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### CORRELATION

Pearson correlation
A Pearson correlation measures the strength of the linear relationship between two continuous variables. A linear relationship is one that can be captured by drawing a straight line on a scatterplot between the two variables of interest. The value of the correlation provides information both about the nature and the strength of the relationship.

Correlations range between -1.0 and 1.0.
The sign of the correlation describes the direction of the relationship. A positive sign indicates that as one variable gets larger the other also tends to get larger, while a negative sign indicates that as one variable gets larger the other tends to get smaller.
The magnitude of the correlation describes the strength of the relationship. The further that a correlation is from zero, the stronger the relationship is between the two variables. A zero correlation would indicate that the two variables aren’t related to each other at all.

Correlations only measure the strength of the linear relationship between the two variables. Sometimes you have a relationship that would be better measured by a curve of some sort rather than a straight line. In this case the correlation coefficient would not provide a very accurate measure of the strength of the relationship. If a line accurately describes the relationship between your two variables, your ability to predict the value of one variable from the value of the other is directly related to the correlation between them. When the points in your scatterplot are all clustered closely about a line your correlation will be large and the accuracy of the predictions will be high. If the points tend to be widely spread your correlation will be small and the accuracy of your predictions will be low.

The Pearson correlation assumes that both of your variables have normal distributions. If this is not the case then you might consider performing a Spearman rank-order correlation instead (described below).

To perform a Pearson correlation in SPSS
Choose Analyze thengoto Correlate thengoto Bivariate.
Move the variables you want to correlate to the Variables box.
Click the OK button.

The output of this analysis will contain the following section.
Correlations. This section contains the correlation matrix of the variables you selected. A variable always has a perfect correlation with itself, so the diagonals of this matrix will always have values of 1. The other cells in the table provide you with the correlation between the variable listed at the top of the column and the variable listed to the left of the row. Below this is a p-value testing whether the correlation differs significantly from zero. Finally, the bottom value in each box is the sample size used to compute the correlation.

Point-biserial correlation
The point-biserial correlation captures the relationship between a dichotomous (two-value) variable and a continuous variable. If the analyst codes the dichotomous variable with values of 0 and 1, and then computes a standard Pearson correlation using this variable, it is mathematically equivalent to the point-biserial correlation. The interpretation of this variable is similar to the interpretation of the Pearson correlation. A positive correlation indicates that group associated with the value of 1 has larger values than the group associated with the value of
0. A negative correlation indicates that group associated with the value of 1 has smaller values than the group associated with the value of 0. A value near zero indicates no relationship between the two variables.

To perform a point-biserial correlation in SPSS.
Make sure your categories are indicated by values of 0 and 1.
Obtain the Pearson correlation between the categorical variable and the continuous variable, as discussed above.

The result of this analysis will include the same sections as discussed in the Pearson correlation section.

Spearman rank correlation
The Spearman rank correlation is a nonparametric equivalent to the Pearson correlation. The Pearson correlation assumes that both of your variables have normal distributions. If this assumption is violated for either of your variables then you may choose to perform a Spearman rank correlation instead. However, the Spearman rank correlation is a less powerful measure of association, so people will commonly choose to use the standard Pearson correlation even when the variables you want to consider are moderately nonnormal. The Spearman Rank correlation is typically preferred over Kenda’s tau, another nonparametric correlation measure, because its scaling is more consistent with the standard Pearson correlation.

To perform a Spearman rank correlation in SPSS
Choose Analyze thengoto Correlate thengoto Bivariate.
Move the variables you want to correlate to the Variables box.
Check the box next to Spearman.
Click the OK button.

The output of this analysis will contain the following section.
Correlations. This section contains the correlation matrix of the variables you selected. The Spearman rank correlations can be interpreted in exactly the same way as you interpret a standard Pearson correlation. Below each correlation SPSS provides a p-value testing whether the correlation is significantly different from zero, and the sample size used to compute the correlation.