[R] Std. errors of intercept and slope

[Spencer Graves]

See especially any discussion of "the matrix formulation of 
regression".  I'm sure this is in many books.  I'm not familiar with the 
recent literature, but I know it is in Draper and Smith, Applied 
Regression Analysis and Box, Hunter and Hunter, Statistics for 
Experimenters. 

Briefly, suppose we write the regression model as y = X b + e, where y 
and e are N x 1 vectors, X is an N x k matrix, and e is a vector of 
normal, independent errors with standard deviation s.e.  Then the least 
squares and maximum likelihood estimate of b is

      b.hat = (inverse(X' X))*(X'y),

and the covariance matrix for b.hat is

      var(b.hat) = s.e^2 * inverse(X'X). 

I apologize if this is too terse for you;  if so, please see any good 
book on regress. 

hope this helps.  spencer graves

Ben Bolker wrote:

>  I'm afraid you're going to have to look it up in a basic statistics 
>textbook.
>
>  Ben Bolker
>
>
>On Fri, 26 Sep 2003, Yao, Minghua wrote:
>
>  
>
>>Thanks, Ben.
>>
>>Could you tell me the formula for calculating this sd., given (x_i, y_i)
>>(i=1,2,...,N)?
>>We only have one intercept and slope for them.
>>
>>-Minghua
>>
>>-----Original Message-----
>>From: Ben Bolker [mailto:bolker at zoo.ufl.edu]
>>Sent: Friday, September 26, 2003 4:34 PM
>>To: Yao, Minghua
>>Cc: R Help (E-mail)
>>Subject: Re: [R] Std. errors of intercept and slope
>>
>>
>>
>>  Since the intercept and slope are estimated parameters, they have 
>>sampling distributions described by their means and standard deviations.  
>>The s.d. tells you the size of the uncertainty in intercept & in slope.
>>
>>  This is a pretty basic stats question -- you need to refer to a standard 
>>textbook or reference material ...
>>
>>  Ben Bolker
>>
>>On Fri, 26 Sep 2003, Yao, Minghua wrote:
>>
>>    
>>
>>>Dear all,
>>>
>>>I have the following output generated by linear regression. Since there is
>>>only one regression intercept and one slope for one set of data, what is
>>>      
>>>
>>the
>>    
>>
>>>meaning of std. error for intercept and that of slope? Thanks in advance.
>>>
>>>Sincerely,
>>>
>>>Minghua
>>>
>>>
>>>      
>>>
>>>>data(thuesen)
>>>>attach(thuesen)
>>>>lm(short.velocity~blood.glucose)
>>>>        
>>>>
>>>Call:
>>>lm(formula = short.velocity ~ blood.glucose)
>>>
>>>Coefficients:
>>>  (Intercept)  blood.glucose  
>>>      1.09781        0.02196  
>>>
>>>      
>>>
>>>>summary(lm(short.velocity~blood.glucose))
>>>>        
>>>>
>>>Call:
>>>lm(formula = short.velocity ~ blood.glucose)
>>>
>>>Residuals:
>>>     Min       1Q   Median       3Q      Max 
>>>-0.40141 -0.14760 -0.02202  0.03001  0.43490 
>>>
>>>Coefficients:
>>>              Estimate Std. Error t value Pr(>|t|)    
>>>(Intercept)    1.09781    0.11748   9.345 6.26e-09 ***
>>>blood.glucose  0.02196    0.01045   2.101   0.0479 *  
>>>---
>>>Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 
>>>
>>>Residual standard error: 0.2167 on 21 degrees of freedom
>>>Multiple R-Squared: 0.1737,     Adjusted R-squared: 0.1343 
>>>F-statistic: 4.414 on 1 and 21 DF,  p-value: 0.0479 
>>>
>>>      
>>>
>>>______________________________________________
>>>R-help at stat.math.ethz.ch mailing list
>>>https://www.stat.math.ethz.ch/mailman/listinfo/r-help
>>>
>>>      
>>>
>>    
>>
>
>  
>