Lecture 8: Analysis of Variance ANOVA, Part 2

Alternative forms of ANOVA (for repeated measures)

Purpose of using ANOVA

• for repeated measures designs

  • A variable is measured repeatedly

  • e.g. drug 1, drug 2, drug 3, and placebo

  • Need three or more points of measures

    • Otherwise use related samples t-test

RM measures and RM ANOVA

• sometimes the way how you measure sometimes is more difficult to make causal inference (e.g. age and problem solving ability)

  • age and problem solving ability is a non-experimental design

RM-ANOVA vs IM-ANOVA

for RM-ANOVA

• same hypothesis as IM-ANOVA

  • Means of the population means are identical

  • H1 is there is at least one difference

  • Then use post-hoc like in IM-ANOVA

• similar but not identical assumptions (not tested)

  • equality of variance

  • observations are not independent (compared to IM)

• Similar F-ratio but slightly difference

Issue with IM-ANOVA that RM-ANOVA can solve

• individual differences between subjects is alway an issue for both parts of the fraction in the F-ratio

• RM-ANOVA focuses on testing the same individual, therefore there is less of in issue within-treatments

RM-ANOVA

• who si participating

• the same individual is participating in all conditions

• - the random, unsystematic error attributed to individual difference is mostly eliminated from the numerator and remove it from the deominator

What variable for RM-ANOVA help control?

• confounding variables (e.g. weird scores)

F-ratio for RM-ANOVA: 2 stage process

• numerator calculations similar

• more work for the denominator

RM-ANOVA notation

• k

• n

• N

• T

• G

• P

• k = number of groups

• n = size of a particular samples

• N is all observations (because there is not more people, just repeated tests for the same participants)

• T = sum of all scores in a single condition

• G = sum of all scores in all conditions

• P = sum of scores for each participant

Stage 1 of RM-ANOVA

SStotal = sum X² - G2 / N

df total = N – 1

Numerator

SSbetween treatments = sum (T² / n) - (G² / N)

dfbetween treatments = k − 1

• thats it for the numerator

Denominator

SSwithin treatments = sum SS inside each treatment

dfwithin treatments = sum df inside each treatment

Stage 2 of RM-ANOVA

• assume SS within = SS between + SS error

  • SSbetween-subjects = sum (P2/k) – G2/N

  • solve for SS error

• Assume df error = (n-1)(k-1)

  • dfbetween-subjects = n – 1 

• Because RM-ANOVA does not have individual differences, between subjects has been removes, and we use SS error and df error for the no treatment effect calculate or MS

F critical value and F ditribution

• F critical value requires the a value, the df error and the df between treatments

Effect size calculations

partial N² =

What is the Two-Factor ANOVA? Compared to IM and RM- ANOVA

• given def for IM, RM, and then two factor

• Give example for two factor

• IM: different groups of observations

  • e.g. phones and attentions (1 factor with three levels)

• RM: more than two points of observation in the same subject

  • Study strategy and performance

    • 1 factor multiple levels but same subject

• Two factors ANOVA is two factor! (2 dependent things you are measuring)

  • gender, level of violence in video games and aggression

    • Factor A: gender

    • Factor B: level of violence

Vocabulary for two-factor ANOVAs

• what are the categories called

  • What is the entire thing, including all categories/level combinations called

• Give a sentence for a two-factor ANOVA

---

• is A a row or column

• is factor B a row or colum

• Cells

  • Matrix

for a 2 by 2 matrix of 4 cells

• A is rows

• B is columns

Multiple conclusions/possibilities with 2 factors

• e.g. of gender, violence and aggression

---

• based on the graph, what can you conclude?

  • Two things

• gender associated with aggression (main effect of gender)

• video game associated with aggression (main effect of video games)

• gender and video games interact in a complex way to influence aggression (interaction)

• Main effect of gender: aggression is higher in males

• Main effect of video games: aggression is higher in violent games

What does interaction mean?

• give an example with aggression and violent video games in the table provided

• Give a famous example

• the cell shows a diff value than predicted by the main effects

• e.g. table

  • If video games increase aggression, it doesn’t in all genders (not predicting as it should)

  • females have no change in aggression while only males do

• Harlow’s study with the monkey and the stuffed mother

  • The milk has a greater effect when comparing time spend with cloth vs wire mother

Practice: The the table provided an interaction?

• no surprises = no interactions

Hypothesis (list all possible scenarios)

For Factor A

• H0: no difference among the means of A: muA1 = muA2

• H1: at least one difference among the means of A

Factor Factor B

• H0: no difference among the means of B: muB1 = muB2

• H1: at least one difference among the means of B

For Interaction (A X B)

• H0: there is not interaction between the factors, The Mean difference between cells are explained by additive effects of the factors

• H1: There is an interaction

calculating the F ratio - for the first two hypotheses

• How do you calculate it (how many do you need)?

• need an F ratio or separate F statistic for both factors

What is the F statistics for the Interactive hypothesis

2 stage process for calculating F stats

• describe each step breifly

• 1: separate variances from between and within treatments

• 2: separate the variances from between into three separate components to then calculate each F ratio

Two-Factor ANOVA notation

• k

• n

• N

• T, T col, T row

• G

• k = number of groups

• n = number of individuals in one particular group

• N is the total number of individuals

• T the total sum of scores in each group: T col is the tot sum of scores in a particular column, T row is the tot sum of scores in a particular row

Stage one calculations

• Six steps

• M, T, and SS for each cell

• calculate T for each row and column

• calculate N, G, and sum(X²)

• calculate for SS tot, SS between and SS within treatments

• df tot = N - 1, df between = numbers of cells - 1, df within = tot df of each cell

• Calculate for the MS within treatment

  • Denominator will be the same for all three F ratios going forward

Stage 2 calculations

• Factor A variance and df

• Factor B variance and df

• Interaction variance and df

• How to calculate for the df for all

• SS A = sum T² row/n row - G²/N

• df A = rows - 1

• SS B = sum T² column/n column - G²/N

• df B = columns - 1

• SS AXB = SS between - SS A - SS B

• df A X B = df between - df A - df B

• Look at all F ratios

How to predict if it will be an interaction for the study?

• if the values in the table cross over

Measuring effect sizes, don’t have to remember