Instructor: Mr. Le Quoc

Members : Pham Thi Huong

Tran Thi Loan

Nguyen Ngoc Linh

Hoang Thanh Hai

ANOVA IN DEPTH – DESIGNS AND EXAMPLES

Table of contents

• Full factorial design

♪ One way ANOVA

♪ Multifactor ANOVA

♪ Repeated measures design

♪ Mixed design

• Non-full factorial (nested) design

♪ Latin square design

♪ Split-plot design

Rationale of ANOVA

• Purpose: test for significant differences between means.

• Why the name analysis of variance? In order to test for statistical significance between means, we

are actually comparing (i.e., analyzing) variances.

• The Partitioning of Sums of Squares Variances can be divided, that is, partitioned. Ex: SS total = SS between-group + SS within-group

• Dependent and independent variables.– Dependent variables: measured = response– Independent variables: controlled = factor

Full factorial designOne way ANOVA

Test for the effect of one factor (independent variable) on the response variable (dependent variable)

Example: Test the effect of 3 books on the participants’ probability to have a lover. 3 levels: Book A (Aggressive Approach), Book M (Moderate Approach) and Book P (Passive Approach). Prepare a sample of participants and assign randomly to each a book. After one month, all participants are tested for results in interactive tests.

Book A Book M Book P

( mark of Love Test of each person)

Full factorial designMultifactor ANOVA

♣ Definition: Test for the effect of multiple factors on the dependent variable simultaneously

♣ Example: Test the effect of both the books (A, M and P) and gender

(Male, Female and Homosexual) on probability to have a lover. Prepare a sample and randomly assign one book to each participant. After one month, all participants are tested for results in interactive tests.

Book A Book M Book P

Gender 1Male

( mean mark of Love Test )

Gender 2 Female

Gender 3Homosexual

Full factorial designMultifactor ANOVA

♣ Advantages: + More realistic

+ More efficient than multiple t-tests

+ Enhance the power ( sensitivity of the test)

♣ Disadvantages - Difficult to be completely randomized

Interaction Effect• Example: Male who read book A and women who read book P tend to have

high score. Conversely, Male reading book P and Female reading book A have the lowest score. Out of people reading book M, homosexual have highest score.

• Note: Interaction effects often override main effects.

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Male Homosexual Female

Mean Score

Book A

Book M

Book P

Full factorial designRepeated measures design

• Definition: An experimental design in which the measurements are taken at two or more points in time on the same set of experimental units.

• Example: Test the effect of 3 books on the participants’ probability

to have a lover measured by the score of a test. 3 levels: Book A, Book M and Book P. Prepare a sample of participants and assign randomly to each a book. After one month, all participants are tested in interactive tests. Then they all must complete a multiple-choice test.

Full factorial designRepeated measures design

• Advantages– Repeated measures required in some research

hypothesis (ex: longitudinal research)– Reduce the error– Economical

• Disadvantages:– Carryover – Progressive effect

Solution: Counter balance

Full factorial designMixed design

• Definition: Combination of both multi-factor and repeated measures design.

• Example: Test the effect of both the books (A, M and P) and gender (Male, Female and Homosexual) on probability to have a lover. Prepare a sample and randomly assign one book to each participants. After one month, all participants are tested in interactive tests. Then they all must complete a multiple-choice test.

• Ad/disadvantage: Same as multi-factor and repeated measures.

Non-full factorial designLatin Square design

• Definition: A Latin Square extends the Randomized Complete Block design to the case in which there are two blocking factors and one treatment. It is used to comparing t treatments in t rows and t columns, where rows and columns represent two blocking factors. The allocation of experimental treatments is such that each treatment occurs exactly once in each row and column.

• Example: Treatment factor: Book (3 levels: A, M, P )

Blocking factor 1: Apperance (3 levels: Beautiful (Handsome), Normal,Ugly)

Blocking factor 2: Gender (3 levels: Male, Female, Homosexual)

Non-full factorial designLatin Square designApperance 1

Beautiful (Handsome)Apperance 2

NormalApperance 3

Ugly

Gender 1Male

Book M Book A Book P

Gender 2 Female

Book A Book P Book M

Gender 3Homosexual

Book P Book M Book A

Non-full factorial designLatin Square design

• When to use : The Latin square design applied when there are repeated

exposures/treatments and two other factors => It’s useful where the experimenter desires to control variation in two different directions

• Advantage: This design avoids the excessive numbers required for full three way

ANOVA => economical

• Disadvantage: The number of levels of blocking factors and treatment factor must

be equal Not reflect the interaction

Non-full factorial designSplit-plot design

• Definition: Some factors of interest may be hard-to-change while the remaining factors are easy-to-vary. the running order of the treatment combination is determined by these “hard-to-change factors”

• Example: Restrict randomization by determining the hard-to-change factor: Location. Choose randomly one of three level of factor “Location”. Within that level, randomly select a participant of whichever gender and randomly assign him/ her a book. Then randomly select another level of “location "and so on. After one month, all participants are tested for results in interactive tests.

Location

Urban

Rural

Mountainous

Male

Female

Homosexual

Book M

Book A

Book P

Non-full factorial designSplit-plot design

Non-full factorial designSplit-plot design

• Advantages: + Increasing precision in estimating certain effects

+ Saving time, money and easy to follow the results

• Disadvantages:

- Sacrificing precision in other effects

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