8210 WK 8 ASSGN

Regression Coefficients

Learning Objective: Interpret correlation and regression coefficients.

Regression Models

In addition to Pearson’s correlation analysis, linear regression is another way to examine the association between two variables. Regression models are quite flexible and, unlike with a single Pearson’s correlation analysis, the researcher can include multiple predictor variables that they wish to associate with the dependent variable in the same regression model. Regression models can also incorporate both categorical and continuous predictors. For the sake of this Skill Builder, however, we will focus on models that have just one (1) single predictor variable (bivariate regression models), and we will focus on cases in which both variables (the predictor and the outcome) are continuous.

Let’s return to our example of the association between students’ and peers’ value for English. We can run a regression model that is comparable to the Pearson’s correlation analysis that we did previously. Take a look below at the regression results from SPSS.

Regression Results

Model Summary

Model	R	R Square	Adjusted R Square	Std. Error of the Estimate
1	.438a	.191	.178	1.35361

a. Predictors: (Constant), peer group's English value.

ANOVAa

Model	Sum of Squares	df	Mean Square	F	Sig.
1	Regression	26.466	1	26.466	14.445	.000b
Residual	111.768	61	1.832	blank	blank
Total	138.234	62	blank	blank	blank

a. Dependent Variable: English value. b. Predictors: (Constant), peer group's English value.

Coefficientsa

Model	Unstandardized Coefficients	Standardized Coefficients	t	Sig.	95% Confidence Interval for B
B	Std. Error	Beta	Lower Bound	Upper Bound
1	(Constant)	1.667	.769	blank	2.168	.034	.129	3.204
peer group's English value	.654	.172	.438	3.801	.000	.310	.998

Legend for Coefficientsa
	Unstandardized regression coefficient
	Standardized regression coefficient
	p-value for the association between students' and peers' English value

The Standardized Regression Coefficient +

Of interest in the output is the standardized regression coefficient for the peer group’s English value and the unstandardized regression coefficient. If you take a look first at the standardized regression coefficient, it should look familiar to you. The value of β = .438 is the same value that we obtained previously for Pearson’s correlation coefficient! That is, of course, not a coincidence. When you have just one (1) predictor in your regression model, the standardized regression coefficient will be the same as the coefficient you would get if you ran a Pearson’s correlation analysis. You can, therefore, interpret standardized regression coefficients in the same way as you would interpret a Pearson’s correlation coefficient.

Unstandardized Regression Coefficient +

Let’s focus now on the unstandardized regression coefficient (B = .654) and think about how to interpret it. Just like when we previously discussed correlation, the unstandardized regression coefficient will have a direction and be either positive or negative:

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

The Strength of the Effect +

In addition to understanding the direction of the effect, we will also, just like with correlation, want to understand the strength of the effect. We can use the standardized regression coefficient to gauge the strength (the strength of the coefficient), using the general rule previously noted for correlation. We can also, however, interpret the strength of the effect by examining the unstandardized regression coefficient. The unstandardized coefficient can be interpreted to indicate that for every one (1) unit increase in the predictor variable, the outcome variable will increase by the amount of the unstandardized coefficient. So, in the output above, we can see that for every one (1) unit increase in peers’ English value, students’ English value increases by .654 units. Recall that English value is measured on a scale of 1 to 7.

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