Multiple Regression

The test you choose depends on level of measurement: Independent VariableDependentVariableTest DichotomousInterval-Ratio Independent Samples t-test Dichotomous NominalNominalCross TabsDichotomous NominalInterval-RatioANOVADichotomous Interval-RatioInterval-RatioBivariate Regression/Correlation Dichotomous Two or More… Interval-Ratio DichotomousInterval-RatioMultiple 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). Independent Variables 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. Independent Variables 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: Children (Y): Education (X 1 ) Income 1=$10K (X 2 ):

Multiple Regression Case: Children (Y): Education (X 1 ) Income 1=$10K (X 2 ): Y X1X1 X2X2 0 Plotted coordinates (1 – 10) for Education, Income and Number of Children

Multiple Regression Case: Children (Y): Education (X 1 ) Income 1=$10K (X 2 ): Y X1X1 X2X2 0 What multiple regression does is fit a plane to these coordinates.

Multiple Regression  Mathematically, that plane is: Y = a + b 1 X 1 + b 2 X 2 a = y-intercept, where X ’ s equal zero b=coefficient or slope for each variable For our problem, SPSS says the equation is: Y = X X 2 Expected # of Children = *Educ -.40*Income  

Multiple Regression  Let ’ s take a moment to reflect… Why do I write the equation: Y = a + b 1 X 1 + b 2 X 2 Whereas KBM often write: Y i = a + b 1 X 1i + b 2 X 2i + e i One is the equation for a prediction, the other is the value of a data point for a person. 

Multiple Regression Y = X X 2 57% of the variation in number of children is explained by education and income! 

Multiple Regression Y = X X 2 r2r2  (Y – Y) 2 -  (Y – Y) 2  (Y – Y) 2  ÷ =.573 

Multiple Regression So what does our equation tell us? Y = X X 2 Expected # of Children = *Educ -.40*Income Try “ plugging in ” some values for your variables. 

Multiple Regression So what does our equation tell us? Y = X X 2 Expected # of Children = *Educ -.40*Income If Education equals:&If Income Equals:Then, children equals: ^

Multiple Regression So what does our equation tell us? Y = X X 2 Expected # of Children = *Educ -.40*Income If Education equals:&If Income Equals:Then, children equals: ^

Multiple Regression So what does our equation tell us? Y = X X 2 Expected # of Children = *Educ -.40*Income If Education equals:&If Income Equals:Then, children equals: ^

Multiple Regression If graphed, holding one variable constant produces a two- dimensional graph for the other variable. Y X 2 = Income b = -.4 Y X 1 = Education 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 RateY = X + 

Multiple Regression  “ Simpson ’ s Paradox ” Education Crime RateY = X 1 + Urbanization (is related to both) Education Crime Rate + + Regression Controlling for Urbanization Education Urbanization Crime Rate - + Y = X 1 +.7X 2  

Multiple Regression Crime Education Original Regression Line Looking at each level of urbanization, new lines Rural Small town Suburban City

Multiple Regression Now… 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. Independent Variables 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). Independent Variables 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. Independent Variables 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  Regression with more than two independent variables is based on fitting a shape to your constellation of data on an multi-dimensional graph.  The shape will be placed so that it minimizes the distance (sum of squared errors) from the shape to every data point.

Multiple Regression  Regression with more than two independent variables is based on fitting a shape to your constellation of data on an multi-dimensional graph.  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 Regression Y X1X1 X2X2 0 Imagining a graph with four dimensions! Y X1X1 X2X2 0 Y X1X1 X2X2 0 Y X1X1 X2X2 0 Y X1X1 X2X2 0

Multiple Regression For our problem, our equation could be: Y = X X X X 4 E(Children) = *Educ -.40*Income + 0.5*Sex *Gender Att. 

Multiple Regression So what does our equation tell us? Y = X X X X 4 E(Children) = *Educ -.40*Income + 0.5*Sex *Gender Att. Education:Income: Sex:Gender Att: Children: ^

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 X 2 = Education Y X 1 = Income b = -.3 b = -.4 Y = X X X X 4 ^

Multiple Regression Y X 3 = Sex Y X 4 = Gender Attitudes 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 = X X X X 4 ^

Multiple Regression: SPSS Model Summary  R 2 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 R 2 test from SPSS to see if adding new variables improves the fit of your model

Multiple Regression: SPSS Model Summary  R The correlation of your actual Y value and the predicted Y value using your model for each person  Adjusted R 2 Explained variation can never go down when new variables are added to a model. Because R 2 can never go down, some statisticians figured out a way to adjust R 2 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 Criteria The BLUE Regression Criteria  Regression 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 Criteria The BLUE Regression Criteria  Violating 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: 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!)

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 (e i ) 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: Multicollinearity Controlling 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 0 Years 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 0 Years 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 different Nomimal VariableDummy Variables Race1. White 1 = White 0 = Not White; 1 = White 2 = Black2. Black 3 = 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 VariableDummy Variables Religion1. Catholic 1 = Catholic 0 = Not Catholic; 1 = Catholic 2 = Protestant2. Protestant 3 = Jewish 0 = Not Prot.; 1 = Protestant 4 = Muslim3. Jewish 5 = Other Religions 0 = Not Jewish; 1 = Jewish 4. Muslim 0 = Not Muslim; 1 = Muslim 5. 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 OneEnter Rest into Regression WhiteBlack 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 OneEnter Rest into Regression CatholicProtestant 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 + b 1 X 1 + b 2 X 2  a = the y-intercept, which in this case is the predicted value of self-esteem for the excluded group, white. b 1 = the slope for variable X 1, black b 2 = the slope for variable X 2, other

Multiple Regression  If our equation were: For Race, with 3 dummies, predicting self-esteem: Y = X 1 – 2X 2 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 X 1, black -2 = the slope for variable X 2, 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 X 1 = 0 and X 2 = 0, they are white; when X 1 = 1 and X 2 = 0, they are black; when X 1 = 0 and X 2 = 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 + b 1 X 1 + b 2 X 2 + b 3 X 3 + b 4 X 4 How would you interpret this? Y = 30 – 4X 1 + 5X 2 – 2X X 4 X 1 = Female X 2 = Black X 3 = Other X 4 = Education  

Multiple Regression How would you interpret this? Y = 30 – 4X 1 + 5X 2 – 2X X 4 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. X 1 = Female X 2 = Black X 3 = Other X 4 = Education 

Multiple Regression How would you interpret this? Y = 30 – 4X 1 + 5X 2 – 2X X 4 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 X 1 = Female X 2 = Black X 3 = Other X 4 = Education 

Multiple Regression How would you interpret this? Y = 30 – 4X 1 + 5X 2 – 2X X 4 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. X 1 = Female X 2 = Black X 3 = Other X 4 = Education  Make sure you get into the habit of saying the slope is the effect of an independent variable “while holding everything else constant.”

Multiple Regression How would you interpret this? Y = 30 – 4X 1 + 5X 2 – 2X X 4 The same regression rules apply… R 2 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. X 1 = Female X 2 = Black X 3 = Other X 4 = 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 X 1 and Y is affected by the value each person has on X 2. For example: Wages (Y) are decreased by being black (X 1 ), and wages (Y) are decreased by being female (X 2 ). However, being a black woman (X 1* X 2 ) 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 women 27k 24k Black 28k 27k 0 1 white black 25k 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 ^ 28k 25k 0 1 men women 27k 26k Black 28k 27k 0 1 white black 25k 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 Regression With the interaction, the equation may be: Wages = 19k - 1k*F +.4k*Educ -.2k*F*Educ Education 30k 20k Wages 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 Regression Standardized 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 Regression Standardized 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 Regression Standardized 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.

Multiple Regression Standardized 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 Regression Nested 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 R 2 ).  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 Regression Nested Models Y = a + b 1 X 1 + b 2 X 2 + b 3 X 3Reduced Model Y = a + b 1 X 1 + b 2 X 2 + b 3 X 3 + b 4 X 4 + b 5 X 5Complete 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 R 2 test (which you can request from SPSS) R 2 c - R 2 r /df=#extra slopes in complete F = 1 - R 2 c / df=#slopes+1 in complete Nested Models

Multiple Regression Nested Models with Change in R 2 Dependent Variable: How often does S attend religious services. Higher values equal more often. Model 1Model 2Female White (W=1)White Black (B=1)BlackAge Education

Multiple Regression Nested Models with Change in R 2 Dependent Variable: How often does S attend religious services. Higher values equal more often.

Multiple Regression Nested Models with Change in R 2 Dependent Variable: How often does S attend religious services. Higher values equal more often.

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.