How to get p-val from correlation coefficient and number of sample?

Hi, all!

We know that in MATLAB, [r,p]=corrcoef(A) could return the correlation coefficient and p-val.

But who knows how p-val is calculated, what is the formulas?

Thanks for your help!

[R. Gokulakannan]

>>m=8+4

Answer 1

[Wayne King]

The p-value is based on a t statistic

t = r* sqrt((n-2)/(1-r^2))

which has n-2 degrees of freedom.

You can find a discussion of the sampling distribution of r both when the null hypothesis is that r=0 and when you are testing for nonzero values of r here:

http://luna.cas.usf.edu/~mbrannic/files/regression/corr1.html

[frewise]

Thank you!

But how we get p-val from t? Is there a workable

formula?

Answer 2

[Wayne King]

??? I told you that the the t above is t-distributed with n-2 degrees of freedoms. Once you know the distribution, the p-value is the probability that you would observe a value at least that extreme in that distribution.

For example, if you had a t-value of 3 and you had 20 samples, then

1-tcdf(3,18)

gives you the probability in the upper tail. Since the t-distribution is symmetric. If you are doing a two-sample hypothesis test, you have to double that probability to get the p-value (you have equal probability in the lower tail)

For example:

rng default
x = randn(18,2);
[r,p] = corrcoef(x);

gives

r =  -0.0242
t = r*sqrt(16/(1-r^2));
2*tcdf(t,16)

which is exactly what MATLAB gives in the p matrix.

[frewise]

Thank you！

Answer 3

[Gregory Pelletier]

Here is how matlab calculates the p-value in corrcoef (see p_check below for the manual calculation of the p-value compared with p from corrcoef):

load hospital
X = [hospital.Weight hospital.BloodPressure];
[R, p] = corrcoef(X) 
N = size(X,1);
t = sqrt(N-2).*R./sqrt(1-R.^2);
s = tcdf(t,N-2);
p_check = 2 * min(s,1-s)
% R =
   % 1.0000e+00   1.5579e-01   2.2269e-01
   % 1.5579e-01   1.0000e+00   5.1184e-01
   % 2.2269e-01   5.1184e-01   1.0000e+00
% p =
   % 1.0000e+00   1.2168e-01   2.5953e-02
   % 1.2168e-01   1.0000e+00   5.2460e-08
   % 2.5953e-02   5.2460e-08   1.0000e+00
% p_check =
            % 0   1.2168e-01   2.5953e-02
   % 1.2168e-01            0   5.2460e-08
   % 2.5953e-02   5.2460e-08            0