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Research Methods I

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    1. Research Methods I Non-orthogonal Multiple Regression

    2. Correlated Independent Variables In the social sciences we will often have situations in which the independent variables are correlated with each other. Education and maternal sensitivity Marital aggression and marital harmony Age and speech rate to predict memory span In the case of a non-orthogonal multiple regression the equation ry.x2 + rY.T2 = RY.XT2 is not true anymore.

    3. Each variables specific contribution Since both variables are correlated we have to find another way to find the specific contribution of each variable. If we calculate the quality of prediction for each variable as we did with orthogonal variables we will get an R square above 1. We will have to find the semi-partial correlation of each variable

    4. Semi-partial correlations We need to isolate the specific part of each independent variable First, predict a given independent variable from the other independent variable The residual of the prediction is by definition uncorrelated with the predictor (it represents the specific part of the independent variable) XT = aX.T + bX.TT TX = aT.X + bT.XX

    5. Semi-partial correlations bX.T = SCPXT / SST (sum of cross-products and sum of squares of T) aX.T = MX bX.T*MT Then we subtract the predicted value from the actual value in order to get the residual The residual is the specific component of this independent variable

    6. Specific Contribution of Each IV We can now calculate the correlation coefficient of the relationship between the specific part of the independent variable and the dependent variable. r2Y.X/T = r2Y.eX.T = (SCPYeX.T)2 / SSY * SSeX.T The results from these calculations will show that they are very different from the total contributions that had been calculated before.

    7. Simple Statistics Variable Label MEDUCM01 Mother's Education neg supcom MPCSIO1S SI Comp: Maternal Pos Caregiving, 1S Pearson Correlation Coefficients, N = 65 Prob > |r| under H0: Rho=0 MEDUCM01 neg supcom MPCSIO1S MEDUCM01 1.00000 -0.19157 0.34782 0.39443 Mother's Education 0.1263 0.0045 0.0011 neg -0.19157 1.00000 -0.49907 -0.28732 0.1263 <.0001 0.0203 supcom 0.34782 -0.49907 1.00000 0.42503 0.0045 <.0001 0.0004

    8. Simple Statistics Partial Partial Variable Variance Std Dev Label MEDUCM01 Mother's Education MPCSIO1S 20.54176 4.53230 SI Comp: Maternal Pos Caregiving, 1S supcom 0.60101 0.77525 neg 0.20982 0.45806 Pearson Partial Correlation Coefficients, N = 65 Prob > |r| under H0: Partial Rho=0 MPCSIO1S supcom neg MPCSIO1S 1.00000 0.33409 -0.23479 SI Comp: Maternal Pos Caregiving, 1S 0.0070 0.0618 supcom 0.33409 1.00000 -0.46994 0.0070 <.0001 neg -0.23479 -0.46994 1.00000 0.0618 <.0001

    9. The REG Procedure Model: MODEL1 Dependent Variable: MPCSIO1S SI Comp: Maternal Pos Caregiving, 1S Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 3 392.92286 130.97429 7.01 0.0004 Error 61 1139.63099 18.68248 Corrected Total 64 1532.55385 Root MSE 4.32232 R-Square 0.2564 Dependent Mean 31.73846 Adj R-Sq 0.2198 Coeff Var 13.61856 Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 20.07294 4.47496 4.49 <.0001 MEDUCM01 Mother's Education 1 0.52776 0.22312 2.37 0.0212 neg 1 -0.98784 1.34682 -0.73 0.4661 supcom 1 1.67892 0.79578 2.11 0.039

    10. SAS program proc corr; var MEDUCM01 neg supcom MPCSIO1S; run; proc corr; var MPCSIO1S supcom neg; partial MEDUCM01; run; proc reg; model MPCSIO1S = MEDUCM01 neg supcom; run;