Spearman Rank Correlation

It calculates the Spearman rank correlation coefficient.

Description

It calculates the Spearman rank correlation coefficient from 2 or more data sets, and the associated t-test and p-values. The code is adapted with major changes from the Numerical Recipes book (http://www.nr.com/)

Example:
>> x = [1 2 3 3 3]';
>> y = [1 2 2 4 3; rand(1,5)]';
>> [r,t,p] = spear(x,y)

>> [r,t,p]=spear(x,y)

r =

    0.8250 -0.6000

t =

    2.5285 -1.2990

p =

    0.0855 0.2848

• Required Products: Statistics Toolbox
• MATLAB release: MATLAB 7 (R14)

[wood ma]

very useful

[Xiang Shi-Ming]

It is the excellent one

[Brad Vines]

Glad to find it.

[Jonathan D. Nelson]

note that input matrixes should be n-by-1 for the list one is comparing with, and n-by-x where there are x rank correlations one wishes to compute. outputs are [correlation, t-value, p-value].

[M Matsuhashi]

Nice to use, but p significance value is incorrect for small N (<10)

[Xiaoyang Tan]

Thanks

[Dan S.]

I got this error using Matlab release 13:

EDU>> spear A B
Warning: Divide by zero.
(Type "warning off MATLAB:divideByZero" to suppress this warning.)
> In C:\matlab_sv13\work\spear.m at line 54
??? Undefined function or variable 'tcdf'.

Error in ==> C:\matlab_sv13\work\spear.m
On line 68 ==> p=2*(1-tcdf(abs(t),N-2));

maybe "tcdf" is a Matlab release 14 script/function?

[S Nishimoto]

Thanks. Note that the script needs Statistic toolbox
as it uses a function "tcdf".
With the toolbox, it can work on Release 13.

[someguy someguy]

err..this is not spearman but pearson cc

[someguy someguy]

please ignore my comments. i was drunk!

[buffalo fbi]

xp

[Linda S.]

The built-in Matlab function corr() can calculate Spearman's rho like so:

[rho, pvalue] = corr(x, y, 'type', 'spearman')

What's a bit confusing is that p-values produced by spear() are different from those from corr():

>> x = [1 2 3 4 5 ]';
>> y = [8 6 7 5 4]';
>> [rho, pvalue] = corr(x, y, 'type', 'spearman')
rho =
   -0.9000
pvalue =
    0.0833
>>
>>
>> [r, t, p] = spear(x,y)
r =
   -0.9000
t =
   -3.5762
p =
    0.0374

Any idea what's going on?

[Peter Li]

Nice implementation straight from Numerical Recipes. A simpler approach with the same result:

function [rho, rhop, rholo, rhoup] = spear(X)
% X is a matrix whose columns are variables
% and whose rows are observations

% Generate the ranks matrix Z for each column
% of X. Tying ranks are averaged as is
% standard.
Y = sort(X);
for col = 1:size(X,2)
    for row = 1:size(X,1)
        Z(X(:,col) == X(row,col),col) = mean(find(Y(:,col) == X(row,col)));
    end
end

[rho, rhop, rholo, rhoup] = corrcoef(Z);

[Benjamin Levy]

A small point: the normal approximation for the probability values is valid for n>30 (Conover, Practical Non-parametric Statistics, 1980, Table A10). I would suggest alerting the reader/user to this in returning the p value. A modification of the code to use the rank-based critical values for Spearman's rho would be trivial (I'm going to do for myself, anyway). For Linda S., the difference in Pearson's r (corr function) and Spearman's rho diminishes as sample size grows so that the discrete ranks-based distribution becomes more and more like a continuous pdf. Try it with synthetic data for yourself.

[Joost de Winter]

Very helpful. Too bad that the function did not automatically handle rows instead of columns, but this was easily corrected.

[Kyaw Tun]

I found this error

??? Undefined command/function 'tcdf'.

Error in ==> spear at 91
p=2*(1-tcdf(abs(t),N-2));

[Sunnia Chai]

It is a helpful and actual-working-fine program. I ran the SPSS cp the result, and it is matched. for tcdf : google it, you will find a shared code. Just download it and add it in your work folder. For input example: x = [1 2 3]';y = [8 2 7]';spear(x,y);

[Sergio Vallina]

It does not handle NAN values. See corrected *.m file below (it handle NAN):

*************************************
function [r,t,p]=spear(x,y)
%Syntax: [r,t,p]=spear(x,y)
%__________________________
%
% Spearman's rank correalation coefficient.
%
% r is the Spearman's rank correlation coefficient.
% t is the t-ratio of r.
% p is the corresponding p-value.
% x is the first data series (column).
% y is the second data series, a matrix which may contain one or multiple
% columns.
%
%
% Reference:
% Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P.(1996):
% Numerical Recipes in C, Cambridge University Press. Page 641.
%
%
% Example:
% x = [1 2 3 3 3]';
% y = [1 2 2 4 3; rand(1,5)]';
% [r,t,p] = spear(x,y)
%
%
% Products Required:
% Statistics Toolbox
%
% Alexandros Leontitsis
% Department of Education
% University of Ioannina
% 45110- Dourouti
% Ioannina
% Greece
%
% University e-mail: [e-mail]
% Lifetime e-mail: [e-mail]
% Homepage: http://www.geocities.com/CapeCanaveral/Lab/1421
%
% 3 Feb 2002.

for i=1:size(y,2)
    
    % quito NaN:
    x=x(:);
    yi=y(:,i);
    
    H=find(x>=0 & yi>=0);
    x=x(H);
    yi=yi(H);
    
    % x and y must have equal number of rows
    if size(x,1)~=size(yi,1)
error('x and y must have equal number of rows.');
    end
    
    % Find the data length
    N = length(x);
    
    % Get the ranks of x
    R = crank(x)';

    % Get the ranks of y
    S = crank(yi)';
    
    % Calculate the correlation coefficient
    r(i) = 1-6*sum((R-S).^2)/N/(N^2-1);
    
end

% Calculate the t statistic
if r == 1 | r == -1
    t = r*inf;
else
    t=r.*sqrt((N-2)./(1-r.^2));
end

% Calculate the p-values
p=2*(1-tcdf(abs(t),N-2));

function r=crank(x)
%Syntax: r=crank(x)
%__________________
%
% Assigns ranks on a data series x.
%
% r is the vector of the ranks
% x is the data series. It must be sorted.
%
%
% Reference:
% Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P.(1996):
% Numerical Recipes in C, Cambridge University Press. Page 642.
%
%
% Alexandros Leontitsis
% Department of Education
% University of Ioannina
% 45110- Dourouti
% Ioannina
% Greece
%
% University e-mail: [e-mail]
% Lifetime e-mail: [e-mail]
% Homepage: http://www.geocities.com/CapeCanaveral/Lab/1421
%
% 3 Feb 2002.

u = unique(x);
[xs,z1] = sort(x);
[z1,z2] = sort(z1);
r = (1:length(x))';
r=r(z2);

for i=1:length(u)
    
    s=find(u(i)==x);
    
    r(s,1) = mean(r(s));
    
end
***********************************

[Stephen Cowen]

tdcf is a function in the matlab Statistics Toolbox

[Priscilla Mabhande]

Good reading, very informative

[Robert Schleicher]

I cross-checked the results with SPSS 14 on a trial basis and got slightly different values. That's because this version doesn't account for tied ranks (which occured in my test), I suppose.

[Lana cfai]

im using matlab2006a, when i type in the coding it appear
"Undefined command/function 'spear'"
can i know why?
please advise..

[Elisabeth Larsson]

Warning if you have missing values! This program gave significant correlation with a column of NaNs!

[Josue Alvarez]

I would like to calculate the spearman correlation of two matrix, how can I?