# Explanation of Pearson correlation changing with the degrees of freedom

I did Pearson correlation and I got this results:

Pearson's product-moment correlation

data:  m$$P.x and m$$P.y
t = 2823.5, df = 852010, p-value < 2.2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
0.9502929 0.9507029
sample estimates:
cor
0.9504983

The elements I got in he previous correlation test are the same as bellow plus additional ones (df is higher)

cor.test(t$$P.x, t$$P.y, method = "pearson", conf.level = 0.95)

Pearson's product-moment correlation

data:  t$$P.x and t$$P.y
t = 2000, df = 2e+06, p-value <2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
0.801 0.801
sample estimates:
cor
0.801

How is this difference in correlation explained by adding more "df"?

I am comparing here two set of p values. (m$$P.x and m$$P.y)

My data looks like this:

> dim(m)
[1] 852688      3
SNP    P.x    P.y
1: rs10000012 0.7563 0.7563
2: rs10000013 0.7007 0.7324
3:  rs1000002 0.1870 0.2263
4: rs10000029 0.9078 0.9078
5:  rs1000003 0.6359 0.6359
6: rs10000033 0.7704 0.7162

...

The degrees of freedom for a correlation is just the number of complete data points (after excluding any missing data) minus two. Your first data set, m has $$852,010 + 2 = 852,012$$ points. The second, t, has around $$2e^6 = 2,000,000$$. You're doing the tests with different data (even if one is a subset of the other), so you're getting different results.