Multiple Regression. The test you choose depends on level of measurement: Independent...

Multiple Regression

Multiple RegressionThe test you choose depends on level of measurement:

Independent Variable Dependent Variable Test

Dichotomous Interval-Ratio Independent Samples t-testDichotomous

Nominal Nominal Cross TabsDichotomous Dichotomous

Nominal Interval-Ratio ANOVADichotomous Dichotomous

Interval-Ratio Interval-Ratio Bivariate Regression/CorrelationDichotomous

Two or More…Interval-Ratio Dichotomous Interval-Ratio Multiple Regression

Multiple Regression Multiple Regression is very popular among

social scientists. Most social phenomena have more than one

cause. It is very difficult to manipulate just one social

variable through experimentation. Social scientists must attempt to model complex

social realities to explain them.

Multiple Regression Multiple Regression allows us to:

Use several variables at once to explain the variation in a continuous dependent variable.

Isolate the unique effect of one variable on the continuous dependent variable while taking into consideration that other variables are affecting it too.

Write a mathematical equation that tells us the overall effects of several variables together and the unique effects of each on a continuous dependent variable.

Control for other variables to demonstrate whether bivariate relationships are spurious

Multiple Regression For example:

A researcher may be interested in the relationship between Education and Income and Number of Children in a family.

Independent Variables

Education

Family Income

Dependent Variable

Number of Children

Multiple Regression

For example: Research Hypothesis: As education of respondents

increases, the number of children in families will decline (negative relationship).

Research Hypothesis: As family income of respondents increases, the number of children in families will decline (negative relationship).


Education

Family Income

Dependent Variable

Number of Children

Multiple Regression

For example: Null Hypothesis: There is no relationship between

education of respondents and the number of children in families.

Null Hypothesis: There is no relationship between family income and the number of children in families.


Education

Family Income

Dependent Variable

Number of Children

Multiple Regression

Bivariate regression is based on fitting a line as close as possible to the plotted coordinates of your data on a two-dimensional graph.

Trivariate regression is based on fitting a plane as close as possible to the plotted coordinates of your data on a three-dimensional graph.

Case: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Children (Y): 2 5 1 9 6 3 0 3 7 7 2 5 1 9 6 3 0 3 7 14 2 5 1 9 6

Education (X1) 12 16 2012 9 18 16 14 9 12 12 10 20 11 9 18 16 14 9 8 12 10 20 11 9

Income 1=$10K (X2): 3 4 9 5 4 12 10 1 4 3 10 4 9 4 4 12 10 6 4 1 10 3 9 2 4

Multiple Regression

Case: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Children (Y): 2 5 1 9 6 3 0 3 7 7 2 5 1 9 6 3 0 3 7 14 2 5 1 9 6

Education (X1) 12 16 2012 9 18 16 14 9 12 12 10 20 11 9 18 16 14 9 8 12 10 20 11 9

Income 1=$10K (X2): 3 4 9 5 4 12 10 1 4 3 10 4 9 4 4 12 10 6 4 1 10 3 9 2 4

Y

X1X2

0

Plotted coordinates (1 – 10) for Education, Income and Number of Children

Multiple Regression

Case: 1 2 3 4 5 6 7 8 9 10

Children (Y): 2 5 1 9 6 3 0 3 7 7

Education (X1) 12 16 2012 9 18 16 14 9 12

Income 1=$10K (X2): 3 4 9 5 4 12 10 1 4 3

Y

X1X2

0

What multiple regression does is fit a plane to these coordinates.

Multiple Regression Mathematically, that plane is:

Y = a + b1X1 + b2X2

a = y-intercept, where X’s equal zero

b=coefficient or slope for each variable

For our problem, SPSS says the equation is:

Y = 11.8 - .36X1 - .40X2

Expected # of Children = 11.8 - .36*Educ - .40*Income

Multiple Regression Let’s take a moment to reflect…

Why do I write the equation:

Y = a + b1X1 + b2X2

Whereas KBM often write:

Yi = a + b1X1i + b2X2i + ei

One is the equation for a prediction, the other is the value of a data point for a person.

Multiple RegressionModel Summary

.757a .573 .534 2.33785Model1

R R SquareAdjustedR Square

Std. Error ofthe Estimate

Predictors: (Constant), Income, Educationa. ANOVAb

161.518 2 80.759 14.776 .000a

120.242 22 5.466

281.760 24

Regression

Residual

Total

Model1

Sum ofSquares df Mean Square F Sig.

Predictors: (Constant), Income, Educationa.

Dependent Variable: Childrenb.

Coefficientsa

11.770 1.734 6.787 .000

-.364 .173 -.412 -2.105 .047

-.403 .194 -.408 -2.084 .049

(Constant)

Education

Income

Model1

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients

t Sig.

Dependent Variable: Childrena.

Y = 11.8 - .36X1 - .40X2

57% of the variation in number of children is explained by education and income!

Multiple RegressionModel Summary

.757a .573 .534 2.33785Model1



Predictors: (Constant), Income, Educationa. ANOVAb

161.518 2 80.759 14.776 .000a

120.242 22 5.466

281.760 24

Regression

Residual

Total

Model1



Dependent Variable: Childrenb.

Coefficientsa

11.770 1.734 6.787 .000

-.364 .173 -.412 -2.105 .047

-.403 .194 -.408 -2.084 .049

(Constant)

Education

Income

Model1

B Std. Error


Beta


t Sig.


Y = 11.8 - .36X1 - .40X2

r2

(Y – Y)2 - (Y – Y)2

(Y – Y)2

161.518 ÷ 261.76 = .573

Multiple RegressionSo what does our equation tell us?

Y = 11.8 - .36X1 - .40X2


Try “plugging in” some values for your variables.

Multiple Regression

So what does our equation tell us?

Y = 11.8 - .36X1 - .40X2


If Education equals:& If Income Equals: Then, children equals: 0 0 11.810 0 8.210 10 4.2 20 10 0.620 11 0.2

^


Y = 11.8 - .36X1 - .40X2


If Education equals:& If Income Equals: Then, children equals:

1 0 11.44

1 1 11.04

1 5 9.44

1 10 7.44

1 15 5.44

^


Y = 11.8 - .36X1 - .40X2


If Education equals:& If Income Equals: Then, children equals:

0 1 11.40

1 1 11.04

5 1 9.60

10 1 7.80

15 1 6.00

^

Multiple Regression

If graphed, holding one variable constant produces a two-dimensional graph for the other variable.

Y

X2 = Income0 15

11.44

5.44

b = -.4

Y

X1 = Education0 15

11.40

6.00

b = -.36

Multiple Regression An interesting effect of controlling for other

variables is “Simpson’s Paradox.” The direction of relationship between two

variables can change when you control for another variable.

Education Crime Rate Y = -51.3 + 1.5X+

Multiple Regression “Simpson’s Paradox”

Education Crime Rate Y = -51.3 + 1.5X1

+

Urbanization (is related to both)

Education

Crime Rate

+

+

Regression Controlling for Urbanization

Education

UrbanizationCrime Rate

-

+Y = 58.9 - .6X1 + .7X2

Multiple Regression

Crime

Education

Original Regression Line

Looking at each level of urbanization, new lines

Rural

Small town

Suburban

City

Multiple RegressionNow… More Variables! The social world is very complex. What happens when you have even more variables?

For example:

A researcher may be interested in the effects of Education, Income, Sex, and Gender Attitudes on Number of Children in a family.


Education

Family Income

Sex

Gender Attitudes

Dependent Variable

Number of Children

Multiple Regression

Research Hypotheses:1. As education of respondents increases, the number of children in

families will decline (negative relationship).

2. As family income of respondents increases, the number of children in families will decline (negative relationship).

3. As one moves from male to female, the number of children in families will increase (positive relationship).

4. As gender attitudes get more conservative, the number of children in families will increase (positive relationship).


Education

Family Income

Sex

Gender Attitudes

Dependent Variable

Number of Children

Multiple Regression

Null Hypotheses:1. There will be no relationship between education of respondents and

the number of children in families.

2. There will be no relationship between family income and the number of children in families.

3. There will be no relationship between sex and number of children.

4. There will be no relationship between gender attitudes and number of children.


Education

Family Income

Sex

Gender Attitudes

Dependent Variable

Number of Children

Multiple Regression

Bivariate regression is based on fitting a line as close as possible to the plotted coordinates of your data on a two-dimensional graph.

Trivariate regression is based on fitting a plane as close as possible to the plotted coordinates of your data on a three-dimensional graph.

Regression with more than two independent variables is based on fitting a shape to your constellation of data on an multi-dimensional graph.

Multiple Regression


The shape will be placed so that it minimizes the distance (sum of squared errors) from the shape to every data point.

Multiple Regression


The shape will be placed so that it minimizes the distance (sum of squared errors) from the shape to every data point.

The shape is no longer a line, but if you hold all other variables constant, it is linear for each independent variable.

Multiple RegressionY

X1X2

0

Imagining a graph with four dimensions!Y

X1X2

0

Y

X1X2

0

Y

X1X2

0

Y

X1X2

0

Multiple RegressionFor our problem, our equation could be:

Y = 7.5 - .30X1 - .40X2 + 0.5X3 + 0.25X4

E(Children) =

7.5 - .30*Educ - .40*Income + 0.5*Sex + 0.25*Gender Att.


Y = 7.5 - .30X1 - .40X2 + 0.5X3 + 0.25X4

E(Children) =

7.5 - .30*Educ - .40*Income + 0.5*Sex + 0.25*Gender Att.

Education: Income: Sex: Gender Att: Children:

10 5 0 0 2.5

10 5 0 5 3.75

10 10 0 5 1.75

10 5 1 0 3.0

10 5 1 5 4.25

^

Multiple Regression

Each variable, holding the other variables constant, has a linear, two-dimensional graph of its relationship with the dependent variable.

Here we hold every other variable constant at “zero.”Y

X2 = Education

Y

X1 = Income0 10 0 10

7.57.5

4.5

3.5

b = -.3b = -.4

Y = 7.5 - .30X1 - .40X2 + 0.5X3 + 0.25X4^

Multiple Regression

Y

X3 = Sex

Y

X4 = Gender Attitudes0 1 0 5

7.5 7.5

8

8.75

Each variable, holding the other variables constant, has a linear, two-dimensional graph of its relationship with the dependent variable.

Here we hold every other variable constant at “zero.”

b = .5

b = .25

Y = 7.5 - .30X1 - .40X2 + 0.5X3 + 0.25X4^

Multiple Regression:SPSS Model Summary R2

TSS – SSE / TSS TSS = Distance from mean to value on Y for each case SSE = Distance from shape to value on Y for each case

Can be interpreted the same for multiple regression—joint explanatory value of all of your variables (or “your model”)

Can request a change in R2 test from SPSS to see if adding new variables improves the fit of your model

Model Summary

.757a .573 .534 2.33785Model1




Multiple Regression:SPSS Model Summary

Model Summary

.757a .573 .534 2.33785Model1




R The correlation of your actual Y value and the predicted Y value using

your model for each person

Adjusted R2

Explained variation can never go down when new variables are added to a model.

Because R2 can never go down, some statisticians figured out a way to adjust R2 by the number of variables in your model.

This is a way of ensuring that your explanatory power is not just a product of throwing in a lot of variables.

Average deviation from the regression shape.

Multiple Regression:BLUE CriteriaThe BLUE Regression CriteriaRegression forces a best-fitting model (a “straight-edges” shape so to speak) onto data (data-points constellation so to speak). If the model (shape) is appropriate for the data (constellation), regression should be used.But how do we know that our “straight-edges” model (shape) is appropriate for the data (constellation)?Criteria for determining whether a regression (straight-edge) model is appropriate for the data (constellation) are nicknamed “BLUE” for best linear unbiased estimate.

Multiple Regression:BLUE CriteriaThe BLUE Regression CriteriaViolating the BLUE assumptions may result in biased estimates or incorrect significance tests. (However, OLS is robust to most violations.)Data (constellation) should meet these criteria:

The relationship between the dependent variable and its predictors is linear

No irrelevant variables are either omitted from or included in the equation. (Good luck!)

All variables are measured without error. (Good luck!)

Multiple Regression:BLUE Criteria

1. The relationship between the dependent variable and its predictors is linear

2. No irrelevant variables are either omitted from or included in the equation. (Good luck!)

3. All variables are measured without error. (Good luck!)

4. The error term (ei) for a single regression equation has the following properties:

Error is normally distributed The mean of the errors is zero The errors are independently distributed with constant variances

(homoscedasticity) Each predictor is uncorrelated with the equation’s error term*

*Omitted variable, IV measurement error, time series missing t – 1 variables affecting IV, simultaneity IV DV

Multiple Regression:MulticollinearityControlling for other variables means finding how one variable affects the dependent variable at each level of the other variables.

So what if two of your independent variables were highly correlated with each other???

Multicollinearity

Income

Age

0Years on Job

Control, Typical

Control, Multicollinear

Multiple Regression

So what if two of your independent variables were highly correlated with each other???

(this is the problem called multicollinearity)

How would one have a relationship independent of the other?

Multicollinearity Income

Age

0Years on Job

As you hold one constant, you in effect hold the other constant!Each variable would have the same value for the dependent variable at each level, so the partial effect on the dependent variable for each may be 0.

Multiple Regression

Some solutions for multicollinearity:

1. Remove some of the variables

2. Create a scale out of repetitive variables (making one variable out of several)

3. Run separate models with each independent variable

Multicollinearity

Multiple Regression Dummy Variables

They are simply dichotomous variables that are entered into regression. They have 0 – 1 coding where 0 = absence of something and 1 = presence of something. E.g., Female (0=M; 1=F) or Southern (0=Non-Southern; 1=Southern).

What are dummy

variables?!

Multiple Regression

But YOU said we

CAN’T do that!

A nominal variable has no rank or order, rendering the numerical coding scheme useless for regression.

Dummy Variables are especially nice because they allow us to use nominal

variables in regression.

Multiple Regression The way you use nominal variables in regression is by

converting them to a series of dummy variables.

Recode into differentNomimal Variable Dummy VariablesRace 1. White1 = White 0 = Not White; 1 = White2 = Black 2. Black3 = Other 0 = Not Black; 1 = Black

3. Other 0 = Not Other; 1 =

Other

Multiple Regression The way you use nominal variables in regression is by converting them to a

series of dummy variables.Recode into different

Nomimal Variable Dummy VariablesReligion 1. Catholic1 = Catholic 0 = Not Catholic; 1 = Catholic2 = Protestant 2. Protestant3 = Jewish 0 = Not Prot.; 1 = Protestant4 = Muslim 3. Jewish5 = Other Religions 0 = Not Jewish; 1 = Jewish

4. Muslim 0 = Not Muslim; 1 = Muslim5. Other Religions 0 = Not Other; 1 = Other

Relig.

Multiple Regression When you need to use a nominal variable in

regression (like race), just convert it to a series of dummy variables.

When you enter the variables into your model, you MUST LEAVE OUT ONE OF THE DUMMIES.

Leave Out One Enter Rest into Regression

White Black

Other

Multiple Regression The reason you MUST LEAVE OUT ONE OF THE

DUMMIES is that regression is mathematically impossible without an excluded group.

If all were in, holding one of them constant would prohibit variation in all the rest.

Leave Out One Enter Rest into Regression

Catholic Protestant

Jewish

Muslim

Other Religion

Multiple Regression The regression equations for dummies will

look the same.For Race, with 3 dummies, predicting self-esteem:

Y = a + b1X1 + b2X2

a = the y-intercept, which in this case is the predicted value of self-esteem for the excluded group, white.

b1 = the slope for variable X1, black

b2 = the slope for variable X2, other

Multiple Regression If our equation were:For Race, with 3 dummies, predicting self-esteem:

Y = 28 + 5X1 – 2X2

a = the y-intercept, which in this case is the predicted value of self-esteem for the excluded group, white.

5 = the slope for variable X1, black

-2 = the slope for variable X2, other

Plugging in values for the dummies tells you each group’s self-esteem average:

White = 28

Black = 33

Other = 26

When cases’ values for X1 = 0 and X2 = 0, they are white;

when X1 = 1 and X2 = 0, they are black;

when X1 = 0 and X2 = 1, they are other.

Multiple Regression Dummy variables can be entered into multiple

regression along with other dichotomous and continuous variables.

For example, you could regress self-esteem on sex, race, and education:

Y = a + b1X1 + b2X2 + b3X3 + b4X4

How would you interpret this?

Y = 30 – 4X1 + 5X2 – 2X3 + 0.3X4

X1 = Female

X2 = Black

X3 = Other

X4 = Education

Multiple RegressionHow would you interpret this?

Y = 30 – 4X1 + 5X2 – 2X3 + 0.3X4

1. Women’s self-esteem is 4 points lower than men’s.

2. Blacks’ self-esteem is 5 points higher than whites’.

3. Others’ self-esteem is 2 points lower than whites’ and consequently 7 points lower than blacks’.

4. Each year of education improves self-esteem by 0.3 units.

X1 = Female

X2 = Black

X3 = Other

X4 = Education


Y = 30 – 4X1 + 5X2 – 2X3 + 0.3X4

Plugging in some select values, we’d get self-esteem for select groups:

White males with 10 years of education = 33 Black males with 10 years of education = 38 Other females with 10 years of education = 27 Other females with 16 years of education = 28.8

X1 = Female

X2 = Black

X3 = Other

X4 = Education


Y = 30 – 4X1 + 5X2 – 2X3 + 0.3X4

The same regression rules apply. The slopes represent the linear relationship of each independent variable in relation to the dependent while holding all other variables constant.

X1 = Female

X2 = Black

X3 = Other

X4 = Education

Make sure you get into the habit of saying the slope is the effect of an independent variable “while holding everything else constant.”


Y = 30 – 4X1 + 5X2 – 2X3 + 0.3X4

The same regression rules apply…

R2 tells you the proportion of variation in your dependent variable that explained by your independent variables

The significance tests tell you whether your null hypotheses are to be rejected or not. If they are rejected, you have a low probability that your sample could have come from a population where the slope equals zero.

X1 = Female

X2 = Black

X3 = Other

X4 = Education

Multiple Regression

Interactions

Another very important concept in multiple regression is “interaction,” where two variables have a joint effect on the dependent variable. The relationship between X1 and Y is affected by the value each person has on X2.

For example:

Wages (Y) are decreased by being black (X1), and wages (Y) are decreased by being female (X2). However, being a black woman (X1* X2) increases wages relative to being a black man.

Multiple Regression

One models for interactions by creating a new variable that is the cross product of the two variables that may be interacting, and placing this variable into the equation with the original two.

Without interaction, male and female slopes create parallel lines, as do black and white.

Wages = 28k - 3k*Black - 1k*Female^

28k

25k

0 1

men

women27k

24k

Black

28k27k

0 1

white

black25k

24k

Female

Multiple Regression

One models for interactions by creating a new variable that is the cross product of the two variables that may be interacting, and placing this variable into the equation with the original two.

With interaction, male and female slopes do not have to be parallel, nor do black and white slopes.

Wages = 28k - 3k*Black - 1k*Female + 2k*Black*Female^

28k25k

0 1

men

women27k

26k

Black

28k27k

0 1

whiteblack25k 26k

Female

Multiple Regression Let’s look at another example… Sex and Education may affect Wages as such:

Wages = 20k - 1k*Female + .3k*Education

But there is reason to think that men get a higher payout for education than women.

With the interaction, the equation may be:

Wages = 19k - 1k*F + .4k*Educ - .2k*F*Educ

^

^

Multiple RegressionWith the interaction, the equation may be:

Wages = 19k - 1k*F + .4k*Educ - .2k*F*Educ

0 10 20 Education

30k

20kWages

men

women

The results show different slopes for the increase in wages for women and men as education increases.

Multiple Regression When one suspects that interactions may be occurring

in the social world, it is appropriate to test for them. To test for an interaction, enter an “interaction term”

into the regression along with the original two variables.

If the interaction slope is significant, you have interaction in the population. Report that!

If the slope is not significant, remove the interaction term from your model.

Multiple RegressionStandardized Coefficients Sometimes you want to know whether one variable

has a larger impact on your dependent variable than another.

If your variables have different units of measure, it is hard to compare their effects.

For example, if wages go up one thousand dollars for each year of education, is that a greater effect than if wages go up five hundred dollars for each year increase in age.

Multiple RegressionStandardized Coefficients So which is better for increasing wages, education or aging? One thing you can do is “standardize” your slopes so that

you can compare the standard deviation increase in your dependent variable for each standard deviation increase in your independent variables.

You might find that Wages go up 0.3 standard deviations for each standard deviation increase in education, but 0.4 standard deviations for each standard deviation increase in age.

Multiple RegressionStandardized Coefficients Recall that standardizing regression coefficients is accomplished

by the formula: b(Sx/Sy)

In the example above, education and income have very comparable effects on number of children.

Each lowers the number of children by .4 standard deviations for a standard deviation increase in each, controlling for the other.

Coefficientsa

11.770 1.734 6.787 .000

-.364 .173 -.412 -2.105 .047

-.403 .194 -.408 -2.084 .049

(Constant)

Education

Income

Model1

B Std. Error


Beta


t Sig.


Multiple RegressionStandardized Coefficients One last note of caution...

It does not make sense to standardize slopes for dichotomous variables.

It makes no sense to refer to standard deviation increases in sex, or in race--these are either 0 or they are 1 only.

Multiple RegressionNested Models “Nested models” refers to starting with a smaller set of

independent variables and adding sets of variables in stages. Keeping the models smaller achieves parsimony, simplest

explanation. Sometimes it makes sense to see whether adding a new set of

variables improves your model’s explanatory power (increases R2).

For example, you know that sex, race, education and age affect wages. Would adding self-esteem and self-efficacy help explain wages even better?

Multiple RegressionNested ModelsY = a + b1X1 + b2X2 + b3X3 Reduced Model

Y = a + b1X1 + b2X2 + b3X3 + b4X4 + b5X5 Complete Model

You should start by seeing whether the coefficients are significant.

Another test, to see if they jointly improve your model, is the change in R2 test (which you can request from SPSS)

R2c - R2

r/df=#extra slopes in complete

F =

1 - R2c / df=#slopes+1 in complete

Nested Models

Multiple RegressionNested Models with Change in R2

Dependent Variable: How often does S attend religious services. Higher values equal more often.Model 1 Model 2Female FemaleWhite (W=1) WhiteBlack (B=1) BlackAge Age

Education


Dependent Variable: How often does S attend religious services. Higher values equal more often.

Model Summary

.232a .054 .052 2.632 .054 38.565 4 2724 .000

.249b .062 .060 2.622 .008 23.763 1 2723 .000

Model1

2



R SquareChange F Change df1 df2 Sig. F Change

Change Statistics

Predictors: (Constant), AGE OF RESPONDENT, female = 1, Black, Whitea.

Predictors: (Constant), AGE OF RESPONDENT, female = 1, Black, White, HIGHEST YEAR OF SCHOOL COMPLETEDb.

ANOVAc

1068.987 4 267.247 38.565 .000a

18876.482 2724 6.930

19945.469 2728

1232.295 5 246.459 35.863 .000b

18713.174 2723 6.872

19945.469 2728

Regression

Residual

Total

Regression

Residual

Total

Model1

2


Predictors: (Constant), AGE OF RESPONDENT, female = 1, Black, Whitea.

Predictors: (Constant), AGE OF RESPONDENT, female = 1, Black, White, HIGHESTYEAR OF SCHOOL COMPLETED

b.

Dependent Variable: HOW OFTEN R ATTENDS RELIGIOUS SERVICESc.


Dependent Variable: How often does S attend religious services. Higher values equal more often.

Coefficientsa

2.298 .232 9.887 .000

.783 .102 .144 7.698 .000

-.116 .205 -.017 -.566 .572

.894 .237 .116 3.779 .000

.019 .003 .124 6.561 .000

1.110 .336 3.302 .001

.777 .101 .143 7.674 .000

-.140 .204 -.021 -.688 .492

.966 .236 .125 4.093 .000

.021 .003 .135 7.157 .000

.084 .017 .092 4.875 .000

(Constant)

female = 1

White

Black

AGE OF RESPONDENT

(Constant)

female = 1

White

Black

AGE OF RESPONDENT

HIGHEST YEAR OFSCHOOL COMPLETED

Model1

2

B Std. Error


Beta


t Sig.

Dependent Variable: HOW OFTEN R ATTENDS RELIGIOUS SERVICESa.

Multiple Regression Females attend services more often than males. Blacks attend services more often than whites and

others. Older persons attend services more often than

younger persons. The more educated a person is, the more often he or

she attends religious services. Education adds to the explanatory power of the

model. Only five to six percent of the variation in religious

service attendance is explained by our models.

Multiple Regression. The test you choose depends on level of measurement: Independent...

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