[R] linear regression and testing the slope

[(Ted Harding)]

On 08-Jul-09 12:29:40, evrim akar wrote:
> Dear All,
> First of all I would like to say I do not have much knowledge
> about this subject, so most of you can find it really easy.
> I am doing a linear regression and I want to test if the slope
> of the curve is 0. R gives the summary statistics:
> 
> Call:
> lm(formula = x ~ s)
> 
> Residuals:
>       Min        1Q    Median        3Q       Max
> -0.025096 -0.020316 -0.001203  0.011658  0.044970
> 
> Coefficients:
>              Estimate Std. Error t value Pr(>|t|)
> (Intercept)  0.005567   0.016950   0.328    0.750
> s           -0.001599   0.002499  -0.640    0.538
> 
> Residual standard error: 0.02621 on 9 degrees of freedom
> Multiple R-squared: 0.04352,    Adjusted R-squared: -0.06276
> F-statistic: 0.4095 on 1 and 9 DF,  p-value: 0.5382
> 
> what is this t-value for? The explanation in the help file was
> unfortunately not clear to me. How can I test my hypotheses that
> if the slope is 0?
> 
> Thank you in advance,
> regards,
> Evrim

The quantity 't' is the estimated value (-0.001599 for the slope 's')
divided by its estimated standard error (0.002499). Taking the values
as reported by the summary:

  t = -0.001599/0.002499 = -0.639856

which R has reported (to 3 significant figures) as -0.640

The "Pr(>|t|)" is the probability, assuming the null hypothesis that
the slope (coefficient of 's') is zero, that data could arise at random
giving rise to a t-value which, in absolute value, would exceed the
absolute value |t| = |-0.639856| = 0.639856 which you got from your
data.

The relevance of this for testing the hypothesis that the slope is 0
is that, if the slope really is 0, then large values (either way) of
the coefficient of 's' (reported by R as "Estimate") are unlikely.
So if you got a value of "Pr(>|t|)" which was small (conventionally
less that 0.05, or 0.01, etc.) then you would have a value so large
that getting a value at least as large as this if the hypothesis
were true would be unlikely. Therefore it would be more plausible
that the null hypothesis was false.

In your case, the "P-value" Pr(>|t|) = 0.538, so you would be more
likely than not to get an estimate at least as deviant from 0 as the
one you did get, if the null hypothesis were true. Hence the data do
not provide grounds for rejecting the null hypothesis.

Note that not having grounds for rejection does not mean that you
must accept it: a "non-signifcant" t-value is not proof that the
null hypothesis is true.

There is a good basic outline of the t-test in the Wikipedia article
"Student's t-test":

  http://en.wikipedia.org/wiki/Student%27s_t-test

Hoping this helps,
Ted.

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E-Mail: (Ted Harding) <Ted.Harding at manchester.ac.uk>
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Date: 08-Jul-09                                       Time: 14:17:52
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