How to compare two correlation factor values? In case I have two models, one of them has correlation factor of 0.7 and the other one has 0.85 how can I compare them? Is it meaningful to say that there is a relative increase of (0.85-0.7)/0.7 = 0.21 in the correlation factor? Or makes more sense to compare the absolute difference: there is an increase of 0.85-0.7 = 0.15 in the correlation?
Also, is an absolute increase of 0.15 good?
 A: 0.7 and 0.85 are both strong correlations. One question is whether they are different at all from a statistical perspective, or whether the observed differences could be just due to random sampling.
Assuming that you have Pearson's correlation coefficients from a sample, then you can construct a confidence interval for their difference based on the general formula for a confidence interval of a normally distributed variable:
$$ CI = X \pm Z_{(1-\alpha/2)} SE $$
where $X$ is the point estimate of the statistic of interest (which in your case will be the difference in the sample correlations), $\alpha$ is the confidence level, $Z_{(1-\alpha/2)}$ is the $(1-\alpha/2)$ percentile of the normal distribution and $SE$ is the standard error of the statistic of interest.
If this interval contains zero then you can conclude that there is no statistically significant difference, at the $\alpha\%$ level, between the two correlations. In other words, putting it simply, it is plausible that the actual difference between the two correlations is zero.
If the confidence interval does not contain zero then you can conclude that they are statistically significant, at the $\alpha\%$ level. Whether this difference is meaningful in a clinical/practical sense is a subjective question that depends on context and can't really be answered statistically. 
Since correlations are bound in the interval $[-1,1]$ and their sampling distribution is not normal, they must be transformed first. This can be achieved using the inverse hyperbolic tangent (atanh) otherwise known as Fisher's Z transform:
$$ Z_1 = \text{atanh}(r) = \frac{1}{2} \ln(\frac{1+r_1}{1-r_1}) $$
$$ Z_2 = \text{atanh}(r) = \frac{1}{2} \ln(\frac{1+r_2}{1-r_2}) $$
where $r_1$ and $r_2$ are your sample correlation coefficients.
The standard error for the difference is
$$ SE_{(Z_1 - Z_2)} = \sqrt {\frac{1}{N_1 - 3}+ \frac{1}{N_2 - 3}} $$
where $Z_1$ and $Z_2$ are the transformed variables and $N_1$ and $N_2$ are the respective sample sizes. Then apply the confidence interval formula:
$$ CI = (Z_1-Z_2) \pm Z_{(1-\alpha/2)} SE_{(Z_1 - Z_2)} $$
If you wanted a 95% confidence interval, this would be:
$$ CI_1 = (Z_1-Z_2) + 1.96 SE_{(Z_1 - Z_2)} $$
$$ CI_2 = (Z_1-Z_2) - 1.96 SE_{(Z_1 - Z_2)} $$
Once the interval has been calculated on the $Z$ scale it can be transformed back to the correlation scale using the $\tanh$ function:
$$ \eta_1 = \tanh(CI_1) = \dfrac{\exp(2 CI_1)-1}{\exp(2 CI_1)+1}$$
$$ \eta_2 = \tanh(CI_2) = \dfrac{\exp(2 CI_2)-1}{\exp(2 CI_2)+1}$$
And then the confidence interval for the difference in correlations is $[\eta_1,\eta_2]$.
