In a test of the correlation coefficient, the null hypothesis posits that there is no relationship between the variables in the broader population. This is formalized using the Greek lowercase rho ($$\rho$$) to represent the population parameter: $$\rho = 0$$. Conversely, the alternative hypothesis asserts that a relationship does exist in the population: $$\rho \neq 0$$. Researchers can specify this as a two-tailed test if they do not expect a specific direction, or a one-tailed test if they theoretically predict the relationship will be strictly positive or strictly negative.

Null and Alternative Hypotheses for the Correlation Coefficient

When utilizing the sample correlation coefficient to compute a $$t$$ score for null hypothesis testing, the degrees of freedom ($$df$$) are determined by subtracting two from the total sample size ($$N$$). This calculation is expressed mathematically as $$df = N - 2$$.

Degrees of Freedom (Correlation Coefficient)

When conducting a test of the correlation coefficient by hand, researchers can use a statistical table to find the critical values of $$r$$ for various sample sizes at a predetermined alpha level, such as $$\alpha = .05$$. If the calculated sample correlation coefficient is more extreme than the critical value retrieved from the table, the result is statistically significant, meaning there is sufficient evidence to reject the null hypothesis and conclude a relationship exists in the population.

Critical Values of r

To illustrate a test of the correlation coefficient, consider a health psychologist investigating the correlation between people's calorie estimates and their weight. Having no directional expectation, she conducts a two-tailed test. For a sample of $$22$$ students, she computes Pearson's $$r$$ as $$-.21$$. Statistical software provides a $$p$$-value of $$.348$$. Because this $$p$$-value is greater than $$.05$$, she retains the null hypothesis, concluding there is no relationship between the variables. Alternatively, computing the test by hand, the degrees of freedom are $$20$$ ($$22 - 2 = 20$$), and a statistical table indicates the critical value is $$.444$$. Because the calculated correlation coefficient ($$-.21$$) is less extreme than this critical value, the $$p$$-value is confirmed to be greater than $$.05$$, leading to the same conclusion to retain the null hypothesis.

Example Test of a Correlation Coefficient

When testing a correlation coefficient, a p-value greater than the alpha threshold indicates that the researcher should retain the null hypothesis. For example, if software reports a p-value of $$.348$$ and the threshold is $$.05$$, the result is not statistically significant and the researcher should conclude that the sample does not provide enough evidence for a population relationship.

Evaluating p-Values in a Correlation Test

When evaluating statistical relationships between quantitative variables, how can the correlation coefficient (Pearson's r) be utilized to determine the p-value in a null hypothesis test?

The test of the correlation coefficient is the appropriate null hypothesis test for evaluating statistical relationships between quantitative variables when Pearson's $$r$$ is used to describe the strength of those relationships. While it is possible to use the sample correlation coefficient to compute a $$t$$ score and proceed with a standard $$t$$-test, the correlation coefficient is mathematically structured so that it can also be treated directly as its own test statistic to find the $$p$$-value.

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A numerical index, commonly represented as Pearson's $$r$$, that measures the direction and strength of a linear relationship between two quantitative variables. The coefficient is bounded between $$-1.00$$ and $$+1.00$$, where values further from zero indicate a tighter, more predictable association, and values near zero indicate a negligible relationship. In research, this coefficient frequently serves as an effect size measure to evaluate how strongly variables are connected in the population.

Correlation Coefficient

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KPU Research Methods in Psychology - 4th American Edition

The sign of a correlation coefficient, whether positive or negative, indicates the direction of the relationship between variables. Based on this directionality, correlations can be classified into three main types: positive, negative, and zero (no correlation).

Types of Correlation

A researcher is examining the statistical association between several pairs of variables. Which of the following numerical values represents the strongest association between two variables?

A correlation matrix is a specific type of table utilized to present the correlation coefficients—typically Pearson’s $$r$$—among several different variables. To avoid redundant information and improve overall readability, only the lower half of the matrix is filled with values, as the upper half would display identical mirror correlations. Furthermore, because any variable's correlation with itself is always a perfect $$+1.00$$, these values along the diagonal are conventionally replaced with dashes.

Correlation Matrix

The computational formula for Pearson's $$r$$ defines it as the mean cross-product of $$z$$ scores. To calculate it, each score for both variables is first transformed into a $$z$$ score. Then, each individual's two $$z$$ scores are multiplied to create a cross-product. The mean of all these cross-products represents Pearson's $$r$$, expressed by the formula: $$r = \frac {\sum(z_xz_y)}{N}$$.

Pearson's r Formula

Restriction of range is a statistical issue that occurs when one or both variables in a correlational study have a limited range of values in the sample relative to the actual population. This artificial limitation can cause a strong overall correlation to appear misleadingly weak or even non-existent. To prevent this, researchers should aim to collect data from subjects spanning a wide range of values for their primary variables.

Restriction of Range

The correlation coefficient indicates the strength of a statistical relationship between two variables. The relationship is considered the weakest when the correlation coefficient is closest to $${}0$$. A coefficient near $${}0$$ signifies a very weak, unpredictable relationship, regardless of whether the sign is positive or negative.

Identifying the Weakest Correlation

Statistical regression is a technique that allows researchers to predict the value of one variable given the value of another variable. While correlation coefficients describe the strength and direction of a relationship, regression utilizes that established relationship to make specific mathematical predictions.

Statistical Regression

Test of the Correlation Coefficient

Scatterplots are highly effective for visually illustrating the varying strengths and directions of Pearson's $$r$$. A perfect negative relationship ($$r = -1.00$$) is depicted as a tight, downward-sloping straight line, whereas a moderate negative relationship ($$r = -0.50$$) shows points loosely clustered around a downward trend. A value of $$r = 0$$ appears as a shapeless cloud of points, indicating no relationship. Conversely, a moderate positive relationship ($$r = +0.50$$) shows a loose upward trend, and a perfect positive relationship ($$r = +1.00$$) features points falling exactly on an upward-sloping diagonal line.

Example of Scatterplots for Pearson's r Values

In psychological research, Cohen provided standard guidelines for interpreting the strength of a statistical relationship measured by Pearson's $$r$$. According to his criteria, $$r$$ values near $$\pm 0.10$$ are considered small, values near $$\pm 0.30$$ are considered medium, and values near $$\pm 0.50$$ are considered large. The positive or negative sign of the coefficient only signifies the direction of the relationship, so an $$r$$ value of $$-0.30$$ is just as strong as a value of $$+0.30$$.

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