8210 WK 8 ASSGN

Correlation Coefficients

Learning Objective:

Interpret correlation and regression coefficients.

Before You Begin

Before reading this Skill Builder, be sure to review the following concepts:

• Steps of hypothesis testing

  Steps of hypothesis testing

• Null hypothesis 

  Null hypothesis 

• Alternative hypotheses

  Alternative hypotheses

• p-value

  p-value

• alpha

  alpha

• Under what circumstances to reject the null hypothesis

  Under what circumstances to reject the null hypothesis

• Effect size

  Effect size

• Practice significance/meaningfulness

  Practice significance/meaningfulness

Correlation Coefficients

Researchers who study adolescent peer relationships are often interested in whether peers shape one another's attitudes toward many different things, including drug and alcohol use, delinquent behavior, and academics. Suppose you are a researcher interested in whether adolescents and their peers shape each other's attitudes toward academics. Although it can be tricky to obtain data that would allow you to examine causal relationships between adolescents’ and peers’ academic attitudes, you can, as a start toward studying this topic, examine whether there are associations between adolescents’ and peers’ academic values. For example, do adolescents who value academics tend to have friends who also value academics?

Pearson’s Correlation

Pearson’s correlation is one method of examining associations among variables. It allows researchers to examine how one variable changes as another variable change. For example, will one variable increase as the other variable increases? Or, will one variable decrease as the other variable increases? 

The table below shows an example of SPSS output from Pearson’s correlation showing the association between adolescents’ value for the subject of English and their peers’ value for English (Loken, 2005). Students were asked to answer questions that would indicate how much value they place on English (e.g., how much they like English) and their peers were also asked the same set of questions (Eccles, et. al., 1983). Scores on the English value variable range from a low of 1 to a high of 7, with higher scores indicating a greater degree of value for English.

References: Loken, E.. Academic achievement in middle schoolers. 2005. Eccles et. al. Achievement and achievement motivation in expectancies, values, and academic behaviors. W.H. Freeman. 1983

Table: SPSS Output from Pearson's Correlation

Empty	English Value	Peer Group's English Value
English value	Pearson Correlation	1	.438**
Sig. (2-tailed)		.000
N	67	63
Peer group's English Value	Pearson Correlation	.438**	1
Sig. (2-tailed)	.000	blank
N	63	63

**. Correlation is significant at the 0.01 level (2-tailed).

Pearson’s correlation is typically used when research scenarios meet these criteria: 

• The researcher wants to examine the association between two variables

  bullet

  The researcher wants to examine the association between two variables

• Both variables can be considered to be continuous

  bullet

  Both variables can be considered to be continuous

Although the relationship between the two variables is not always linear, we will only focus on linear associations in this Skill Builder. While we will not focus on scatter plots here, examining a scatter plot is an effective way to see if the association is linear or curvilinear and to gauge the strength and direction of the association between the variables.

The Strength and Direction of Correlation Coefficients

When researchers examine the correlation coefficients in their SPSS output, how do they interpret them to discern the strength and direction of the association between the two variables? First, when we examine our SPSS output, we should think about the null hypothesis that we are testing, and we will need to decide whether or not to reject the null hypothesis. The null and alternative hypotheses for a Pearson’s correlation test can be written as:

Null: p = 0

Alternative: p ≠ 0

The null hypothesis states that there is no association between the variables; that is, the Pearson’s correlation coefficient is equal to zero. The alternative hypothesis, on the other hand, specifies that there is an association between the variables – that the Pearson’s correlation coefficient is not equal to zero. In our example of students’ and peers’ English value, SPSS is indicating a p-value of .000 (see the “sig. (2-tailed)” value in the table below). If we assume that alpha was set at .05, we would reject the null hypothesis and conclude that the results are consistent with there being an association between students’ and peers’ value for English.

Correlations

able with columns for english value and peer group's english value. The rows are as follows: row 1, english value, pearson correlation, 1, .438. A callout refers to the value 1 and reads: standardized regression coefficient for age; the closer this value is to 1, the stronger the effect size. A callout points to .438 and reads: pearson's correlation between students' english value and peers' english value. Row 2, english value, sig. (2-tailed), blank, .000. A callout points to .000 and reads: p-value. Row 3, english value, n, 67, 63. A callout points to 63 and reads: the number of participants used to calculate the correlation coefficient of .438. Row 4, peer group's english value, pearson correlation, .438, 1. Row 5, peer group's english value, sign. (2-tailed), .000, blank. Row 6, peer group's english value, n, 63, 63. Note that for the value .438, correlation is significant at the 0.01 level (2-tailed)." title="Table with columns for english value and peer group's english value. The rows are as follows: row 1, english value, pearson correlation, 1, .438. A callout refers to the value 1 and reads: standardized regression coefficient for age; the closer this value is to 1, the stronger the effect size. A callout points to .438 and reads: pearson's correlation between students' english value and peers' english value. Row 2, english value, sig. (2-tailed), blank, .000. A callout points to .000 and reads: p-value. Row 3, english value, n, 67, 63. A callout points to 63 and reads: the number of participants used to calculate the correlation coefficient of .438. Row 4, peer group's english value, pearson correlation, .438, 1. Row 5, peer group's english value, sign. (2-tailed), .000, blank. Row 6, peer group's english value, n, 63, 63. Note that for the value .438, correlation is significant at the 0.01 level (2-tailed).

SPSS Output: Students' and Peers' English Value

Simply looking at the p-value in order to interpret correlation results will not be sufficient, however. Now that we have concluded that there is evidence of an association between the variables, we need to figure out the direction and the strength of the association between the two variables. This helps researchers understand the nature of the relationship between the variables and get a sense of the effect size for the correlation results, which will help with understanding the practical significance, or meaningfulness, of the results.   

The correlation value will indicate two possible directions for the association, based on whether the value in our output is positive (e.g., .438) or negative (e.g., -.438):

• Positive: As one variable increases, the other variable increases

  bullet

  Positive: As one variable increases, the other variable increases

• Negative: As one variable increases, the other variable decreases

  bullet

  Negative: As one variable increases, the other variable decreases

Based on the correlation value, researchers can also discern whether the association between the variables is weak, moderate, or strong. The correlation value will range from -1 to 1. In order to assess the strength of the association, we want to pay attention to the absolute value of the correlation coefficient. In other words, a .4 correlation coefficient and a -.4 correlation coefficient will indicate the same strength of association.

General Rule for Interpreting the Correlation Coefficient

Although different sources give slightly different information about assessing the strength of a correlation coefficient, we can use this as a general rule:

• 1

  .8 to 1: very strong

  1

  .8 to 1: very strong

• 2

  .6 to .8: strong

  2

  .6 to .8: strong

• 3

  .4 to .6: moderate

  3

  .4 to .6: moderate

• 4

  .2 to .4: weak

  4

  .2 to .4: weak

• 5

  0 to .2: very weak to no relationship

  5

  0 to .2: very weak to no relationship

Again, note that we are interested in the absolute value of the correlation coefficient. So, for example, if our correlation coefficient were to be -.8, we would conclude that there is a very strong relationship between the variables.

In the Students' and Peers' English Value SPSS output above, our Pearson’s correlation value is .438. If we were reporting this result in APA style, we would say that there is a moderate positive relationship between students’ and peers’ English value, r  (61) = .44, p <.01. The correlation coefficient does not have a negative sign in front of it so it would be considered to be positive and, according to the general rule above the association falls in the .4 to .6 range, so it would be considered to be of moderate strength. The positive relationship indicates that as students’ value for English increases, so does their peers’ value for English. In other words, students and their peers show some degree of similarity in the value they place on English, which is probably not surprising.  

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