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Stages of Analysis for factorial designs
- –StageI: Descriptive statistics and Assumptions
- –StageII: Global test for effect (ANOVA)
- –StageIII: Size of Effect (Partial Eta2)
- –StageIV: Nature of Effect
What is a factorial design?
- –A design with two or more factors (IVs)
- -this provides researchers with the flexibility to develop studies that address scientific questions questions that could not be answered by a single design using a single factor.
- –Every level of every factor is paired with every level of every other factor
Two Experiment Single IV Design:
How are Independent variables tested?
- –Can’t explore interactions between variables
- –Requires many participants
What is a factor?
An independent variable
What is a level
A condition of a Factor
What is a cell?
A combination of levels from multiple factors
What is the main effect of a factorial design?
- •The effect of a single factor
- •One per IV
What is the interaction effect of a factorial design
- •The effect of one factor may change at different levels of your other factors
- •One per combination of IV
- •The effect of an IV on the DV averaged across the other IV
Are all effects independent Of each other in a Factorial Design?
What do Non-Parallel lines on a factoral design indicate?
a potential interaction
What does it mean if lines are parallel in a factorial design?
What does it mean if Lines Cross for an interaction in a factorial design?
A has effect at only one level of B. Main effects are to be suspected but not 100%
what does it mean if lines cross for a magnitude interaction in a factorial design?
Effect of A is larger at some levels of B. Main Effects are generally correct
what does is mean if lines cross for a Cross-over Interaction in a factorial design?
Effect of A reverses based on B. Can’t trust main effects!
what is a Factorial Design (Two-Way)?
- 2 x 2
- •Two Factors
- •Two levels of each factor
- •Four cells
What all can you find in a factorial design (two-way)?
- -Main Effect of A
- -Main Effect of B
- -Interaction of A x B
What all can you find in a factorial design (two-way) 2 x 3
- •Two Factors
- •Two levels of factor A
- •Three levels of factor B
- •Six cells
- -main effect of A, main effect of B, interaction of AxB
What is a Factorial Design (Three-Way)
- –2 x 2 x 2
- •Three Factors
- –Two levels of factor A
- –Two levels of factor B
- –Two levels of factor C
- •Eight total cells
what interactions can be made in a factorial design (three-way)
- -two-way interactions:
- A x B
- B x C
- A x C
- -Three way interactions:
- A x B x C
Analysis Strategy: Factorial ANOVA
- •Stage I
- –Descriptive statistics for all main effects and interactions
- •Assumptions (Independence, Normality, HOV)
- •Stage II
- –Separate global tests of each main effect and interaction
- •Stage III
- –Nature of the effect for each main effect and interaction
- •Stage IV
- –Size of effect for each main effect and interaction
- •Partial η2
Computing Factorial ANOVA (2 x 2)
- •Partition Degrees of Freedom
- •Construct Mean Squares
- •Construct F
What do you do if you have a significant F-test?
- –Post hoc/aPriori testing (Stage III)
- –Effect size (Stage IV)
What do you do if you do not get a significant F-test?
–Stop: there is no effect to explore
How do you find the nature of the effect for factorial designs for 2 conditions?
Visual Examination of means
How do you find the nature of the effect for factorial designs for more than 2 conditions?
Post Hoc or Planned comparisons
Do interactions generally trump main effects?
How do you find the nature of the effect for two-way interactions?
- –Through Simple Main Effects:
- •The effect of one factor at a single level of another factor
- –Effect of Level of Sound (B) in the random condition (A1)
- –Effect of Level of Sound (B) in the Fixed condition (A2)
- •t-tests or ANOVA to confirm differences
- •Multiple simple main effects exist- one for each condition of your IVs
How do you find the size of the effect?
- •Partial η2(Eta Squared)
- •Full η2
- •ω2 (Omega Squared)
What are the Types of Interactions & Analyses
- •2 x 2
- –Two IV’s, Two conditions each
- •2 x 3
- –Two IV’s, Two conditions in A and Three Conditions in B
- •2 x 2 x 2
- –Three IV’s, two conditions in each
What are the simple main effects for a 2 x 2 interaction
–The effect of one IV at each level of the other IV
How can you control for experimentwise type I error of ANOVA?
What is a repeated-measures design?
- -a design in which a single sample of individuals is measured more than once on the same dependent variable.
- -Compares participant to themselves across time rather than to other participants (independent measures)
Advantages of repeated measure design
- -you can Measure effect of time (development) on participants. Allows for real effects to be shown within that individual
- -Conserve Participants. Dont need as many overall bc you are using the same people across all levels
- -Test relationships where a control group may be unethical (pre &post test of learned material throughout a course)
- -Control for Individual Differences. Allows us to identify and remove variability due to participant from the analysis.
What are advantages of repeated measure design compared to Independent Measures Designs?
- –Requires fewer participants
- –Study change across time (can demonstrate development rather than cohort effects)
- –More Statistical Power due to control for Individual Differences
What are disadvantages of repeated measure design compared to Independent Measures Designs
- –Order Effects due to multiple observations (Practice, Fatigue, Carryover)
- –Change may be due to Time rather than IV
- -Carry over effect (ex. how liberal is obama, clinton. may compare clinintion in relation to obama) <-confound
- -Participants are only a blank slate once. Try to put time inbetween each study to bring back the blank state mentality
What is counterbalancing in repeated measure designs
- –Manipulation of the order of presentation of the conditions
- Order 1: Verbal then Auditory
- Order 2: Auditory then Verbal
- -Average across both orders
- –Controls for time and order effects
Disadvantages to counterbalancing
–Not all designs can be counterbalanced (ie. Pre-Post & Developmental studies)
If there is no counterbalancing & no random assignment then...
there is no causality!
Partitioning of Variance: Repeated Measures Design
effect of IV divided by sampling error (no difference between IV factor bc youre using all the same people)
Partitioning of Variance: independent measures
effect of IV divided by sampling error + individual differences
independent measures compare means between conditions, whereas repeated measures
compute the different score between conditions (con.1 minus condition 2)
what is the sign of there is no effect of the IV for a related samples t test?
Ho: μD = 0
-fail to reject null
what is the sign of there is an effect of the IV for a related samples t test?
H1: μD ≠ 0
what is the hypothesis test steps for a related samples t test?
- Step 1: State your hypotheses
- –Ho: μD = 0
- –H1: μD ≠ 0
- Step 2: Locate your Critical Region
- –t distribution
- –Two tailed
- –DF = N -1
- –α= .05
- Step 3: Calculate t
- Step 4: Make a Decision
Related Samples t Test: Assumptions
- -Observations within each treatment condition must be independent
- -Population distribution of Difference Scores must be normal
- -NO assumption of HOV
Related Samples t-test: Effect Size
Cohen's d or r2
never do repeated measures as a ____ way design
what does it mean if your "t" is negative in a related samples t test: hypothesis test?
negative means they improved bc its pre test minus post, so if they get a negative it means the number got better.
the related samples t-test in comparison to the independent samples t test
the independent samples t-test is smaller and not as powerful
Related Samples Design w/ 3 conditions: Omnibus Test
Related Samples Design w/ 3 conditions: Nature of effect
- -polynomial contrasts
- -pairwise comparisons
Related Samples Design w/ 3 conditions: size of effect
partial eta squared
Repeated Measures ANOVA: Stages of Analysis
- Step1: Partition Variance
- -stage 1: partition total into between & within treatments
- -stage 2: parition w/in treatments & between subjects & error
- Step2: Determine Degrees of Freedom
- Step3: Calculate Mean Squares
- Step4: Calculate F
What is G?
sum of all raw scores
What is Sphericity?
- –The variances of the differences between the conditions should be approximately equal
- –Violations lower statistical power (i.e. increase type II error rates)
- –ANOVA is robust to all but major departures from Sphericity
- -no HOV assumption
- -error variability must be consistant across same person (Probability of same mistake made over time)
- -must make sure sphericity isnt violated
Detecting violations in sphericity
- –Visual inspection of variances of difference scores
- –Mauchly’s Test (SPSS)
- -Reject H0 indicates violation of assumption
- *greater than .05 than not violating sphericity
what is the most preferred correction for violation of sphericity?
- -low power with large violations of sphericity
what do you do once a violation of sphericity is deteced?
- -use a correction to DF for F (greenhouse-geisser or huynh-feldt)
- -use an omnibus test that does not assume sphericity (multivariate analysis of variance (MANOVA))
- -if all 3 scores differ then something is wrong
- **chop df down to make is harder to reject null
what omnibus test doesnt assume sphericity?
multivariate analysis of variance (MANOVA)
What are the ordinal IV's
what are the nominal IV's
pairwise comparisons (related samples t-tests, bonferroni correction)
What is a mixed design?
- A factorial design with…
- -Atleast one Independent-measures IV
- -At least one Related-Samples IV
- -Involves all assumptions of both types of ANOVA
- -Involves all nature of effect and size of effect issues of both types
Why do we use mixed designs?
- -to Explore the effect of an Independent Measures variable & a repeated-measures variable
- -Test for Order Effects
- -Control for potential confounds due to time in pure within subjects designs
What are typical mixed designs? (example)
- Doratings of attractiveness…
- -Change with width of face?
- -Change based on context?
- -Change as an interaction of both?
when interpreting mixed designs, which effects are interesting?
- potentially all three are interesting
- -main effects(A and B)
- - A x B
whats a 2 x 3 blocking design?
- -Evaluate the effect of the Within-Subjects IV across blocks of potentially confounding variables (Rotation, Latin-Square)
- -Does the order of presentation influence ratings of the candidates?
Whats a split-plot design?
- -Within-subjects IV tests for effect of manipulation
- -Between Subjects IV: controls for effect of time
- -Control: no treatment, only time
- -Experimental:Treatment + Time
Interpreting Mixed Designs: Split Plot
- Within Subjects Main Effect (time): Not Interesting. Averages across both experimental and control conditions
- -Between-subjects Main Effect: Not Interesting. Combines pre and post test conditions: No effect of treatment
- -Interaction: Interesting. Can indicate differential improvement based on treatment
Mixed Designs: Partitioning of Variance
- -Both effect and error contain variability due to independent and repeated measures sources
- -Both must be partitioned
mixed desigins: patitioning effect between treatment
- Main Effects
- IV A: Treatment (Independent)
- IV B: Time (related)
- -Interaction Effect: Ax B: Differences in the effect of time for each treatment (mixed)
mixed designs: Partitioning error within treatment
- -Error: Sampling Error
- -Between Subjects: Individual Differences
mixed designs: partitioning of variance, testing your effects
- Independent Measures Main Effect
- Effect: MSA
- Error: MSWT
- Within-subjects Main Effect
- Effect: MSB
- Error: MSerror
mixed designs: partitioning of variance, testing your effects
- -Interaction uses within-subjects error term
- -Effect: MSAB
- -Error: MSerror
issues to remember with mixed designs
- Issues of Sphericity exist for all effects that involve a repeated measures and k > 2 IV
- Partial eta2 rather than eta2
- Nature of effect options differ based on type of IV:
- -Between Subjects: Post-Hocs, planned comparisons, polynomial contrasts
- -Within-subjects: planned comparisons, polynomial contrasts
- -Interactions: Simple main effects, Simple Interaction Effects
partitioning of within treatments variance with repeated measures design
within treatments = error plus between subjects
What does a large F indicate?
That the difference between treatment are greater than would be expected by chance or error alone.
related samples t statistic
The single sample t statistic applied to a sample of difference scores (D values) and the corresponding population of difference scores.
The effects of participating in one treatment that may influence the scores in the following treatment
between subjects variability
The differences that exist from one subject to another.
between treatments variability
The differences that exist from one treatment to another (a measure of mean differences).
Unexplained, unsystematic differences that are not caused by any known factor.
A ratio of two variances: between-treatments variance in the numerator and chance/error in the denominator.
Systematic differences that are caused by changing treatment conditions.
within treatments variability
The differences that exist inside each treatment condition.
what is an interaction
Mean differences that cannot be explained by the main effects of the two factors. An interaction exists when the effects of one factor depend on the levels of the second factor.
what is a main effect?
The overall mean differences between the levels of one factor. When the data are organized in a matrix, the main effects are the mean differences among the rows (or among the columns).
what is matrix & cells
A two-dimensional table is a matrix and each box in the table is called a cell.