one-way ANOVA: 2 groups

IV vs DV

IV vs DV

IV: independent variable/factor/treatment variable

DV: dependent variable/value/score

one-way ANOVA within-subjects design

the same people that experience all the levels/factors

single-factor (one-way) designs

involve a singe independent variable with two or more levels

• one-way independent groups design

• one-way design with repeated measures

factorial designs

involve more than one IV with two or more levels

• e.g., two-way independent-groups designs

• experimental design with 2 factors w/ 2 levels each= 2X2 factorial design—means that there are 4 unique combinations

how do you calculate the number of unique combinations when you have more than one factor?

multiply the number of levels of each individual factor

• e.g., two-way ANOVA w/ factor 1 having 4 levels and factor 2 having 3 levels

  • 4X3= 12 unique combinations (# of cells)

marginal means

the mean of a condition/column on a vertical

row mean

the mean of a condition/row on a horizontal level

what is the purpose of one-way ANOVA

to test whether the means of k (>2) populations significantly differ

what is k?

it is the number of levels a factor has

• i.e., the number of groups

example of null and alternative hypothesis for a one-way ANOVA

H0: u1=u2=….=uk

H1: not all u’s are the same (at least one of the means is different)

identify the IV and DV from the following description:

suppose additional readings were assigned thinking that they will increase interest in stats. people are randomly assigned to one of three experimental conditions that differ in terms of reading material—Stat 1+2: one of two stats books intended for the general public; NoStat: control condition

IV: book assigned—there are three levels: Stat 1, Stat 2+ NoStat

DV: self-reported interest in taking more stats courses

prior requirements/assumptions of one-way ANOVA

• the population distribution of the DV is normal within each group

• homogeneity of variance assumption

• independence of observations (equal chance of being included in the study and no clusters in the samples)

homogeneity of variance assumption

the variance of the population distributions are equal for each group

how can we turn a one-way anova into separate t-tests?

separate the null hypothesis into individual ones and do a t-test for each one

• e.g., H0: u1=u2=u3

  • H01: u1=u2

  • H02: u1=u3

  • H03: u2=u3

• this amounts to 3 t-tests

familywise Type 1 error rate

it is defined as the probability of making at least one Type 1 error in the family of tests if the null hypotheses are true.

C

the number of tests

if you have a family of t-tests where c=3, what would the equations for Pr (no familywise errors) and aFW be

Pr= (1-a)(1-a)(1-a)= (1-a)c

aFW= 1–Pr= 1– (1–a)c

• e.g., aFW= 1–(1–.05)3= 1–.857= .143

  • .143 is the familywise Type 1 error rate

  • if the null hypothesis is true, there is a .143 chance of having one false positive in the family of 3 t tests vs one-way ANOVA which would keep it at .05

overall F-test function

answering the main question of if the H0 is false

what to do if the F-test indicates the null hypothesis as false?

post-hoc tests are used to look at pairs of groups and find which group(s) specifically is different from the others

basic concepts of ANOVA

• divides the observed variance of the DV into parts resulting that can be explained by group membership/the model and parts that aren’t accounted for (aka residual variance)

• assesses the relative magnitude of the different parts of variance

• examines whether a particular part of the variance is greater than expectation under the H0

the two sources of variance

• the model (MSm)

• the residual variance (MSr)

  • MS= “mean” of sum of squared deviations

type of variance explained by the model (MSm)

variance between groups due to the IV or different treatments/levels of a factor

• MSb: between-group mean square/variance

type of variance explained by the residual (MSr)

within each group, there’s some random variation in the scores for the subjects

• MSw: within-group mean square/variance

F ratio

stat calculated to assess relative magnitude of the two different parts of the variance

formula for the F-stat/ratio

F = MSm/MSr

how do group means affect MSm, MSr and F-stat?

if group means differ from each other, MSm>MSr= F tends to be large

if Fstat>Fcrit: H0 is rejected

on what scales does the F distribution vary in shape?

dfM: between group/model df

dfR: within group/residual df

how is the F distribution skewed?

it is right-skewed, meaning the left side is high and it sloped down the more you go to the left

finding F critical/value on the table

use your alpha, given dfM and dfR

• dfM: look on top

• dfR: look on the side

• find square of scores that lines up with your dfM and dfR and check the Fcrit that aligns with your alpha

relationship between p-value and a level

p-value<a, null hypothesis can be rejected

finding your F ratio using a t value (and vice versa)

e.g., number of groups is 2—F= t²

note about eta squared

+vely biased because it overestimates the amount of variance explained in the DV by the IVs

note about omega squared

it is unbiased and is reported even if it is -ve

what is considered small, medium and large for omega squared

small= .01; medium= .06; large= .14

ANOVA report template (APA style)

1. 1–2 sentence overview of analyses

   1. IV and DV stated conceptually

2. F-test results

3. patter of mean differences

   1. if significant differences were found

4. conceptual conclusion

what is included when reporting the F-test results?

the associated df, statistic and p-value is included, as well as effect size measures

• e.g., F(1,8)= 16.774, p<.05, w²= .61.