Understanding the Pearson Correlation Coefficient (r)
The Pearson product-moment correlation coefficient (r) assesses the degree that quantitative variables are linearly related in a sample. Each individual or case must have scores on two quantitative variables (i.e., continuous variables measured on the interval or ratio scales). The significance test for r evaluates whether there is a linear relationship between the two variables in the population. The appropriate correlation coefficient depends on the scales of measurement of the two variables being correlated.
There are two assumptions underlying the significance test associated with a Pearson correlation coefficient between two variables. Assumption 1: The variables are bivariately normally distributed. If the variables are bivariately normally distributed, each variable is normally distributed ignoring the other variable and each variable is normally distributed at all levels of the other variable. If the bivariate normality assumption is met, the only type of statistical relationship that can exist between two variables is a linear relationship. However, if the assumption is violated, a non-linear relationship may exist. It is important to determine if a non-linear relationship exists between two variables before describing the results using the Pearson correlation coefficient. Non-linearity can be assessed visually by examining a scatterplot of the data points. Assumption 2: The cases represent a random sample from the population and the scores on variables for one case are independent of scores on the variables for other cases. The significance test for a Pearson correlation coefficient is not robust to violations of the independence assumption. If this assumption is violated, the correlation significance test should not be computed.
SPSS© computes the Pearson correlation coefficient, an index of effect size. The index ranges in value from -1.00 to +1.00. This coefficient indicates the degree that low or high scores on one variable tend to go with low or high scores on another variable. A score on a variable is a low (or high) score to the extent that it falls below (or above) the mean score on that variable. As with all effect size indices, there is no good answer to the question, “What value indicates a strong relationship between two variables?” What is large or small depends on the discipline within which the research question is being asked.
If one variable is thought of as the predictor and another variable as the criterion, we can square the correlation coefficient to interpret the strength of the relationship. The square of the correlation (r2) gives the proportion of criterion variance that is accounted for by its linear relationship with the predictor. In other words, the square of the correlation coefficient equals the proportion of the total variance in Y that can be associated with the variance in X. The square of the correlation coefficient is called the coefficient of determination.
Conducting Pearson Correlation Coefficients
Open the data file
then click Bivariate
You will see the Bivariate Correlations dialog box.
Select the variables of interest
You can double-click each variable to bring them into the Variables box since there is only a single option to move to. b.
You can hold down the Ctrl key, click the three desired variables and click ► to move them to the Variable box. 4.
Make sure Pearson is selected in the Correlation Coefficients area. 5.
Make sure the Two-tailed option is selected in the Test of Significance box (unless you have some a priori reason to select one-tailed). 6.
Click Options. You will see the Bivariate Correlations: Options dialog box. 7.
Click Means and standard deviations in the Statistics box. 8.
(For Total Sample information) click OK.
You should be in the Output1 – SPSS Viewer and are now ready to examine the output. 9b.
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