Partial Correlation

A partial correlation is the same as a Pearson's bivariate correlation, except that you add a control variable. 

The control variable must be continuous, and the independent and dependent variables must both be continuous. 

You perform and interpret the hypothesis test the same as for a Pearson's bivariate correlation. The hypotheses are:

Ho: There is no relationship between IV and DV, controlling for CV.  r = 0

H1: There is a relationship between IV and DV, controlling for CV.  r ≠ 0 

*Fill in what IV, DV, and CV are in the above hypotheses.

Example 1:

Dependent Variable = income; measured in dollars

Independent Variable = tv hours; measured in daily hours watched

Control Variable (CV) = education; measured in years

Hypotheses:

Null: There is no relationship between the number of TV hours watched daily and income, adjusting for the number of years of education.  r = 0

Research: There is a relationship between the number of TV hours watched daily and income, adjusting for the number of years of education.  r ≠ 0

When we control for the effect of education on the relationship between number of TV hours watched daily and income, we find the following by doing a partial correlation with the GSS2000 data:

r = -.11, p = .000

p is less than alpha. Reject null.

There is a weak, negative relationship between the number of TV hours watched daily and income when we control for the effect of education.  As the number of daily TV viewing hours goes up, income goes down for people of any level of education (r = -.11, p = .000). Or, as income increases the number of daily TV viewing hours goes down.

r2 = .11 * .11 = .0121

When controlling for the effect of education,  the number of TV hours watched daily explains 1% of the variation in income.  Or,  when controlling for the effect of education,  income explains 1% of the variation in the number of TV hours watched daily. 

Example 2: Do on the Computer

^{I think that the number of hours that people work per week influences how many times they have sex, holding SEI constant.} 

DV = sex frequency (treat as continuous)

IV = number of hours worked (continuous)

CV = SEI (continuous)

3 variables, all continuous. Analysis = partial correlation 

^{Null: The number of hours that people work per week does not influence how many times people have sex, holding SEI constant.  r = 0} 

^{Research: The number of hours that people work per week influences how many times they have sex, holding SEI constant.  r} ≠ ⁰ 

r = .06, p = .02

p is less than alpha. Reject null. 

^{The number of hours that people work per week influences how many times they have sex, holding SEI constant.} 

^{There is a weak positive correlation between the number of hours that people work per week and how many times they have sex, holding SEI constant. As the number of hours that people work per week increases, their sex frequency increases, holding SEI constant.} 

<sup>r*r = .06* .06 = .004

The number of hours that people work per week explains .4% of the variation in sex frequency, holding SEI constant.</sup>  

Example 3: In Class Exercise

I think the number of children that people have is determined, in part, by the amount of education they have, controlling for current income.

number of children (childs)
education (educ)
income (rincom98)

Answer:

All continuous.  Analysis = partial correlation

Null: Years of education does not influence the number of children that people have, when adjusting for income.  r = 0

Research: Years of education does influence the number of children that people have, when adjusting for income.  r ≠ 0

r = -.19, p=.000

p is less than alpha. Reject null.  Years of education does influence the number of children that people have, when adjusting for income.

There is a negative linear correlation between years of education and the number of children that people have, when adjusting for income.  As education increases, number of children decreases.

r*r = .19 * .19 = .04

Years of education explains 4% of the variation in the number of children that people have, when adjusting for income.  

Example 4: Take Home Example

I think that age influences how well students do in college statistics courses, controlling for the number of previous math or statistics courses taken.

age = age in years
performance in statistics = average score in course on a scale of 0 to 100
previous math or statistics courses = total number of math or statistics courses previously taken for college credit

r = .18, p = .36

Answers:

DV = performance in statistics, continuous
IV = age, continuous
CV = number of previous stats or math courses, continuous

All continuous.  Analysis = partial correlation

Ho: Age does not influence how well students do in college statistics courses, controlling for the number of previous math or statistics courses taken.   r = 0

H1: Age influences how well students do in college statistics courses, controlling for the number of previous math or statistics courses taken.   r ≠ 0

p is greater than alpha.  Accept null.

Age does not influence how well students do in college statistics courses, controlling for the number of previous math or statistics courses taken. 

Do not interpret r or r²