what is an ancova?

One-way Analysis of Covariance(ANCOVA)

Conceptual Tutorial

First

How did we get here?

Consider a similar problem

A pizza café owner wants to know which type of high school athlete she should market to.

A pizza café owner wants to know which type of high school athlete she should market to. Should she market to football, basketball or soccer players?

A pizza café owner wants to know which type of high school athlete she should market to. Should she market to football, basketball or soccer players? So she measures the ounces of pizza eaten by 12 football, 12 basketball, and 12 soccer players in one sitting.

Here are the raw data:Football Players Basketball Players Soccer Players

29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten14 oz. of pizza eaten 13 oz. of pizza eaten 15 oz. of pizza eaten27 oz. of pizza eaten 36 oz. of pizza eaten 23 oz. of pizza eaten27 oz. of pizza eaten 29 oz. of pizza eaten 26 oz. of pizza eaten28 oz. of pizza eaten 27 oz. of pizza eaten 17 oz. of pizza eaten27 oz. of pizza eaten 31 oz. of pizza eaten 25 oz. of pizza eaten32 oz. of pizza eaten 33 oz. of pizza eaten 14 oz. of pizza eaten13 oz. of pizza eaten 32 oz. of pizza eaten 29 oz. of pizza eaten35 oz. of pizza eaten 15 oz. of pizza eaten 22 oz. of pizza eaten32 oz. of pizza eaten 30 oz. of pizza eaten 30 oz. of pizza eaten17 oz. of pizza eaten 26 oz. of pizza eaten 25 oz. of pizza eaten

The owner wondered how much these athletes like pizza to begin with and how that might affect the results.

The owner wondered how much these athletes like pizza to begin with and how that might affect the results. She surveyed them prior to their eating the pizza.

The Survey

On a scale of 1 to 10 how much do you like pizza?

1 2 3 4 5 6 7 8 9 10

Here were the results:Football Basketball Soccer

7.0 3.0 7.55.0 8.0 4.53.5 4.5 3.59.0 9.5 6.07.0 6.5 6.08.0 7.0 4.56.5 7.5 6.07.5 9.0 1.52.5 8.5 6.59.0 4.0 5.08.0 7.5 5.55.0 8.0 4.0

Based on this information, let’s determine how we got here.

Here is the problem again:

A pizza café owner wants to know which type of high school athlete she should market to, by

comparing how many ounces of pizza are consumed across all three athlete groups.

She will control for pizza preference.

The Problem: A pizza café owner wants to know which type of high school athlete she should market to, by comparing how many ounces of pizza are consumed across all three athlete groups. She will control for pizza preference.

Is This:

Inferential Descriptiveor



Is This:



Is This:


Based on the data set of 36 athletes, this is a sample from which the owner would like to make generalizations about potential athlete customers.


Is This Question of:

Relationship Differenceor


Is This Question of:

Relationship Differenceor

Because the owner wants to compare groups differences, we are dealing with DIFFERENCE.


Inferential Descriptive

InferentialDescriptive


Is The Distribution:

Normal Not Normalor


Is The Distribution:

Normal Not Normalor

After graphing each column we find that the distributions are mostly normal.

Football Players Basketball Players Soccer Players29 oz. of pizza eaten 15 oz. of pizza eaten 32 oz. of pizza eaten

24 oz. of pizza eaten 28 oz. of pizza eaten 27 oz. of pizza eaten














Normal Not Normal


Are the Data:

Scaled? (ratio/interval/ordinal)

Categorical?(ordinal)

or


Are the Data:

Scaled? (ratio/interval/ordinal)

Categorical?(ordinal)

or

The data is interval (ounces of Pizza)
















Normal Not Normal

Scaled Categorical


Is there:

1 DV 2 or more DVor




Normal Not Normal

Scaled Categorical

1 DV 2 or more DV


Is there:

1 IV 2 or more IVsor

[Type of Athlete is the only Independent Variable (IV)]




Normal Not Normal

Scaled Categorical

1 DV 2 or more DV

1 IV 2 or more IV


Is there:

1 IV Level 2 or more IV Levelsor




Normal Not Normal

Scaled Categorical

1 DV 2 or more DV

1 IV 2 or more IV

2 or more IV Levels1 IV Level


Are the Samples:

Repeated Independentor


Are the Samples:

Repeated Independentor

No individual is in more than one group




Normal Not Normal

Scaled Categorical

1 DV 2 or more DV

1 IV 2 or more IV


IndependentRepeated


Is There:

A Covariate Not a Covariateor




Normal Not Normal

Scaled Categorical

1 DV 2 or more DV

1 IV 2 or more IV


IndependentRepeated

A Covariate Not a Covariate

Now that we know how we got here, let’s consider what Analysis of Covariance is.

Now that we know how we got here, let’s consider what Analysis of Covariance is.First, . . . what is covariance?

Now that we know how we got here, let’s consider what Analysis of Covariance is.First, . . . what is covariance?As you know, variance is a statistic that helps you determine how much the data in a distribution varies.


6 75

Number of Pizza Slices

eaten by Basketball

Players

Not much variance


6 75


eaten by Basketball

Players

Not much variance

6 754 8 932 10


eaten by Soccer Players

A lot of variance

Covariance is a statistic that helps us determine how much two distributions that have some relationship covary.

Let’s imagine that students take a math test and their ordered scores look like this.

Let’s imagine that students take a math test and their ordered scores look like this.

Student Test Scores

Bambi 98

Belle 92

Billy 84

Boston 77

Bryne 73

Bubba 68

Etc

Let’s imagine that students take a math test and their ordered scores look like this. Then let’s imagine they take a math anxiety survey.

Student Test Scores

Bambi 98

Belle 92

Billy 84

Boston 77

Bryne 73

Bubba 68

Etc

Let’s imagine that students take a math test and their ordered scores look like this. Then let’s imagine they take a math anxiety survey.

Student Test Scores

Bambi 98

Belle 92

Billy 84

Boston 77

Bryne 73

Bubba 68

Etc

Math Anxiety Scores

6

5

4

4

3

2

Etc.

How much do they covary?

Student Test Scores

Bambi 98

Belle 92

Billy 84

Boston 77

Bryne 73

Bubba 68

Etc

Math Anxiety Scores

6

5

4

4

3

2

Etc.

Student Test Scores

Bambi 98

Belle 92

Billy 84

Boston 77

Bryne 73

Bubba 68

Etc

Math Anxiety Scores

6

5

4

4

3

2

Etc.

Bambi has the highest Math Test Score and the highest Math Anxiety Score

Student Test Scores

Bambi 98

Belle 92

Billy 84

Boston 77

Bryne 73

Bubba 68

Etc

Math Anxiety Scores

6

5

4

4

3

2

Etc.

Belle has the 2nd highest Math Test Score and the 2nd highest Math Anxiety Score

Student Test Scores

Bambi 98

Belle 92

Billy 84

Boston 77

Bryne 73

Bubba 68

Etc

Math Anxiety Scores

6

5

4

4

3

2

Etc.

Billy has the 3rd highest Math Test Score and the 3rd highest Math Anxiety Score

Student Test Scores

Bambi 98

Belle 92

Billy 84

Boston 77

Bryne 73

Bubba 68

Etc

Math Anxiety Scores

6

5

4

4

3

2

Etc.

Boston has the 4th highest Math Test Score and the 4th highest Math Anxiety Score

Student Test Scores

Bambi 98

Belle 92

Billy 84

Boston 77

Bryne 73

Bubba 68

Etc

Math Anxiety Scores

6

5

4

4

3

2

Etc.

Bryne has the 5th highest Math Test Score and the 5th highest Math Anxiety Score

Student Test Scores

Bambi 98

Belle 92

Billy 84

Boston 77

Bryne 73

Bubba 68

Etc

Math Anxiety Scores

6

5

4

4

3

2

Etc.

Bubba has the 6th highest Math Test Score and the 6th highest Math Anxiety Score

Student Test Scores

Bambi 98

Belle 92

Billy 84

Boston 77

Bryne 73

Bubba 68

Etc

Math Anxiety Scores

6

5

4

4

3

2

Etc.

These two data sets perfectly covary.

Student Test Scores

Bambi 98

Belle 92

Billy 84

Boston 77

Bryne 73

Bubba 68

Etc

Math Anxiety Scores

6

5

4

4

3

2

Etc.

This means as one changes the other changes.


Student Test Scores

Bambi 98

Belle 92

Billy 84

Boston 77

Bryne 73

Bubba 68

Etc

Math Anxiety Scores

6

5

4

4

3

2

Etc.


Either in the same direction


Student Test Scores

Bambi 98

Belle 92

Billy 84

Boston 77

Bryne 73

Bubba 68

Etc

Math Anxiety Scores

6

5

4

4

3

2

Etc.


Or opposite directions

Math Anxiety Scores

2

3

4

4

5

6

Etc.


Covariance is a statistic that describes that relationship.

Covariance is a statistic that describes that relationship. The larger the covariance statistic (either positive or negative), the more the two samples covary.

Covariance is a statistic that describes that relationship. The larger the covariance statistic (either positive or negative), the more the two samples covary. Let’s demonstrate how to calculate covariance by hand.

Covariance is a statistic that describes that relationship. The larger the covariance statistic (either positive or negative), the more the two samples covary. Let’s demonstrate how to calculate covariance by hand.

(Although most statistical software can do it for you automatically.)

Here is the data set

StudentTest

ScoresMath Anxiety

Scores Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2

iX iY

Here is the sum of each column:

StudentTest

ScoresMath Anxiety


Sum 492 24

iX iY

StudentTest

ScoresMath Anxiety


Sum 492 24Mean 82 4

iX iY

And the mean:

StudentTest

ScoresMath Anxiety


Sum 492 24Mean 82 4

iX iY

Remember – Covariance can only be computed between two or more variables (e.g., test questions, test scores, ect.) with scores across each variable for each

person.

Here is the formula for covariance:

i iX Y

XY

X Y

N

With an explanation for each value:

i iX Y

XY

X Y

N

i iX Y

XY

X Y

N

Anxiety Scores

i iX Y

XY

X Y

N

Test Scores

i iX Y

XY

X Y

N

Each Test Scores

i iX Y

XY

X Y

N

i iX Y

XY

X Y

N

Or in this case is the mean for Test

Scores (82)

i iX Y

XY

X Y

N

i iX Y

XY

X Y

N

Each Anxiety Score

i iX Y

XY

X Y

N

i iX Y

XY

X Y

N

Or in this case is the mean for

Anxiety Scores (24)

i iX Y

XY

X Y

N

Let the Covariance Calculations Begin!

Student

StudentTest

ScoresMath

Anxiety Bambi 98 5Belle 92 6Billy 84 4Boston 77 4Bryne 73 3Bubba 68 2

Sum 492 24Mean 82 4

iX iY

i XX Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY

i XX Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY

Deviation between each math score and the mean.

i XX Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY

98


i XX Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY

98 - 82


i XX Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY

16


i XX Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY

1692 - 82


i XX Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY

161084 - 82


i XX Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY

1610277 - 82


i XX Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY

16102-573 - 82


i XX Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY

16102-5-968 - 82


i XX Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY

16102-5-9-14


Student

StudentTest

ScoresMath

Anxiety Bambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14

Sum 492 24Mean 82 4

iX iY i XX Student

StudentTest

ScoresBambi 98 5 16Belle 92 6 10Billy 84 4 2Boston 77 4 -5Bryne 73 3 -9Bubba 68 2 -14

Sum 492 24Mean 82 4

iX iY i XX

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX Student

StudentTest


Sum 492 24Mean 82 4

iX iY i XX

Deviation between each Anxiety score and its mean.

i YY

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX Student

StudentTest


Sum 492 24Mean 82 4

iX iY i XX i YY

5


Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX Student

StudentTest


Sum 492 24Mean 82 4

iX iY i XX i YY

5 - 4


Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX Student

StudentTest


Sum 492 24Mean 82 4

iX iY i XX i YY

1


Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX Student

StudentTest


Sum 492 24Mean 82 4

iX iY i XX i YY

16 - 4


Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX Student

StudentTest


Sum 492 24Mean 82 4

iX iY i XX i YY

124 - 4


Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX Student

StudentTest


Sum 492 24Mean 82 4

iX iY i XX i YY

1204 - 4


Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX Student

StudentTest


Sum 492 24Mean 82 4

iX iY i XX i YY

12003 - 4


Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX Student

StudentTest


Sum 492 24Mean 82 4

iX iY i XX i YY

1200

-12 - 4


Student

StudentTest

ScoresMath

Anxiety Bambi 98 5 16 1Belle 92 6 10 2Billy 84 4 2 0Boston 77 4 -5 0Bryne 73 3 -9 -1Bubba 68 2 -14 -2

Sum 492 24Mean 82 4

iX iY i XX i YY Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX i iX YX Y

Multiply the two paired Deviations to get what is called “the cross

product”

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4


16


product”

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4


16 x 1


product”

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4


16


product”

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4


1610 x 2


product”

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4


16202 x 0


product”

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4


16200

-5 x 0


product”

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4


162000

-9 x -1


product”

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4


1620009

-14 x -2


product”

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4


1620009

28


product”

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX

1620009

28

Student

Student Cross ProductsBambi 98 5 16 1 16Belle 92 6 10 2 20Billy 84 4 2 0 0Boston 77 4 -5 0 0Bryne 73 3 -9 -1 9Bubba 68 2 -14 -2 28

Sum 492 24Mean 82 4

iX iY i XX i YY i iX YX Y

Here’s the covariance equation again:

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX

1620009

28

Student


Sum 492 24Mean 82 4


Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX

1620009

28

Student


Sum 492 24Mean 82 4


i iX Y

XY

X Y

N

i iX Y

XY

X Y

N

So far we have calculated a portion of the numerator of this equation.

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX

1620009

28

Student


Sum 492 24Mean 82 4


i iX Y

XY

X Y

N

Now we will sum or add up the cross products.

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX

1620009

28

Student


Sum 492 24Mean 82 4


i iX Y

XY

X Y

N


Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX

1620009

28

Student


Sum 492 24Mean 82 4


Add up

i iX Y

XY

X Y

N


Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX

1620009

28

Student


Sum 492 24Mean 82 4


73

Add up

i iX Y

XY

X Y

N


Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX

1620009

28

Student


Sum 492 24Mean 82 4


73

i iX Y

XY

X Y

N

Then we divide the result by the number of subjects, which in this case is (6)

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX

1620009

28

Student


Sum 492 24Mean 82 4


73

i iX Y

XY

X Y

N


Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX

1620009

28

Student


Sum 492 24Mean 82 4


73 / 6

i iX Y

XY

X Y

N


Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX

1620009

28

Student


Sum 492 24Mean 82 4


73 / 6 = 12.2

i iX Y

XY

X Y

N

Covariance = 12.2

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX

1620009

28

Student


Sum 492 24Mean 82 4


73 / 6 = 12.2

12.2

Let’s see what the covariance looks like when the direction of the data goes in the opposite direction:

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY

BEFORE

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iYStudent

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY

BEFORE AFTER

First we calculate the deviations from the mean:

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY

First we calculate the deviations from the mean:

Student

StudentTest

ScoresMath

Anxiety Test

ScoresMath

Anxiety Bambi 98 2 16 - 2Belle 92 3 10 - 1Billy 84 4 2 0Boston 77 4 - 5 0Bryne 73 6 - 9 2Bubba 68 5 - 14 1

Sum 492 24Mean 82 4

iX iY i XX i YY

We now compute the Cross Products

Student

StudentTest

ScoresMath

Anxiety Test

ScoresMath

Anxiety Bambi 98 2 16 - 2Belle 92 3 10 - 1Billy 84 4 2 0Boston 77 4 - 5 0Bryne 73 6 - 9 2Bubba 68 5 - 14 1

Sum 492 24Mean 82 4

iX iY i XX i YY

x =x =x = x =x =x =

We now compute the Cross Products

Student

StudentTest

ScoresMath

Anxiety Test

ScoresMath

Anxiety Cross ProductsBambi 98 2 16 - 2 -32Belle 92 3 10 - 1 -10Billy 84 4 2 0 0Boston 77 4 - 5 0 0Bryne 73 6 - 9 2 -18Bubba 68 5 - 14 1 -14

Sum 492 24Mean 82 4


We sum the cross products and then divide it by the number of students

Student

StudentTest

ScoresMath

Anxiety Test

ScoresMath


Sum 492 24Mean 82 4



Student

StudentTest

ScoresMath

Anxiety Test

ScoresMath


Sum 492 24Mean 82 4


-73


Student

StudentTest

ScoresMath

Anxiety Test

ScoresMath


Sum 492 24Mean 82 4


-73 / 6


Student

StudentTest

ScoresMath

Anxiety Test

ScoresMath


Sum 492 24Mean 82 4


-73 / 6 = -12.2

This is the covariance!

Student

StudentTest

ScoresMath

Anxiety Test

ScoresMath


Sum 492 24Mean 82 4


-73 / 6 = -12.2


Student

StudentTest

ScoresMath

Anxiety Test

ScoresMath


Sum 492 24Mean 82 4


-73 / 6 = -12.2

Notice that when there is a negative relationship between two variables


Student

StudentTest

ScoresMath

Anxiety Test

ScoresMath


Sum 492 24Mean 82 4


-73 / 6 = -12.2

Notice that when there is a negative relationship between two variables

The covariance is negative

On the other hand, when the relationship between two variables is positive . . .

On the other hand, when the relationship between two variables is positive . . . The covariance will be positive.

On the other hand, when the relationship between two variables is positive . . . The covariance will be positive.

Student

StudentTest

ScoresMath


Sum 492 24Mean 82 4


StudentTest

ScoresMath


Sum 492 24Mean 82 4

iX iY i XX

1620009

28

Student


Sum 492 24Mean 82 4


73 / 6 = 12.2

Why is this important to know?

Why is this important to know?Especially in light of the question we are trying to answer?

Computing covariance helps us-




Meaning that we want to know what the difference between the three groups would be if we took away all of the covariance between pizza preference and amount of ounces of pizza eaten.



If we did not take out the covariance, then a bunch of soccer players may like pizza not because they are soccer players (our research question) but because they just LOVE PIZZA (not our research question).



Because their love of pizza is not what we are testing, we will control for it by computing covariance and see how much of the fact that they are soccer players really affects the amount of ounces of pizza they eat.

So, let’s begin by running a One-way ANOVA without removing the covariance between pizza preference and ounces eaten by athlete type.

Here’s the data













Here are the results of the one-way ANOVA for this data set:

Sums of Squares df Mean Square F-Ratio

Between Groups 38.9 2 19.4 0.4Within Groups (error) 1587.4 33 48.1Total 1626.3













Here are the results of the one-way ANOVA for this data set:

Sums of Squares df Mean Square F-Ratio

Between Groups 38.9 2 19.4 0.4Within Groups (error) 1587.4 33 48.1Total 1626.3













As you will recall, an F-ratio 1 or lower with any ANOVA method is not significant.

Then

After calculating the covariance between pizza preference and ounces of pizza eaten in one sitting, we find that there is a positive relationship.


Pizza Preference (scale 1-10)Football Basketball Soccer

7.0 3.0 7.55.0 8.0 4.53.5 4.5 3.59.0 9.5 6.07.0 6.5 6.08.0 7.0 4.56.5 7.5 6.07.5 9.0 1.52.5 8.5 6.59.0 4.0 5.08.0 7.5 5.55.0 8.0 4.0














Pizza Preference (scale 1-10)Football Basketball Soccer

7.0 3.0 7.55.0 8.0 4.53.5 4.5 3.59.0 9.5 6.07.0 6.5 6.08.0 7.0 4.56.5 7.5 6.07.5 9.0 1.52.5 8.5 6.59.0 4.0 5.08.0 7.5 5.55.0 8.0 4.0













Covariance = 12.1

After running the Analysis of Covariance on the data and partialling out pizza reference, here is the resulting ANOVA table:


SS df MS FAdjusted means (BG) 74.5 2 37.2 3.8Adjusted error (WG) 314.1 32 9.8

Adjusted total 388.6 34




Adjusted means – after we took out the covariance between the two variables: Type of Athlete and Pizza Preference (the covariate)




Notice the F-ratio is larger making it more likely to be

significant.

Let’s compare the F-ratio for just the ANOVA


SS df MS F-RatioBetween Groups 38.9 2 19.4 0.4Within Groups (error) 1587.4 33 48.1Total 1626.3

Before


With the ANCOVA


Before


With the ANCOVA


Before

SS df MS F-RatioAdjusted means (BG) 74.5 2 37.2 3.8Adjusted error (WG) 314.1 32 9.8


After


With the ANCOVA


Before

SS df MS F-RatioAdjusted means (BG) 74.5 2 37.2 3.8Adjusted error (WG) 314.1 32 9.8


After

So we would conclude that there is a significant difference between football, basketball & soccer players in terms of the ounces of pizza they eat, that is, when we control for pizza preference.

We can even adjust the original means for amount of ounces of pizza eaten, after controlling for preference.


Athlete Football Basketball Soccer

Means 25.4 26.3 23.8

Original Means



Means 25.4 26.3 23.8

Original Means


Means 24.3 23.8 27.3

Adjusted Means (after controlling for the covariance)



Means 25.4 26.3 23.8

Original Means


Means 24.3 23.8 27.3


Notice that after controlling for pizza preference, the mean for

Basketball players drops



Means 25.4 26.3 23.8

Original Means


Means 24.3 23.8 27.3


And the mean for Soccer players

INCREASES!



Means 25.4 26.3 23.8

Original Means


Means 24.3 23.8 27.3


That’s the Power of ANCOVA!

Important note,The more the covariate (pizza preference) covaries with the independent variable (type of athlete) . .

Important note,The more the covariate (pizza preference) covaries with the independent variable (type of athlete) . . . the bigger the adjustment will be between original and adjusted means.


Meaning they share a larger covariance value (either positive or negative).



Means 25.4 26.3 23.8

Original Means


Means 24.3 23.8 27.3




Means 25.4 26.3 23.8

Original Means


Means 24.3 23.8 27.3


Big Adjustments

One final note

In this case the covariate (the thing we were controlling for) was a continuous variable like

• Ounces of pizza eaten• Time it takes to eat pizza• The weight of each athlete.

In this case the covariate (the thing we were controlling for) was a continuous variable like • Ounces of pizza eaten• Time it takes to eat pizza• The weight of each athlete.

In this case the covariate (the thing we were controlling for) was a continuous variable like • Ounces of pizza eaten• Time it takes to eat pizza• The weight of each athlete.But it also can be categorical (one or the other)

In this case the covariate (the thing we were controlling for) was a continuous variable like • Ounces of pizza eaten• Time it takes to eat pizza• The weight of each athlete.But it also can be categorical (one or the other)• Year in School (Sophomores, Juniors, or Seniors)• Gender (Male or Female)• Religious Affiliation (Muslim, Catholic, etc.)

In summary

Analysis of Covariance is a powerful tool that makes it possible to control for any variable that is not of interest (eg. pizza preference)

Analysis of Covariance is a powerful tool that makes it possible to control for any variable that is not of interest (eg. pizza preference) in order to see the true effect of the variable of interest (type of athlete) on a dependent variable of interest (ounces of pizza eaten)

There are more complex methods such as Factorial ANCOVA, Repeated measures ANCOVA and Multivariate ANCOVA.

There are more complex methods such as Factorial ANCOVA, Repeated measures ANCOVA and Multivariate ANCOVA.

This presentation gives you the conceptual foundation necessary to understand the Analysis of Covariance elements of these methods.