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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 variable’s 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;