Factorial ANOVA
What are the sources of variance in MULTIfactorial ANOVA?
-Some variance attributable to the IV's (their main effects) -Some variance attributable to their interaction effects (e.g. Beatles produced music together they never could have produced alone, whole is greater/different than the sum of its parts) -Error variance (experimental error, individual differences (except IDs are also partitioned in repeated/within ANOVA)
How are simple effects used to explore significant interactions for between participants' designs?
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**Have idea in head of Bonferroni corrected alpha values: 0.5/2= 0.5/3= 0.5/4=
0.5/2= 0.5/3= 0.5/4=
What is ANCOVA and when is it used?
Analysis of covariance=
**How to work out what kind of design an ANOVA has! =Need to work on, working out ANOVA designs.
BETWEEN:When different subjects are used for the levels of a factor, the factor is called a between-subjects factor or a between-subjects variable. The term "between subjects" reflects the fact that comparisons are between different groups of subjects. All factors between subjects= a between subjects design WITHIN:When the same subjects are used for the levels of a factor, the factor is called a within-subjects factor or a within-subjects variable. Within-subjects variables are sometimes referred to as repeated-measures variables since there are repeated measurements of the same subjects. Each subject tested within each condition REPEATED MEASURES AND "WITHIN" ARE SYNONYMOUS= SAY "REPEATED MEASURES", OR SAY BOTH! (purely within participants could also potentially be quasi-experimental(?) so say repeated to avoid ambiguity REPEATED MEASURES: same participants perform every condition All factors within subjects= a within subjects design MIXED: At least one factor between, at least one factor within= mixed design Practice these: http://onlinestatbook.com/2/analysis_of_variance/anova_designs.html
What three kinds of graphs can you use to check if the data you are planning to do multifactorial ANOVA analysis on is normally distributed?
Boxplots Stem and leaf plots or histograms (grouped frequency plot)
Family wise error rate
Controlling family wise error: There is a general consensus that the three principal effects (i.e., the two main effects and the interaction) are planned tests and do not require error correction. They are evaluated at a conventional significance level, such as α= .05. The more IVs you have in a multifactorial ANOVA, the greater the chance of type 1 error, e.g. if four IVs (A,B,C,D), you are testing 15 different effects against their corresponding null hypotheses (main effects of A,B,C,D plus 11 interactions between these IVs) In doing this you are drastically increasing the family wise error rare, and thus increasing the probability of making a type 1 error The more simple effects you calculate, the higher your family-wise error will be: should be selective in your simple effects calculations: Your hypothesis(prediction about how cell means will differ (based on previous lit)) should guide which simple effects calculations you conduct (a priori comparisons, but using t-tests!) If your comparisons are post hoc (or a priori, but using t tests and making many comparisons) then use Bonferroni's correction!
If you find a significant interaction in ANOVA, what should you do?
Explore that interaction further! How should you do that? Simple effect tests: corrected T-tests!
What are the benefits of using factorial ANOVA designs? (4)
Factorial designs can test the effect of 2 or more FACTORS on 1 DV at the same time. Enables us to find out if there is an INTERACTION between the two factors! 1)Two factor design moves one step closer to reality- testing the effects of two IVs on a DV simultaneously. [Manipulation of a single IV with all other variables held constant is criticised for its extreme separation from reality- in life we are affected by several influences together at any one time]. 2) By manipulating more than one factor in an experiment, we get to see the ways in which one factor INTERACTS with another E.g. comparing effects of caffeine on driving performance, against placebo, after five hours' sleep and after none. (2X2 DESIGN) 3) Use of two factors is often demanded by the research question, but often simply convenient- two experiments in one, plus the interaction effects. E.g. Coffee on driving, sleep on driving, and interaction between the two. 4) Statistically: type 1 errors are more efficiently accounted for than they would be by running several experiments each with a .05 probability of type 1 error. Overall, can help us discover more general, nuanced principles e.g. After five hours sleep caffeine improved performance, compared with the placebo for a full two hours of driving, whereas after no sleep, caffeine only improved performance for the first 30 minutes, but performance deteriorated markedly after that. Can draw out more useful general/applicable conclusions from this/ factorial designs!
** Rounding decimals.
First, find the digit in the tenths place. This is the place you want to round: 1.-0-482 Now look one place to the right: 1.0-4-82 4 is less than 5, so you will round down. Remove all the digits to the right of the tenths place: 1.0482 → 1.0 1.04 is too low to round to 1.1 (1DP): forget the rest! You start rounding from one place before the place you need (to round to) 1.0 is equivalent to 1. https://www.ixl.com/math/grade-5/round-decimals
What kind of follow-up tests should you use for within-subjects factorial ANOVA?
For follow up testing: WITHIN participants; use RELATED T-tests rather than independent T tests
How to abbreviate factorial ANOVA designs(and how to refer to them: x factors = x Way Anova)
FxF 3x3= two factors, three levels of each factor 2x3x2 ANOVA 3 factors, two levels, three levels Two way ANOVA= compares effects of two factors on 1 DV - No matter how many levels there are in each factor, we will (at most) just find a main effect for each factor, and the interaction between them Three way ANOVA= compares effects of three factors on 1 DV See p331 dancey
Commenting on confidence intervals
How much overlap is there between CISCO?
ANOVA analysis of two within participants factors
If more than two conditions in any of the within-factors IVs, have to check whether the assumption of sphericity has been violated: Mauchley's test of sphericity Sphericity= an assumption of within participant's ANOVA
How can you tell from looking at a graph if there is an interaction effect (see ppt and p 334 Dancey for egs) ?
If the lines are not parallel (especially if they cross each other!) that indicates an interaction! (but not whether or not the interaction is significant!) If the lines are parallel, that only indicates a main effect; that one IV facilitated/enabled greater DV values than the other
What are the three main types of multifactorial ANOVA?
Independent/BETWEEN Groups ANOVA= the analysis of unrelated designs, a design where all factors contain independent samples, Repeated Measures/WITHIN Groups ANOVA= Only related factors are involved (repeated measures or matched pairs) Mixed Design ANOVA= ANOVA analysis where both unrelated and related factors are involved
**How would you find the total number of conditions/levels in a factorial design? E.g. 2x3
It's factorial! (multiplication, multiply it! 2x3=6 conditions/cell means or, draw the table to help visualise it
Give all the sources of variance for: two way between ANOVA, factors A and B and three way between ANOVA, factors A,B,C
Main effect A Main Effect B Interaction A and B Error Main effect A Main effect B Main effect C Interaction A and B Interaction A and C Interaction B and C Interaction ABC Error (IDs. experimental errror) (see pie chart p329 Dancey!)
Define main effect
Main effect= the overall effects of each of the IVs (factors) on the DV For there to be a significant main effect (as opposed to that factor being involved in an interaction effect), it would have to have a global effect; influence grades/scores across ALL conditions
Explain marginal means
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 (see diagram!) What is their importance?
What is MANOVA and when is it used?
Non-parametric ANOVA equivalent=
Split-plot/Mixed ANOVA design
One within-participants factor, one between-participants factor. Eg. P's revised either in exam hall or sitting room (within participants), but everyone sat an exam in the exam hall, and an exam in the sitting room (between participants) In split-plot design, have one error term for within-participants part of the analysis, and one error term for the between participants part of the analysis. -The interaction between factors is in the within within-participants print-out, as it has a within-participants component, i.e. one of the interaction terms (exam conditions) is within-participants. Simple effects analyses: because both between and within participants variables, will need to use both the independent and related t-tests. To run the two related t-tests comparing the two expan conditions with each of the revision conditions, you will need to split the file.
When is partial eta squared useful, and when is d useful?
Partial eta squared= global measure of magnitude of effect D= magnitude of difference between two conditions
How does multifactorial ANOVA differ with within-participant factors?
Same asssumptions as for between-subjects ANOVA, plus assumption of sphericty (ONLY IF MORE THAN TWO CONDITIONS FOR A WITHIN-SUBJECTS IV) Constant source of error due to having same participants in different conditions is subtracted from the error variance, as a result reducing the error term ("partialling out"; we do this because one assumption of the statistical tests (not earlier covered) is that the data from each condition should be independent of all other conditions. In order to do this, the consistent effects of participants across all conditions (e.g. those who tend to perform well will do so over all conditions) are removed statistically, so that the conditions will effectively be independent of each other and analysis can continue Extra table: "Within Subjects Effects": because same participants in each condition, we are able to calculate the degree of error associated with each effect (main/interaction) , whereas in the completely between-participants analysis we are able to calculate only the overall amount of error. Can test each main effect and interaction against its own error term. In a within-participants design, ANOVA analyses each main effect as if it were a one-way ANOVA. Calculates overall amount of variability associated with the main effect (including all sources of variance, including error.) ANOVA then subtracts from this overall variance the amount of variability that can be attributed to the main effect, and the amount of variability that can be attributed to the constant effect of participants. The error term= the remaining variability; the variance that is unaccounted for. For follow up testing: WITHIN participants; use RELATED T-tests rather than independent T tests
Define "interaction effect"
Significant effect, where effect of one factor is different across levels of another factor When the effect of one factor changes across the different levels of another factor With an interaction, one variable behaves differently in the context of each variable
Define simple effect (sometimes called simple main effects) How would you calculate them?
THE SIMPLE PICTURE OF JUST MAIN EFFECTS If you do get a significant interaction, you can find out what is happening in each of your conditions by analysing the simple effects. Where you find a difference between simple effects, you have spotted an interaction! **Simple effects show the difference between any 2 conditions of 1 IV in one of the conditions of the other IV *** Simple effect analyses are equivalent to t-tests, but involve the calculation of F values, and you can get SPSS to calculate them for you, but this is v complex, so instead, use t-tests!! Simple effects= a comparison of two cell means of 1 IV, within one condition of another IV (see diag!) The more simple effects you calculate, the higher your family wise errors will be. You should therefore be selective in your simple effects calculations: Your hypothesis(prediction about how cell means will differ (based on previous lit)) should guide which simple effects calculations you conduct (a priori comparisons, but using t-tests!) If your comparisons are post hoc (or a priori, but using t tests and making many comparisons) then use Bonferroni's correction!
Name an alternative to one-way ANOVA for independent groups (where ANOVA assumptions not met) ? Name an alternative to Repeated measures ANOVA? What would be suitable follow-up tests to use instead of T-tests for each respectively?
To be used when your data does not meet ANOVA assumptions (e.g. not normally distributed and small sample) Not as powerful as parametric tests Based on 'ranked' data rather than means, so not skewed by extreme scores Kruskal-Wallis: alternative to one-way ANOVA for independent groups Friedman's ANOVA: alterative to RM ANOVA Use pairwise comparisons to follow up if significant (reducing alpha level): Mann-Whitney Tests - independent groups Wilcoxon Test - related groups
What are the sources of variance in two way ANOVA? (main effects and interactions)
Variance due to Factor 1 Variance due to Factor 2 Variance due to the interaction between these factors Error variance ('within-groups' variance) Need to report the F value (with associated d of f and p value) for both of the "main effects" and the "interaction"
What are the assumptions of multifactorial ANOVA? (remember, it is a parametric test)
normal distribution homogeneity of variance (one way of telling = similar SD values, e.g. 3.97, 3.73, 3.09, 4.22) If more than two conditions in any of the within-factors IVs, have to check whether the assumption of sphericity has been violated: Mauchley's test of sphericity Sphericity= an assumption of within participant's ANOVA If Mauchley's test of sphericity 's sig is 0.5 or lower, then this means the assumption of sphericity is violated, use Greenhouse Geisser
Have a go at describing each of the following designs (in terms of IVs) 6x2 ANOVA 3x3x3 ANOVA 4x2x4x2 ANOVA 2x2x2x2x2 ANOVA
one IV with six conditions, one IV with two conditions three IVs, each with three conditions four IVs, two with four conditions and two with two conditions five IVs, each with two conditions IVs/factors levels/conditions