The larger the sample size, the lower the value of r at which a significant result occurs. For small samples it is possible to have a high correlation coefficient which is not significant and for large samples it is possible to have a small correlation coefficient which is significant. Thus it is important to look at the value of r as well as the p-value.

A small coefficient does not necessarily imply a lack of a relationship any more than a large coefficient implies a strong relationship. It depends on the type of relationship and the data used to characterize it.

You also need to speak to the actual effect size. Effect size measures either measure the sizes of associations or the sizes of differences. You already know the most common effect-size measure, as the correlation/regression coefficients r and R are actually measures of effect size. Because r covers the whole range of relationship strengths, from no relationship whatsoever (zero) to a perfect relationship (1, or -1), it is telling us exactly how large the relationship really is between the variables we've studied -- and is independent of how many people were tested. Cohen provided rules of thumb for interpreting these effect sizes, suggesting that an r of |.1| represents a 'small' effect size, |.3| represents a 'medium' effect size and |.5| represents a 'large' effect size. 

The larger the sample size, the lower the value of r at which a significant result occurs. For small samples it is possible to have a high correlation coefficient which is not significant and for large samples it is possible to have a small correlation coefficient which is significant. Thus it is important to look at the value of r as well as the p-value.

A small coefficient does not necessarily imply a lack of a relationship any more than a large coefficient implies a strong relationship. It depends on the type of relationship and the data used to characterize it.

You also need to speak to the actual effect size. Effect size measures either measure the sizes of associations or the sizes of differences. You already know the most common effect-size measure, as the correlation/regression coefficients r and R are actually measures of effect size. Because r covers the whole range of relationship strengths, from no relationship whatsoever (zero) to a perfect relationship (1, or -1), it is telling us exactly how large the relationship really is between the variables we've studied -- and is independent of how many people were tested. Cohen provided rules of thumb for interpreting these effect sizes, suggesting that an r of |.1| represents a 'small' effect size, |.3| represents a 'medium' effect size and |.5| represents a 'large' effect size. 

Just as the statistics suggest, you can say that the effect is statistically significant but small. You certainly cannot say there is "no effect", unless you can show that the effect is significantly smaller than the smallest possible meaningful effect size (using e.g. equivalence hypothesis testing [http://daniellakens.blogspot.hk/2016/05/absence-of-evidence-is-not-evidence-of.html] or the "small telescopes" method [http://datacolada.org/wp-content/uploads/2015/05/Small-Telescopes-Published.pdf]), or show that your study did not have sufficient power to meaningfully detect effects of that size (http://www.stat.columbia.edu/~gelman/research/published/retropower20.pdf).

What value of R2 counts as "too small to be meaningful" depends on your field of study and your research question. In some contexts, R2 of .1 may be too negligible to be of meaningful practical importance; in other contexts, it may be big enough. Without some practical, theory-based predictions about how big an effect "should" be, then it's difficult to say whether any given R2 value is or is not "too" small.

In this situation (low R**2 and high p-value) I would assume that other parameters are active, beyond the one you looked at. As stated in the previous answer, a high p-value should not be ignored. A low R**2 means that your factor does not explain much of the existing variation. This might be due to excessive random variation, due to other parameters or (to a lower extend) to a substantial non-linearity. In practice you will often see all three of the "issues" at the same time.

It is highly possible to  receive  a highly significant   F  -  Statistic   with  a very low   RSQ  (R2) because   the   sample  size is really large. Moreover the partial  regression  coefficient    b  could be   close  to  zero   such as 0.0001 and highly significant. In this case I recommend that you better discuss the practical  meaning in addition to the statistical  significance   by relating the findings of your study to the findings of previous  research that you have used in building your regression model

If the intended purpose behind the regression models is to predict the dependent variable, then you may stress on R-square value. On the other hand, if the objective is to establish the statistically influential variables, then p value is to be given more weightage.

Regards

Ashu

As was nicely stated in the first post, it's better practice to place emphasis on the magnitude of the R^2 value (or any effect size for that matter), rather than whether it significantly deviates from zero. I have an admittedly extreme bias against NHST. But in my opinion, it's of little use to place faith in the p-value at all. With large enough samples, even the smallest of correlations can be found significant. On the flip-side, sampling variability can lead to significant and spurious correlations with small samples. I'd like to add that context is an important consideration when reporting effect size as well.  I don't subscribe to the typically accepted "rules of thumb", because an R^2 of .02 can be an important finding in one setting and meaningless in another.

Sure, There is a different between P-vale and R2 , we interest by the value of R , not the value of  R2 Suppose:

(1)R = -1 this means completely inverse relation and R here is minimum low , meaning one of the variable effect strongly on the second variable and P-Value will be smaller than 0.05 too much may be approaching to 0 e.g. 0.0001 so there is a strongly  effective while R2=1 and R=-1

(2) R = 0.6 is strongly increment relation on the other hand, the corresponding P-Value will be greater than 0.05  meaning there are not effective between variables although  R2 = 0.36 more lower