[R] Std. errors of intercept and slope
Spencer Graves
spencer.graves at pdf.com
Sat Sep 27 15:33:57 CEST 2003
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
>>>
>>>
>>>
>>
>>
>
>
>
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