Week 7 Linear Regression Exercises

Week 7 Linear Regression Exercises

Simple Regression

Research Question: Does the number of hours worked per week (workweek) predict family income (income)?

Using Polit2SetA data set, run a simple regression using Family Income (income) as the outcome variable (Y) and Number of Hours Worked per Week (workweek) as the independent variable (X). When conducting any regression analysis, the dependent (outcome) variables is always (Y) and is placed on the y-axis, and the independent (predictor) variable is always (X) and is placed on the x-axis.

Follow these steps when using SPSS:

1. Open Polit2SetA data set.
2. Click on Analyze, then click on Regression, then Linear.
3. Move the dependent variable (income) in the box labeled “Dependent” by clicking the arrow button. The dependent variable is a continuous variable.
4. Move the independent variable (workweek) into the box labeled “Independent.”
5. Click on the Statisticsbutton (right side of box) and click on Descriptives, Estimates, Confidence Interval (should be 95%), and Model Fit, then click on Continue.
6. Click on OK.

Assignment: Through analysis of the SPSS output, answer the following questions.

1. What is the total sample size?
2. What is the mean income and mean number of hours worked?
3. What is the correlation coefficient between the outcome and predictor variables? Is it significant? How would you describe the strength and direction of the relationship?
4. What it the value of R squared (coefficient of determination)? Interpret the value.
5. Interpret the standard error of the estimate? What information does this value provide to the researcher?
6. The model fit is determined by the ANOVA table results (F statistic = 37.226, 1,376 degrees of freedom, and the p value is .001). Based on these results, does the model fit the data? Briefly explain. (Hint: A significant finding indicates good model fit.)
7. Based on the coefficients, what is the value of the y-intercept (point at which the line of best fit crosses the y-axis)?
8. Based on the output, write out the regression equation for predicting family income.
9. Using the regression equation, what is the predicted monthly family income for women working 35 hours per week?
10. Using the regression equation, what is the predicted monthly family income for women working 20 hours per week?

Multiple Regression

Assignment: In this assignment we are trying to predict CES-D score (depression) in women. The research question is: How well do age, educational attainment, employment, abuse, and poor health predict depression?

Using Polit2SetC data set, run a multiple regression using CES-D Score (cesd) as the outcome variable (Y) and respondent’s age (age), educational attainment (educatn), currently employed (worknow), number, types of abuse (nabuse), and poor health (poorhlth) as the independent variables (X). When conducting any regression analysis, the dependent (outcome) variables is always (Y) and is placed on the y-axis, and the independent (predictor) variable is always (X) and is placed on the x-axis.

Follow these steps when using SPSS:

1. Open Polit2SetC data set.

2. Click on Analyze,then click on Regression, then Linear.

3. Move the dependent variable, CES-D Score (cesd) into the box labeled “Dependent” by clicking on the arrow button. The dependent variable is a continuous variable.

4. Move the independent variables (age, educatn, worknow, and poorhlth) into the box labeled “Independent.” This is the first block of variables to be entered into the analysis (block 1 of 1). Click on the bottom (top right of independent box), marked “Next”; this will give you another box to enter the next block of indepdent variables (block 2 of 2). Here you are to enter (nabuse). Note: Be sure the Method box states “Enter”.

5. Click on the Statistics button (right side of box) and click on Descriptives, Estimates, Confidence Interval (should be 95%), R square change, and Model Fit, and then click on Continue.

6. Click on OK.

Assignment: (When answering all questions, use the data on the coefficients panel from Model 2).

1. Analyze the data from the SPSS output and write a paragraph summarizing the findings. (Use the example in the SPSS output file as a guide for your write-up.)
2. Which of the predictors were significant predictors in the model?
3. Which of the predictors was the most relevant predictor in the model?
4. Interpret the unstandardized coefficents for educational attainment and poor health.
5. If you wanted to predict a woman’s current CES-D score based on the analysis, what would the unstandardized regression equation be? Include unstandardized coefficients in the equation.

ANOVA^b

Model		Sum of Squares	df		Mean Square	F	Sig.
1	Regression	3.068E7		1	3.068E7	37.226	.000a
	Residual	3.099E8		376	824241.002
	Total	3.406E8		377

a.Predictors: (Constant), Hours worked per week in current job

b.Dependent Variable: Family income prior month, all sources

Coefficients^a

Model		Unstandardized	Standardized			95.0% Confidence Interval
		Coefficients	Coefficients			for B
		B	Std. Error	Beta	t	Sig.	Lower Bound	Upper Bound
1	(Constant)	711.651	135.155		5.265	.000	445.896	977.405
	Hours worked per week	23.083	3.783	.300	6.101	.000	15.644	30.523
	in current job

a. Dependent Variable: Family income prior month, all sources

Part II: Multiple Regression SPSS Output

This part is going to begin with an example that has been interpreted for you. Analyze the output provided and read the interpretation of the data so that you will have an understanding of what you will do for the multiple regression assignment.

Descriptive Statistics

	Mean	Std. Deviation	N
CES-D Score	18.5231	11.90747	156
CESD Score, Wave 1	17.6987	11.40935	156
Number types of abuse	.83	1.203	156

Correlations

			CESD Score,	Number types
		CES-D Score	Wave 1	of abuse
Pearson Correlation	CES-D Score	1.000	.412	.347
	CESD Score, Wave 1	.412	1.000	.187
	Number types of abuse	.347	.187	1.000
Sig. (1-tailed)	CES-D Score	.	.000	.000
	CESD Score, Wave 1	.000	.	.010
	Number types of abuse	.000	.010	.

N	CES-D Score	156	156	156
	CESD Score, Wave 1	156	156	156
	Number types of abuse	156	156	156

Model Summary

Model						Change Statistics
			Adjusted R	Std. Error of	R Square
	R	R Square	Square	the Estimate	Change	F Change	df1		df2	Sig. F Change
1	.412a	.170	.164	10.88446	.170	31.506	1		154	.000
2	.496b	.246	.236	10.41016	.076	15.352	1		153	.000

a.Predictors: (Constant), CESD Score, Wave 1

b.Predictors: (Constant), CESD Score, Wave 1, Number types of abuse

ANOVA^c

Model		Sum of Squares	df		Mean Square	F	Sig.
1	Regression	3732.507		1	3732.507	31.506	.000a
	Residual	18244.613		154	118.472
	Total	21977.120		155
2	Regression	5396.278		2	2698.139	24.897	.000b
	Residual	16580.842		153	108.372
	Total	21977.120		155

a.Predictors: (Constant), CESD Score, Wave 1

b.Predictors: (Constant), CESD Score, Wave 1, Number types of abuse

c. Dependent Variable: CES-D Score

Coefficients^a

Model		Unstandardized		Standardized					95.0% Confidence Interval for
			Coefficients		Coefficients						B
		B		Std. Error	Beta	t		Sig.		Lower Bound		Upper Bound
1	(Constant)	10.911		1.612		6.768	.000	7.726	14.095
	CESD Score, Wave 1		.430		.077	.412	5.613	.000	.279	.581
2	(Constant)		9.584		1.579		6.071		.000		6.465		12.702
	CESD Score, Wave 1		.376		.075	.360	5.035		.000		.228		.523
	Number types of		2.772		.707	.280	3.918		.000		1.374		4.170
	abuse

a. Dependent Variable: CES-D Score

In the regression example, we were statistically controlling for women’s level of depression 2 years earlier and attempting to determine if recent abuse experiences affected current levels of depression, earlier depression held constant.

The correlation between CES-D scores in the two waves of data collection was moderate and positive, r = .412. You can see this value in the Model Summary panel—the value of R in the first step is the bivariate correlation (i.e., r) between the two CES-D scores. Yes, R2 was statistically significant at p < .001in both steps of the regression analysis, as shown in the ANOVA panel. Using R2 increased from .170 in the first model to .246 when the abuse variable was added. The R2 change (increase) of .076 (7.6%) was significant at p<.001, as shown in the Model Summary panel, under change Statistics. This indicates that even when prior levels of depression were held constant, recent abuse accounted for a significant amount of variation in current depression scores. The availability of longitudinal data does not “prove” that abuse experiences affected the women’s level of depression, but it does offer greater supportive evidence than cross-sectional data. If we wanted to predict current CES-D scores, using prior CES-D scores and abuse experiences as predictors, the unstandardized regression equation would be as follows: Y’= 9.584 + .376 (cesdwav1) + 2.772 (nabuse). This information comes from the panel labeled Coefficients.

In terms of the independent variables there are two coefficients on the panel labeled coefficients. The first is the unstandardized coefficients (b-values) which represent the individual contribution of each predictor to the model. The b-value for number types of abuse (2.772) tells us about the relationship between CES-D score (Dependent variable) and number type of abuse (independent variable). These values are used when making predictions and they tell us to what degree the independent variable affects the outcome when the effects of all other variables in the equation are held constant. For example, the interpretation of number, types of abuse is as follows: For each unit increase in the number, types of abuse, the CES-D score (depression) increases by 2.772 units. The increase is dependent on the units that the variable is measured in. So, for each additional type of abuse reported the CES-D depression score will increase by 2.772 points. Always check the value in the significance column to determine if the variables are making a significant contribution to the model.

The second coefficient reported is the standardized Beta coefficient. The standardized coefficient tells us the number of standard deviations that the dependent variable will change as a result of one standard deviation change in the independent variable. The standardized coefficient is typically used

to permit the researcher to understand which of the independent variables is most important in explaining the dependent variable. In the above example, the CES-D score, Wave 1 has a Beta coefficient of .360 and the number, types of abuse has a Beta coefficient of .280. This indicates that the CES-D score, Wave 1 is the most significant predictor in the model and makes the strongest unique contribution to explaining the dependent variable. Note: When you are determining the most significant predictor ignore the negative sign if one exists. So, a predictor with a Beta of -.96 is stronger than a Beta of .55.

SPSS Output for Multiple Regression Assignment

Descriptive Statistics

	Mean	Std. Deviation	N
CES-D Score	18.5815	11.78965	939
Respondent's age at time of	36.54749	6.234511	939
interview
Educational attainment	1.57	.584	939
Currently employed?	.45	.498	939
Poor health self rating	.06	.247	939
Number types of abuse	.85	1.160	939

Correlations

			Respondent'				Number
		CES-D	s age at time	Educational	Currently	Poor health	types of
		Score	of interview	attainment	employed?	self rating	abuse
Pearson	CES-D Score	1.000	.061	-.155	-.220	.270	.370
Correlation	Respondent's age at	.061	1.000	.065	-.077	.140	-.020
	time of interview
	Educational attainment	-.155	.065	1.000	.060	-.074	-.026
	Currently employed?	-.220	-.077	.060	1.000	-.162	-.073
	Poor health self rating	.270	.140	-.074	-.162	1.000	.095
	Number types of abuse	.370	-.020	-.026	-.073	.095	1.000
Sig. (1-tailed)	CES-D Score	.	.031	.000	.000	.000	.000
	Respondent's age at	.031	.	.023	.009	.000	.272
	time of interview
	Educational attainment	.000	.023	.	.032	.012	.215
	Currently employed?	.000	.009	.032	.	.000	.012
	Poor health self rating	.000	.000	.012	.000	.	.002
	Number types of abuse	.000	.272	.215	.012	.002	.
N	CES-D Score	939	939	939	939	939	939
	Respondent's age at	939	939	939	939	939	939
	time of interview
	Educational attainment	939	939	939	939	939	939

	Currently employed?	939	939	939	939	939	939
	Poor health self rating	939	939	939	939	939	939
	Number types of abuse	939	939	939	939	939	939

Model Summary

Model						Change Statistics
			Adjusted R	Std. Error of	R Square					Sig. F
	R	R Square	Square	the Estimate	Change	F Change	df1		df2	Change
1	.348a	.121	.117	11.07693	.121	32.148	4		934	.000
2	.483b	.233	.229	10.34980	.112	136.849	1		933	.000

a. Predictors: (Constant), Poor health self rating, Educational attainment, Respondent's age at time of interview, Currently employed?

b. Predictors: (Constant), Poor health self rating, Educational attainment, Respondent's age at time of interview, Currently employed?, Number types of abuse

ANOVA^c

Model		Sum of Squares	df		Mean Square	F	Sig.
1	Regression	15777.841		4	3944.460	32.148	.000a
	Residual	114600.356		934	122.698
	Total	130378.197		938
2	Regression	30436.854		5	6087.371	56.829	.000b
	Residual	99941.343		933	107.118
	Total	130378.197		938

a. Predictors: (Constant), Poor health self rating, Educational attainment, Respondent's age at time of interview, Currently employed?

b. Predictors: (Constant), Poor health self rating, Educational attainment, Respondent's age at time of interview, Currently employed?, Number types of abuse

c. Dependent Variable: CES-D Score

Coefficients^a

Model		Unstandardized	Standardized			95.0% Confidence Interval for
		Coefficients	Coefficients				B
		B	Std. Error	Beta	t	Sig.	Lower Bound		Upper Bound
1	(Constant)	22.182	2.351		9.434	.000	17.567		26.796
	Respondent's age at	.045	.059	.024	.767	.443	-.070		.161
	time of interview
	Educational attainment	-2.608	.624	-.129	-4.179	.000	-3.832		-1.383
	Currently employed?	-4.092	.738	-.173	-5.544	.000	-5.540		-2.643
	Poor health self rating	10.928	1.503	.229	7.270	.000	7.978		13.878
2	(Constant)	18.165	2.224		8.169	.000	13.801		22.528
	Respondent's age at	.068	.055	.036	1.240	.215	-.040		.176
	time of interview
	Educational attainment	-2.518	.583	-.125	-4.318	.000	-3.663		-1.374
	Currently employed?	-3.605	.691	-.152	-5.219	.000	-4.961		-2.250
	Poor health self rating	9.496	1.410	.199	6.735	.000	6.729		12.263
	Number types of abuse	3.432	.293	.338	11.698	.000	2.856		4.008

a. Dependent Variable: CES-D Score