[R] Linear Regression

[David Winsemius]

On Oct 12, 2010, at 9:01 AM, Vittorio Colagrande wrote:

> Dear R-group,
>
> We have begun to use it for teaching Statistics. In this context we  
> have run into a problem with linear regression
>
> where we found the results of are confusing.
>
> Specifically, considering the data:
>
> x=c(4,5,6,3,7,8,10,14,13,15,6,7,8,10,11,4,5,17,12,11)
> y=c(rep(7,20))
> #and settings
> regress=lm(y~x)
> summary(regress) gives the following results:
>
>             Estimate Std. Error    t value Pr(>|t|)
> (Intercept)  7.000e+00  8.623e-17  8.118e+16   <2e-16 ***
> x           -1.116e-17  8.956e-18 -1.247e+00    0.229
> ---
> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
>
> Residual standard error: 1.565e-16 on 18 degrees of freedom
> Multiple R-squared: 0.6416,     Adjusted R-squared: 0.6217
>
>
>
> Other statistical packages respond that the analysis can not be  
> done. We think that the results of R-squared  does not seem to  
> express the variability of y explained by x. We would greatly  
> appreciate any clarification you could provide.

It is expressing the degree to which the estimate of the intercept  
"explains" the tendency of the data to be away from the null  
hypothesis of y=0. Consider what you get from two (equivalent to each  
other) lm calls:

 > regress2=lm( I(y-7)~x )
 > summary(regress2)

Call:
lm(formula = I(y - 7) ~ x)

Residuals:
    Min     1Q Median     3Q    Max
      0      0      0      0      0

Coefficients:
             Estimate Std. Error t value Pr(>|t|)
(Intercept)        0          0      NA       NA
x                  0          0      NA       NA

Residual standard error: 0 on 18 degrees of freedom
Multiple R-squared:   NaN,	Adjusted R-squared:   NaN
F-statistic:   NaN on 1 and 18 DF,  p-value: NA

 > y2=y-7
 > regress2=lm( y2~x )
 > summary(regress2)

Call:
lm(formula = y2 ~ x)

Residuals:
    Min     1Q Median     3Q    Max
      0      0      0      0      0

Coefficients:
             Estimate Std. Error t value Pr(>|t|)
(Intercept)        0          0      NA       NA
x                  0          0      NA       NA

Residual standard error: 0 on 18 degrees of freedom
Multiple R-squared:   NaN,	Adjusted R-squared:   NaN
F-statistic:   NaN on 1 and 18 DF,  p-value: NA


>

-- 

David Winsemius, MD
West Hartford, CT