Correlation
Case I, X, Y are both normally distributed:
,
then 
Case II, A is normal, B is binary (B=0 or 1),
the correlation(A B) :
Just treat B as a continuous
variable and calculate the Pearson corr of A and B.
Test the significance of this correlation:
Note 
, or 
where t is the 2-sample t-statistics for
testing means under equal variance hypothesis.
Case III, A is normal, B is a categorical
variable with more than 2 categories.
Then 
, where 
is the F-statistics for testing means in ANOVA.
Case IV, if both A, B are binary,
|
B\ A |
1 |
0 |
|
1 |
N11 |
N12 |
|
0 |
N21 |
N22 |
Let 
then define corr(A,B)
as
, (analogous to Case I)
Note 
, where Q is the Pearson 
for testing the association of A, B.
Case V, if both A, B are categorical with more
I and J categories respectively,
Then 
Case VI, both X, Y are non-normal variables.
Let R(xi)
and R(yi) be the rank variables w.r.t.
X and Y, and di= R(xi)-R(yi),
then define Spearman rank correlation as
It can be shown that
.
Eg. 兩個鑑賞家評鑑 10件作品
When n>10, a mimic t-test as
in case II can be applied.
Case VII Altenative of Spearman
rank correlation for small n Kendal's �� :
Step1: rank x and y,
Step2, sort x
Step3, count the disarray of any
of the 
ordered pair
of y, say, d pairs,
Step4, 
e.g.
|
|
a |
b |
c |
d |
e |
|
|
a |
b |
c |
d |
e |
|
x |
83 |
72 |
65 |
79 |
86 |
�� |
x |
2 |
4 |
5 |
3 |
1 |
|
y |
82 |
70 |
74 |
87 |
92 |
|
y |
3 |
5 |
4 |
2 |
1 |
|
|
|
e |
a |
d |
b |
c |
|
�� |
x |
1 |
2 |
3 |
4 |
5 |
|
|
y |
1 |
3 |
2 |
5 |
4 |
D=2, so 
Case VIII 
are non-normal variables: Kendall Coefficient of
Concordance
E.g. k個鑑賞家評鑑 10件作品 (k dependent samples with size n)
Let 
be the rank of ith
subject within sample j, 
be the sum of the rank of subject i
across k sample, and 
be
the mean of 
. Then the coef. is
,
where
,