Introduction to bivariate statistics: Covariance and Correlation Basic concept of association of two interval/ratio variables
Basics of concept of “association” Income and education are associated Religiosity and political conservatism are associated Illicit drug use and GPA are associated Authoritarian parenting and conservative gender role attitudes are associated
One question, three answers Is drug use and GPA associated? Case 1: GPA is measured as actual GPA. Drug use is measured by categorizing those who ever used and those who never used. Case 2: GPA is measured as actual GPA. Drug use is measured by asking the number of times used drugs in the last month. Case 3: GPA is measured in 4 intervals: 0-.99; 1-1.99; 2-2.99, 3+. Drug use is measured by categorizing those who ever used and those who never used.
How do we measure association? If two interval-ratio level variables Correlation (or covariance) If an interval level and nominal variable T-test - example Variants of correlation If two nominal-ordinal variables Chi-square test - example
Covariance
Understanding the covariance Covariance should represent the degree of association To the extent that differences from the mean are in the same direction, covariance should be larger The stronger the association is, the larger the covariance should be
Application ID Education Income Education: Comparison to the mean Income: Comparison to the mean 1 8 15000 -4.4 -14300 2 12 23000 -0.4 -6300 3 11 16000 -1.4 -13300 4 18 65000 5.6 35700 5 16 39000 3.6 9700 6 11000 -18300 7 14 28000 1.6 -1300 38000 8700 9 32000 2700 10 26000 -3.4 -3300 Mean 12.4 29300
Covariance Two variables, x and y. When one is above the mean, is the other one also above the mean? x & y are the variables that we are interested in xi & yi are the values of the variables for one individual n is the sample size
Example
REFERENCE TO G&W P. 468-469 strength and direction of the association.
How about IQ and Income?
Inspect the association
Which association is stronger?
Compute covariance (again)
Comparison Covariance between education and income is: 46,088.89 Covariance between IQ and income is: 72,977.78
Comparison Unit of measurement of covariance between education and income is: Years of education dollars Unit of measurement of covariance between IQ and income is: IQ points dollars QUESTION: When we need something in universal units, how do we measure it? (think t- test, think effect size)
How do we compare Try to express both quantities in terms of a single unit of analysis CORRELATION Divide covariance by the standard deviation of each variable
Education and Income
IQ and Income
Comparing correlations Correlation between education and income is: .83 Correlation between IQ and income is: .54
Correlation - formula Two variables, x and y. When one is above the mean, is the other one also above the mean? x & y are the variables that we are interested in xi & yi are the values of the variables for one individual n is the sample size
Hypothesis testing with a correlation The test is similar to a t-test. (G&W 482-484) Null hypothesis: r=0 Alternative hypothesis: One tail or two tail
IMPORTANT INTERPRETATION of r G&W p. 470 – 15.2
Conditions that make r small! G&W p.478-479 – read and ask questions if needed.