Chapter 9: Notes
Statistical Inferences Concering One/Two Correlation Coefficients
Statistical Tests involving single Correlation Coefficient
A sample drawn from a population, of which the correlation coefficient
between two variables is computed. From this computed correlation, we
want to make an inference to the population’s correlation between the two
variables.
The null hypothesis in this case, is that there is zero correlation between the
two variables of interest in the population, i.e. Ho :  = 0.00.
Two ways to determine whether to reject or fail-to-reject null hypothesis:
1.
Using the correlation coefficient as a calculated value
The correlation coefficient is used as calculated value, which it will
compare with a correlation coefficient critical value. This value is
obtained from tables where correlation coefficients are used as
entries to determine the critical value.
If the correlation coefficient is equal or greater than the critical
value, null hypothesis is rejected.
2.
Using t-test
Alternatively, a t-test could be used. Correlation coefficient is put
into a t-formula to yield a calculated value referred to as t. This is
compared against a critical t-value (located from a t-table). If t is
equal or greater than the critical t-value, the null hypothesis is
rejected.
Regardless of which method used, the same decision should be
arrived at.
Correlation coefficient so far mostly referred to Pearson’s product-moment
but can include other types: Spearman’s, biserial etc.
Test on Many Single Correlation Coefficients
This refers to multiple tests conducted for the multiple single correlation
coefficients. These coefficients are sometimes shown as a correlation matrix.
Each test is treated separately, but as with multiple one-way tests, inflated
Type I Error is a risk. Bonferroni adjustment technique could be used.
Test of Reliability and Validity Coefficients
Reliability and validity tests often rely on correlation coefficients. Hence,
researchers often apply a statistical test to determine whether or not their
reliability and validity coefficients are significant. Again, the null hypothesis
set up in this case is that their correlation coefficient is zero.
Statistically Comparing Two Correlation Coefficients
Refer to textbook Pg. 221 Figure 9.2
Two possible scenarios:
1. Two population, two same variables in each (e.g. x and y). One
sample drawn from each population. For each sample, compute r xy .
The inference is that there is no difference between the correlation
coefficient (of x and y) of each population.
One example: Two population, male and female. Sample drawn.
Within male sample, correlation between x and y computed. Within
female, correlation between x and y computed. The null hypothesis is
that for the two populations, the correlation between x and y in male
population is the same as the correlation between x and y in female
population.
2. One population, correlation between x and y computed, Correlation
between x and z computed. Null hypothesis is that in the population,
correlation between x and y is the same as correlation between x and
z.
Cautions
1.
Relationship strength, Effect size and power
Give credit for calculating strength of association, effect size or
post hoc power analysis to see if there are practical significance.
2.
Linearity and Homoscedasticity
For Pearson’s correlation coefficient, two assumptions:
a. Linearity of data
b. Homoscedasticity of data (means variance of one variable is
independent from the value of the other variable)
Best way to check: scatter diagrams.
If diagrams not provided, researcher should at least talk about it.
3.
Causality
Credit to those who said correlation does not mean causality.
4.
Attenuation
If measuring instruments are not perfect, the correlation
coefficient will be underestimated.
One way to rectify is: if researchers have info concerning the
reliabilities of the measuring instruments, correlation-forattenuation adjustment can be made to increase the correlation
coefficient.
5.
Fisher’s r-to-z Transformation
Some kind of transformation.