Interpreting the relation between p-value and rho value

Question

I am applying Spearman correlation test on two different variables. I'm getting results of p-value and rho as follows:

Case    p-value                 rho
1       0.0004077509            0.6129457645
2       0.0059893012            0.4978762712
3       0.0091239476            -0.1592959273
4       0.1368309165            0.7281007785
5       2.22722438947104E-08    0.7456261445

I have confusion about interpreting these results. Is it correct to say that, in the first case, there is a statistically significant and strong correlation between the two sets since the p-value < 0.05 and rho is high? It's confusing how I can relate the p-value and rho when interpreting the relation?

My understanding so far is that when the p-value < 0.05 and rho is higher (i.e., > 0.6) that means the correlation between the two variables is strong and positive. If my understanding is true, what about other cases such as the third case? I couldn't really find a proper interpretation for such a case. Can you please clarify that to me? BTW, I provided the 5 cases that confused me.

Answer 1

Assuming you used the standard test that comes with this correlation ranking, the null hypothesis of your test is that $H_0:\rho=0$. The alternative hypothesis of your test is $H_1: \rho \ne 0$. Having a low p-value is strong evidence against the null hypothesis and in favor of the alternative hypothesis.

Should you decide on a cut-off p-value of 0.05 than p-values below that will cause you to reject the null hypothesis and accept the alternative instead.

Applying that reasoning to case 3, you would say that $\rho \ne 0$. The fact that it's negative only implies that there is a negative correlation, that is, if the one variable goes up, the other goes down.

For case 4 you have a pretty strong correlation, but the p-value is above 0.05. That means that you do NOT have very strong evidence against $H_0:\rho=0$. So even though the correlation itself is relatively strong, the value you found is still prety likely to occur if $\rho=0$