Overview of
Correlation & Regression
Bivariate Correlation and Regression
Bivariate correlation and regression evaluate the degree of
relationship between two quantitative variables.
Pearson
Correlation (r), the most commonly used bivariate correlation
technique, measures the association between two quantitative
variables without distinction between the independent and
dependent variables (e.g., What is the relationship between SAT
scores and freshman college GPA?).
In contrast, bivariate
regression utilizes the relationship between the independent and
dependent variables to predict the score of the dependent
variable from the independent variable (e.g., To what degree do
SAT scores [IV] predict freshman college GPA [DV]?).
When to use bivariate correlation/regression?
1
IV (quantitative)
relationship/prediction
1
DV (quantitative)
Multiple Regression
Multiple regression identifies the best combination of
predictors (IVs) of the dependent variable.
Consequently it is
used when there are several independent quantitative variables
and one dependent quantitative variable (e.g., Which combination
of risk taking behaviors [amount of alcohol use, drug use, sexual
activity, and violence—IVs] best predicts the amount of suicide
behavior [DV] among adolescents?).
To produce the best
combination of predictors of the dependent variable, a sequential
multiple regression selects independent variables, one at a time,
by their ability to account for the most variance in the
dependent variable.
As a variable is selected and entered into
the group of predictors, the relationship between the group of
predictors and the dependent variables is reassessed.
When no
more variables are left that explain a significant amount of
variance in the dependent variable, then the regression model is
complete.
When to use multiple regression?
2+
IV (quantitative)
relationship/prediction
1
Source:
DV (quantitative)
Mertler, C. A., & Vannatta, R. A. (2005). Advanced and
multivariate statistical methods: Practical application
and interpretation (3rd ed.). Los Angeles, CA: Pyrczak.
Bivariate
I.
Correlation
Research Questions…

Generically-Stated Research Question:



II.


Examples of Appropriately-Stated Research Questions:

What is the relationship between students’ on-task
behavior and academic achievement?

What is the relationship between instructors’ assessment
knowledge and the passage rates in their courses?

What is the relationship between salary and years of
teaching service?
Examples of Inappropriately-Stated Research Questions:

What is the effect of students’ on-task behavior on their
academic achievement?

What is the impact of instructors’ assessment knowledge
on the passage rates in their courses?
Sampling & Data…
Samples should always be selected randomly (probability
samples)

Allows for generalization of results to larger population

Unless the goal is only descriptive in nature (e.g.,
action research)
Data must be quantitative

III.

What is the relationship between Variable A and Variable
B?
Scale of measurement (i.e., nominal, ordinal, interval,
ratio) does not necessarily matter, as their exist
numerous types of correlation coefficients that can be
calculated
Data Analysis & Interpretation…
Analysis involves the calculation of a correlation
coefficient (i.e., a quantitative measure of a
relationship)

Most common is a Pearson correlation coefficient (r)—
correlation between two interval variables

Numerous others exist for various combinations of
variables…

However, all are interpreted in similar manner; range
from –1.00 to +1.00 (some range from 0.00 to +1.00)

General rule of thumb for interpretation…

Sample output from SPSS…
Value of
coefficient
P-value
(significanc
e)
Sample
size
Symmetrical
matrix
Multiple
I.
Regression
Research Questions…

Generically-Stated Research Question:



Examples of Appropriately-Stated Research Questions:

Which student demographic variables best predict academic
achievement?

What combination of student demographic variables (i.e.,
SES, ethnicity, gender, education status of
mother/father, family income, age, and birth order) best
predicts academic achievement in college chemistry
courses?
Examples of Inappropriately-Stated Research Questions:

II.


Samples should always be selected randomly (probability
samples)

Allows for generalization of results to larger population

Unless the goal is only descriptive in nature (usually
not the case for MR)
Data must be quantitative
III.

What is the relationship between student demographic
variables and academic achievement?
Sampling & Data…


Which combination of variables from a larger set of IVs
best predicts the DV?
Scale of measurement (i.e., nominal, ordinal, interval,
ratio) should be interval or ratio (others will work, but
perhaps result in less clear interpretation)
Data Analysis & Interpretation…
Analytic process…

Multiple regression selects independent variables, one at
a time, by their ability to account for the most variance
in the dependent variable (rank order IVs)

As a variable is selected and entered into the group of
predictors, the relationship between the group of
predictors and the dependent variable is reassessed

When no more variables are left that explain a
significant amount of variance in the dependent variable
(i.e., that contribute significantly to the model), then
the regression model is complete
Example multiple regression…


IVs —
o
BEGINNING SALARY
o
JOB SENIORITY
o
AGE
o
WORK EXPERIENCE
o
SEX & RACE CLASSIFICATION
DV —
o
CURRENT SALARY

Sample output from SPSS…
Four
(progressive)
regression
models, based
on rank
ordering
Multiple
Correlation:
Pearson
correlation
between predicted
and actual DV
scores
Contribution
in variance at
each step
(model)
Variance in
DV accounted
for by IVs
Test of
significance
of each
models’
predictabili
ty
Variables
included
in each
model
Tests of sig.
for
individual
variables
Coefficients
used to develop
regression
(prediction)
equation
ZCurrentSalary = (.845)ZBeginningSalary + (-.145)ZWorkExperience +
(.097)ZJobSeniority + (-.088)ZSexRace