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Correlations

The following bivariate correlations are covered in this unit:

Pearson correlation (parametric), works with scale data that meet parametric assumptions.

Spearman correlation (nonparametric), works with ranked data (ordinal) or scale data that violate the parametric assumptions.

Point biserial correlation (nonparametric) – examines the relationship between a binomial variable and scale variable.

Symbols Used to Report Correlations

The lowercase r is used to report the Pearson correlation and point biserial correlation.

rho is used for Spearman correlations and sometimes uses the lowercase Greek character (rho).

also represents the parameter designation for correlation. So, write it like this:

rho =

Correlation Coefficients

Correlation coefficients can have values between –1 and 1.

A value of -1 means you have a perfect negative relationship between your variables (also called an inverse relationship). In an inverse relationship, one variable increases in magnitude while the other decreases. A correlation of 1 is a perfect positive relationship; both variables increase or decrease in magnitude in the same direction).

These relationships are very rare. A correlation of zero (0) means there is no relationship between the variables. The closer the correlation is to –1 or 1, the stronger the relationship.

Scatter Plots

Scatter plots are a good tool to understand correlations.

The plot between variable a and variable b will be very indistinct (essentially a blob) for a zero correlation, whereas all the point will form up in a line for correlation of –1 or 1.

If you have data that are outliers, they will also stand out in the scatter plot. Your data might not be linear— it can be U shaped or have an exponential curve. Seeing these curves may indicate a transformation is needed to get a meaningful correlation.

Reading Reports that use Correlations

When you read reports that use correlations you will often see adjectives describing the strength of the relationship such as weak, moderate, strong.

There is no specific rule at what value the correlation becomes weak, moderate or strong. Basically a value of 0.25 or less is generally considered weak, 0.5 is moderate, and 0.75 is strong. A better way to consider the strength of a correlation is to use the coefficient of determination (r2).

The square of the correlation coefficient tells the proportion or percentage of variance in variable a explained by variable b or vice-versa.

The table below illustrates:

r

r2

Explained percentage

1

1

100

0.95

0.9

90

0.9

0.81

81

0.8

0.64

64

0.75

0.56

56

0.7

0.49

49

0.6

0.36

36

0.5

0.25

25

0.4

0.16

16

0.3

0.09

9

0.25

0.06

6

0.2

0.04

4

0.1

0.01

1

The strength of the correlation and your interpretation of it is fluid. In biological and behavioral measures that are used in public health bivariate correlations are often influenced by other things that are known or unknown.

An example of the interpretation could be such that a correlation of 0.35, on the face might seem to be a weak correlation, still can be meaningful for your data. Weak correlations can still tell you something in the bigger picture. Correlation really needs to be looked at in the context of your study.

Statistical Significance

You will also be looking at the statistical significance of the correlation. Don’t let the significance sway your interpretation of the results. The magnitude and direction are more important.

Generally speaking a weak correlation will not be statistically significant, but as with other inferential statistics the size of your sample will influence at what level of r it becomes significant.

Hypotheses for Correlation

The following are examples of hypotheses for correlation:

Ho : r = 0

H1; r > 0

H1: r < 0

H1: r 0

3

1

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