Multiple OLS Regression

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1 Multiple OLS Regression Ronet Bachman, Ph.D. Presented by Justice Research and Statistics Association 12/8/2016 Justice Research and Statistics Association th Street, NW, Third Floor Washington, DC 20001

2 Training and Technical Assistance Webinar Series This webinar is being audio cast via the speakers on your computer and via teleconference. To access the audio stream information, select audio and audio conference from the menu bar. This will display the call information and the button to access the audio stream. If you have speakers or headphones for your computer there is no need to call in, simply select call using computer. Justice Research and Statistics Association th Street, NW, Third Floor Washington, DC 20001

3 Training and Technical Assistance Webinar Series This webinar is being audio cast via the speakers on your computer and via teleconference. To access the audio stream information, select audio and audio conference from the menu bar. This will display the call information and the button to access the audio stream. If you have speakers or headphones for your computer there is no need to call in, simply select call using computer. Justice Research and Statistics Association th Street, NW, Third Floor Washington, DC 20001

4 Training and Technical Assistance Webinar Series All telephones have been muted. If you would like to ask a question please use the chat feature unless instructed otherwise. Please remember to select Host, Presenter or Panelists Justice Research and Statistics Association th Street, NW, Third Floor Washington, DC 20001

5 The beauty of statistical control Presented by Ronet Bachman, PhD University of Delaware To Justice Research and Statistics Association

6 Establish Relationship Between X and Y Establish Correct Time Order (X precedes Y) Establish Non-Spuriousness a spurious relationship is one that is really a false relationship caused by a third or fourth variable. Experimental Design Statistical Control Enter Multivariate Equations Multiple Regression Analyses

7

8 But we live in a Multivariate World!! Age SES Victimization Gender

9 Simple Bivariate Crosstabs Delinquency (DV) Low Medium High Gender IV Female % % % 680 Male % % %

10 Multivariate Models allow us to Statistically control for 3 rd, 4 th, kth, variables - Controlling for Parental Supervision in Partial Crosstab Table Weak Parental Supervision Female Male Delinquency Low Medium High 26% 28% 46% % 27% 49% 592 Strong Parental Supervision Delinquency 764 Female Male Low Medium High 54% 26% 20% % 30% 24%

11 Multiple OLS Regression Equations for the Population and Sample A simple extension of the bivariate model!

12 Assumptions for OLS Multiple Regression: (1) It is assumed that the data were randomly selected. (2) It is assumed that all populations are normally distributed. (3) We must assume that the data are continuous, that they are measured at the interval or ratio level. (4) We must assume that the nature of the relationship between the dependent and each of the independent variables is linear. (5) It is assumed that the error term ( ) is independent of and therefore uncorrelated with each of the independent or x variables, that it is normally distributed, and has an expected value of 0 and constant variance across all levels of x. This is referred to as the assumption of homoscedasticity. (6) The new assumption in the multivariate regression model is that the independent variables are independent of or uncorrelated with one another. Having independent variables that are highly correlated is referred to as the problem of multicollinearity.

13 Partial Regression Slopes b b 1 2 sy ryx1 ( ryx2)( rx1x2) = 2 sx1 1 r x1x2 sy ryx2 ( ryx1)( rx1x2) = 2 sx2 1 r x1x2 Notice that models control for variation between X1 and x2 as well as that between x1 and y and x2 and y Where: b 1 = the partial slope of x 1 on y b 2 = the partial slope of x 2 on y s y = the standard deviation of y s 1 = the standard deviation of the first independent variable (x 1 ) s 2 = the standard deviation of the second independent variable (x 2 ) r y1 = the bivariate correlation between y and x 1 r y2 = the bivariate correlation between y and x 2 r 12 = the bivariate correlation between x 1 and x 2

14 Model Variables Entered/Removed b Variables Entered 1 certainty of punishment a a. All requested variables entered. Variables Removed. Enter b. Dependent Variable: time 1 delinquency scale Method Model R R Square Model Summary Adjusted R Square Std. Error of the Estimate a a. Predictors: (Constant), certainty of punishment ANOVA b Model Sum of Squares df Mean Square F Sig. 1 Regression a Residual Total a. Predictors: (Constant), certainty of punishment b. Dependent Variable: time 1 delinquency scale Model Coefficients a Unstandardized Coefficients Standardized Coefficients B Std. Error Beta 1 (Constant) certainty of punishment a. Dependent Variable: time 1 delinquency scale t Sig. y= (x)

15 Model Variables Entered/Removed b Variables Entered 1 sex of respondent a a. All requested variables entered. Variables Removed. Enter b. Dependent Variable: time 1 delinquency scale Method Model R R Square Model Summary Adjusted R Square Std. Error of the Estimate a a. Predictors: (Constant), sex of respondent ANOVA b Model Sum of Squares df Mean Square F Sig. 1 Regression a Residual Total a. Predictors: (Constant), sex of respondent b. Dependent Variable: time 1 delinquency scale Model Coefficients a Unstandardized Coefficients Standardized Coefficients B Std. Error Beta 1 (Constant) sex of respondent a. Dependent Variable: time 1 delinquency scale y = (x) t Sig.

16 Model Variables Entered/Removed b Variables Entered 1 certainty of punishment, sex of respondent a a. All requested variables entered. Variables Removed. Enter Method b. Dependent Variable: time 1 delinquency scale Multiple r Multiple Coefficient of Determination Model R R Square Model Summary Adjusted R Square Std. Error of the Estimate a a. Predictors: (Constant), certainty of punishment, sex of respondent This test is now important ANOVA b Model Sum of Squares df Mean Square F Sig. 1 Regression a Residual Total a. Predictors: (Constant), certainty of punishment, sex of respondent b. Dependent Variable: time 1 delinquency scale Model Coefficients a Unstandardized Coefficients Standardized Coefficients B Std. Error Beta 1 (Constant) sex of respondent certainty of punishment a. Dependent Variable: time 1 delinquency scale y = (x1) (x2) t Sig. Null hypothesis tests For each slope

17 What happens when both IV s are included in the model? Model Variables Entered/Removed b Variables Entered 1 certainty of punishment, gender of respondent a Variables Removed. Enter Method b. Dependent Variable: time 1 delinquency scale Multiple r Multiple Coefficient of Determination Model R R Square Model Summary Adjusted R Square Std. Error of the Estimate a H0: No relationship between any of the IVs and the DV OR ρ = 0. ANOVA b Model Sum of Squares df Mean Square F Sig. 1 Regression a Residual Total Model Coefficients a Unstandardized Coefficients Standardized Coefficients B Std. Error Beta 1 (Constant) gender of respondent certainty of punishment a. Dependent Variable: time 1 delinquency scale Null hypothesis tests For each slope Delinquency (y) = (x1) (x2) H 0 : No relationship between gender and delinquency after perceptions of risk are controlled, OR β 1 =0. H 0 : No relationship between perceptions of risk and delinquency after gender is controlled, OR β 2 =0. t Sig.

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