**ANCOVA **

*ANCOVA*
is a procedure like ANOVA except researchers can study the effects of one or
more **independent variables** on a **dependent variable** after adjusting for other
variables, called *covariates*, which
were not a primary focus of the study. The letter *C* in ANCOVA stands for *covariate*.
There can be several covariates in a study. In testing for differences among
groups experiencing different leadership styles, we could study the effects on employee
satisfaction after adjusting for a covariate of years of employment. A key word
in ANCOVA studies is *adjusting*.
Analysts adjust the scores based on information about the covariate before
testing for significant differences.

Basic features of an ANCOVA:

Independent or grouping Variable = 1 or more

Dependent or criterion Variable = 1

Covariates = 1 or more

An *F *test indicates significance overall and for specific effects or relationships.

A commonly reported measure of effect size is eta squared.

A *p *value reveals the probability of a significant relationship-- one that is not due to chance factors.

**Read more about ANCOVA in the following books.**

**Applied Statistics Concepts for Counselors on AMAZON or GOOGLE**

*Creating Surveys*** ****on ****AMAZON **** or ****GOOGLE Worldwide**

**MORE about ANCOVA and COVARIANCE**

Analysis of Covariance

Geoffrey W. Sutton, Ph.D.

The
analysis of covariance is a research strategy that is based upon a two or more groups design that yields at least interval data and could be analyzed using ANOVA. We
use the term ANCOVA as an acronym. The letter *C* in the acronym represents a covariate. We usually refer to the
covariate as a CV. A covariate is a variable that is significantly correlated
with the dependent variable.

**In
experimental research**, the covariate helps reduce error variance and makes the *F*-test more sensitive to any main or
interaction effects. The correlation between the covariate and the DV allows
for the removal of the effects of a CV on a DV and represents a known source of
systematic bias.

I**n nonexperimental research**, researchers can use the covariate
to statistically remove the influence of a variable to help equate groups that
could not be formed by random assignment or to better understand another
relationships of interest.

**A third purpose** is to examine group differences by
controlling for the influence of a DV when there are several DVs in an
analysis. This latter use is known as multivariate analysis of variance or,
MANCOVA.

You
can use more than one CV in a research design or ANCOVA procedure. However, if
there is a correlation greater than *r*
= .80 between two CVs, you should use
only one of the CVs because they appear to be measuring a lot of the same
variance.

As
with all statistical procedures, there are several assumptions to meet. The
first three are basic assumptions for ANOVA and the next three are additional
assumptions for ANCOVA.

1. All data are from random samples and
independent of other data.

2. The scores on the DV are normally
distributed in the population.

3. The distributions of scores on the DV
have equal variances.

Additional
assumptions for ANCOVA

4. There is a linear relationship
between the CV and the DV.

5. The slope for the regression line
(for the CV) is the same in each group.

6. The CV has high reliability and was
measured without error.

**Research questions
and hypotheses**

We
generally use a key phrase to identify the CV in a research study. That key
phrase can be **controlling for** or **adjusted for**. Here are some examples.

1. What is the difference between memory
scores for people with right and left hemisphere stroke *when adjusted for *age?

2. What is the effect of a marriage
enrichment program *when controlling for*
years of marriage?

The
research hypothesis states there is an effect or a difference when adjusting
for the CV and the null hypothesis assumes the usual no difference, or no
effect result, when adjusting for the CV.

Following is a hypothesis based on
question number one.

H1: When adjusting for age, there is a
significant difference between the means on verbal memory between patients who
experience right and left hemisphere strokes.

H_{0}1: When adjusting for age,
there is no significant difference between the means on verbal memory in the
population between patients who experienced right and left hemisphere strokes (*p* < .05).

**The research
method with a CV**

We
would follow usual procedures for delivering the IV (independent variable) or measuring a QIV (quasi-independent variable) along
with the DVs. We would consider what variables might affect the DVs and collect
data to measure those CVs. After all the data have been entered into our
database, we would obtain the descriptive statistics. Next, make any
adjustments to the data and calculate correlations between the measured
variables. Those variables that were not the primary focus of the experiment or
study will be entered as CVs in the ANOVA procedure if they are highly
correlated with the DVs. We will perform the usual post hoc analyses, if
applicable.

**Results**

When
interpreting the results of an ANCOVA, we will refer to the adjusted means.
SPSS reports the results of the analysis. In the *Test of Between Subjects Effects *table, SPSS reports the CV along
with an *F*-test. If the CV made a
significant contribution to the analysis, the *p*-value for the CV will be less than .05 (or your preferred level of significance). The output will also
include adjusted and unadjusted means. When reporting the results, you should
report both sets of means. In a small study, the means can be reported in a
paragraph. In a larger study, the means should be placed in a table.

Example of ANCOVA reporting for a
fictitious study.

IV = communication skills training vs. a no skills training control (2
groups)

DV = some measure of communication on a
continuous scale

CV = years of employment-- a continuous scale

Workplace communication skills training for employees significantly improved positive statements when adjusted for years of employment, *F*(2,38) = 4.56,* p* = .03, eta^{2} = .42, Observed Power = .67.

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