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Assignment: Testing for Multiple Regression

You had the chance earlier in the course to practice with multiple regression and obtain peer feedback. Now, it is time once again to put all of that good practice to use and answer a social research question with multiple regression. As you begin the Assignment, be sure and pay close attention to the assumptions of the test. Specifically, make sure the variables are metric level variables.

Part 1

To prepare for this Part 1 of your Assignment:

· Review this week 9 and 10 Learning Resources and media program related to multiple regression.

· Using the SPSS software, open the Afrobarometer dataset or the High School Longitudinal Study dataset (whichever you choose) found in the Learning Resources for this week.

· Based on the dataset you chose, construct a research question that can be answered with a multiple regression analysis.

· Once you perform your multiple regression analysis, review Chapter 11 of the Wagner text to understand how to copy and paste your output into your Word document.

For this Part 1 Assignment:

Write a 1- to 2-page analysis of your multiple regression results for each research question. In your analysis, display the data for the output. Based on your results, provide an explanation of what the implications of social change might be.

Use proper APA format, citations, and referencing for your analysis, research question, and display of output.

Week Nine: Multiple Regressions

Posted on: Friday, July 22, 2022 9:31:03 AM EDT

As social scientists, we frequently have questions that require the use of multiple predictor variables. Moreover, we often want to include control variables (i.e., workforce experience, knowledge, education, etc.) in our model. Multiple regression allows the researcher to build on bivariate regression by including all of the important predictor and control variables in the same model. This, in turn, assists in reducing error and provides a better explanation of the complex social world.

Example: a local school system is trying to mitigate poor attendance. The researchers may look at several, possible interventions. In the end, a study may find a combination of interventions will work better than any single one. This finding is a typical product of multiple regression. In addition, because combinations of data may need to combined, a researcher can infer. The word is a power word in social sciences as it empowers a researcher to synthesize and speculate based upon responsible use of data.

In the end, having concluded your analysis of a regression, what has been learned? In two sentences or less what can you share with others?

Introduction

Consider the two research studies and the challenges they presented.

In a classic research study by Ellen Langer and Judith Rodin (1976), elderly individuals who resided in nursing homes were assigned to one of two groups; one group was granted much more personal control and the other much less. For example, in the first group, individuals were given latitude to have a say in how their furniture was arranged and given a plant to take care of if they wished. The results were striking in that the group granted greater personal control showed much better health outcomes than the group given less control.  

In this study, each group of participants lived on a different floor of the nursing home, and researchers randomly selected which floor would be placed into which group (Langer & Rodin, 1976). The two floors were similar in pre-existing characteristics that might impact their health outcomes. The researchers were, therefore, able to control, at the start, other factors besides being in one of the two groups that might relate to health outcomes. That is, control in this study was achieved through the research design.

Langer, E., & Rodin, J. The effects of choice and enhanced personal responsibility for the aged: A field experiment in an institutional setting. Journal of Personality and Social Psychology.34/ 2. 1976.

In many research studies, however, it is simply not feasible to design a study that sufficiently controls other factors besides the independent variable that might help to explain the dependent variable. Researchers often must find statistical methods of controlling for confounding variables when conducting correlational research.

Consider research conducted by Angela Duckworth and her colleagues (2007). Duckworth focused on a construct called grit, which she has found to be related to achievement. She and her colleagues define grit as “perseverance and passion for long-term goals.” Duckworth was interested in what types of individual qualities might be related to achievement besides academic ability such as that measured by IQ tests. Why is it that people of about the same level of intelligence differ in their degree of achievement and their degree of persevering in tasks that are difficult?  

Previous research on personality factors linked to achievement had already supported the idea that those who are more conscientious show advantageous achievement. In one study, therefore, Duckworth wanted to examine whether grit offered a unique predictive ability of educational attainment over and above the personality trait of conscientiousness and other personality traits (Duckworth, Peterson, Matthews, & Kelly, 2007)[2]. That is, she wanted to control for certain personality traits to better assess whether grit showed the unique prediction of educational attainment. What if those who were higher in grit showed higher educational attainment simply because of other factors such as other personality traits and not the characteristic of grit per se?

In this type of research, it would be quite difficult to simply control for preexisting differences in personality between participants using a strategy like a random assignment. For example, it would be difficult to assign some to have high grit randomly and some to have low grit as grit is a personal attribute. Duckworth and her colleagues, therefore, controlled for personality factors when conducting their statistical analyses to examine if grit is associated with educational attainment beyond other personality factors. Note: They did, in fact, find that grit was related to educational attainment even after controlling for other personality traits (Duckworth et al., 2007).

Duckworth, A. L., Peterson, C., Matthews, M. D., & Kelly, D. R. Grit: Perseverance and passion for long-term goals. Journal of Personality and Social Psychology. 92/6. 2007.

The focus of this Skill Builder is on the purpose of control variables in regression models and on how to interpret regression results when there is more than one (1) predictor in the model. 

Multiple Regression Models

Topic 2 of 4

Learning Objective: Interpret regression results when the regression model has more than one predictor.

The Purpose of Control Variables

A control variable in a statistical model is a variable that we are attempting to “hold constant” while we examine the association among other variables in our model. In essence, we want to know if our independent variable of interest (e.g., grit) is associated with our dependent variable after factoring in other variables (e.g., personality factors) that could also be related to the dependent variable.

In addition to our independent variables of interest, in regression models, we also include control variables as predictor variables because we suspect the control variable is related to our outcome variable and could explain the association between our independent variable and the outcome.

For example, suppose we want to understand factors that might predict an individual's income. Education level seems like an obvious predictor variable that we would want to examine as it is probably a predictor of income. Might there be other variables, however, that would predict income besides education? And, if we find that education level is associated with income, could it partially be because those with more education are also likely to be older and more accomplished/established and therefore earn more money?  For this reason, we probably want to include age as a control variable in our regression model predicting income with education.

Take a look at the output below from SPSS, which shows the results of a regression model based on data from the 2004 General Social Survey ( http://sda.berkeley.edu/archive.htm  ). Use an alpha value of .05 to interpret the results.

Model Summary A model summary showing the results of a regression model based on data from the 2004 General Social Survey. Highest year of school completed, age of respondent. 

Model	R	R Square	Adjusted R Square	Std. Error of the Estimate
1	.256a	.066	.064	2.276

a. Predictors: (Constant), HIGHEST YEAR OF SCHOOL COMPLETED, AGE OF RESPONDENT.

Interpreting the Regression Coefficient

So, how would we interpret the regression coefficient in this model for education level, if we are controlling for age? Researchers would say that holding age constant, education level has a weak, positive association with income, β = .21, p < .05. Recall that a positive association indicates that as education level increases, the income also increases. Another way to say the same thing is to say that education level predicts income above and beyond an individual's age.  

Recall, too, that we need to look at the p-value for each predictor in the model in order to discern whether the predictor shows a statistically significant association with the outcome variable and that we can use the standardized regression coefficients to gauge the effect size for each predictor. In our results below, we can see that each predictor, age, and education level is statistically significant as the p-value is less than the alpha value of .05.

Coefficientsa Table of coefficients showing both unstandardized coefficients and standardized coefficients for age of respondent, highest year of school completed. 

Model	Unstandardized Coefficients	Standardized Coefficients	t	Sig.
	B	Std. Error	Beta
1	(Constant)	6.535	.534	blank	12.245	.000
	AGE OF RESPONDENT	.025	.007	.120	3.703	.000
	HIGHEST YEAR OF SCHOOL COMPLETED	.202	.031	.209	6.470	.000

Legend for Coefficientsa
	Standardized regression coefficient for age; the closer this value is to 1, the stronger the effect size.
	p-value for age
	Standardized regression coefficient for education level;  the closer this value is to 1, the stronger the effect size.
	p-value for educational level

If you look at the standardized regression coefficients, you can see that each predictor shows a weak relationship with income, as each predictor has a standardized regression coefficient that is about .1 or .2; stronger effects would be indicated if the regression coefficients had values closer to 1. Of the two predictors, the education level has a greater value for its standardized regression coefficient, indicating that it is a stronger predictor of income than age.

R-squared

Aside from looking at the individual regression coefficients and the p-values, another thing to note when you are discussing your multiple regression results is the R-squared value. R-squared is an important statistic that tells you the proportion of variability in the dependent variable that is accounted for by your model. In other words, it tells you how good of a job your predictors are doing at predicting your outcome variable. The R-squared value ranges from 0 to 1 and can be expressed as a percent.  In the output shown below, you can see that the R-squared value is .066.  

Model Summary

Model	R	R Square	Adjusted R Square	Std. Error of the Estimate
1	.256a	.066	.064	2.276

a. Predictors: (Constant), HIGHEST YEAR OF SCHOOL COMPLETED, AGE OF RESPONDENT.

Numbered divider 1

Consider the following scenario when answering the question below.

Using the SPSS output above, and assuming an alpha level of .05, suppose we wanted to control for education level instead of age this time around.

Hint: Look at the p -value in the “sig.” column of the output for age. Is that value less than the alpha value of .05?

How would we interpret the results if we were interested in predicting income with age while controlling for education level?

Holding education level constant, an increase in age predicts an increase in income.

Education level is a stronger predictor of income than age, indicating that age is not related to income after controlling for education level

Age does not predict income after controlling for education level.

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How Predictors Are Related to the Dependent Variable?

The above question emphasizes the fact that regardless of whether the researcher is thinking of age or education level as the control variable, the mathematical interpretation of how the predictors are related to the dependent variable does not change. When we interpret the coefficient for one predictor in the model, it is always in the context of holding the other variables “constant,” regardless of which variable, conceptually, we are thinking of as a control variable.

Sometimes, researchers include multiple predictors in a model and are not thinking of any of them, conceptually, as control variables. They are simply interested in how the predictors, together, are related to the outcome variable, or they may be interested in seeing which predictor variables show the strongest relationships with the outcome.

Numbered divider 2

Consider the following scenario when answering the question below.

Suppose we wanted to predict the number of slices of pepperoni pizza people ate at a party based on how many slices their friends ate. Suppose we also gathered data on three (3) additional variables: individual's mood, how much they like pepperoni, and how hungry they reported being when they arrived at the party. Take a look at the correlation results below from SPSS, which is based on fictitious data.

Correlations

Blank	number of slices	positive mood	friends' number of slices	like pepperoni	hunger
number of slices	Pearson Correlation	1	.036	.791**	.806**	-.080
	Sign. (2-tailed)	blank	.864	.000	.000	.702
	N	25	25	25	25	25
positive move	Pearson Correlation	.036	1	-.106	.174	.238
	Sig. (2-tailed)	.864	blank	.613	.406	.253
	N	25	25	25	25	25
friends' number of slices	Pearson Correlation	.791**	-.106	1	.638**	-.139
	Sig. (2-tailed)	.000	.613	blank	.001	.507
	N	25	25	25	25	25
like pepperoni	Pearson Correlation	.806**	-.174	.638**	1	-.198
	Sig. (2-tailed)	.000	.613	blank	.001	.507
	N	25	25	25	25	25
hunger	Pearson Correlation	-.080	.238	-.139	-.198	1
	Sig. (2-tailed)	.702	.253	.507	.343	blank
	N	25	25	25	25	25

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

Hint: Take a look at whether each variable is associated with the outcome variable of number of slices.  Is the association between the variable and number of slices statistically significant?

Which of these three (3) variables would be most logical to control for in the regression model?

Hunger

How much individuals reported liking pepperoni

Positive mood

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Numbered divider 3

In the output shown below, which is based on predicting income with age and education level, the R-squared is .066. 

Model Summary

Model	R	R Square	Adjusted R Square	Std. Error of the Estimate
1	.256a	.066	.064	2.276

a. Predictors: (Constant), HIGHEST YEAR OF SCHOOL COMPLETED, AGE OF RESPONDENT.

Hint: Remember that to convert a decimal to a percent, you will need to move the decimal point two places to the right.

Which of the following is the appropriate interpretation of the R-squared value? 

Age and education level account for 6.6% of the variability in income.

Income accounts for 6.6% of the variability in age and education level

Age and education level account for 66% of the variability in income.

Income accounts for 66% of the variability in age and education level completed.

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Topic 3 - 

Module Summary and Quiz

Interpreting Regression Models with Dummy-Coded Variables

by Robin KouvarasRobin Kouvaras

Topic 1 of 5

Learning Objective: Interpret regression models with dummy-coded variables.

Before You Begin

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

· Familiarity with multiple regression models

· How to interpret regression coefficients for continuous predictor variables

· How to identify categorical vs. continuous variables

· p-values

· alpha

Introduction

Scenario: Linking Marital Status to Religiosity

Suppose a researcher is interested in how marital status is linked to religiosity. For example, the researcher might wonder if individuals who are married are different from those with other marital statuses in their degree of religiosity. Although the researcher could use a one-way ANOVA to examine this type of research question, she is also interested in later adding additional continuous predictors of religiosity and has, therefore, opted to use a multiple regression model for her analyses.

The researcher decides to use the  2004 General Social Survey data  for her research question. There is a variable in the data set that specifies individuals’ current marital status, and the researcher wants to focus on three marital status groups: married, divorced, and never married. For the outcome variable, the researcher chooses frequency of religious attendance, which has values that range from 0 to 8, with higher values indicating more frequent religious attendance:

Frequency of Religious Attendance Values
0 =	Never
1 =	Less than once a year
2 =	Once a year
3 =	Several times a year
4 =	Once a month
5 =	2-3 times a month
6 =	Nearly every week
7 =	Every week
8 =	More than once a week

On the face of it, it seems that the regression model would have one (1) predictor: marital status. Recall, however, that the unstandardized regression coefficient for a predictor indicates how much the dependent variable changes for every one (1) unit increase in the predictor variable. So, if you put marital status into a regression model, the coefficient would indicate how much religious attendance increases for every one (1) unit increase in marital status. 

Note, though, that marital status is a categorical variable, so it is not clear what a one- (1) unit increase would mean, particularly if there are more than two (2) groups.

When you change values on a categorical variable (i.e., you go from married to divorced), you are switching categories rather than “climbing a scale” as you would be doing for a continuous variable. For a continuous variable like height, for example, it would be clear that a one (1) unit increase corresponds to, for example, a 1-inch increase. 

So, the main point here is that if you simply enter a categorical variable “as is” into a regression model as a predictor, the regression coefficient that corresponds to that variable may not be very meaningful. For this reason, researchers typically use dummy coding for categorical predictor variables in regression models.

How to Create Dummy-Coded Variables

by Robin KouvarasRobin Kouvaras

Topic 2 of 5

Learning Objective: Interpret regression models with dummy-coded variables.

How to Create Dummy-Coded Variables

Dummy-coded variables are created by only using the values of 0 and 1. The general rule used for dummy coding is that you need one (1) fewer dummy-coded variables than you have groups (# total groups – 1). So, for our variable of marital status, we would need two (2) dummy-coded variables because we have chosen to focus on three (3) marital status groups (3 – 2 = 1). The group for which we do not create a dummy-coded variable is typically called the reference category. Often the reference category will be the one that researchers want to compare to other groups. For our research, we might choose “married” as our reference category if we want to compare non-married individuals to married individuals.  

Before we conduct our regression analyses in SPSS, then, we will need to create two (2) dummy-coded variables for marital status:

1. one variable for the divorced group

2. one variable for the never-married group

We will use a 1 to indicate membership to that category (e.g., to indicate that someone is divorced for the “divorced” dummy-coded variable) and 0 to indicate non-membership.  

The table below shows how we would dummy-code our marital status variables.

Notice the Following

If the original value for an individual’s marital status is a 1 (indicating married), that individual would have a 0 for the “divorced” variable and a 0 for the “never married” variable. This is because they are not a “member” of either of these groups, they are not divorced, and they are not in the never-married category. This same logic holds for the remaining two (2) values of marital status. If an individual is divorced, they get a 1 for the divorced group, for example, and a 0 for the never-married group.

Also, note that each individual in the data set will have a value (either a 0 or a 1) for each dummy-coded variable that the researcher creates. 

Suppose the researcher decides to add an additional marital status group (separated), so that she now has the following marital status groups: married, divorced, never married, and separated.

Hint: Count the number of groups you have and subtract 1.

 How many dummy-coded variables would the researcher need to create for her regression model?

5

2

3

4

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Week Ten: Dummy Variables

Posted on: Friday, July 29, 2022 9:50:18 AM EDT

Sometimes, by creative constructs [drafting and using responsible assumptions] a researcher can manipulate data sets to provide more insights  [dummy variables] .

In social science, many of the predictor variables a researcher may want to use are inherently quantitative and measured categorically (i.e., race, gender, political party affiliation, etc.). During week 10, you will learn how to use categorical variables within multiple regression models.

Having now discussed the benefits of multiple regression, we have been reticent about what can go wrong in our models. For models to provide accurate estimates, we must adhere to a set of assumptions. You have had plenty of opportunity to interpret coefficients for metric variables in regression models. Using and interpreting categorical variables takes just a little bit of extra practice. In this Discussion, you will have the opportunity to practice how to recode categorical variables [ dummy ] so they can be used in a regression model and how to properly interpret the coefficients.

A  dummy variable  is a numerical variable used within regression analyses to represent subgroups of the sample within a study. In research design, a dummy variable is often used to distinguish different treatment groups. In the simplest case, we would use a (0,1) dummy variable where a person is given a value of 0 if in the control group or a 1 if in the treated group. Dummy variables are useful because they enable a single regression equation to represent multiple groups: meaning no need to write out separate equation models for each subgroup.

Further, social scientists often need to work with categorical variables in which the different values have no real numerical relationship with each other. Examples include variables for race, political affiliation, or marital status. If you have a variable for political affiliation with possible responses including Democrat, Independent, and Republican, it obviously doesn't make sense to assign values of (1 - 3) and interpret, by error, that a Republican is somehow three times more politically affiliated then a Democrat. The solution is to use a dummy variable(s) with only two values, zero and one. By creating a variable called "Republican" and assign the group  a 1 indicating, simply, members are "Republican" and all others within the study are not.

The decision to code a level is often arbitrary but must be responsible [makes sense]. The level which is not coded is the category to which all other categories will be compared. As such, often the biggest group will be the not-coded category. For example, often "Caucasian" will be the not-coded group if the race of most participants in the sample. Following, if you have a variable called "Asian", the coefficient on the "Asian" variable in your regression will show the effect being Asian rather than Caucasian has on your dependent variable.

Interpreting the Coefficients for Dummy-Coded Variables

by Robin KouvarasRobin Kouvaras

Topic 3 of 5

Learning Objective: Interpret regression models with dummy-coded variables.

How to Interpret Regression Results

Now that you are familiar with how to create dummy-coded variables, we will discuss how to interpret your regression results. Below is the SPSS output using the marital status groups to predict the frequency of religious attendance using multiple regression. Below the regression output, there is also the SPSS output that shows the mean for religious attendance for each of the marital status groups.

SPSS output using the marital status groups to predict the frequency of religious attendance using multiple regression.

Coefficientsa

Model	Unstandardized Coefficients	Standardized Coefficients	t	Sig.
	B	Std. Error	Beta
1	(Constant)	4.328	.095	blank	45.627	.000
	Divorced	-1.239	.206	-.166	-6.009	.000
	Never Married	-1.190	.174	-.189	-6.825	.000

Legend for Coefficientsa
	p-value for the Never Married predictor variable.
	p-value for the Divorced predictor variable.

SPSS output that shows the mean for religious attendance for each of the marital status groups.

Descriptives HOW OFTEN R ATTENDS RELIGIOUS SERVICES

Blank	N	Mean	Std. Deviation	Std. Error	95% Confidence Interval for the Mean	Minimum	Maximum
					Lower Bound	Upper Bound
MARRIED	789	4.33	2.731	.097	4.14	4.52	0	8
DIVORCED	212	3.09	2.687	.185	2.73	3.45	0	8
NEVER MARRIED	332	3.14	2.484	.136	2.87	3.41	0	8
Total	1333	3.83	2.728	.075	3.69	3.98	0	8

Let’s focus on the unstandardized regression coefficients in the output. Each coefficient will indicate how that particular group compares to the reference category (e.g., married) on the dependent variable. The coefficient reflects the comparison between the mean value of the dependent variable for the reference category and the mean value for the group represented by that particular coefficient. For example, first, take a look at the unstandardized regression coefficient for “divorced” (-1.239). This value reflects how the divorced group compares to the married group on religious attendance and indicates that the mean religious attendance for the divorced group is 1.239 units lower than that for the married group. 

A few more things about the output:

· bullet

If you subtract the mean for divorced (3.09) from the mean for married (4.33), you can see that you get the absolute value of the coefficient for the divorced variable: 4.33 – 3.09 = 1.24.  (If you round 1.239, you get 1.24.)  

· bullet

If the value had been positive (1.239 instead of -1.239), it would indicate that the divorced group had a higher mean than the married group on the dependent variable.  

· bullet

Similar to when you are interpreting the coefficients for continuous predictor variables in a regression model, the difference between the reference category and the indicated group is only considered to be statistically significant if the p-value is less than alpha.  In our results above, if we assume an alpha of .05 (or even .01), each predictor would be statistically significant, indicating that each group (divorced, never married) differs from the reference category of married on the dependent variable.

· bullet

Also similar to when you are interpreting the coefficients for continuous predictor variables in a regression model, you can use the absolute value of the standardized regression coefficients to gauge the effect size for each variable; values closer to 0 indicate weaker effects, and values closer to 1 indicate stronger effects.

Hint: Remember that the unstandardized regression coefficients reflect a comparison to the reference category about the mean value of the outcome variable.

Take a look now at the unstandardized regression coefficient for never married (-1.19). What would be an appropriate interpretation of this value?

The never married group mean for religious attendance is 1.19 units lower than the mean for the divorced group.

The never married group mean for religious attendance is 1.19 units higher than the mean for the divorced group.

The never married mean is 1.19 units lower than the married group mean for the dependent variable.

The never married mean is 1.19 units lower than the married group mean for the independent variable

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Topic 4 - 

Module Summary and Quiz

Summary

For categorical predictor variables in regression models, researchers typically create dummy-coded variables. There will be one (1) fewer dummy-coded variables than there are a total number of categories, and each dummy-coded variable will have a value of 0 or 1. The unstandardized regression coefficient for a dummy-coded variable will indicate how that group mean on the dependent variable compares to the group mean for the reference category.   

References

https://go.openathens.net/redirector/waldenu.edu?url=https://dx.doi.org/10.4135/9781412985604.n5

https://go.openathens.net/redirector/waldenu.edu?url=https://dx.doi.org/10.4135/9781412985604

https://go.openathens.net/redirector/waldenu.edu?url=https://methods.sagepub.com/book/regression-diagnostics/n8.xml

https://go.openathens.net/redirector/waldenu.edu?url=https://dx.doi.org/10.4135/9781412985604.n6

https://go.openathens.net/redirector/waldenu.edu?url=https://dx.doi.org/10.4135/9781412985604.n7

https://go.openathens.net/redirector/waldenu.edu?url=https://dx.doi.org/10.4135/9781412985604.n4

Part 2

To prepare for this Part 2 of your Assignment:

· Review Warner’s Chapter 12 and Chapter 2 of the Wagner course text and the media program found in this week’s Learning Resources and consider the use of dummy variables.

· Using the SPSS software, open the Afrobarometer dataset or the High School Longitudinal Study dataset (whichever you choose) found in this week’s Learning Resources.

· Consider the following:

· Create a research question with metric variables and one variable that requires dummy coding. Estimate the model and report results. Note: You are expected to perform regression diagnostics and report that as well.

· Once you perform your analysis, review Chapter 11 of the Wagner text to understand how to copy and paste your output into your Word document.

For this Part 2 Assignment:

Write a 2- to 3-page analysis of your multiple regression using dummy variables results for each research question. In your analysis, display the data for the output. Based on your results, provide an explanation of what the implications of social change might be.

Use proper APA format, citations, and referencing for your analysis, research question, and display of output.

By Day 7

Submit Parts 1 and 2 of your Assignment: Testing for Multiple Regression.

Submission and Grading Information

Wagner, III, W. E. (2020). Using IBM® SPSS® statistics for research methods and social science statistics (7th ed.). Thousand Oaks, CA: Sage Publications.

· Chapter 2, “Transforming Variables” 

· Chapter 11, “Editing Output” (previously read in Week 2, 3, 4, 5. 6, 7, 8, and 9)