FISHERINV

Application:

Let's say a market researcher is studying the relationship between the number of ads viewed by a consumer and the amount of money they spend on a product. They collect data from 50 consumers and find a sample correlation coefficient (r) of 0.65. They want to calculate a 95% confidence interval for this correlation.

Step 1: Fisher Transformation

The first step is to transform the correlation coefficient (r=0.65) into a z-score using the Fisher transformation.

$z=0.5\ln \frac{1+0.65}{1-0.65}=0.5\ln \frac{1.65}{0.35}=0.5\ln (4.714)=0.5\times 1.551=0.7755$

Step 2: Calculate the Confidence Interval for the z-score

The standard error of the z-score is calculated as:

$S{E}_z=\frac{1}{\sqrt{N-3}}$, where N is the sample size.

$S{E}_z=\frac{1}{\sqrt{50-3}}=\frac{1}{\sqrt{47}}=\frac{1}{6.855}=0.1459$

For a 95% confidence interval, the critical value (z-critical) is 1.96.

The confidence interval for the z-score is:

$z_{lower}=z-z_{critical}\times S{E}_z=0.7755-1.96\times 0.1459=0.7755-0.2860=0.4895$

$z_{upper}=z+z_{critical}\times S{E}_z=0.7755+1.96\times 0.1459=0.7755+0.2860=1.0615$

Step 3: Use FISHERINV to Convert Back to Correlation Coefficients

Now, we use the FISHERINV function to convert the z-scores back to correlation coefficients.

$r_{lower}=\mathrm{FISHERINV}(z_{lower})=\mathrm{FISHERINV}(0.4895)$

$r_{upper}=\mathrm{FISHERINV}(z_{upper})=\mathrm{FISHERINV}(1.0615)$

The formula for FISHERINV is:

$\mathrm{FISHERINV}(z)=\frac{e^{2z}-1}{e^{2z}+1}$

$r_{lower}=\frac{e^{2\times 0.4895}-1}{e^{2\times 0.4895}+1}=\frac{e^{0.979}-1}{e^{0.979}+1}=\frac{2.66179-1}{2.66179+1}=\frac{1.66179}{3.66179}=0.4538$

$r_{upper}=\frac{e^{2\times 1.0615}-1}{e^{2\times 1.0615}+1}=\frac{e^{2.123}-1}{e^{2.123}+1}=\frac{8.356-1}{8.356+1}=\frac{7.356}{9.356}=0.7862$

Results Summary Table: