2-way independent ANOVA - factorial ANOVA Flashcards
Factorial ANOVAs
- are used to test for differences when we have more than one IV
- each IV has 2 or more levels
the 3 broad factorial ANOVA designs
- all IVs are between-subjects (independent)
- all IVs are within-subjects (repeated measures)
- a mixture of between-subjects and within-subjects IVs (mixed)
factor
IV and factor are interchangeable
- ANOVAs with one or more IV are called factorial ANOVAs
factorial ANOVAs can include:
- 2-way independent ANOVA, 2-way repeated measures ANOVA, 3-way repeated measures, 2-way mixed
- number denotes the number of IVs/factors
- SPSS refers to IVs as factor
Ivs/factors with at least 2 levels
e.g. 2x2 ANOVA: 2 IVs/factors, each with 2 levels
e.g. 4x2x2 - 3 Ivs/ one with 4 levels, one with 2, another with 2
numbers represent levels of each factor
(main effect number of levels first, secondary effect levels after)
note
we will only be looking at designs with 2 Ivs
- for interpreting results refer back to plot of data to help
Factorial 2*2 independent ANOVA: what are the IVs and levels
- gender: 2 levels: female and male
- texture: 2 levels: chunk and tablet
- every level of IV found in combination with every level of other IV
4 conditions
- each participant only takes part in one condition
what does a 2-way factorial ANOVA tell us… INTERACTIVE + MAIN EFFECTS
- gender effect
- texture effect
… there are our MAIN EFFECTS
does gender effects depend on texture…. INTERACTION
whenever you look at a factorial ANOVA (as you will never have more than 2 IVs) always consider these 3 points
what does a factorial ANOVA control
familywise error rate (type I error)
- also tells us about interaction effects
interaction effect
the dependency of one factor on another factor
factorial ANOVA hypothesis testing
- calculate F for each effect (two main effects AND interaction)
- determine whether we can reject the null for each effect (no difference)
examples of hypothesis
2-way independent ANOVA: partitioning variance
variance between I levels: IV1, IV2, interaction
variance within IV levels: error (individual differences & experimental error)
F ratio
(variance due to error alone is always the same)
- 3 F values
3-way independent ANOVA : partitioning variance
don’t need to memorize
- harder to find significant results
- rate of type II errors increases
interaction effects - what does it mean?
- combined effects of multiple IVs on the DV
- significant interaction indicates the effect of manipulating one IV depends on level of other IV
e.g. gender differences in enjoyment of chocolate depend on the texture of the chocolate…. women and me report same satisfaction when the chocolate is a chunk, but women report higher satisfaction when chocolate is a tablet
marginal means
- average of the 2 means for each level of IV
- show main effects
- no main effects if marginal means are close/overlap
(picture only shows marginal mean for gender, do the same for texture )
cell means
to show interaction between IV levels
- you can have an interaction without having main effects
(- if lines are parallel no interaction)(if lines are not parallel there is an interaction )
2-way independent ANOVA example
factorial ANOVA specific rules
- first IV referred to is ‘main IV’
- second IV referred to is ‘secondary IV’
e.g.’also… whether any influence of sport exercise is DEPENDENT on gender’(gender is secondary IV)
ANOVA: means for factorial independent ANOVA
cell means: mean scores for each condition
marginal means: means scores for single IV levels (ignoring other IV)
DATA INPUT SPSS: 2-way independent ANOVA
- every participant gets their own row
- for independent groups you need separate columns for each IV (not level! this is specified in the value in the column) and DV
- assign values for each level of IV
e.g. 1 column for sport experience (main effect), 1 column for gender (secondary IV), 1 column for positivity rating (DV)
plotting data
- main IV on horizontal axis
- secondary IV in defined clusters
- DV on vertical axis
assumptions independent 2-way ANOVA
- normality: the DV should be normally distributed
- homogeneity of variance: the variance in the DV, within each condition, should be (reasonably) equivalent
- equivalent sample size: sample size within each condition should be roughly equal
- independence of observations: scores within each condition should be independent
how do we check homogeneity of variance
Levene’s test - assumption of equal variances
- tests the null hypothesis and assumes homogeneity
- p > .05 we don’t need a correction, not violated
- no correction if violated for two-way independent ANOVA (welch for one way)!
- in our course we won’t ever have a violation for Levene’s
note on parametric equivalent for factorial ANOVA
there is no alternative
- but factorial ANOVAs are robust enough that we don’t need to
- if data violates assumptions seriously we can just simplify the design
2-way independent ANOVA output
d.f. for model: effect/interaction row
d.f. for residual : error row
…if p < .05 F is significant (effect)
e.g. F(2,54) = 39.51, p < .001 (sport experience, main effect)
partial eta squared - effect size measure in SPSS
- check test of between-subjects effects table (even if independent)
- only takes into account the variance from one IV at a time
reported as .338, = 34% of variance we see in people’s positivity ratings is explained by the interation of the way they experience sport and their gender
eta squared vs partial eta squared
eta squared: proportion of total variance attributable to the factor
- sum of squares for model / sum of squares total
partial eta squared: proportion of total variance attributable to the factor, partially (excluding) variance due to other factors (see example in photo)
- sum of squares for model / sum of squares for model + sum of squares for residual
post hoc test for 2-way independent ANOVA
- if main effect (e.g. sport experience) is significant, we reject the null
- now we need to find out which level the effect of type of sport experience on positivity lies
- in MULTIPLE COMPARISONS TABLE
example in photo
- if p < .05 it is a significant effect
- solo vs. watch, no significant effect
- you can refer to bar chart to see which level is higher then
e.g. team is significantly higher than solo
when are post hoc tests relevant
- when main effect of IV is significant
- and when IV has MORE than 2 levels ( we wouldn’t look at them for gender)
note: cohen’s d not reported with factorial designs
interaction output
if p < .05, Iv is dependent on Iv level
e.g. sport experience is dependent on gender
simple effects on 2-way independent ANOVA
simple effects follow on from interactions that are significant
the effect of an IV at a single level of another IV (comparison of cell means (conditions))
- to determine whether simple effects are significant, we need to conduct t-test between individual cell means (only when interaction is significant)
simple effects on SPSS
- you need to run individual t-tests
- correct for multiple comparisons
- Bonferroni correction(*): dived 0.05 (alpha level) by the number of comparisons (controls for type 1 errors)
- don’t forget to check Levene’s
- cohen’s d alongside this
-* important in write up to calculate this
our by hand cohen’s d formula
write up factorial ANOVA: design
- IV: its levels, sunject design, steps taken to eliminate confounds (e.g. counterbalancing, random allocation)
- DV
factorial ANOVA results: step one: descriptive statistics in the table
- measure of central tendency: mean
- measure of spread: standard deviation
- interval estimate: CIs for upper and lower limit
factorial ANOVA: step 2: results write up + post hoc
- reference descriptive statistics table
- ANOVA test used
- main effect and if its significant and reported as (primary IV): F(dfM, dfR) = _.__, p = .___, partial eta sqaured.
- post hoc if significant and if IV has more than 2 levels, Tukey HSD for between, Bonferroni for within
- p-values for each post hoc, if significant clarify direction of differences between IVs, if not just state no significance
- then report for secondry IV in same format, clarify direction if significant, but no need for post hoc tests either way
factorial ANOVA: results write up: step 3: interaction
- test statistic and clarification of significance
Format: F(dfM, dfR) = _.__, p = .___, partial eta squared. - state how many INDEPENDENT t-test were conducted to investigate the interaction and why e.g. to compare positivity ratings across the three different sport experiences (primary IV) separately for each gender (secondary IV)
- based on these n pairwise comparisons of simple effects, we applied bonferonni correction criterion for significance of p < X(remember to check levene’s - read from equal variance not assumed if violated) (significant if p < (less than) alpha from Bonferonni correction)
- report results of each simple effect as Format: (t(df) = _.__, p = .___, d = .)
- report direction if significant, attempt to group results e.g. where effect of main IV is consistent for both levels of secondary IV, those where main effect IV was different from secondary IV, and those where no effect was found
discussion factorial ANOVA
- summary to main IV effect in research question
- but mention how secondary effect (if dependent) might change this / how main IV dependent on this
number of t-tests conducted
number of levels of primary IV x number of levels of secondary IV
- as different p’s in different groups
