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When discussing ordinary correlations we looked at tests for the null hypothesis that the ordinary correlation is equal to zero, against the alternative that it is not equal to zero. If that null hypothesis is rejected, then we look at confidence intervals for the ordinary correlation. Similar objectives can be considered for the partial correlation.

First consider testing the null hypothesis that a partial correlation is equal to zero against the alternative that it is not equal to zero. This is expressed below:

\(H_0: \rho_{jk\textbf{.x}}=0\) against \(H_a: \rho_{jk\textbf{.X}}\ne 0\)

Here we will use a test statistic that is similar to the one we used for an ordinary correlation. This test statistic is shown below:

\[t = r_{jk\textbf{.x}}\sqrt{\frac{n-2-k}{1-r^2_{jk\textbf{.x}}}} \dot{\sim} t_{n-2-k}\]

The only difference between this and the previous one is what appears in the numerator of the radical. Before we just took n - 2. Here we take n - 2 - k, where k is the number of variables upon which we are conditioning. In our Adult Intelligence data we conditioned on two variables so k would be equal to 2 in this case.

Under the null hypothesis this test statistic will be approximately t-distributed, also with n - 2 - k degrees of freedom.

We would reject Ho if the absolute value of the test statistic exceeded the critical value from the t-table evaluated at α over 2:

\[|t| > t_{n-2-k, \alpha/2}\].

Example: Wechsler Adult Intelligence Data

For the Wechsler Adult Intelligence Data we found a partial correlation of 0.711879, which we enter into the expression for the test statistic as shown below:

\[t = 0.711879 \sqrt{\frac{37-2-2}{1-0.711879^2}}=5.82\]

The sample size is 37, along with the 2 variables upon which we are conditioning is also substituted in. Carry out the math and we get a test statistic of 5.82 as shown above.

Here we want to compare this value to a t-distribution with 33 degrees of freedom for an α = 0.01 level test. Therefore, we are going to look at the critical value for 0.005 in the table (because 33 does not appear use the closest df that does not exceed 33 which is 30).  In this case it is 2.75, meaning that t(df,1-α/2) = t(33,0.995) is 2.75. NOTE: Some text tables provide the right tail probability (the graph at the top will have the area in the right tail shaded in) while other texts will provide a table with the cumulative probability - the graph will be shaded in to the left. The concept is the same. For example, if alpha was 0.01 then using the first text you would look under 0.005 and in the second text look under 0.995.

Because 5.82 > 2.75 = t(33,0.995), we can reject the null hypothesis, Ho at the α = 0.01 level and conclude that there is a significant partial correlation between these two variables. In particular we would include that this partial correlation is positive indicating that even after taking into account Arithmetic and Picture Completion, there is a positive association between Information and Similarities.

Confidence Interval for the partial correlation - \(\rho_{jk\textbf{.x}}\)

The procedure here is very similar to the procedure we used for ordinary correlation.

Step 1: Compute the Fisher's transformation of the partial correlation using the same formula as before.

\[z_{jk} = \frac{1}{2}\log \frac{1+r_{jk\textbf{.X}}}{1-r_{jk\textbf{.X}}}\]

In this case, for a large n, this Fisher transform variable will be possibly normally distributed. The mean is equal to the Fisher transform for the population value for this partial correlation, and variance equal to 1 over n-3-k.

\(z_{jk}\)  \(\dot{\sim}\)  \(N \left( \frac{1}{2}\log \frac{1+\rho_{jk\textbf{.X}}}{1-\rho_{jk\textbf{.X}}}, \frac{1}{n-3-k}\right)\)

Step 2: Compute a (1 - α) × 100% confidence interval for the Fisher transform correlation. This expression is shown below:

\[ \frac{1}{2}\log \frac{1+\rho_{jk\textbf{.X}}}{1-\rho_{jk\textbf{.X}}}\]

This yields the bounds Zl and Zu as before.

\[\left(\underset{Z_l}{\underbrace{Z_{jk}-\frac{Z_{\alpha/2}}{\sqrt{n-3-k}}}}, \underset{Z_U}{\underbrace{Z_{jk}+\frac{Z_{\alpha/2}}{\sqrt{n-3-k}}}}\right)\]

Step 3: Back transform to obtain the desired confidence interval for the partial correlation - \(\rho_{jk\textbf{.X}}\)

\[\left(\frac{e^{2Z_l}-1}{e^{2Z_l}+1}, \frac{e^{2Z_U}-1}{e^{2Z_U}+1}\right)\]

Worked Example: Wechsler Adult Intelligence Data

The confidence interval is calculated substituting in the results from the Wechsler Adult Intelligence Data into the appropriate steps below:

Step 1: Compute the Fisher transform:

\[Z_{12} = \frac{1}{2}\log \frac{1+r_{12.34}}{1-r_{12.34}} = \frac{1}{2} \log \frac{1+0.711879}{1-0.711879} = 0.89098\]

Step 2: Compute the 95% confidence interval for \( \frac{1}{2}\log \frac{1+\rho_{12.34}}{1-\rho_{12.34}}\) :

\[\begin{array}{lllllll}Z_l & = & Z_{12}-Z_{0.025}/\sqrt{n-3-k} & = & 0.89098 - \frac{1.96}{\sqrt{37-3-2}} & = & 0.5445\\ Z_U & = & Z_{12}+Z_{0.025}/\sqrt{n-3-k} & = & 0.89098 + \frac{1.96}{\sqrt{37-3-2}} & = & 1.2375\end{array}\]

Step 3: Back-transform to obtain the 95% confidence interval for \(\rho_{12.34}\) :

\[\left(\frac{\exp\{2Z_l\}-1}{\exp\{2Z_l\}+1}, \frac{\exp\{2Z_U\}-1}{\exp\{2Z_U\}+1}\right)\]

\[\left(\frac{\exp\{2\times 0.5445\}-1}{\exp\{2\times 0.5445\}+1}, \frac{\exp\{2\times 1.2375\}-1}{\exp\{2\times 1.2375\}+1}\right)\]

\[(0.4964, 0.8447)\]

Based on this result, we can conclude that we are 95% confident that the interval (0.4964, 0.8447) contains the partial correlation between Information and Similarities scores given scores on Arithmetic and Picture Completion.