ANCOVA

Analysis of
Covariance (ANCOVA) is a blend of ANOVA that analyzes the difference of means
of a continuous response variable across levels of a categorical predictor,
controlling for the effect of covariates that are of less interest.

ANCOVA can explain
the within-group error variance and hence reduce it. It tries to explain the
unexplained variance () in ANOVA in terms
of covariates. Additionally, ANCOVA helps to eliminate the influence on results
by confounders. The power of the ANCOVA test would increase when entering a
strong covariate and be reduced by having a weak covariate.

**Assumptions**

For the ANCOVA test,
the predictor is categorical, while the response variable and covariates are
continuous.

There are five assumptions of ANCOVA:

- Linearity: the
covariates and the response variable are linearly related.

- Homogeneity of
variance: the error is a random variable with zero mean and equal variance
across all subpopulations.

- Homogeneity of
regression slopes: the slopes of regression lines are equal across all
subpopulations.

- Independence:
observations are independent, which means errors are uncorrelated.

- Normality: The
errors follow a normal distribution.

**Model**

Suppose a researcher
is studying the relationship between the decrease in a person’s blood pressure
(DBP) and personal and environmental variables, to which they are exposed.
Knowledge from observational studies and laboratory findings suggest that the
decrease in a person’s blood pressure is related to their age and a drug
believed to reduce blood pressure.

Then the model can be
written as

The main effects of
drug treatment on the decrease of blood pressure are tested, controlling for
the effect of age.

**Example Code in SAS**

/* Analysis of
Covariance --------------------------------------*/

data DrugTest;

input
Drug $ PreTreatment PostTreatment @@;

datalines;

A 11 6 A 8 0 A 5 2 A
14 8 A 19 11

A 6 4 A 10
13 A 6 1 A
11 8 A 3 0

D 6 0 D 6 2 D 7 3 D 8 1 D
18 18

D 8 4 D 19
14 D 8 9 D 5 1 D
15 9

F 16
13 F 13 10 F 11 18 F 9 5 F 21 23

F 16
12 F 12 5 F
12 16 F 7 1 F 12 20

;

proc glm data=DrugTest;

class Drug;

model PostTreatment = Drug PreTreatment / solution;

lsmeans Drug / stderr pdiff cov out=adjmeans;

run;

proc print data=adjmeans;

run;

/* Visualize the
Fitted Analysis of Covariance Model -----------*/

ods graphics on;

proc glm data=DrugTest plot=meanplot(cl);

class Drug;

model PostTreatment = Drug PreTreatment;

lsmeans Drug / pdiff;

run;

ods graphics off;

**Example Code in R**

install.packages("jmv")

*# load
library jmv for function "ancova"*

**library**(jmv)

*# input
data*

drug <- c("A", "A", "A", "A", "A", "A", "A", "A", "A", "A",

"D", "D", "D", "D", "D", "D", "D", "D", "D", "D",

"F", "F", "F", "F", "F", "F", "F", "F", "F", "F")

pre_treatment <- c(11, 8, 5, 14, 19, 6, 10, 6, 11, 3,

6, 6, 7, 8, 18, 8, 19, 8, 5, 15,

16, 13, 11, 9, 21, 16, 12, 12, 7, 12)

post_treatment <- c(6, 0, 2, 8, 11, 4, 13, 1, 8, 0,

0, 2, 3, 1, 18, 4, 14, 9, 1, 9,

13, 10, 18, 5, 23, 12, 5, 16, 1, 20)

*# identify
drug as a categorical variable*

drug <- factor(drug)

*# create
a data frame*

df <- data.frame(drug, pre_treatment, post_treatment)

*# ANCOVA*

ancova(formula = post_treatment ~ drug + pre_treatment,
data=df)

References

1. Stephanie
Glen. "ANCOVA: Analysis of Covariance" From StatisticsHowTo.com:
Elementary Statistics for the rest of us!
https://www.statisticshowto.com/ancova/

2. Keppel,
G. (1991). Design and analysis: A researcher's handbook (3rd ed.). Englewood
Cliffs: Prentice-Hall, Inc.

3. Tabachnick, B. G.; Fidell,
L. S. (2007). Using Multivariate Statistics (5th ed.). Boston: Pearson
Education.

4. Example
48.4 Analysis of Covariance. (n.d.). SAS® Help Center. Retrieved November 3,
2021, from
https://documentation.sas.com/doc/en/pgmsascdc/9.4_3.3/statug/statug_glm_examples04.htm