On Correlation Coefficients and Sample Size

New Relevant Tool: https://irthoughts.wordpress.com/2018/09/14/regression-correlation-calculator-updates-and-improvements/

Today I updated my Tutorial on Correlation Coefficients to include a new section on the effect of sample size on the significance of correlation coefficients. This was motivated by some comments from search engine marketers on correlation strengths. (http://searchenginewatch.com/3641002). The new material might help those interested in learning whether a reported correlation coefficient is statistically different from zero. It is given below. Enjoy it.

The problem with correlation strength scales is that these say nothing about how the size of a sample impacts the significance of a correlation coefficient. This is a very important issue that is now addressed.

Consider three different correlation coefficients: 0.50, 0.35, and 0.17. Assume that we want to test that there is no significant relationship between the two variables at hand. The null hypothesis (H0) to be tested is that these r values are not statistically different from zero (rho = 0). How to proceed?

As recommended by Stevens (17), for rho = 0, H0 can be tested using a two tailed (i.e.,two sided) t-test at a given confidence level, usually at a 95% level. If t_calculated ≥ t_table, H0 is rejected. However, if t_calculated < t_table H0 is not rejected and there is no significant correlation between variables.

Here t_calculated is computed as r/SEr = r*SQRT[((n – 2)/(1 – r²))] while t_table values are obtained from the literature (http://en.wikipedia.org/wiki/Student%27s_t-distribution#Table_of_selected_values ). Table 2 summarizes the result of testing the null hypothesis at different sample size values.

Table 2. H0 tests at different sample sizes; two-tailed, 95% confidence.
n	df = n – 2	r	SEr	t(calc)	t (0.95)	Reject (H0 : rho = 0)?
5	3	0.50	0.50	1.000	3.182	don’t reject
10	8	0.50	0.31	1.633	2.306	don’t reject
12	10	0.50	0.27	1.826	2.228	don’t reject
14	12	0.50	0.25	2.000	2.179	don’t reject
20	18	0.50	0.20	2.449	2.101	reject
30	28	0.50	0.16	3.055	2.048	reject
40	38	0.50	0.14	3.559	2.024	reject
50	48	0.50	0.13	4.000	2.011	reject
5	3	0.35	0.54	0.647	3.182	don’t reject
10	8	0.35	0.33	1.057	2.306	don’t reject
12	10	0.35	0.30	1.182	2.228	don’t reject
14	12	0.35	0.27	1.294	2.179	don’t reject
20	18	0.35	0.22	1.585	2.101	don’t reject
30	28	0.35	0.18	1.977	2.048	don’t reject
40	38	0.35	0.15	2.303	2.024	reject
50	48	0.35	0.14	2.589	2.011	reject
5	3	0.17	0.57	0.299	3.182	don’t reject
10	8	0.17	0.35	0.488	2.306	don’t reject
12	10	0.17	0.31	0.546	2.228	don’t reject
14	12	0.17	0.28	0.598	2.179	don’t reject
20	18	0.17	0.23	0.732	2.101	don’t reject
30	28	0.17	0.19	0.913	2.048	don’t reject
40	38	0.17	0.16	1.063	2.024	don’t reject
50	48	0.17	0.14	1.195	2.011	don’t reject

The table addresses at which size level an r value is high enough to be statistically significant.

For n = 14, all three r values (0.50, 0.35, and 0.17) are not statistically different from zero.

For n = 30, r = 0.50 is statistically different from zero while r = 0.35 and r = 0.17 are not.

Conversely, r = 0.50 is not statistically different from zero when n is equal or less than 14 while r = 0.35 is not different from zero when n is equal or less than 30.

Finally, r = 0.17 is not statistically different from zero at any of the sample sizes tested.

Related post: https://irthoughts.wordpress.com/2016/04/18/virus-evolution-citation/