WEEK 6: Two-Way ANOVA Flashcards
Factorial ANOVA/ 2-Way ANOVA lecture
This week:
> From 1 way ANOVA to 2 way ANOVA
2-way between participants factorial ANOVA: a conceptual approach
Following up significant ANOVA results
There is no non-parametric version of the factorial ANOVA
2-way – 2 factors, between participants, conceptual approach in this lecture
Learning Objectives:
- Understand and be able to explain when a factorial ANOVA would be used
- Be able to report the results of a 2x2 factorial ANOVA
- Understand and be able to explain main effects in a factorial ANOVA
- Understand and be able to explain the interaction in a factorial ANOVA
From the beginning RECAP
> In t-tests we have two conditions i.g. experimental condition and control, and one DV (thing we are measuring)
> In a one-way ANOVA we have 1 factor with 3 conditions, and still one DV/ thing we are measuring
e. g. teaching, on three levels (informal, semi structured, structured) and we measure academic outcomes
- In a one-way ANOVA we could also have a factor with 5 levels…
e. g. the factor is drunkenness and the levels are: sober, tipsy, drunk, wasted and kaput
- DV – A&E admissions at these levels
So what is a Factorial ANOVA/ 2 Way ANOVA
> Factorial ANOVA
What happens if you’re drunk AND you experience different levels of teaching
So… there are two factors: teaching style and drunkenness
- There are levels for both of the factors
There is still one DV – academic outcomes
Example 2: What if you’re drunk and on drugs whilst driving
Factors - Drink, Drugs
Levels - drink – alcohol vs water
- drugs – caffeine vs placebo
There are different ppts in each group, nobody has had experience of more than one condition – between ppts design, need a lot of participants
What could we predict in this exp.?
Guys on the placebo and water will probably do better in a driving simulator than those on alcohol and caffeine
How a factorial ANOVA works…
MAIN EFFECTS
marginal means
So there are 2 factors e.g. drinking and drug intake, each with different levels e.g. drunkenness and caffeine or placebo, and a common DV.
MAIN EFFECTS
As well as looking at each of the different groups of ppts, we also look at the ppts as a whole group within each level of one of our factors e.g. the people in the alcohol condition as a whole, ignoring the fact that some of them have had caffeine and some have had a placebo… called the main effect (of drink)
- Would do the same for the drug level…
This would be the main effect of drug
The marginal means are the means of each (group 1 and 2 pictured)
to help visualise this imagine a table with the levels of the factors in against each other. We take the horizontal groups and the vertical groups and calculate means (these means are called MARGINAL MEANS). So everyone in alcohol, water, caffeine and placebo. To find the main effects of each substance.
Interpreting the outcomes
Can use bar graphs or multiple-line graphs to interpret the differences between the differences between groups in instances of errors made
We can think about our main effects in many different ways:
Bar graph
Line graph
Actual values in a table
It’s your preference, whatever works for you, the line graph is usual for interpreting the whole lot – marginal means and the means for each of the groups (the values taken from the ends of the lines).
Using line graphs to interpret main effects
We can also use line graphs to see where differences might lie when looking at main effects
The guys that had some caffeine performed better overall than the guys who had a placebo.
In order to calculate the main effects, we need to calculate the marginal mean for our groups within two levels of one factor
If we ignore the fact that some ppts drank alcohol and others water, and we wanted to look at the mean score of the caffeine group and the placebo group we would need to… take the value at either end of the line, add then up, divide by two, in order to calculate the mean value for the particular group – would end up in the middle of the line
So the caffeine group would make on average 15 errors whereas the placebo group made about 22 errors. This is our ‘main effect’ of DRUG.
What is a marginal mean?
(In a design with two factors) the marginal means for one factor are the means for that factor averaged across all levels of the other factor. E.g. with the drink/ drug example, the marginal mean for drink would be the means of both the drug levels added up and divided.
How a factorial ANOVA works…
INTERACTION
This is where we are looking for a difference between the groups 1, 2, 3, 4 (alcohol + caffeine, alcohol + placebo, water + caffeine, water + placebo). Again, the means are calculated. We group the things we are comparing by category, so its between 1 and 2 and 1 and 3 etc (the lines on the table go across and up, not diagonal). –>
Looking for a difference between the factor and the level
“The effect of Alcohol on Caffeine compared to Placebo will be different to the effect of Water on Caffeine compared to a Placebo”
Interaction effects –
Is there a difference between the difference between these groups?
Is that an illustration of a different pattern of response in the data depending on the factor and the level?
(Crossing over lines) This pattern shows evidence of an interaction because there is a different pattern in the data for the 2 levels of factor one, depending on the 2 levels of factor 2. Interaction effects occur when the effect of one variable depends on the value of another variable e.g as one thing changes so does the other.
(Lines would cross over at some point) You can get all different kinds of interaction outcomes that look a bit different. Here there is an interaction, theres a much larger difference between alcohol and water in the caffeine level of the drug condition than in the placebo level. Imagining that the lines carry on, at some point they would cross.
(Lines would never cross over)
In this example however, there is no interaction (There may be a main effect, as there’s a difference between the water and the alcohol levels) but no inter as the pattern of response between caffeine and placebo conditions is the same – no difference between the differences, no crossing over.
Simple effects tests
IF THERE IS A SIGNIFICANT INTERACTION YOU NEED TO DO FOLLOW UP TESTS…
These follow up tests are called simple effects tests. They are a specific type of post-hoc test. Follow up tests for a main effect with 2 levels are not required (see next lecture), if there is a main effect with three levels, then simple effects tests are needed.
So, again, we are looking for differences between the differences between the points at the end of the lines - interested in exploring the interactions between the factors.
Need to do Bonferroni corrected t-tests to work out whats going on with the interactions.
Different kinds of factorial ANOVAs
Factorial ANOVAs can have lots of levels and lots of factors, today we looked at a 2x2. But they come in all shapes and sizes.
Highly unlikely to see a 432*2 ANOVA
- Logistical nightmare
- Need lots of participants
- Very expensive
- Difficult to explain results
Reporting a factorial ANOVA
Outline
- Main effect of Drink (Factor #1)
- Report the ‘marginal means’
- F(df1,df2) = [F value], p = [p value]
- Main effect of Drug (Factor #2)
- Report the ‘marginal means’
- F(df1,df2) = [F value], p = [p value]
- Interaction of Factors
- F(df1,df2) = [F value], p = [p value]
> If the interaction is significant do the t tests and adjust the p value cut off using a Bonferroni Correction
If the interaction is not significant don’t do the t tests
If you have a significant main effect in a 2x2 ANOVA there is no need for post hoc tests
Reporting a factorial ANOVA
Example write up
A 2x2 between participants ANOVA with 1 Factor of [Factor #1] ([Level 1 & Level 2]) and a second factor of [Factor #2] ([Level 1 & Level 2]) was conducted.
The results showed that there was a significant main effect of [Factor #1] on [DV] (F(df1, df2) = [F value], p = [p value]). Overall [Factor #1 Level 1] had a higher [DV] (Mean = [Factor #1 Level 1 marginal mean value]) than [Factor #1 Level 2] (Mean = [Factor #1 Level 2 marginal mean value]).
There was also significant main effect of [Factor #2] on [DV] (F(df1, df2) = [F value], p = [p value]). Overall [Factor #2 Level 1] had a higher [DV] (Mean = [Factor #2 Level 1 marginal mean value]) than [Factor #2 Level 2] (Mean = [Factor #2 Level 2 marginal mean value]).
In addition there was a significant interaction between [Factor #1 and Factor #2] on [DV] (F(df1, df2) = [F value], p = [p value]).
Bonferroni corrected post-hoc t-tests showed that [Factor 1 level 2 with Factor 2 level 1] participants were less socially anxious (M = ____, SD = _____) compared to [Factor 1 level 2 with Factor 2 level 2] participants (M = ___, SD = _____; t(df) = [t value], p = [p value]), whereas [Factor 1 level 1 with Factor 2 level 2] demonstrated less social anxiety (M = _____, SD = ____) than [Factor 1 level 1 with Factor 2 level 1] (M = _____, SD = ______; t(df) = [t value], p = [p value]).
The contrast between [Factor 2 level 2 with Factor 1 level 1] and [Factor 2 level 2 with Factor 1 level 2] was also significant (t(df) = [t value], p = [p value]) and demonstrated that [Factor 2 level 2 with Factor 1 level 1] are less socially anxious than [Factor 2 level 2 with Factor 1 level 2]. However, the contrast between [Factor 2 level 1 with Factor 1 level 1 and level 2] was not significant (t(df) = [t value], p = [p value]) .
This suggests that _________________________
ONCE YOU HAVE REPORTED THE MEANS THERE’S NO NEED TO DO IT AGAIN - PUT THEM IN A TABLE FOR EASE
