When conducting correlation analyses by two independent groups of different sample sizes, typically, a comparison between the two correlations is examined. This is recommended when the correlations are conducted on the same variables by two different groups, and if both correlations are found to be statistically significant. The way to do this is by transforming the correlation coefficient values, or r values, into z scores. This transformation, also known as Fisher’s r to z transformation, is done so that the z scores can be compared and analyzed for statistical significance by determining the observed z test statistic.
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With the observed z test statistic (zobserved) at a set alpha level (level of significance), statistical significance can be assessed. SPSS does not conduct this analysis, and so alternatively, this can be done by hand or an online calculator. The first step is to run the correlation analyses between the two independent groups and determine their correlation coefficients (r); any negative signs can be ignored. The next step is to note, or write down, the sample sizes per each independent group. Then, using a statistical chart with z values and calculator, or an online calculator, determine the z values (z1 and z2) that correspond to the correlation coefficients (r). Usually, with an online calculator, significance is also calculated once you enter in the two correlation values and different sample sizes (N1 and N2). But in lieu of the online calculator, and with just a statistical chart with z values and calculator, the following formula is implemented:
Zobserved = (z1 – z2) / (square root of [ (1 / N1 – 3) + (1 / N2 – 3) ]
Once the observed z value has been determined, statistical significance can be assessed by checking to see if the observed value is greater than the critical value. For example, if the observed value was zobserved = -1.97 and your level of significance is set at .05, which indicates that the critical value is ± 1.96, your zobserved falls into the rejection region and is greater than your critical value; thus, statistical significance. You can reject the null hypothesis that the two correlations are not significantly different.
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