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 design One 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 ABook MBook P ( mark of Love Test of each person)

Full factorial design Multifactor 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 ABook MBook P Gender 1 Male ( mean mark of Love Test ) Gender 2 Female Gender 3 Homosexual

Full factorial design Multifactor 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.

Full factorial design Repeated 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 design Repeated 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 design Mixed 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 design Latin 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 design Latin Square design Apperance 1 Beautiful (Handsome) Apperance 2 Normal Apperance 3 Ugly Gender 1 Male Book MBook ABook P Gender 2 Female Book ABook PBook M Gender 3 Homosexual Book PBook MBook A

Non-full factorial design Latin 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 design Split-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 design Split-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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