### Linear Statistical Models: Regression

### Partial and Semipartial Correlation

**Experimental Control**

Controling variances by using equal groups.
Randomly assigning subjects to groups.
Conditions for all groups identical except for independent variable.
**Statistical Control**

Control variance by removing unwanted variance from other variables.
control by partialing.
**Partial Correlation**

**r**_{12.3} is the correlation between variables 1 and 2 with variable 3 removed from both
variables. To illustrate this, run separate regressions using **X**_{3} as the
independent variable and **X**_{1} and **X**_{2} as dependent variables. Next,
compute residuals for regression...

**Venn Diagram of Partial Correlation**

**More Partial Correlation**

**Example**

**Let r**_{12} = 0.7; r_{13} = 0.6; r_{23} = 0.9.

**Using Multiple Correlations**

Thus, squared partial correlations represent the ratio of incremental variance to the
residual variance.
**Example**

**Let R**^{2}_{1.23} = .4947 and R^{2}_{1.3} = .6^{2} = .36
(from the above example)

**Higher Order Partial Correlation**

and so on...

**Suppressor Variable**

A special case when the partial correlation is larger than the zero-order correlation.
Zero or close to 0 correlation with the dependent variable.
Correlated with one or more independent variables.
Serves to suppress or control irrelevant variance.
**Example**

**Let r**_{12} = 0.3; r_{13} = 0.0; r_{23} = 0.5.

**Let r**^{2}_{12} = 0.09; r^{2}_{12.3} = 0.12;

Note: **r**^{2}_{12.3} is greater than **r**^{2}_{12} even
though **r**_{13} = 0.
Standardized regressions Coefficients: **β**_{2} = .4
and **β**_{3} = -.2
Note: suppressor variable receives a negative coefficient.
**Causal Relationships**

Partial correlation as a control method must be predicated on a sound theoretical framework.
Routine presentation of all higher-order partial correlations is a sign that theory
explaining the relationship among the variables is missing (Gordon, 1968).
With only three variables there are many possible causal models.

**Measurement Error**

Measurement error leads to biased estimates of zero-order and partial correlations.
Measurement error attenuates zero-order correlations.
**Correction for Attenuation**

Example

**When r**_{12} = .7; r_{11} = r_{22} = .8

**Correction for Attenuation in Partial Correlation**

Partial correlation when correcting for unreliability in **X**_{3} only.

**Example**

**When r**_{12} = .7; r_{13} = .5; r_{23} = .6; r_{33} = .8;**
and r**_{12.3} = .58

**Another Example**

**When r**_{12} = .7; r_{13} = .8; r_{23} = .7; r_{33} = .8;**
and r**_{12.3} = .33

**Correction for Attenuation in Partial Correlation**

Partial correlation when correcting for unreliability in all measures.

**Example**

**When r**_{12} = .7; r_{13} = .5; r_{23} = .6;
r_{11} = r_{22} = r_{33} = .8;**
and r**_{12.3} = .58

**Semipartial Correlation**

AKA -- part correlation.
**r**_{1(2.3)} is the correlation between variables 1 and 2
with variation from variable 3 removed only from variable 2.
Compute residuals for **X**_{2}
using **X**_{3} as the independent variable...

**Venn Diagram of Semipartial Correlation**

**Semipartial Correlation Formulas**

Thus, the squared semipartial correlation represents the proportion of variance of the dependent
variable accounted for by a given independent variable after another variable has already been
taken into account.
**Example**

**r**_{12} = .7; r_{13} = .6; r_{23} = .9

**Remember**

**r**^{2}_{1(2.3)} does not necessarily equal r^{2}_{1(3.2)}
since r^{2}_{1(2.3)} = R^{2}_{1.23} - R^{2}_{1.3}

**
and r**^{2}_{1(3.2)} = R^{2}_{1.23} - R^{2}_{1.2}

**Higher Order Semipartials**

**Regression & Semipartials**

When independent variables are uncorrelated:

When independent variables are correlated:

**Multiple Partial Correlation**

**R**^{2}_{1.23(4)} is the squared multiple correlation of variable 1 with variables 2 and 3 after
the variation due to variable 4 has been partial out from all variables.

**Multiple Semipartial Correlation**

**R**^{2}_{1(23.4)} is the squared multiple correlation of variable 1 with variables 2 and 3 after
the variation due to variable 4 has been partial out only from variables 2 and 3.

**Partial & Semipartial Correlation in Stata**

**pcorr write read math science female**
(obs=200)
Partial and semipartial correlations of write with
Partial Semipartial Partial Semipartial Significance
Variable | Corr. Corr. Corr.^2 Corr.^2 Value
------------+-----------------------------------------------------------------
read | 0.2567 0.1756 0.0659 0.0308 0.0003
math | 0.2925 0.2022 0.0855 0.0409 0.0000
science | 0.2792 0.1922 0.0779 0.0369 0.0001
female | 0.4239 0.3094 0.1797 0.0957 0.0000
**partcorr write read, part(math science female)** /* findit partcorr */
Response Variable: write
Predictor Variable(s): read
Partial Variable(s): math science female
Number of obs: 200
Coef Coef Squared
Mutiple Correlation = .59677648 .35614217
Partial Correlation = .25672403 .06590723
Semipartial Correlation = .17559606 .03083398
**partcorr write read math, part(science female)**
Response Variable: write
Predictor Variable(s): read math
Partial Variable(s): science female
Number of obs: 200
Coef Coef Squared
Mutiple Correlation = .66641064 .44410315
Partial Correlation = .47497762 .22560374
Semipartial Correlation = .35680815 .12731206

Linear Statistical Models Course

Phil Ender, 24sep10, 28Jan98