Analysis of Covariance (ANCOVA)
Comparing simple linear regression lines
Analysis of covariance (ANCOVA) combines features of
simple linear regression
with one-way analysis of variance
Both a quantitative variable X and an ANOVA grouping variable
are used to describe the measurement (Y) variable.
Simple linear regression fits a straight line to X-Y data.
One-way analysis of variance fits a mean to each group.
One-way analysis of covariance fits a straight line to
each group of X-Y data, such that the slopes of the
lines are all equal.
This fitted model may then be used
to test the null hypotheses:
The slopes are in fact equal.
Given that the slopes are equal, the intercepts for each line
are also equal. This is the test for whether the group
(treatment) means are equal, after making the adjustment
for X.
Given that the slopes are equal, the slopes are all equal to 0.
Assumptions:
The simple linear function Y[ij] = M + m[i] + b*(X[ij]-Xbar) + e[i] is the correct model,
where Y[ij] is the jth observed value of Y
in group i, X[ij] is the jth observed
value of X in group i, M is an overall mean for all the Y
values, m[i] is treatment effect for group i,
Xbar is the overall average of the X values,
and e[i] is the error term.
Equivalently, the expected value
of Y for a given value of X is M + m[i] + b*(X[ij]-Xbar).
The slope is b, the amount by which the expected
value of Y increases when X increases by a unit amount.
The X variable (predictor variable) values are fixed
(i.e., X is not a random variable).
The Y variable (response variable) observations are
independent.
Y[ij] is
normally distributed
with the same variance as the e[i].
For a given value of X[ij] in group i, the variable Y has constant mean.
The X variable is also known as a covariate, or
as a concomitant variable.
Analysis of covariance controls for X to make a more precise
test of whether the treatment (group) means (intercepts) are equal.
It can also be used to study the linear relationship
between X and Y for each group.
Guidance:
Ways to detect before performing the
analysis of covariance whether your data violate any
assumptions.
Ways to examine analysis of covariance results to detect
assumption violations.
Possible alternatives if your data or
analysis of covariance results indicate assumption violations.
To properly analyze and interpret the
results of analysis of covariance (ANCOVA), you should be familiar with the following terms and
concepts:
Failure to understand and properly apply
analysis of covariance (ANCOVA) may result in
drawing erroneous conclusions from your data.
If you are not familiar with these terms and concepts, you may wish to
consult with a statistician.
You may also want to consult the following references:
Brownlee, K. A. 1965. Statistical Theory and Methodology
in Science and Engineering. New York: John Wiley & Sons.
Neter, J., Wasserman, W., and Kutner, M.H. 1990. Applied
Linear Statistical Models. 3rd ed. Homewood, IL: Irwin.
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