Split plot anova slide

Split Plot
Analysis of Variance Designs
PSYCHOLOGY 3800, LAB 003

•  two-way ANOVA assignment feedback (short version)
•  split plot ANOVA overview
•  example analysis
•  example output of overall effects
•  example post hoc investigation
•  assignment
In Lab Today…

Assignment #4: Feedback
•  keep your report organized
o  introduction
o  assumptions
o  interaction + simple main effects
o  main effect 1 (self-esteem)
o  main effect 2 (difficulty) + post hoc tests
o  conclusion
•  tell a story
o  make clear what you are seeing
o  describe effects
o  write concluding sentences

Description of Effects
•  be specific about the nature of the effect observed
Example
No significant main effect was found when comparing the low self-esteem group (M
= 30.67, SE = 1.79) to the high self-esteem group (M = 31.23, SE = 1.09), F(1, 54)
= 0.23, ns, η2 = 00, power = .08.
…need to add to this to clarify the effect…
Therefore, participants in low self-esteem groups did not differ significantly from
those in the high self-esteem on their test performance.

•  example two-way ANOVA results section posted on lab blog
http://psych3800g.tumblr.com/post/44614034425/two-way-anova-example-report

•  extension of the completely randomized factorial design
o  two or more factors (independent variables)
o  each factor has multiple levels
o  measuring one dependent variable
o  can have main effects and interaction
o  interested in differences between means
•  main design difference
o  two-way ANOVA: both factors are independent
o  split plot ANOVA: one factor is independent, one is correlated
Split Plot ANOVA: Overview

•  independent factor = between-subjects factor
o  composed of 2 (or more) levels of completely different people
•  correlated factor = within-subjects factor
o  composed of 2 (or more) levels that consist of the same people (repeated)
What kind of study would use this design?
Split Plot ANOVA: Overview

Does a participant’s level of amusement after watching different types of ‘80s action
movies change depending on their level of sleep deprivation?
1.  obtain a sample of participants
2.  randomly assign participants to either sleep deprivation or lots of sleep condition
3.  have each person watch 3 different ‘80s action movies
4.  record participants’ amusement level after each movie
(1 = not at all amused, 5 = extremely amused)
Research question:
Procedure:
Split Plot ANOVA: Example

Sleep status Action movie from 1980s
Ghostbusters The Terminator Indiana Jones
Sleep deprived
Tons of sleep
Each participant experiences only one combination of variables.
Two-Way Factorial Design

Sleep deprived
Tons of sleep
Split Plot Design
Participants get assigned to a group, and experience all levels of second factor in that group.

Sleep deprived 4.105 3.975 2.425
Tons of sleep 3.540 3.555 2.970
3.823 3.765 2.698
3.502
3.355
Split Plot ANOVA: Effects
The types of values that we can calculate are similar to those obtained via a
two-way factorial design (in a two-way ANOVA)…

main effect of sleep
(if significant, know that these means differ
significantly)
Tons of sleep 3.540 3.555 2.970
3.823 3.765 2.698
3.502
3.355

Tons of sleep 3.540 3.555 2.970
main effect of movies
(if significant, know that at least two of
these means differ significantly)
3.823 3.765 2.698
3.502
3.355

Tons of sleep 3.540 3.555 2.970
interaction effect
(if significant, know that cell means differ
significantly)
From here, we must decide on an approach to interpreting interaction effect.
(same as in two-way ANOVA)

Tons of sleep
3.540 3.555 2.970
Interpreting the Interaction: Option #1
o simple main effects of movie at each level of sleep
Sleep deprived: G vs. T G vs. I T vs. I
Tons of Sleep: G vs. T G vs. I T vs. I
6 comparisons

Tons of sleep
3.540 3.555 2.970
Interpreting the Interaction: Option #2
o simple main effects of sleep at each level of movie
Ghostbusters: sleep deprived vs. tons of sleep
3 comparisonsTerminator: sleep deprived vs. tons of sleep
Indiana Jones: sleep deprived vs. tons of sleep

As in two-way ANOVA, we are interested in three main hypotheses:
1.  interaction hypotheses
  H0: a significant interaction does not exist between sleep status and movies
  HA: a significant interaction exists between sleep status and movies
2.  main effect for variable 1 (sleep status)
  H0: µDEPRIVED = µLOTS
  HA: µDEPRIVED ≠ µLOTS
3.  main effect for variable 1 (movie)
  H0: µGHOSTBUSTERS = µTERMINATOR = µINDIANA
  HA: at least two means differ significantly
Split Plot ANOVA: Hypotheses

1.  independent random sampling
2.  normality
3.  homogeneity of variance (2 parts)
Split Plot ANOVA: Assumptions

Homogeneity of Variance
•  Levene’s test (F)
o  between-groups variances are homogenous (as in previous tests)
o  e.g., is variance of the DV (amusement scores) equal for both for sleep-deprived
versus sleep-affluent people at each movie?
• Mauchly’s test (W)
o  circularity of the pooled variance-covariance matrix
o  variances of difference scores are the same (as in repeated ANOVA)
o  regardless of results, always report Greenhouse-Geisser corrected values
Significant results suggest that assumption has been violated (applicable to both tests).
Split Plot ANOVA: Assumptions

Split Plot ANOVA:
Example Analysis in SPSS

with-subjects factor
(as labeled)
between-subjects factor
(1 = deprived, 2 = lots)
scores on DV, 1-5
(level of amusement)
40 cases
First participant:
-randomly assigned
to “lots of sleep”
condition
-rated Ghostbusters
and Terminator as
highly amusing,
Indiana Jones as low
Split Plot ANOVA: Example Data
*ignore the 5th column for now

Analyze  General Linear Model  Repeated Measures
specify repeated (within-subjects) factor name and
number of levels, click “Add”
when info added (as shown) click “Define”
Split Plot ANOVA: SPSS Analysis

define each level of within-subjects variable as in a repeated measures ANOVA
(select level on right, click on corresponding movie on left, click )

define the between-subjects variable by moving Sleep_Status into the Between-Subjects Factor(s) box
(click on variable in left panel, click on beside Between-Subjects box)

Options Menu
provides Levene’s test output
for between-subjects factor
(Sleep Status)
gives descriptive values
for within-subjects factor
(Movies) and interaction

Plots Menu
request both types of plots to
help you decide in which way
you would like to frame/interpret
the interaction

Once all selections have been made, click “OK” to run the analyses.

Split Plot ANOVA:
Example Output for Overall Effects

Descriptive Statistics
Split Plot ANOVA: SPSS Output
•  these are the cell means representing the effect of one variable at each level of the
other (will use when assessing interaction)
•  standard errors are not provided and so will have to be calculated by hand:
€
SE =
s
n

Descriptive Statistics
•  these are the group means for the movie levels (one mean for each movie) and will
be assessed when we examine the main effects of the within-subjects factor
•  standard errors are not provided and so will have to be calculated by hand:
€
SE =
s
n

Descriptive Statistics: Obtaining Data for the Between-Subjects Factor
•  need to create a single variable that represents the mean enthusiasm for each
participant, collapsed across the movies (average movie scores per participant)
•  I have already done this for you in your data file

Analyze  Descriptive Statistics  Explore
specify that the averaged scores represent your DV, which you are examining at each level of your
Sleep Status IV (request statistics only)

•  these are the group means for the sleep
levels (one mean for each sleep group)
and will be assessed when we examine
the main effects of the between-subjects
factor
•  standard errors are provided

Mauchly’s W = 0.863, χ2(2) = 6.388, p < .05
* significant effect = assumption of circularity has been violated
Test of Assumptions: Mauchly’s Test

Test of Assumptions: Levene’s Test
Ghostbusters: Levene F(1, 38) = 1.048, ns
Terminator: Levene F(1, 38) = 0.427, ns
Indiana Jones: Levene F(1, 38) = 1.867, ns
Equal variances are assumed on the DV for the sleep groups
at each level of the within-subjects (repeated) variable.

F(2, 66) = 5.652, p < .01, η2 = .129, power = .806
•  significant interaction exists between movies and sleep status
•  proceed with simple main effects
Omnibus Test: Interaction

F(2, 66) = 24.928, p < .001, η2 = .396, power = 1.000
•  significant main effect exists for the repeated variable of Movies
•  proceed with post hoc tests (Tukey’s HSD)
Omnibus Test: Within-Subjects Effects (Movie)

Omnibus Test: Between-Subjects Effects (Sleep Status)
F(1, 38) = 0.405, ns, η2 = .011, power = .095
•  no significant main effect for sleep status exists
•  no Tukey’s HSD post hoc tests required

So far, we know:
•  significant interaction between level of sleep and movie type
•  significant within-subjects main effect for movie type
•  non-significant between-subjects main effect for level of sleep
Next steps:
•  investigation of simple main effects for interaction
•  post hoc tests (Tukey’s HSD) for main effect of movies

Split Plot ANOVA:
Post Hoc Analyses

Split Plot ANOVA: Post Hoc Analyses
Interaction 1: Simple Main Effects of Movie at Each Level of Sleep
i.e. simple main effects of within-subjects factor at each level of between-subjects factor
0
1
2
3
4
5
Sleep deprived Lots of Sleep
MeanAmusementRating
Sleep Status

Step 1: Run a MANOVA using the syntax option in SPSS
File  New  Syntax
levels of within-subjects factor (movie) between-subjects factor (sleep) with coding
name of within-subjects factor (number of levels) comparing means of within-subject factor (movie)
at first level of sleep-status (sleep-deprived)

movies at sleep-deprived: F(2, 76) = 27.13, p < .001
movies at tons-of-sleep: F(2, 76) = 3.45, p < .05
Reading the very bottom table of the MANOVA output…
at least two movie means differ significantly at each sleep level  proceed
with Tukey’s HSD to pinpoint differences

Step 2: follow up with separate Tukey’s HSD analyses for applicable levels using the
POSTHOC program (use sphericity assumed values, no pooled error term)
Example: SME of Movies at Sleep Deprived
from “Tests of Within-
Subjects Effects” table
Report as:
q(k, dferror)  q(3, 76)
k = # of means compared
dferror = same as what you
gave to POSTHOC
*compare against q-critical
from tables

Interaction 2 : Simple Main Effects of Sleep at Each Level of Movie
i.e. simple main effects of between-subjects factor at each level of within-subjects factor
0
1
2
3
4
5
Ghostbusters Terminator Indiana Jones
MeanAmusementRating
Action Movie from the 1980s

Analyze  Compare Means  One-Way ANOVA
Step 1: run a one-way ANOVA in SPSS for all variables

•  the output is not the results of our simple main effects analysis (sorry!)
•  this is a way for us to get the necessary info to calculate our needed statistics
•  pull out the Mean Square and df values for the “Between Groups” effects
(the rest of the info is meaningless)

Option #1: by hand
€
Pooled MSerror =
SSerror1 + SSerror2
dferror1 + dferror2
€
Pooled dferror =
(SSerror1 + SSerror2)2
SSerror1
2
dferror1
+
SSerror2
2
dferror2
where…
SSerror1 = sums of squares for error, sphericity assumed
(Tests of Within-Subjects Effects table)
dferror1 = degrees of freedom for error, sphericity assumed
(Tests of Within-Subjects Effects table)
SSerror2 = sums of squares for error
(Tests of Between-Subjects Effects table)
dferror2 = degrees of freedom for error
(Tests of Between-Subjects Effects table)
Step 2: Calculate pooled error terms for the analysis

Option #2: POSTHOC program
•  enter in all cell means (found in your descriptive values output)
•  specify the nature of the sample (group size, all groups equal)
•  identify two Mean Square Error (MSE) values and their degrees of freedom (df)
 MSE1 and df1 Tests of Within-Subjects Effects table, Error section,
sphericity assumed value
 MSE2 and df2 Tests of Between-Subjects Effects table, Error row

this is your pooled
error term
(when reporting it
in your results, the
convention is to
round it down…
so, it would be 93)
Option #2: POSTHOC program

Step 3: Calculate F-obtained values for each comparison by hand
€
F =
MSBG
MSerror
where…
MSBG = between-groups MS value of interest
(one-way ANOVA output)
MSerror = pooled MS error value
(from POSTHOC or by hand)
F(dfBG, dferror) = calculated value
where…
dfBG = between-groups df value of interest
(one-way ANOVA output)
dferror = pooled df error value, rounded down
(from POSTHOC or by hand)

Step 3: Calculate F-obtained values for each comparison by hand
€
F =
MSBG
MSerror
=
3.192
.960333
= 3.324
Example: sleep-deprived versus lots-of-sleep after watching Ghostbusters
F(1, 93) = 3.324

Step 3: Compare F-obtained values against F-critical values to determine significance
•  to find F-critical: use tables or website suggested by Dr. McRae
•  for F-tables: use the same degrees of freedom in looking up your critical value as
you do when reporting your obtained value:
F(dfBG, dferror) = critical value
•  reject H0 (and conclude significant difference) if F-obtained > F-critical

•  use POST HOC program to output qobtained values for all comparisons
•  enter sphericity-assumed data (as in repeated ANOVA)
•  no pooled error term needed
•  compare qobtained values against qcritical value from tables
q (k, dferror) = obtained or critical value
where: k number of within-subjects factor levels
dferror error df from Tests of Within-Subjects Effects table
(sphericity assumed)
…also include significance info for qobtained
Main Effect: Within-Subjects (Repeated) Variable

•  3-page report in APA-style
•  two main sections:
1)  response to question #1 (part A in point-form, part B in sentence form)
2)  APA-style results section describing overall results
•  all output
o  SPSS output (Split Plot analysis, One-Way ANOVA, MANOVA, descriptives)
o  POST HOC output (post hoc tests for any significant main effects, post hoc tests
for simple main effects when needed)
Assignment: What to Submit

Assignment: What to Report for Question 1
Within-Subjects Variable at Each Level of Between-Subjects Variable
(movies at each level of sleep status)
•  simple main effect of movies for sleep deprived participants
  MG = 4.11, MT = 3.98, MI = 2.42
  F(2, 76) = 27.13, p < .001
  q(3, 76) = 0.72, ns (Ghostbusters = Terminator)
q(3, 76) = 9.42, p < .01 (Ghostbusters > Indiana Jones)
q(3, 76) = 8.69, p < .01 (Terminator > Indiana Jones)
•  simple main effect of movies for tons-of-sleep participants
  MG = 3.54, MT = 3.56, MI = 2.97
  etc…
Example: Method 1 of Interpreting Interaction

Assignment: What to Report for Question 1
Between-Subjects Variable at Each Level of Within-Subjects Variable
(sleep status at each level of movie)
•  simple main effect of sleep status for Ghostbusters
  MDEPRIVED = 4.11, MLOTS = 3.54
  F(1, 93) = 3.324, p < .05
•  simple main effect of movies for Terminator
  MDEPRIVED = 3.98, MLOTS = 3.56
  etc…
Example: Method 2 of Interpreting Interaction
Don’t forget to address part B!

•  introductory paragraph
o  general overview of study
o  provide design being used (split plot)
o  identify IVs (and levels)  specify which is between-subjects, within-subjects
o  identify DV
•  tests of assumptions
o  Levene’s test for between-subjects factor (all F-values applicable)
o  Mauchly’s test for within-subjects factor
o  write a concluding sentence for each test, stating what we can conclude on
basis of results
Assignment: What to Report in Results Section

•  interaction effect
o  report F-statistics (with df and p-value), effect size, power
o  if significant, report one set of simple main effects (repeat of question 1);
no critical values need to be included in your report)
o  provide interpretation and caution regarding interpretation of main effects
•  main effect for within-subjects variable (Concept)
o  descriptive values
o  if significant, post hoc tests (with qcritical and qobtained values)
o  provide interpretation
•  main effect for between-subjects variable (groups)
o  if significant, post hoc tests (with qcritical and qobtained values)
o  provide interpretation and caution if interaction significant
•  general conclusion
Assignment: What to Report in Results Section

•  use Greenhouse Geisser corrected values for the ANOVA (where applicable…
within-subjects and interaction) but use sphericity assumed values for post hoc
tests
•  use the appropriate error term in reporting your results
o  Tests of Within-Subjects Effects: overall interaction (adjusted), overall within-
subjects main effect (adjusted), post hoc for within-subjects main effect
(unadjusted)
o  Tests of Between-Subjects Effects: overall between-subjects main effect,
post hoc for between-subjects main effect
o  pooled error term (POST HOC program or hand calculation): SME of
between-subjects factor at leach level of within-subjects factor
o  MANOVA output: SME of within-subjects factor at each level of between-
subjects factor
Helpful Hints

Good Friday: Update
•  Friday lab section has been re-scheduled as follows:
Wednesday, March 27, 2013
10:30 AM – 12:30 PM
SSC 1020 (computer lab in basement of SSC)
•  will be sending out a general e-mail outlining who I have attending which lab
section (mine, Sarah’s, Paul’s)
 if there is an error or if you have changed your preference, let me know
 I will be forward a list of student names to Sarah/Paul, so they know
how many students to expect
•  deadline for the assignment that week will be Thursday, March 28, 5:00 PM
 can submit earlier

Make sure you have and understand all output
Before you leave lab today!

Split plot anova slide

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