Student's t test comes from the normality of the estimated coefficients. The standard formula is for testing whether βk = a, where a a fixed real number and βk is the real parameter (the value of which is unknown, since only the estimated βk* parameter is known). Because of normality:

βk* ~ N(βk, σ2)

Which, under the hypothesis of equality to a (which we are testing) means that the following transform follows the normal distribution:

(βk* - a) / σ2 ~ N(0,1)

As before, σ is unknown (because we don't have access to the underlying distribution of the error, only the observed one) and we must examine the same equation with σ* instead (which, if you remember, is the empirical standard deviation, which you computed from your observation set). I don't have the justification right here, but it follows a Student law with n-p degrees of liberty (n the number of observations, p the number of parameters):

T* = (βk* - a) / σ*2 ~ Tn-p

Therefore, you have estimated an empirical T (your t-test value), and you must now compare it against the distribution function of the Student law to determine how likely it is to observe your particular empirical measure.

EDIT: and σ is the standard deviation of the parameter βk*, not of the entire model, as implied by the first equation above.

Quote:Original post by visage
So my question here is, without knowing T* or a, is there a way to find one of them?

Well, finding the t-value without knowing a is silly. The reason is that the t-value allows you to determine how likely it is for the real-world (unknown) coefficient βk to be equal to a given a. That is, you (the stastician) choose any a you wish, and you get the t-value for the hypothesis βk = a (and since the Student law distribution is known, you know how likely it is for you to observe the empirical t-value if the hypothesis were true, so that you may reject it). Long stories short, you don't have to find or compute a, you have to choose it. The t-value is computed as mentioned above:

T*(a) = (βk* - a) / σ*2

The typical t-value, the one displayed in most statistical analysis programs, is the one for a = 0, because it's very frequent for statisticians to choose a = 0 (since a zero coefficient means there is no impact of the descriptive variable on the explained variable, which is something interesting to know). However, there are as many t-values as there are possible values for a (and you can choose any real you wish).

In the case of a = 0, the t-value becomes equal to:

T*(0) = βk* / σ*2

That is, coefficient over squared standard error (yes, the standard error is usually the empirical standard deviation of the noise in your model, as opposed to the theoretical one, noted σ).