Question

In Problem \(12.8\), you used the States database to analyze the bivariate relationships between region (South/Non-South), education (percentage of the population with a college degree), and infant mortality rates. Now, you will take education and region as the independent variables and look at their combined effects on infant mortality rates. \- Click Analyze \(\rightarrow\) Regression \(\rightarrow\) Linear. \- Move InfantMort to the "Dependent" window and College and SthDUMMY to the "Independent" window. \- Click Statistics and check "Descriptives" Click Continue to return to the "Linear Regression" screen. \- Click OK a. State the unstandardized multiple regression equation. $$ Y=+X_{1}+X_{2} $$ b. State the standardized multiple regression equation. What is the direction of each relationship? Which independent variable had the stronger effect on InfantMort? $$ Z y=Z_{1}+Z_{2} $$ c. Report the value of \(R^{2}\). What percentage of the variance in InfantMort is explained by the two independent variables combined? How does this compare to the amount of the variance explained by each independent variable alone? (HINTS: \(\mathrm{R}^{2}\) is in the "Model Summary" box and you can compute \(\mathrm{r}^{2}\) from the r's in the "Correlations" window of the output.)

Short answer

The regression equations are derived from the coefficients in the analysis output. Compare Beta values to determine the stronger effect. R² shows combined variance explained.

Step-by-step solution

Open Linear Regression Tool

Click Analyze → Regression → Linear in your statistical software.

Set Dependent Variable

Move 'InfantMort' to the 'Dependent' window. This represents the infant mortality rates.

Set Independent Variables

Move 'College' and 'SthDUMMY' to the 'Independent' window. 'College' is the education variable and 'SthDUMMY' represents the region.

Select Descriptive Statistics

Click 'Statistics' and check 'Descriptives'. Then click 'Continue' to return to the 'Linear Regression' screen.

Run the Regression

Click 'OK' to run the linear regression analysis.

Find Unstandardized Regression Equation

From the output, locate the coefficients for 'College' and 'SthDUMMY'. Use the intercept and these coefficients to write the unstandardized multiple regression equation, which will be in the form of: y = b_0 + (b_1 * College) + (b_2 * SthDUMMY)

Find Standardized Regression Equation

Identify the standardized coefficients (Beta values) for 'College' and 'SthDUMMY' in the output. The standardized multiple regression equation will be:Z_y = (Beta_1 * Z_College) + (Beta_2 * Z_SthDUMMY)The direction of each relationship (positive or negative) is shown by the sign of each Beta coefficient.

Determine Stronger Independent Variable

Compare the absolute values of the Beta coefficients to determine which independent variable had a stronger effect on 'InfantMort'.

Report R² Value

Find the R² value in the 'Model Summary' box of the output. This represents the percentage of variance in 'InfantMort' explained by the combined independent variables.

Compute r² for Individual Variables

Locate the correlation coefficients (r) for 'College' and 'SthDUMMY' in the 'Correlations' window. Compute r² for each independent variable separately to compare with the combined R².

Key concepts

bivariate relationships

When studying relationships between two variables, we delve into what is called bivariate relationships. In problem 12.8, the bivariate relationships involve region (South/Non-South), education (percentage of the population with a college degree), and infant mortality rates. Here, we are interested in observing how each of these variables interacts or correlates with infant mortality rates.
For example, one might ask whether having a college degree (education) influences the infant mortality rate, or if being in the South region has any effect. By exploring these bivariate relationships, one can identify patterns or correlations that might indicate potential causes or contributing factors.

independent variables

In multiple regression analysis, the variables considered to have an influence on another variable are known as independent variables. For this exercise, the independent variables are 'College' (representing the percentage of the population with a college degree) and 'SthDUMMY' (a dummy variable for region, indicating whether the region is South or Non-South).
The independent variables are critical because they provide the basis for understanding how different factors might impact the subject of interest, which in this case is the 'InfantMort' (infant mortality rates). By including multiple independent variables in the regression model, we can assess their combined effects and determine how they individually and collectively influence the dependent variable.

dependent variable

The dependent variable is the outcome or the main subject that we are trying to predict or explain in our analysis. In this scenario, the dependent variable is 'InfantMort', representing the infant mortality rates.
The dependent variable is influenced by the independent variables, and our goal in multiple regression analysis is to quantify this influence. Essentially, we want to see how changes in 'College' (education) and 'SthDUMMY' (region) affect the rates of infant mortality. Understanding the dependent variable helps in setting up the regression equation and interpreting the results of the analysis.

standardized coefficients

Standardized coefficients, also known as Beta coefficients, are used in regression analysis to make the variables comparable. They are obtained by standardizing the independent variables (converting them to a common scale).
In the standardized multiple regression equation, these coefficients indicate the relative strength and direction of the relationships between the independent variables and the dependent variable. For instance, in our exercise, we expressed this as:
$$Z_y = (Beta_1 * Z_College) + (Beta_2 * Z_SthDUMMY)$$
Here, the Beta coefficients (Beta_1 and Beta_2) tell us how many standard deviations the dependent variable 'InfantMort' will change for a one standard deviation change in the independent variables 'College' and 'SthDUMMY' respectively.
Standardized coefficients are particularly useful because they allow us to identify which independent variable has the stronger effect on the dependent variable by comparing their absolute values. By doing so, we can conclude which factor, education or region, has a more significant impact on infant mortality rates.