I am not sure why you want to do this, but here is an idea using Mplus.

Enter the standardized data and build a model in which Z causes X and Y where one parameter in your model is the correlation in disturbances of X and Y (i.e., the partial r; you would need to standardize the disturbances of X and Y too). Define the correlation of X and Y using Mplus's option defining new parameters. Then define the difference between the partial (which is in your model) and the newly defined parameter which is the raw correlation. Then you should be able to bootstrap that difference which gives you a test of the difference.

Independent two sample t-test is suitable for this problem.

Ho: No difference between zero order and first order partial correlations

Ha: there is difference between zero order and first order partial correlations

you can use chi square test also for this problem. If you want any further information about formula refer statistics by levin and rubin book.

@ Venkata Palakiti

Thanks for your reply

But I do not understand why you suggest independent sample t test. The zero order correlation (say between X and Y) is obtained on the same sample on which you compute the first order partial correlation after controlling the third variable (say Z). Thus rxy and rxy.z are from the same sample. In such situation the correlated sample Z test fro significance of difference appears to be more suitable.

But the problem is that in the textbooks such correlated sample tests have been described for two correlations from the same sample obtained between different pairs of variables with one variable common to both. For instance the difference between rxy and rxz. In the present situation the correlations are not only from the same sample but also between the same variables (xy) with only difference that in one the effect of third variable has been partialed out but not in the other.

It is likely that I may not have stated the question clearly and as a result you though that independent sample t test, generally described for testing the difference between two correlations obtained between same set of variables, may also apply here.

I am not sure why you want to do this, but here is an idea using Mplus.

Enter the standardized data and build a model in which Z causes X and Y where one parameter in your model is the correlation in disturbances of X and Y (i.e., the partial r; you would need to standardize the disturbances of X and Y too). Define the correlation of X and Y using Mplus's option defining new parameters. Then define the difference between the partial (which is in your model) and the newly defined parameter which is the raw correlation. Then you should be able to bootstrap that difference which gives you a test of the difference.

Thanks Prof. Kenny for your very insightful way of handling the Problem using the Mplus. Though, we do not have access to Mplus, I have got the idea I'll try to do it using AMOS with user defined estimates. The only problem that I see with AMOS is that it will not allow to correlate the two endogenous variables X and Y.

Anyway, this suggestion is quite helpful and hope I'll be able to do it.

As far as the purpose of doing this is concerned, I am interested in explaining the relationship between substance abuse and mental health problems. Our hypothesis is that certain emotion related difficulties and deficits may be the potential cause of both substance abuse and mental health problems and thus if the role of such emotional difficulties is statistically controlled the relationship between substance abuse and mental health should become either non-significant or reduce significantly.

If you think that the same issue may be addressed in a better way using other statistical approach then kindly let us know.

Further, can we also consider it (i.e., when the third variable is causally linked to both the predictor and the outcome variable) as a form of mediation.

I am asking this to you as you categorically mention on your website (with logical explanations) that partial correlation can not be used to test mediated relationship. I agree with this but for the case where the third variable is caused by the predictor and is also a cause of the outcome variable. However, I personally think that the conventional approach of testing such mediation (when third variable is the cause of both the IV and DV) may not be appropriate In this situation the partial correlation may be better alternative.

Thanking you once again.

Kind regards

I suggest you read the MacKinnon et al. paper (Equivalence of mediation, confounding, and suppression) in which they show that spuriousness or confounding which is what you have can be treated as mediation. Thus, the "indirect effect" between SA and MH can be viewed as the reduction in the effect.

Hello Rakesh, I have stumbled into your questions, and I hope this still might help.

First, you must convert each of the r values into z values.

Use "Statistical power analysis for the behavioral sciences" from Jacob Cohen for that purpose.

Go to Conversion Table 4.2.1 (I send you a PDF file), and search at the horizontal the lowest value correlation, follow the same line until you find the value of the highest correlation, which give you the "q" value in the vertical (this is the effect size). In page 115 you see the meaning of this effect size.

Next go to the Power Table 4.3.1 (alpha = 0.01), where you intersect your "q" and your "n", getting the power of the test. You can use the other Tables for alphas of 0.05.

For instance:

Your zero order correlation = 0.755, and your first order = 0.743.

In Table 4.2.1 q = 0.10 (a small effect size).

In Table 4.3.5, for a q = 0.10, and a alpha = 0.05, and n = 40; you get a low power test (7%).

Another example:

Your first order correlation = 0.755, and your first order = 0.38.

In Table 4.2.1, q = 0.6 (a large effect size).

In Table 4.3.5, for a q = 0.6, and a alpha = 0.05, and n = 40; you get a high power test (73%).

References:

1. Pestana, M. H., & Gageiro, J. N. (2008). Análise de Dados para Ciências Sociais. A complementaridade do SPSS (5th ed.). Lisboa: Ed. Sílabo

2. Cohen, J. (1988). Statistical power analysis for the behavioural sciences (2nd ed.). Hillsdale, New Jersey: Lawrence Erlbaum Associates.

Here is the Cohen's Tables PDF file.

My notations are in Portuguese, but not an hindrance.

In regard to Professor Kenny's comment that confounding can be treated as  mediation, what do we do when the two variables that are both affected by the confounding variable have no causal relation to each other?  Doesn't this make it impossible to distinguish between them according to which is IV and which is DV?  Then how do we compute the regression coefficients necessary for a test of the significance of the confounding indirect effect?  The example of confounding in the MacKinnon et al. paper that Professor Kenny mentions (two measures of athletic ability and age as the confounder)  shows one way to test for the significance of the confounding effect, but I did not see how they decided which measure of athletic ability was the IV and which the DV.  In the case considered here, is there some assumed causal connection between  SA and MH that makes one unambiguously the IV and the other the DV?

For people seeking a solution to the op's question, I have created an R package called "zeroEQpart" which performs this test and can be found within the CRAN repository.

The package works for zero order versus both partial and semi-partial correlations of types Pearson, Kendall, and Spearman. Confidence intervals can also be determined.