LECTURE 10: CORRELATION AS DESCRIPTION AND INFERENCE
Overview of Correlation and Regression
A Simple Model
Weight (X1) ------------- + ---------------
Body Fat (Y)
Height (X2) -------------
+ ---------------
Correlation “r” = Strength of the relation of weight and fat
(+.61)
Sig “r”
= Probability the r in sample from pop with r of zero (.000)
Regression “b” = Slope line describing weight and fat (. 18)
Sig “b”
= Probability the b in sample from pop with b of zero (.001)
A Plot of Height and Body Fat
Correlation “r” = Strength of the relation of height and fat (-.025)
Sig “r” = Probability the r in sample from pop with r of zero (.69)
Regression “b” = Slope line describing height and fat (-.70)
Sig “b”
= Probability the b in sample from pop with b of zero (.75)
Note:
* ‘Variation’ in each variable
* How scatter plots are most useful when using measures of
continuous variables (interval and ratio, yet sometimes ordinal
* The utility in thinking in terms of dependent and independent
variables
* The ability to look for “goofy” data points in plots
* Correlation provides information on the strength of a relation
* Regression provides information on the “line” summarizing the
relation
Usually the “line” is a straight line. Thus, linear regression
Other relations can be examined
* Correlations allow one to compare apples and oranges
For example, for the above model the correlation between X1 Y is .613
The correlation between X2 and Y is -.025
Therefore, Weight (X1) has the strongest association with Body Fat (Y)
Correlation in More Detail
In performing a correlation one obtains two types of information
1. Substantive information on the relation between two variables
e.g., an “r” of .613 for the correlation between weight and body
fat (X1 & Y)
2. Inferential information to test the null hypothesis that the true
correlation in the population from which the sample was drawn
is zero ( e.g., a probability of drawing a sample with an “r
greater than.613 or less than -.613 from a population in which
the true “r” is zero.
R2 and Variance Explained
Idea of variance explained
Correlation Matrix
The Limitations of tests of statistical significance
Substantive and Statistical Significance are two different things
“For large sample sizes, even very small correlation coefficients
will have small observed significance levels. Statistically
significant doesn’t mean important or useful.”
Spearman Rank Order correlation
For ordinal data (Order in a race and time training)
(Birth order and I.Q. test)
Partial Correlation Coefficients
Other examples
Ice Cream, Summer, Crime
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