6,58.6,65.3,93.4,101.1,119.6)
> reduced <- lm(y~x)
> full <- lm(y~x + I(x^2))
> anova(full)
Analysis of Variance Table

Response: y
          Df Sum Sq Mean Sq F value    Pr(>F)    
x          1 6145.0  6145.0 276.047 7.687e-05 ***
I(x^2)     1   13.6    13.6   0.611    0.4781    
Residuals  4   89.0    22.3                      
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 
> anova(reduced)
Analysis of Variance Table

Response: y
          Df Sum Sq Mean Sq F value    Pr(>F)    
x          1 6145.0  6145.0  299.34 1.182e-05 ***
Residuals  5  102.6    20.5                

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#
# "By Hand"
#
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> SSEC <- 89.0
> SSER <- 102.6
> n <- 7
> k <- 2
> g <- 1
> Fstat <- ((SSER - SSEC)/(k-g))/(SSEC/(n - k - 1)) 
> Fstat
[1] 0.611236
> 1 - pf(Fstat,1,4)
[1] 0.4780113

###################################################################
#
# by R
#
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> anova(reduced,full)
Analysis of Variance Table

Model 1: y ~ x
Model 2: y ~ x + I(x^2)
  Res.Df     RSS Df Sum of Sq     F Pr(>F)
1      5 102.643                          
2      4  89.042  1    13.601 0.611 0.4781

####################################################################
#
#  Why?
#
####################################################################

> summary(full)

Call:
lm(formula = y ~ x + I(x^2))

Residuals:
      1       2       3       4       5       6       7 
-0.2833  0.9143  1.3071 -6.4048  6.4786 -1.8429 -0.1690 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)  30.8833     4.1183   7.499  0.00169 **
x            12.4000     3.2149   3.857  0.01819 * 
I(x^2)        0.4024     0.5148   0.782  0.47810        <------------- ?????
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 4.718 on 4 degrees of freedom
Multiple R-squared: 0.9857,     Adjusted R-squared: 0.9786 
F-statistic: 138.3 on 2 and 4 DF,  p-value: 0.0002031