Question

Load the GSS2012 data set and analyze the relationships between church attendance (the dependent or \(Y\) variable), and two independent variables: age and number of children. You analyzed some of these relationships in Problem \(12.7 .\) Now, we will use both partial correlation and multiple regression and correlation to further examine these relationships. \- Zero-order correlations. \- Click Analyze \(\rightarrow\) Correlate \(\rightarrow\) Bivariate. \- Enter attend, age, and childs in the Variables: window. \- Click OK. \- Partial Correlation Analysis: Is the relationship between attend and age affected by childs? \- Click Analyze \(\rightarrow\) Correlate \(\rightarrow\) Partial. \- Enter attend and age in the "Variables:" window and childs in the "Controlling for:" window. \- Click OK. \- Partial Correlation Analysis: Is the relationship between attend and childs affected by age? \- Click Analyze \(\rightarrow\) Correlate \(\rightarrow\) Partial. \- Enter attend and childs in the "Variables:" window and age in the "Controlling for:" window. \- Click OK. \- Multiple Regression and Correlation: What are the effects of childs and age on attend? \- Click Analyze \(\rightarrow\) Regression \(\rightarrow\) Linear. \- Move attend to the "Dependent" window and age and childs to the "Independent" window \- Click Statistics and check "Descriptives". Click Continue to retum to the "Linear Regression" screen. \- Click OK. a. Summarize the results for the partial correlation analysis in a paragraph in which you report the value of the zero-order and partial correlations for all relationships. Does the relationship between attend and age seem to be direct? How about the relationship between attend and childs? b. State the unstandardized multiple regression equation. (HINT: The values for a and b are in the "Coefficients" box of the output, under the column labeled B. The value in the first row is a and the values in the second and third rows are the slopes or b.) $$ Y=+X_{1}+X_{2} $$ c. State the standardized multiple regression equation. What is the direction of each relationship? Which independent variable had the stronger effect on attend? (HINT: The beta-weights are in the "Coefficients" box, under "Standardized Coefficients" and "Beta"). $$ Z_{y}=Z_{1}+Z_{2} $$ d. Report the value of \(R^{2}\). What percentage of the variance in attend 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: \(R^{2}\) is in the "Model Summary" box. You can compute \(r^{2}\) from the r's in the "Correlations" window of the output.)

Short answer

Zero-order and partial correlations show the directness of relationships. Unstandardized and standardized equations define model predictions. \(R^{2}\) indicates the explained variance by combined predictors.

Step-by-step solution

Load the Data Set

Load the GSS2012 data set in your statistical software.

Calculate Zero-Order Correlations

Click Analyze \(\rightarrow\) Correlate \(\rightarrow\) Bivariate. Enter attend, age, and childs in the Variables window, then click OK.

Perform Partial Correlation Analysis Between Attend and Age

Click Analyze \(\rightarrow\) Correlate \(\rightarrow\) Partial. Enter attend and age in the Variables window and childs in the Controlling for window. Click OK.

Perform Partial Correlation Analysis Between Attend and Childs

Click Analyze \(\rightarrow\) Correlate \(\rightarrow\) Partial. Enter attend and childs in the Variables window and age in the Controlling for window. Click OK.

Conduct Multiple Regression and Correlation

Click Analyze \(\rightarrow\) Regression \(\rightarrow\) Linear. Move attend to the Dependent window and age and childs to the Independent window. Click Statistics and check Descriptives, then click Continue. Finally, click OK.

Analyze Partial Correlation Results

Summarize the zero-order and partial correlations for the relationships between attend and age, and attend and childs in a paragraph. Discuss whether the relationships seem to be direct or influenced by other variables.

Write Unstandardized Multiple Regression Equation

Find the values for a and b in the Coefficients box under the column labeled B. Write the equation in the form \(Y=a + b_1X_{1} + b_2X_{2}\).

Write Standardized Multiple Regression Equation

Find the beta-weights in the Coefficients box, under Standardized Coefficients and Beta. Write the equation in the form \(Z_y = \beta_1Z_{1} + \beta_2Z_{2}\). Discuss the direction of each relationship and which independent variable had the stronger effect on attend.

Report the Value of \(R^{2}\)

Find \(R^{2}\) value in the Model Summary box. Report the percentage of the variance in attend explained by the two independent variables combined and compare it to the variance explained by each independent variable alone.

Key concepts

church attendance analysis

In this analysis, we explore how church attendance is influenced by age and the number of children. Our aim is to understand these relationships in a more nuanced way, using both partial correlation and multiple regression techniques. This allows us to see not just simple relationships but to control for other variables and get a clearer picture of how these factors interact. The data used for this analysis is from the GSS2012 data set, which is a valuable resource for such statistical investigations.

multiple regression correlation

Multiple regression correlation helps us understand how multiple variables together influence a single outcome. In this case, we look at how both age and the number of children affect church attendance.

To do this, we specify church attendance as the dependent variable and age and number of children as independent variables. By doing this, we can see the combined effect of age and children on church attendance, and determine which of these has a stronger influence when both are considered together.
This method is particularly useful when we suspect that more than one factor might be influencing our outcome.

zero-order correlations

Zero-order correlations represent the straightforward relationships between pairs of variables without considering any other variables. Initially, we compute the zero-order correlations between church attendance, age, and the number of children.

This step helps us see the direct relationships before adding more complexity to our analysis. It’s a critical first step to understand the simple associations. For example, we might find that there is a certain correlation between age and church attendance, and between the number of children and church attendance.

independent variables influence

Understanding how independent variables such as age and number of children influence church attendance is crucial. Partial correlations allow us to investigate these influences in depth by controlling for the effect of one variable while examining the relationship between the other two. For instance, we check if the relationship between age and church attendance remains when we account for the number of children, and vice versa.
This helps us understand if the observed relationships are genuine or if they are CAUSED BY another variable.

GSS2012 data set

The GSS2012 data set is a comprehensive collection of data that includes various demographic and behavioral variables. It serves as a robust foundation for carrying out statistical analysis. We use it in this exercise to draw meaningful conclusions about the relationships between church attendance and other variables.

In our case, the focus is on examining church attendance in relation to age and the number of children.
This data set provides real-world numbers, ensuring that our analysis and conclusions are relevant and accurate.

statistical analysis steps

The process of conducting a thorough statistical analysis involves several vital steps:

• 1. Data Preparation: Load the GSS2012 data set in your statistical software.
• 2. Zero-Order Correlations: Calculate the basic correlations between church attendance, age, and number of children.
• 3. Partial Correlation Analysis: Explore how these relationships change when controlling for one variable at a time.
• 4. Multiple Regression: Use multiple regression to understand the combined effects of age and number of children on church attendance.
• 5. Interpret Results: Summarize and interpret the results from both zero-order and partial correlations and from the multiple regression outputs.

These steps ensure a comprehensive analysis, providing clear insights into complex relationships among the variables.