Comparing three or more means (ANOVA)

COMPARING THREE OR MORE

MEANSPrepared by:

Miss MJ B. Eduria

OUTLINEI. Compare Three or More Means

1. ANOVA

a. One Factor Anova

b. Two Factor Anova

2. ANCOVA

a. Concept of Covariate

b. Selecting Covariates

c. One Factor Ancova

d. Two Factor Ancova

• The analysis of variance, popularly known as the ANOVA, can be used in cases where there are more than two groups.

• A procedure of comparing more than two groups - independent variable: smoking status non smoking, onepack a day, two pack a day

- dependent variable: number of coughs per day

K = number of conditions (in this case, 3)

ANOVA (Analysis of Variance)

Example 1: A. Determine the Exam performance differed based on

test anxiety levels among the students Independent Variable: TEST ANXIETY Levels a. low stressed students b. medium stressed students c. high stressed students Dependent Variable: Exam Performance (measured from 0-100)

ONE WAY ANOVA•The ONE-WAY ANOVA is used to determine whether there are any significant differences between the means of three or more independent (unrelated) groups.

TWO-WAY ANOVA

•A TWO-WAY ANOVA is useful when we desire to compare the effect of multiple levels of two factors and we have multiple observations at each level.•One-Way ANOVA compares three or more levels of one factor. But some experiments involve two factors each with multiple levels in which case it is appropriate to use Two-Way ANOVA.

ONE WAY ANOVA VS TWO WAY ANOVA

One way ANOVA Two way ANOVA

• Has one independent variable (1 factor) with >2 conditions conditions

• Condition, Level, Treatment

Example: for a brand of cola (factor)Level: Coke,Pepsi, RC Cola

Independent Variables = factors

• Has 2 Independent Variables (2 Factors)

-each can have multiple conditions

Example: Two Independent Variable (IV’s) - IV1: Brand; and IV2: Calories - Three Levels of Brand: Coke, Pepsi, RC Cola - Two Levels of Calories: Regular, Diet

ANALYSIS OF

COVARIANCE

•ANCOVAThe obvious difference between ANOVA and ANCOVA is the the letter "C", which stands for 'covariance'. • Like ANOVA, "Analysis of Covariance" (ANCOVA) has a single continuous response variable. •Unlike ANOVA, ANCOVA compares a response variable by both a factor and a continuous independent variable (e.g. comparing test score by both 'level of education' and 'number of hours spent studying'). • The term for the continuous independent variable (IV) used in ANCOVA is "covariate".

WHEN TO USE ANCOVA?

• ANCOVA is used in experimental studies when researchers want to remove the effects of some antecedent variable. For example, pretest scores are used as covariates in pretest posttest experimental designs. • ANCOVA is also used in non-experimental research, such as surveys or nonrandom samples, or in quasi-experiments when subjects cannot be assigned randomly to control and experimental groups. Although fairly common, the use of ANCOVA for non-experimental research is controversial (Vogt, 1999)

EXAMPLE

• A One-way ANCOVA was conducted to determine a statistically significant difference between [name levels of the independent variables] on (dependent variable) controlling for [name the covariate].

SPECIFIC EXAMPLE•A One-way ANCOVA was conducted to determine a statistically significant difference between football, basketball, and basketball players on the number of slices of pizza eaten in one sitting controlling for weight.

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

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: Inferent

ialDescript

iveor

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.


Is This Question of: Relations

hipDifferen

ceor

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


Inferential Descriptive

InferentialDescriptive


Is The Distribution:

NormalNot

Normalor



NormalNot

Normalor

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

Football Players

Basketball Players

Soccer Players

29 oz. of pizza eaten






































Normal Not Normal

or

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

Football Players

Basketball Players

Soccer Players






































Inferential

Descriptive

Inferential

Descriptive

NormalNot

Normal


Are the Data:

Scaled? (ratio/interval/ordinal

)

Categorical?(ordinal)

or

Are the Data:


)


or

The data is interval (ounces of Pizza)

Football Players

Basketball Players

Soccer Players





































Are the Data:


)


or

The data is interval (ounces of Pizza)

Football Players

Basketball Players

Soccer Players






































Inferential

Descriptive

Inferential

Descriptive

NormalNot

Normal

Scaled Categorical


Is there:

1 DV2 or more

DVor


Is there:

1 DV2 or more

DVor


Is there:

1 DV2 or more

DVor




Normal Not Normal

Scaled Categorical

1 DV 2 or more DV


Is there:

1 IV2 or more

IVsor

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


Is there:

1 IV2 or more

IVsor



Is there:

1 IV2 or more

IVsor



InferentialDescriptiv

e


e

NormalNot

Normal

ScaledCategoric

al

1 DV2 or more

DV

1 IV2 or more

IV


Is there:

1 IV Level2 or more IV

Levelsor


Is there:


Levelsor


Is there:


Levelsor


Inferential

Descriptive

Inferential

Descriptive

NormalNot

Normal

ScaledCategori

cal

1 DV2 or

more DV

1 IV2 or

more IV

2 or more IV Levels

1 IV Level


Are the Samples:

RepeatedIndepend

entor


Are the Samples:

RepeatedIndepend

entor

No individual is in more than one group


Are the Samples:

RepeatedIndepend

entor

No individual is in more than one group



e


e

NormalNot

Normal

ScaledCategorica

l

1 DV2 or more

DV

1 IV2 or more

IV

2 or more IV Levels

1 IV Level

Independent

Repeated


Is There:

A CovariateNot a

Covariateor


Is There:

A CovariateNot a

Covariateor


Is There:

A CovariateNot a

Covariateor




Normal Not Normal

Scaled Categorical

1 DV 2 or more DV

1 IV 2 or more IV

2 or more IV Levels

1 IV Level

IndependentRepeated

A CovariateNot 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.

Covariance is a measure of linear association between two variables, (i.e. how much a change in one variable is linearly associated with a change in another variable

https://en.wikipedia.org/wiki/Covariance

There is a significant effect of athlete type on number of slides of pizza eaten in one sitting after controlling for athlete weight, F(2, 26) = 4.83, p < .05

Independent Variable with assumed

levels –football, basketball,

soccer players


Dependent Variable


Covariate


Between Groups

Degrees of Freedom


Within Groups

Degrees of Freedom


F ratio adjusted for

covariate


Statistical Significance adjusted for

covariate

SOURCES

http://wikieducator.org/images/f/fb/Stats_11_ANOVA.pdf

http://www.statsmakemecry.com/smmctheblog/stats-soup-anova-ancova-manova-mancova

http://www.theanalysisfactor.com/confusing-statistical-terms-5-covariate/

http://www.slideshare.net/plummer48/reporting-an-ancova-1?related=1










Education

Comparing three or more means (ANOVA)