This document discusses analysis of covariance (ANCOVA) and provides an example to illustrate its use. ANCOVA involves comparing group means after controlling for a continuous covariate variable. The example analyzes data from an experiment testing four glue formulations, with tensile strength as the dependent variable and thickness as the covariate. ANCOVA is conducted since thickness is related to strength. The results show the covariate (thickness) has a significant effect on strength, but the factor (formulation) does not have a significant effect on strength after controlling for thickness. The adjusted group means from ANCOVA are closer together than the unadjusted means, indicating ANCOVA was necessary to properly analyze the data.

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WAQAS AHMED FAROOQUI
Lecturer of Biostatistics & Research Associate
Department of Research
Dow University of Health Sciences
Email: waqas.ahmed@duhs.edu.pk
Review: Completely Randomized Design ‐
CRD (one‐factor design)
The Completely Randomized design (CRD) randomly divides the
experimental units into “a” groups of size “n” and randomly assigns a
treatment to each group.
CRD (one-way ANOVA) is the generalized form of two-sample
Independent t-test.
Experimental units are relatively homogeneous.
Treatments are assigned to experimental units at random.
Each treatment is replicated the same number of times
(balanced design).
Review ‐ Completely Randomized Design
• Experimental Design ‐ Completely randomized design (CRD)
• Sampling Design ‐ One‐way classification design
CRD Model: ij
i
ij
y 

 


0
1



t
i
i
so
mean
overall
from
deviation
as
defines
are
Effects
Treatment 
yij = the jth observation in the ith treatment
µ= overall mean
αi= the effect of the ith treatment
eij = random error NID (0, σ2) – Normal Probability Plot of Residual
Review – Correlation & Regression
• Correlation –
Relationship (Nature – Positive/Negative & Strength – Weak/Moderate/Strong)
• Regression – Linear Relationship & Estimation
Correlation Formula:
2 2
2 2
x y
x y
n
r ; 1 1
( x ) ( y )
x . y
n n
r

   
   
 
   
   
   
 

 
 
bX
a
ŷ 

 
2
2
b ;
x y
x y
n a y b x
x
x
n

  

 



Linear Regression Model:

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INTRODUCTION:
The Analysis of covariance (ANCOVA) is another technique that occasionally
useful for improving the precision of an experiment.
Suppose in a experiment with a response variable Y there is another variable,
such as X, and that Y is linearly dependent to X.
Further more X cannot be controlled by the experimenter but can be observed
along with Y. The variable X is called covariate.
The ANCOVA involves adjusting the observed response variable for the effect
of covariate.
Analysis of Covariance (ANCOVA) is a statistical test related to ANOVA
It tests whether there is a significant difference between groups after
controlling for variance explained by a covariate
A covariate is a continuous variable that correlates with the dependent variable
Analysis of Covariance
ANCOVA is the combination of linear regression model and the model
employed in the ANOVA
Can be used to compare g treatments, after controlling for quantitative factor
believed to be related to response (e.g. pre‐treatment score)
Can be used to compare regression equations among g groups (e.g. common
slopes and/or intercepts)
Advantage:
Analysis of Covariance
Example
Nineteen rats are assigned at random among four experimental groups. Each
group is fed a different diet. The data are rat body weights, in kilograms, after
being raised on these diets. We wish to ask whether rat weights are the same for
all four diets.
Fed at Different Diet
Feed 1 Feed 2 Feed 3
Weight Age Weight Age Weight Age
60.8 3 68.7 4 61.9 3
67.0 3.5 67.7 2 64.2 3.7
65.0 3.2 75.0 3 63.1 3.5
68.6 3.3 73.3 5.5 66.7 4
61.7 1.2 71.8 5 60.3 2.9
Question?
In determining how different Diets will affect a weight, it
may be essential to take age into account.
ANOVA
One way
ANCOVA
with single factor and a
covariate
Involves one qualitative variable
(factor), One continuous (Y,
response, dependent) variable
Involves one qualitative variable (factor)
, one continuous( Y, response,
dependent) variable and an other
continuous variable (X, covariate)

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Analysis of Covariance Model
Assume that there is a linear relationship between the response and the
covariate, an appropriate statistical model is
..
1, 2,.....,
( )
1, 2,....,
ij i ij ij
i a
Y X X
j n
   


     


Where
Y ij is the j th observation on the response variable taken under the i th
treatment or level of the single factor.
X ij is the measurement made on the covariate, is the mean of the X ij
µ is the overall mean
τ is the effect of the i th treatment.
β is the linear regression coefficient(common slop) indicating the dependency
of Yij on X ij
ε ij are random error component.
..
X
We assume that ε ij are NID(0, σ2 ), that the slop β ≠ 0 and the relationship
between Yij and X ij is linear, that the regression coefficients for each treatment
are identical, that the treatment effects sum to zero, and the covariates X ij is
not affected by the treatment.
Notice immediately previous equation, that the analysis of covariance model is
a combination of the models employed in analysis of variance and regression.
That is we have treatment effect {τ} as in a single factor analysis of variance
and a regression coefficient β as in regression analysis.
We can rewrite previous model as
1, 2 , ..... ,
'
1, 2 , .... ,
ij i ij ij
i a
Y X
j n
   


    


Where µ` is a constant not equal to the overall mean, which for this model is
..
X
 
 
Analysis of Covariance Model (Cont..)
Assumptions for ANCOVA
ANOVA Assumptions:
• The response variable (data) are randomly sampled (Run Test)
• Group are normally distributed
(Kalmogrov Simrnov (ni≥50)/ Shapiro Walk (ni<50) Test)
• Variance is equal between groups (Homogeneity of Variance Test)
• All measurements are independent (No Trend or Funnel Shape)
• The residuals (error) are normally distributed (Q‐Q Plot for Error)
Also, for ANCOVA:
Linearity: relationship between covariate and dependent variable is linear
Independence : covariate and treatment effect are independent (treatment do
not effect covariates X because technique remove the effect of variation in)
Homogeneity: Assuming that there is a common slope for the covariate effect
with all levels of the experimental factor regression slops
Hypotheses for ANCOVA
• H0 and H1 need to be stated slightly differently for an ANCOVA than a
regular ANOVA
• H0: the group means are equal after controlling for the covariate
• H1: the group means are not equal after controlling for the covariate
The model allows to estimate the factor effects through estimation of the µ.
One can also test for significant differences in the effects due to the factor
levels by testing
0 1 2 3
: . . . .
:
a
a i
H
H a t l e a s t t w o d i f f e r
   

   
Testing for Treatment Effect

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Testing for Regression Coefficient
Likewise, one can estimate the effect of the covariate on the response through
the estimation of β. One can also test whether the covariate has a statistically
significant effect on the response by testing
0 : 0
: 0
a
H
H




If we fail to reject null hypothesis then we no need to run ANCOVA
But if null hypothesis is rejected then the adjustment provided by the
ANCOVA is necessary
. . . ..
ˆ( )
i i i
Adjusted Y Y X X

  
23‐14
Example
Four different formulations of an industrial glue are
being tested. The tensile strength (response) of the glue
is known to be related to the thickness as applied. Five
observations on strength (Y) in pounds, and thickness
(X) in 0.01 inches are made for each formulation.
Here:
• There are t=4 treatments (formulations of glue).
• Covariate X is thickness of applied glue.
• Each treatment is replicated n=5 times at different
values of X.
Formulation
Strength
(Pounds)
Thickness
(Inches)
1 46.5 13
1 45.9 14
1 49.8 12
1 46.1 12
1 44.3 14
2 48.7 12
2 49.0 10
2 50.1 11
2 48.5 12
2 45.2 14
3 46.3 15
3 47.1 14
3 48.9 11
3 48.2 11
3 50.3 10
4 44.7 16
4 43.0 15
4 51.0 10
4 48.1 12
4 46.8 11
First we look is there any relationship between strength (Y) and thickness (X).
For this purpose we made scatter plot between strength (Y) and thickness (X).
There is a strong suggestion of a linear relationship between strength and thickness, and
seems appropriate to remove the effect of thickness on strength by ANCOVA
Analysis of Covariance (SPSS Data) ANCOVA.sav

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Analysis of Covariance (Scatter Plot) ANCOVA.sav
Now we run ANCOVA
WHY ?
Because there is relationship between dependent variable
(strength) and a covariate (thickness)
Analysis of Covariance (ANCOVA) ANCOVA.sav
These are the unadjusted means of strength (dependent
variable)
Analysis of Covariance (ANCOVA)

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Also Effect size for Formulation is 0.103 (10%) so we can say there is no large
contribution of different formulation levels in increasing or decreasing glue
strength.
0 1 2 3 4
:
:
a i
H
H a t l e a s t t w o d i f f e r
   

  
Testing for Formulation
Level of significance : α = 0.05
We will reject null hypothesis if Fcal > Ftab or P‐value ≤ α
Results : P‐value = 0.640
Analysis of Covariance (ANCOVA)
Since P‐value for Formulation is 0.640 so we can not reject the null hypothesis
of no effect of formulation, hence there is no strong evidence that the industrial
glue by formulation differ in strength.
Confirmation of Non significance of formulation by Post Hoc
(LSD) tests
Non of the pair is significant confirming the over all non significance of factor formulation.
Also Effect size for thickness is 0.685 (68.5%) so we can say there is a large
contribution of thickness in increasing or decreasing glue strength.
Testing for Regression Coefficient ( Slope ; β1)
Level of significance : α = 0.05
We will reject null hypothesis if Fcal > Ftab or P‐value ≤ α
Results : P‐value = 0.000 (< 0.001)
Analysis of Covariance (ANCOVA)
Since P‐value for thickness is less than 0.001 so we reject the null hypothesis
β1=0, hence there is a linear relationship between strength and thickness.
0 1
1
: 0
: 0
a
H
H




Adjusted means of strength (dependent variable)
Adjusted means of Dependent variable
These adjusted means can also be obtained manually by using the equation
. . . ..
ˆ( )
i i i
Adjusted Y Y X X

  
unadjusted
Now the adjustment provided by the ANCOVA is necessary.

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. . . ..
ˆ( )
i i i
Adjusted Y Y X X

  
Unadjusted
Unadjusted Mean Strength Adjusted Mean Strength
46.520 47.045
48.300 47.680
48.160 47.921
46.720 47.054
Comparing the adjusted mean with unadjusted mean, we note that the adjusted
means are much closer together, another indication that the ANCOVA was
necessary.
THANK YOU