sw388r7 data analysis & computers ii slide 1 multiple regression – basic relationships purpose...

SW388R7Data Analysis

& Computers II

Slide 1

Multiple Regression – Basic Relationships

Purpose of multiple regression

Different types of multiple regression

Standard multiple regression

Hierarchical multiple regression

Stepwise multiple regression

Steps in solving regression problems


& Computers II

Slide 2

Purpose of multiple regression

The purpose of multiple regression is to analyze the relationship between metric or dichotomous independent variables and a metric dependent variable.

If there is a relationship, using the information in the independent variables will improve our accuracy in predicting values for the dependent variable.


& Computers II

Slide 3

Types of multiple regression

There are three types of multiple regression, each of which is designed to answer a different question: Standard multiple regression is used to

evaluate the relationships between a set of independent variables and a dependent variable.

Hierarchical, or sequential, regression is used to examine the relationships between a set of independent variables and a dependent variable, after controlling for the effects of some other independent variables on the dependent variable.

Stepwise, or statistical, regression is used to identify the subset of independent variables that has the strongest relationship to a dependent variable.


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Slide 4

Standard multiple regression

In standard multiple regression, all of the independent variables are entered into the regression equation at the same time

Multiple R and R² measure the strength of the relationship between the set of independent variables and the dependent variable. An F test is used to determine if the relationship can be generalized to the population represented by the sample.

A t-test is used to evaluate the individual relationship between each independent variable and the dependent variable.


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Slide 5

Hierarchical multiple regression

In hierarchical multiple regression, the independent variables are entered in two stages.

In the first stage, the independent variables that we want to control for are entered into the regression. In the second stage, the independent variables whose relationship we want to examine after the controls are entered.

A statistical test of the change in R² from the first stage is used to evaluate the importance of the variables entered in the second stage.


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Slide 6

Stepwise multiple regression

Stepwise regression is designed to find the most parsimonious set of predictors that are most effective in predicting the dependent variable.

Variables are added to the regression equation one at a time, using the statistical criterion of maximizing the R² of the included variables.

When none of the possible addition can make a statistically significant improvement in R², the analysis stops.


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Slide 7

Problem 1 - standard multiple regression

In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing data, violation of assumptions, or outliers, and that the split sample validation will confirm the generalizability of the results. Use a level of significance of 0.05.

The variables "strength of affiliation" [reliten] and "frequency of prayer" [pray] have a strong relationship to the variable "frequency of attendance at religious services" [attend].

Survey respondents who were less strongly affiliated with their religion attended religious services less often. Survey respondents who prayed less often attended religious services less often.

1. True 2. True with caution 3. False 4. Inappropriate application of a statistic


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Slide 8

Dissecting problem 1 - 1

1. In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing data, violation of assumptions, or outliers, and that the split sample validation will confirm the generalizability of the results. Use a level of significance of 0.05.




When a problem states that there is a relationship between some independent variables and a dependent variable, we do standard multiple regression.

The variables listed first in the problem statement are the independent variables (ivs): "strength of affiliation" [reliten] and "frequency of prayer" [pray]

The variable that is related to is the dependent variable (dv): "frequency of attendance at religious services" [attend].


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In order for a problem to be true, we will have find:• a statistically significant relationship

between the ivs and the dv • a relationship of the correct strength

The relationship of each of the independent variables to the dependent variable must be statistically significant and interpreted correctly.


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Slide 10

Request a standard multiple regression

To compute a multiple regression in SPSS, select the Regression | Linear command from the Analyze menu.


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Specify the variables and selection method

First, move the dependent variable attend to the Dependent text box.

Second, move the independent variables reliten and pray to the Independent(s) list box.

Third, select the method for entering the variables into the analysis from the drop down Method menu. In this example, we accept the default of Enter for direct entry of all variables, which produces a standard multiple regression.Fourth, click on the

Statistics… button to specify the statistics options that we want.


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Slide 12

Specify the statistics output options

Second, mark the checkboxes for Model Fit and Descriptives.

Third, click on the Continue button to close the dialog box.

First, mark the checkboxes for Estimates on the Regression Coefficients panel.


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Slide 13

Request the regression output

Click on the OK button to request the regression output.


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LEVEL OF MEASUREMENT

Multiple regression requires that the dependent variable be metric and the independent variables be metric or dichotomous. "Frequency of attendance at religious services" [attend] is an ordinal level variable, which satisfies the level of measurement requirement if we follow the convention of treating ordinal level variables as metric variables. Since some data analysts do not agree with this convention, a note of caution should be included in our interpretation.

"Strength of affiliation" [reliten] and "frequency of prayer" [pray] are ordinal level variables. If we follow the convention of treating ordinal level variables as metric variables, the level of measurement requirement for multiple regression analysis is satisfied. Since some data analysts do not agree with this convention, a note of caution should be included in our interpretation.


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Slide 15

Descriptive Statistics

3.15 2.653 113

2.12 1.084 113

2.90 1.575 113

HOW OFTEN R ATTENDSRELIGIOUS SERVICES

STRENGTH OFAFFILIATION

HOW OFTEN DOES RPRAY

Mean Std. Deviation N

SAMPLE SIZE

The minimum ratio of valid cases to independent variables for multiple regression is 5 to 1. With 113 valid cases and 2 independent variables, the ratio for this analysis is 56.5 to 1, which satisfies the minimum requirement.

In addition, the ratio of 56.5 to 1 satisfies the preferred ratio of 15 to 1.


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Slide 16

ANOVAb

374.757 2 187.379 49.824 .000a

413.685 110 3.761

788.442 112

Regression

Residual

Total

Model1

Sum ofSquares df Mean Square F Sig.

Predictors: (Constant), HOW OFTEN DOES R PRAY, STRENGTH OF AFFILIATIONa.

Dependent Variable: HOW OFTEN R ATTENDS RELIGIOUS SERVICESb.

OVERALL RELATIONSHIP BETWEEN INDEPENDENT AND DEPENDENT

VARIABLES - 1

The probability of the F statistic (49.824) for the overall regression relationship is <0.001, less than or equal to the level of significance of 0.05. We reject the null hypothesis that there is no relationship between the set of independent variables and the dependent variable (R² = 0). We support the research hypothesis that there is a statistically significant relationship between the set of independent variables and the dependent variable.


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Model Summary

.689a .475 .466 1.939Model1

R R SquareAdjustedR Square

Std. Error ofthe Estimate

Predictors: (Constant), HOW OFTEN DOES R PRAY,STRENGTH OF AFFILIATION

a.


VARIABLES - 2

The Multiple R for the relationship between the set of independent variables and the dependent variable is 0.689, which would be characterized as strong using the rule of thumb than a correlation less than or equal to 0.20 is characterized as very weak; greater than 0.20 and less than or equal to 0.40 is weak; greater than 0.40 and less than or equal to 0.60 is moderate; greater than 0.60 and less than or equal to 0.80 is strong; and greater than 0.80 is very strong.


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RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO

DEPENDENT VARIABLE - 1

Coefficientsa

7.167 .442 16.206 .000

-1.138 .194 -.465 -5.857 .000

-.554 .134 -.329 -4.145 .000

(Constant)


HOW OFTENDOES R PRAY

Model1

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients

t Sig.

Dependent Variable: HOW OFTEN R ATTENDS RELIGIOUS SERVICESa.

For the independent variable strength of affiliation, the probability of the t statistic (-5.857) for the b coefficient is <0.001 which is less than or equal to the level of significance of 0.05. We reject the null hypothesis that the slope associated with strength of affiliation is equal to zero (b = 0) and conclude that there is a statistically significant relationship between strength of affiliation and frequency of attendance at religious services.


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Slide 19

Coefficientsa

7.167 .442 16.206 .000

-1.138 .194 -.465 -5.857 .000

-.554 .134 -.329 -4.145 .000

(Constant)



Model1

B Std. Error


Beta


t Sig.




The b coefficient associated with strength of affiliation (-1.138) is negative, indicating an inverse relationship in which higher numeric values for strength of affiliation are associated with lower numeric values for frequency of attendance at religious services.

Since both variables are ordinal level, we will have to look at the coding for each before we can make a correct interpretation. For ordinal level variables the numeric codes can be associated with labels in ascending or descending order.


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Slide 20



The independent variable strength of affiliation is an ordinal variable that is coded so that higher numeric values are associated with survey respondents who were less strongly affiliated with their religion.


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The dependent variable frequency of attendance at religious services is also an ordinal variable. It is coded so that lower numeric values are associated with survey respondents who attended religious services less often.

Therefore, the negative value of b implies that survey respondents who were less strongly affiliated with their religion attended religious services less often.


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Coefficientsa

7.167 .442 16.206 .000

-1.138 .194 -.465 -5.857 .000

-.554 .134 -.329 -4.145 .000

(Constant)



Model1

B Std. Error


Beta


t Sig.


For the independent variable frequency of prayer, the probability of the t statistic (-4.145) for the b coefficient is <0.001 which is less than or equal to the level of significance of 0.05. We reject the null hypothesis that the slope associated with frequency of prayer is equal to zero (b = 0) and conclude that there is a statistically significant relationship between frequency of prayer and frequency of attendance at religious services.


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Slide 23

Coefficientsa

7.167 .442 16.206 .000

-1.138 .194 -.465 -5.857 .000

-.554 .134 -.329 -4.145 .000

(Constant)



Model1

B Std. Error


Beta


t Sig.




The b coefficient associated with how often does r pray (-0.554) is negative, indicating an inverse relationship in which higher numeric values for how often does r pray are associated with lower numeric values for frequency of attendance at religious services.

Since both variables are ordinal level, we will have to look at the coding for each before we can make a correct interpretation. For ordinal level variables the numeric codes can be associated with labels in ascending or descending order.


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Slide 24



The independent variable frequency of prayer is an ordinal variable that is coded so that higher numeric values are associated with survey respondents who prayed less often.


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Slide 25



The dependent variable frequency of attendance at religious services is also an ordinal variable. It is coded so that lower numeric values are associated with survey respondents who attended religious services less often.

Therefore, the negative value of b implies that survey respondents who prayed less often attended religious services less often.


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Slide 26

Answer to problem 1

The independent and dependent variables were metric (ordinal).

The ratio of cases to independent variables was 56.5 to 1.

The overall relationship was statistically significant and its strength was characterized correctly.

The b coefficient for all variables was statistically significant and the direction of the relationships were characterized correctly.

The answer to the question is true with caution. The caution is added because of the ordinal variables.


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Slide 27

Problem 2 – hierarchical regression

In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing data, violation of assumptions, or outliers, and that the split sample validation will confirm the generalizability of the results. Use a level of significance of 0.05.

After controlling for the effects of the variables "age" [age] and "sex" [sex], the addition of the variables "happiness of marriage" [hapmar], "condition of health" [health], and "attitude toward life" [life] reduces the error in predicting "general happiness" [happy] by 36.1%.

After controlling for age and sex, the variables happiness of marriage, condition of health, and attitude toward life each make an individual contribution to reducing the error in predicting general happiness. Survey respondents who were less happy with their marriages were less happy overall. Survey respondents who said they were not as healthy were less happy overall. Survey respondents who felt life was less exciting were less happy overall.



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The variables that we add in after the control variables are the independent variables that we think will have a statistical relationship to the dependent variable: "happiness of marriage" [hapmar], "condition of health" [health], and "attitude toward life" [life]

The variables listed first in the problem statement are the independent variables (ivs) whose effect we want to control before we test for the relationship: "age"[age] and "sex" [sex],

The variable that to be predicted or related to is the dependent variable (dv): "general happiness" [happy]


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Slide 29






The relationship between each of the independent variables entered after the control variables and the dependent variable must be statistically significant and interpreted correctly.

In order for a problem to be true, the relationship between the added variables and the dependent variable must be statistically significant, and the strength of the relationship after including the control variables must be correctly stated.

We are generally not interested in whether or not the control variables have a statistically significant relationship to the dependent variables.


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Slide 30

Request a hierarchical multiple regression



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Slide 31

Specify independent variables to control for

First, move the dependent variable happy to the Dependent text box.

Second, move the independent variables to control for age and sex to the Independent(s) list box.

Third, select the method for entering the variables into the analysis from the drop down Method menu. In this example, we accept the default of Enter for direct entry of all variables in the first block which will force the controls into the regression.

Fourth, click on the Next button to tell SPSS to add another block of variables to the regression analysis.


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Slide 32

Add the other independent variables

First, move the other independent variables hapmar, health and life to the Independent(s) list box for block 2.

Second, click on the Statistics… button to specify the statistics options that we want.

SPSS identifies that we will now be adding variables to a second block.


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Slide 33


Second, mark the checkboxes for Model Fit, Descriptives, and R squared change.

The R squared change statistic will tell us whether or not the variables added after the controls have a relationship to the dependent variable.




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Slide 35


Multiple regression requires that the dependent variable be metric and the independent variables be metric or dichotomous. "General happiness" [happy] is an ordinal level variable, which satisfies the level of measurement requirement if we follow the convention of treating ordinal level variables as metric variables. Since some data analysts do not agree with this convention, a note of caution should be included in our interpretation.

"Age" [age] is an interval level variable, which satisfies the level of measurement requirements for multiple regression analysis.

"Happiness of marriage" [hapmar], "condition of health" [health], and "attitude toward life" [life] are ordinal level variables. If we follow the convention of treating ordinal level variables as metric variables, the level of measurement requirement for multiple regression analysis is satisfied. Since some data analysts do not agree with this convention, a note of caution should be included in our interpretation.

"Sex" [sex] is a dichotomous or dummy-coded nominal variable which may be included in multiple regression analysis.


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Slide 36


1.63 .626 90

45.50 15.221 90

1.61 .490 90

1.42 .540 90

1.80 .810 90

1.49 .525 90

GENERAL HAPPINESS

AGE OF RESPONDENT

RESPONDENTS SEX

HAPPINESS OFMARRIAGE

CONDITION OF HEALTH

IS LIFE EXCITING ORDULL


SAMPLE SIZE

The minimum ratio of valid cases to independent variables for multiple regression is 5 to 1. With 90 valid cases and 5 independent variables, the ratio for this analysis is 18.0 to 1, which satisfies the minimum requirement.

In addition, the ratio of 18.0 to 1 satisfies the preferred ratio of 15 to 1.


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Slide 37

ANOVAc

.006 2 .003 .007 .993a

34.894 87 .401

34.900 89

12.601 5 2.520 9.493 .000b

22.299 84 .265

34.900 89

Regression

Residual

Total

Regression

Residual

Total

Model1

2


Predictors: (Constant), RESPONDENTS SEX, AGE OF RESPONDENTa.

Predictors: (Constant), RESPONDENTS SEX, AGE OF RESPONDENT, IS LIFEEXCITING OR DULL, HAPPINESS OF MARRIAGE, CONDITION OF HEALTH

b.

Dependent Variable: GENERAL HAPPINESSc.


VARIABLES

The probability of the F statistic (9.493) for the overall regression relationship for all indpendent variables is <0.001, less than or equal to the level of significance of 0.05. We reject the null hypothesis that there is no relationship between the set of all independent variables and the dependent variable (R² = 0). We support the research hypothesis that there is a statistically significant relationship between the set of all independent variables and the dependent variable.


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Slide 38

Model Summary

.013a .000 -.023 .633 .000 .007 2 87 .993

.601b .361 .323 .515 .361 15.814 3 84 .000

Model1

2



R SquareChange F Change df1 df2 Sig. F Change

Change Statistics


Predictors: (Constant), RESPONDENTS SEX, AGE OF RESPONDENT, IS LIFE EXCITING OR DULL, HAPPINESS OFMARRIAGE, CONDITION OF HEALTH

b.

REDUCTION IN ERROR IN PREDICTING DEPENDENT VARIABLE - 1

The R Square Change statistic for the increase in R² associated with the added variables (happiness of marriage, condition of health, and attitude toward life) is 0.361. Using a proportional reduction in error interpretation for R², information provided by the added variables reduces our error in predicting general happiness by 36.1%.


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Slide 39

Model Summary

.013a .000 -.023 .633 .000 .007 2 87 .993

.601b .361 .323 .515 .361 15.814 3 84 .000

Model1

2



R SquareChange F Change df1 df2 Sig. F Change

Change Statistics


Predictors: (Constant), RESPONDENTS SEX, AGE OF RESPONDENT, IS LIFE EXCITING OR DULL, HAPPINESS OFMARRIAGE, CONDITION OF HEALTH

b.

REDUCTION IN ERROR IN PREDICTING DEPENDENT VARIABLE - 2

The probability of the F statistic (15.814) for the change in R² associated with the addition of the predictor variables to the regression analysis containing the control variables is <0.001, less than or equal to the level of significance of 0.05. We reject the null hypothesis that there is no improvement in the relationship between the set of independent variables and the dependent variable when the predictors are added (R² Change = 0).

We support the research hypothesis that there is a statistically significant improvement in the relationship between the set of independent variables and the dependent variable.


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Slide 40

Coefficientsa

1.594 .341 4.677 .000

.000 .005 .012 .107 .915

.011 .140 .008 .078 .938

.432 .341 1.268 .208

-.001 .004 -.035 -.385 .701

-.013 .115 -.010 -.113 .911

.599 .104 .517 5.741 .000

.101 .072 .131 1.408 .163

.170 .108 .142 1.570 .120

(Constant)

AGE OF RESPONDENT

RESPONDENTS SEX

(Constant)

AGE OF RESPONDENT

RESPONDENTS SEX


CONDITION OF HEALTH


Model1

2

B Std. Error


Beta


t Sig.

Dependent Variable: GENERAL HAPPINESSa.

RELATIONSHIP OF ADDED INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 1

If there is a relationship between each added individual independent variable and the dependent variable, the probability of the statistical test of the b coefficient (slope of the regression line) will be less than or equal to the level of significance. The null hypothesis for this test states that b is equal to zero, indicating a flat regression line and no relationship.

If we reject the null hypothesis and find that there is a relationship between the variables, the sign of the b coefficient indicates the direction of the relationship for the data values. If b is greater than or equal to zero, the relationship is positive or direct. If b is less than zero, the relationship is negative or inverse. If the variable is dichotomous or ordinal, the direction of the coding must be taken into account to make a correct interpretation.


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Slide 41

Coefficientsa

1.594 .341 4.677 .000

.000 .005 .012 .107 .915

.011 .140 .008 .078 .938

.432 .341 1.268 .208

-.001 .004 -.035 -.385 .701

-.013 .115 -.010 -.113 .911

.599 .104 .517 5.741 .000

.101 .072 .131 1.408 .163

.170 .108 .142 1.570 .120

(Constant)

AGE OF RESPONDENT

RESPONDENTS SEX

(Constant)

AGE OF RESPONDENT

RESPONDENTS SEX


CONDITION OF HEALTH


Model1

2

B Std. Error


Beta


t Sig.



For the independent variable happiness of marriage, the probability of the t statistic (5.741) for the b coefficient is <0.001 which is less than or equal to the level of significance of 0.05.

We reject the null hypothesis that the slope associated with happiness of marriage is equal to zero (b = 0) and conclude that there is a statistically significant relationship between happiness of marriage and general happiness.


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Slide 42

Coefficientsa

1.594 .341 4.677 .000

.000 .005 .012 .107 .915

.011 .140 .008 .078 .938

.432 .341 1.268 .208

-.001 .004 -.035 -.385 .701

-.013 .115 -.010 -.113 .911

.599 .104 .517 5.741 .000

.101 .072 .131 1.408 .163

.170 .108 .142 1.570 .120

(Constant)

AGE OF RESPONDENT

RESPONDENTS SEX

(Constant)

AGE OF RESPONDENT

RESPONDENTS SEX


CONDITION OF HEALTH


Model1

2

B Std. Error


Beta


t Sig.



The b coefficient associated with happiness of marriage (0.599) is positive, indicating a direct relationship in which higher numeric values for happiness of marriage are associated with higher numeric values for general happiness.


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Slide 43


The independent variable happiness of marriage is an ordinal variable that is coded so that higher numeric values are associated with survey respondents who were less happy with their marriages.


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Slide 44


The dependent variable general happiness is also an ordinal variable. It is coded so that higher numeric values are associated with survey respondents who were less happy overall.

Therefore, the positive value of b implies that survey respondents who were less happy with their marriages were less happy overall.


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Slide 45

Coefficientsa

1.594 .341 4.677 .000

.000 .005 .012 .107 .915

.011 .140 .008 .078 .938

.432 .341 1.268 .208

-.001 .004 -.035 -.385 .701

-.013 .115 -.010 -.113 .911

.599 .104 .517 5.741 .000

.101 .072 .131 1.408 .163

.170 .108 .142 1.570 .120

(Constant)

AGE OF RESPONDENT

RESPONDENTS SEX

(Constant)

AGE OF RESPONDENT

RESPONDENTS SEX


CONDITION OF HEALTH


Model1

2

B Std. Error


Beta


t Sig.



For the independent variable condition of health, the probability of the t statistic (1.408) for the b coefficient is 0.163 which is greater than the level of significance of 0.05. We fail to reject the null hypothesis that the slope associated with condition of health is equal to zero (b = 0) and conclude that there is not a statistically significant relationship between condition of health and general happiness. The statement in the problem that "survey respondents who said they were not as healthy were less happy overall" is incorrect.


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Slide 46

Answer to problem 2

The independent and dependent variables were metric or dichotomous. Some are ordinal.


The overall relationship was statistically significant and its strength was characterized correctly.

The change in R2 associated with adding the second block of variables was statistically significant and correctly interpreted.

The b coefficient for happiness of marriage was statistically significant and correctly interpreted. The b coefficient for condition of health was not statistically significant. We cannot conclude that there was a relationship between condition of health and general happiness.

The answer to the question is false.


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Slide 47

Problem 3 – Stepwise Regression


From the list of variables "number of hours worked in the past week" [hrs1], "occupational prestige score" [prestg80], "highest year of school completed" [educ], and "highest academic degree" [degree], the best predictors of "total family income" [income98] are "highest academic degree" [degree] and "occupational prestige score" [prestg80]. Highest academic degree and occupational prestige score have a moderate relationship to total family income.

The most important predictor of total family income is occupational prestige score. The second most important predictor of total family income is highest academic degree.

Survey respondents who had higher academic degrees had higher total family incomes. Survey respondents who had more prestigious occupations had higher total family incomes.



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Slide 48







The best predictors are the variables that will be meet the statistical criteria for inclusion in the model.

The variables listed first in the problem statement are the independent variables from which the computer will select the best subset using statistical criteria.

The variable that to be predicted or related to is the dependent variable (dv): "total family income" [income98]


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Slide 49







In order for a problem to be true, we will have find:• a statistically significant relationship

between the included ivs and the dv • a relationship of the correct strength

The importance of the variables is provided by the stepwise order of entry of the variable into the regression analysis.


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Slide 50







The relationship between each of the independent variables entered after the control variables and the dependent variable must be statistically significant and interpreted correctly.

Since statistical significance of a variable's contribution toward explaining the variance in the dependent variable is almost always used as the criteria for inclusion, the statistical significance of the relationships is usually assured.


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Slide 51

Request a stepwise multiple regression



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Slide 52

Specify variables and method for selecting variables

First, move the dependent variable income98 to the Dependent text box.

Second, move the independent variables to control for hrs1, prestg80, educ, and degree to the Independent(s) list box.

Third, select the Stepwise method for entering the variables into the analysis from the drop down Method menu.


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Slide 53

Open statistics options dialog box

First, click on the Statistics… button to specify the statistics options that we want.


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Slide 54


Second, mark the checkboxes for Model Fit and Descriptives.




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Slide 55




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Slide 56


Multiple regression requires that the dependent variable be metric and the independent variables be metric or dichotomous. "Total family income" [income98] is an ordinal level variable, which satisfies the level of measurement requirement if we follow the convention of treating ordinal level variables as metric variables. Since some data analysts do not agree with this convention, a note of caution should be included in our interpretation.

"Number of hours worked in the past week" [hrs1], "occupational prestige score" [prestg80], and "highest year of school completed" [educ] are interval level variables, which satisfies the level of measurement requirements for multiple regression analysis.

"Highest academic degree" [degree] is an ordinal level variable. If we follow the convention of treating ordinal level variables as metric variables, the level of measurement requirement for multiple regression analysis is satisfied. Since some data analysts do not agree with this convention, a note of caution should be included in our interpretation.


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Slide 57


17.06 4.130 151

41.45 12.076 151

45.64 14.183 151

14.00 2.587 151

1.74 1.159 151

TOTAL FAMILY INCOME

NUMBER OF HOURSWORKED LAST WEEK

RS OCCUPATIONALPRESTIGE SCORE (1980)

HIGHEST YEAR OFSCHOOL COMPLETED

RS HIGHEST DEGREE


SAMPLE SIZE

The minimum ratio of valid cases to independent variables for stepwise multiple regression is 5 to 1. With 151 valid cases and 4 independent variables, the ratio for this analysis is 37.75 to 1, which satisfies the minimum requirement.

However, the ratio of 37.75 to 1 does not satisfy the preferred ratio of 50 to 1. A caution should be added to the interpretation of the analysis and a split sample validation should be conducted.


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Slide 58

Variables Entered/Removeda

RSHIGHESTDEGREE

.

Stepwise(Criteria:Probability-of-F-to-enter <=.050,Probability-of-F-to-remove >=.100).

RSOCCUPATIONALPRESTIGESCORE (1980)

.


Model1

2

VariablesEntered

VariablesRemoved Method

Dependent Variable: TOTAL FAMILY INCOMEa.

RELATIONSHIP BETWEEN BEST PREDICTORS AND THE DEPENDENT

VARIABLE - 1

The best subset of predictors for total family income included the independent variables: highest academic degree and occupational prestige score.


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Slide 59

ANOVAc

620.049 1 620.049 47.661 .000a

1938.415 149 13.009

2558.464 150

722.947 2 361.473 29.146 .000b

1835.517 148 12.402

2558.464 150

Regression

Residual

Total

Regression

Residual

Total

Model1

2


Predictors: (Constant), RS HIGHEST DEGREEa.

Predictors: (Constant), RS HIGHEST DEGREE, RS OCCUPATIONAL PRESTIGESCORE (1980)

b.

Dependent Variable: TOTAL FAMILY INCOMEc.


VARIABLE - 2

The probability of the F statistic (29.146) for the regression relationship which includes these variables is <0.001, less than or equal to the level of significance of 0.05. We reject the null hypothesis that there is no relationship between the best subset of independent variables and the dependent variable (R² = 0). We support the research hypothesis that there is a statistically significant relationship between the best subset of independent variables and the dependent variable.


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Slide 60

Model Summary

.492a .242 .237 3.607

.532b .283 .273 3.522

Model1

2



Predictors: (Constant), RS HIGHEST DEGREEa.

Predictors: (Constant), RS HIGHEST DEGREE, RSOCCUPATIONAL PRESTIGE SCORE (1980)

b.


VARIABLE - 3

The Multiple R for the relationship between the subset of independent variables that best predict the dependent variable is 0.532, which would be characterized as moderate using the rule of thumb than a correlation less than or equal to 0.20 is characterized as very weak; greater than 0.20 and less than or equal to 0.40 is weak; greater than 0.40 and less than or equal to 0.60 is moderate; greater than 0.60 and less than or equal to 0.80 is strong; and greater than 0.80 is very strong.


& Computers II

Slide 61

Variables Entered/Removeda

RSHIGHESTDEGREE

.


RSOCCUPATIONALPRESTIGESCORE (1980)

.


Model1

2

VariablesEntered

VariablesRemoved Method

Dependent Variable: TOTAL FAMILY INCOMEa.


VARIABLE - 4

Based on the table of "Variables Entered/ Removed," the most important predictor of total family income is highest academic degree.

The second most important predictor of total family income is occupational prestige score.

The importance of the predictors stated in the problem is not correct.


& Computers II

Slide 62

Answer to problem 3

The independent and dependent variables were metric, interval or ordinal.


The relationship of the included variables was statistically significant and the strength of the relationship was characterized correctly.

However, the order of entry, or importance, was not stated correctly in the problem.

The answer to the question is false.


& Computers II

Slide 63

Standard multiple regression - 1

The following is a guide to the decision process for answering problems about standard multiple regression analysis:

Yes

Dependent variable metric?Independent variables metric or dichotomous?

Yes

No

Ratio of cases to independent variables at least 5 to 1?

Yes

No

Yes

Probability of ANOVA test of regression less than/equal to level of significance?

Yes

NoFalse

Inappropriate application of a statistic



& Computers II

Slide 64


Yes

Strength of relationship for included variables interpreted correctly?

Yes

No

Probability of relationship between each IV and DV <= level of significance?

Yes

No

Yes

Direction of relationship between each IV and DV interpreted correctly?

Yes

NoFalse

False

False


& Computers II

Slide 65


No

Any independent variable or dependent variable ordinal level of measurement?

Yes

Yes

Ratio of cases to independent variables at preferred sample size of at least 15 to 1?

No

True

True with caution

True with caution


& Computers II

Slide 66

Hierarchical regression - 1

The following is a guide to the decision process for answering problems about hierarchical regression analysis:

Yes


Yes

No


Yes

No

Yes


Yes

NoFalse




& Computers II

Slide 67


Yes

Probability of F test of for change in R² less than or equal to level of significance?

Yes

No

Change in R² correctly reported and interpreted?

Yes

No

Probability of relationship between each IV added after controls and DV less than or equal to level of significance?

Yes

NoFalse

False

False


& Computers II

Slide 68


No


Yes

Yes


No

True

True with caution

True with caution

Direction of relationship between each IV added after controls and DV interpreted correctly?

Yes

NoFalse


& Computers II

Slide 69

Stepwise regression - 1

The following is a guide to the decision process for answering problems about stepwise regression analysis:

Yes


Yes

No


Yes

No

Yes

Is the list of independent variables selected for inclusion correct?

Yes

NoFalse




& Computers II

Slide 70


Yes


Yes

No

Strength of relationship for included variables interpreted correctly?

Yes

No

Yes

Is the stated order of importance independent variables correct?

Yes

NoFalse

False

False


& Computers II

Slide 71


Yes

Probability of relationship between each included IV and DV less than or equal to level of significance?

Yes

NoFalse

Direction of relationship between each included IV and DV interpreted correctly?

Yes

NoFalse


& Computers II

Slide 72


No


Yes


Yes

No

True

True with caution

True with caution

Yes

sw388r7 data analysis & computers ii slide 1 multiple regression – basic relationships purpose...

Documents

multiple regression

regression equation

regression problems

time multiple r

included variables

metric dependent variable

strongest relationship

individual relationship