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Analysis of Regression Models and Interactions in Various Statistical Contexts

Study on the application of regression models and interaction effects.

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    Analysis of Regression Models and Interactions in Various Statistical Contexts 5.24 Expert testimony in homicide trials of battered women. Refer to the Duke Journal of Gender Law and Policy (Summer 2003) study of the impact of expert testimony on the outcome of homicide trials involving battered woman syndrome, Exercise 5.3 (264). Recall that multiple regression was employed to model the likelihood of changing a verdict from not guilty to guilty after deliberations , y, as function of juror gender (male or female) and expert testimony given (yes or no). a. Write a main efforts model for E(y) as a function of gender and expert testimony. Interpret the β coefficients in the model. The main efforts model for E(y) as a function of gender and expert testimony is ( ) 0 1 1 2 2 3 3 E y x x x     = + + + Interpretation of model parameters 0 11 1 2 1 2 3 1 3 2 1 which is mean of the combination of base levels , for any level B ( 1, 2) , for any level B ( 1, 2) , for any level F ( 1, 2,3) j j j j j j i i i j j i            = = − = = − = = − = Interaction Model with two qualitative independent variables, first one at three levels F1,F2,F3 and second one at two levels B1,B2 Then the equation is ( ) 0 1 1 2 2 3 3 4 1 3 5 2 3 1 1 2 2 3 3 4 1 3 5 2 3 1 1 2 3 The main effect term of F is & main effect term of B is ,interaction term is , Dummy variables of x , , are defined in the same way as E y x x x x x x x x x x x x x x x x and x            = + + + + + + for the main effects model. Now interpretation of model parameters

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    ( ) ( ) ( ) ( ) 0 11 1 2 1 2 3 1 3 2 1 4 22 12 21 11 5 32 12 31 11 which is mean of the combination of base levels , for any level B ( 1, 2) , for any level B ( 1, 2) , for any level F ( 1, 2,3) j j j j j j i i i j j i                      = = − = = − = = − = = − − − = − − − The study or the impact of expert testimony on the outcome of homicide trials involving battered woman syndrome. The multiple regressions was applied for modeling the likelihood or changing a verdict from not guilty to guilty after deliberations ( y) as a function of juror gender male or female and expert testimony given. Now writing the main effects for model for E(y) as a function of gender and expert testimony and interpreting the  coefficients in this model. The main effects model is ( ) 0 1 1 2 2 E y x x    = + + Where 1 1 1 2 2 2 is the main effect of and is the main effect of x x x x   Let ( ) 1 1 2 2 1( );0 ; 1 if has the value'yes' x if x is female otherwise x x = = Now interpreting the  coefficients of the model 0 11 1 2 1 2 3 1 which is mean of the combination of base levels , for any level B ( 1, 2) , for any level B ( 1, 2) j j j j j j j j         = = − = = − = The interpretations of 1 1 ( 1, 2) is that there are two values for which is male and female j x j x = And 2 2 ( 1, 2) is that there are two values for which is 'yes' or 'no' j x j x = b) Write an interaction model for E(y) as a function of gender and expert testimony. Interpret the β coefficients in the model. Now write an interaction effects model for E(y) as a function of gender and expert testimony and interpreting the  coefficients in the model. The main effect model is ( ) 0 1 1 2 2 3 1 2 E y x x x x     = + + + + Here 1 2 x x denoted an interaction effects shown below.

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    1 2 1 2 1 2 1 2 1) " " " " 2) " " " " 3) " " " " 4) " " " " x Female and x yes x male and x yes x Female and x no x male and x no = = = = = = = = Now interested in interpreting the  coefficients in an interaction model. ( ) ( ) 0 0 1 0 2 0 3 0 0 0 1 2 2 where is average effect when ' ' and ' ' is the average effect when the variable characterstic is ' ' F Y F Y y x male x no x yes                = = − = − = − − − = = = C) Based on data collected on individual juror votes from past trials, the article reported that “when expert testimony was present, women jurors were more likely than men to change a verdict from not guilty to guilty after deliberations.” Assume that when no expert testimony was present, male jurors were more likely than women to change a verdict from not guilty to guilty after deliberations. Which model, part a or part b, hypothesizes the relationships reported in the article? Illustrate the model with a sketch Consider the data collected on individual juror votes from the past trails. The report was when the expert testimony presents, the women jurors were more likely than men to change a verdict from not guilty to guilty after deliberations. We are interested in determining the model part (a) or part (b) hypothesizes the relationship reported in this problem. The value of 3  is to be >0 , because 2 1 x = . We have the women jurors were more likely than men to change a verdict from not guilty to guilty after deliberations and when 2 0 x = the men jurors were more likely to change the verdict. In this case, 0 y  , which the likelihood probability . since 0 0   We know that the women jurors are more likely to change their verdict from not guilty to guilty when the expert testimony was present. That is when 1 2 and is one x x the expected probability or the likelihood ratio for this event is greater than the expected probability or the likelihood ration when 1 2 =0 and =1 x x Also men jurors are more likely to change their verdict from non - guilty to guilty when the expert testimony was absent.

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