An interesting question. Correlation coefficient of even less than 0.60 could be significant , depending upon degree of freedom or r value at given number of observation. More is the value of r , higher is the degree of correlation between the two variables or more than two variables...either linear or quadratic relationship..

Anoop Kumar Srivastava

Thank so much sir for your response.

It's true that we can tell that the higher level of r or r squared values representing more correlation between the two variables. But is there any lower limit of r or r squared values below which we can say the two variables are not correlated though the p value is significant?

I doubt, we can interpret like this, r value is dependent upon degree of freedom. Usually even 0.60 is a significant value at higher degree of freedom , at 5% or 2% sig. level

I doubt, we can interpret like this, r value is dependent upon degree of freedom. Usually even 0.60 is a significant value at higher degree of freedom , at 5% or 1% sig.level

Anoop Kumar Srivastava

Thank you Sir.

Krishna -

I would not put so much emphasis on r-square, or R-square, as this measure is too vulnerable to factors which can greatly change its value. See the following from Penn State:

https://online.stat.psu.edu/stat462/node/98/ 

(I looked it over, and I think you need to be aware that a quadratic regression, though curved, is still considered to be linear regression.)

They say this is "...the most often misused and misunderstood..." statistical measure. I almost agree. I think p-values may be even worse.

Notice that problems Penn State noted can often be found by looking at a graph. Often a single statistic cannot tell you everything that a graph can. If you see something in it and are concerned that your sample size is too small and that you may be over-interpreting a graph, I suggest that if you cannot obtain more data, you can leave out a little in a rotating manner, to see if the graph changes.

Checking a regression model fit can be done through "graphical residual analysis." You can plot the predicted y values on the x-axis, and the estimated residuals on the y-axis. You might check for overfitting to a specific data set by a "cross-validation" of some kind.

But I would not rely on various r-square levels as a method of delineation.

Also, it sounds like you might be looking at various independent variables, one at a time, with your dependent variable. But when you do multiple linear regression, you may find that different independent variables work better together than others, or in different combinations. Again I suggest a graphical residual analysis.

Cheers - Jim