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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?
 Seperately
 –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?
yes

What do NonParallel lines on a factoral design indicate?
a potential interaction

What does it mean if lines are parallel in a factorial design?
no interaction

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 Crossover Interaction in a factorial design?
Effect of A reverses based on B. Can’t trust main effects!

what is a Factorial Design (TwoWay)?
 2 x 2
 •Two Factors
 •Two levels of each factor
 •Four cells

What all can you find in a factorial design (twoway)?
 Main Effect of A
 Main Effect of B
 Interaction of A x B

What all can you find in a factorial design (twoway) 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 (ThreeWay)
 –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 (threeway)
 twoway 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)
 •PartitionVariance
 •Partition Degrees of Freedom
 •Construct Mean Squares
 •Construct F

What do you do if you have a significant Ftest?
 –Post hoc/aPriori testing (Stage III)
 –Effect size (Stage IV)

What do you do if you do not get a significant Ftest?
–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?
yes!

How do you find the nature of the effect for twoway 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)
 •ttests 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?
Bonferroni

What is a repeatedmeasures 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. PrePost & 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
reject null

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 ttest: Effect Size
Cohen's d or r2

never do repeated measures as a ____ way design
one way

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 ttest in comparison to the independent samples t test
the independent samples ttest is smaller and not as powerful

Related Samples Design w/ 3 conditions: Omnibus Test
–RepeatedMeasures ANOVA

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?
 greenhousegeisser
 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 (greenhousegeisser or huynhfeldt)
 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
polynomial contrasts

what are the nominal IV's
pairwise comparisons (related samples ttests, bonferroni correction)

What is a mixed design?
 A factorial design with…
 Atleast one Independentmeasures IV
 At least one RelatedSamples 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 repeatedmeasures 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 WithinSubjects IV across blocks of potentially confounding variables (Rotation, LatinSquare)
 Does the order of presentation influence ratings of the candidates?

Whats a splitplot design?
 Withinsubjects 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
 Betweensubjects 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
 Withinsubjects Main Effect
 Effect: MSB
 Error: MSerror

mixed designs: partitioning of variance, testing your effects
 Interaction
 Interaction uses withinsubjects 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: PostHocs, planned comparisons, polynomial contrasts
 Withinsubjects: 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.

Order effects
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).

error variability
Unexplained, unsystematic differences that are not caused by any known factor.

Fratio
A ratio of two variances: betweentreatments variance in the numerator and chance/error in the denominator.

treatment effect
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 twodimensional table is a matrix and each box in the table is called a cell.

