**Correlation Coefficients**

**Pearson
correlation coefficient:**

If the variables fit normal distribution, *select the method as “ Pearson”*, Pearson product-moment correlation
coefficient (ρ) will be calculated, which is calculated as the covariance
of the variables divided by their standard deviations.

*If
the variables did not fit normal distribution, two methods “Spearman (rank
based)” or “Kendall (rank based)” can be applied**.*

**Spearman rank based correlation
coefficient:**

The two variables will be ranked first. Tied values are assigned a rank equal to the average of their positions in the ascending order of the values. The correlation (ρ) will be calculated (same formula as above) based on ranked value after that.

**Kendall rank correlation coefficient**:

Kendall rank correlation
coefficient, commonly referred to as **Kendall's
tau (τ) coefficient**, measures the portion of ranks that match between
two data sets.

Let (*x*_{1}, *y*_{1}),
(*x*_{2}, *y*_{2}), …, (*x _{n}*,

The Kendall
τ coefficient is defined as:

The
denominator is the total number of pairs, so the coefficient must be in the
range −1 ≤ *τ* ≤ 1.

·
If the agreement
between the two rankings is perfect (i.e., the two rankings are the same) the coefficient
has value 1.

·
If the
disagreement between the two rankings is perfect (i.e., one ranking is the
reverse of the other) the coefficient has value −1.

·
If *X* and *Y*
are independent, then we would expect the coefficient to be approximately zero.

Below is the **sample
input window**

*In the above
example, only one list of variables in “Selected variable(s)” box was given, no
“With variables (Options)” list, Empower will calculate correlation for any
possible pairs of 2 variables. If “With
variables” is given, Empower will calculate correlation between each of
“Selected variables” with each of the “With variables”.*

Below is the **sample
output** **and explanation** of the
above model:

Correlation

Pearson's product-moment correlation

data: AGE and SBP

t
= 15.0454, df = 793, p-value
< 2.2e-16

alternative
hypothesis: true correlation is not equal to 0

95
percent confidence interval:

0.4153127 0.5236118

sample
estimates:

cor

0.4712365

Pearson's product-moment correlation

data: AGE and DBP

t
= 9.7845, df = 793, p-value
< 2.2e-16

alternative
hypothesis: true correlation is not equal to 0

95
percent confidence interval:

0.2647198 0.3888680

sample
estimates:

cor

0.3282105

Pearson's product-moment correlation

data: BMI and SBP

t
= 0.2441, df = 793, p-value
= 0.8072

alternative
hypothesis: true correlation is not equal to 0

95
percent confidence interval:

-0.06089938 0.07815404

sample
estimates:

cor

0.00866924

Pearson's product-moment correlation

data: BMI and DBP

t
= 0.1835, df = 793, p-value
= 0.8545

alternative
hypothesis: true correlation is not equal to 0

95
percent confidence interval:

-0.06304556 0.07601237

sample
estimates:

cor

0.006514901

In the above output, Person’s correlation coefficient
is calculated. The output includes the value (sample estimate) of correlation
coefficient, its 95% confidence interval, and statistical significance test
(p-value). For example, the correlation
between AGE and SBP is 0.4712365, 95% CI is 0.4153127 – 0.5236118, P value is
<2.2e-16.

**In addition,
Empower also output scatter plots for each pair of variables:**