unit 6- split plot anova

Split Plot Analysis of Variance Designs

PSYCHOLOGY 3800, LAB 002

•  two-way ANOVA assignment feedback (link)

•  split plot ANOVA overview

•  example analysis

•  example output of overall effects

•  example post hoc investigation

•  assignment

In Lab Today…

Assignment #4: Feedback

Summary of commonly made errors available on lab blog: http://uwo3800g.tumblr.com/post/78252831240/assignment-4-commonly-made-errors

Split Plot ANOVA: Overview

•  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


•  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?


Does a participant’s level of amusement after watching different types of ‘80s action movies change depending on their level of sleep deprivation?

Research question:

Variables of Interest:

Split Plot ANOVA: Example

Independent variables (i.e. factors):

(A) between-subjects: sleep group (2 levels) (B) within-subjects: ’80s action movie condition (3 levels)

Dependent variable:

self-reported level of amusement (5-point Likert scale, high scores reflecting greater level of amusement)

Sleep status Action movie from 1980s

Ghostbusters The Terminator Indiana Jones

Sleep deprived

Tons of sleep

Split Plot ANOVA: Example

Each participant experiences only one combination of variables.

Two-Way Factorial Design



Sleep deprived

Tons of sleep

Split Plot ANOVA: Example 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 comparisons Terminator: sleep deprived vs. tons of sleepIndiana Jones: sleep deprived vs. tons of sleep

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 for effects involving the within-subjects variable

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: Values for the Interaction

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 =sn

Recall that we can do the SE calculations quickly in Excel: http://uwo3800g.tumblr.com/post/78007754575/calculating-standard-error-in-excel

Descriptive Statistics: Values for Within-Subjects Effects


•  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 =sn


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 (will be the case for the assignment as well)



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 = 0.841, χ2(2) = 6.388, p < .05

•  significant effect = assumption of circularity has been violated •  apply Greenhouse-Geisser correction to subsequent analyses involving within-subjects effects (would do this even with a non-significant finding)

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 (at least two movie means differ significantly) •  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


Post Hoc for Main Effect: Within-Subjects (Repeated) Variable

•  use POST HOC program to output qobtained values for all comparisons •  enter sphericity-assumed data, as you did in the repeated ANOVA unit •  no pooled error term needed (use error term from Test of Within-Subjects Effects table)

Ghostbusters vs. Terminator q(3, 76) = 0.394, ns Ghostbusters vs. Indiana Jones q(3, 76) = 8.827, p < .01 Terminator vs. Indiana Jones q(3, 76) = 8.433, p < .01

# of levels in factor sphericity assumed dferror

*critical values found using online calculator: http://vassarstats.net/tabs.html

critical values:

.05 = 3.39

.01 = 4.25


Post Hoc for Main Effect: Between-Subjects (Non-Repeated) Variable

•  not needed in our investigation due to non-significant test of main effects •  had test of main effects been significant: would still not have been necessary because we only have to levels defining our between-subjects variable (immediately would known which two means differ significantly)

*critical values found using online calculator: http://vassarstats.net/tabs.html

Had we had a significant main effect for a between-subjects variable with 3+ levels: proceed with post hoc tests using POSTHOC program enter the means outputted via the “Explore” analysis do not request a pooled error term enter the Mean Square and df for error found in the “Tests of Between Subjects Effects” table compare outputted q-obtained values to q-critical values to determine significance

q(df1, df2) = obtained value

# of levels in factor dferror from “Tests of Between-Subjects Effects” table


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

Mea

n A

mus

emen

t Rat

ing

Sleep Status


Step 1: Run a MANOVA using the syntax option in SPSS

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)

Interaction 1: Simple Main Effects of Movie at Each Level of Sleep

*syntax has been given to you for your assignment


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 of condition means using the POSTHOC program (sphericity assumed values, no pooled error term)


•  use sphericity assumed Mean Square and df values for error from the “Tests of Within-Subjects Effects” table •  critical values found using online calculator: http://vassarstats.net/tabs.html

Sleep Deprived: G vs. T: q(3, 76) = 0.724, ns G vs. I: q(3, 76) = 9.418, p < .01 T vs. I: q(3, 76) = 8.694, p < .01

Sleep Affluent: G vs. T: q(3, 76) = 0.111, ns G vs. I: q(3, 76) = 3.176, ns T vs. I: q(3, 76) = 3.288, ns

# levels in within-subjects factor dferror


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

Mea

n A

mus

emen

t Rat

ing

Action Movie from the 1980s

This approach is a little trickier because we cannot use a MANOVA to output F-values for each movie level. We can, however, calculate these F-values using other results that we can obtain through SPSS.


Analyze Compare Means One-Way ANOVA

Interaction 2 : Simple Main Effects of Sleep at Each Level of Movie

Step 1: run a one-way ANOVA in SPSS for all variables

Move the variables representing your levels of the repeated factor to the “Dependent List”. Move the variable representing the non-repeated factor to the “Factor” section.


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



•  enter in all cell means (found in your descriptive values output)

•  specify the nature of the sample (group size, all groups equal)

•  state that you would like to calculate a pooled error term

•  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

•  output the q-values (may need this later, so it’s good to have)


Step 2: Calculate pooled error terms for the analysis using the POSTHOC program



Step 2: Calculate pooled error terms for the analysis using the POSTHOC program

pooled Mean Square error term pooled degrees of freedom for error (round up)


Step 3: Calculate F-obtained values for each comparison by hand


€

F =MSBGMSerror

where…

MSBG = between-groups MS value of interest (one-way ANOVA output)

MSerror = pooled MS error value (from POSTHOC)

F(dfBG, dferror) = calculated value

where…

dfBG = between-groups df value of interest (one-way ANOVA output)

dferror = pooled df error value, rounded up (from POSTHOC)

Calculating the F-obtained: Reporting the F-obtained:


Step 3: Calculate F-obtained values for each comparison by hand


Numerator values…

Denominator value… (as outputted by POSTHOC program)

€

FGhostbusters =MSBGMSerror

=3.192

.9603333= 3.324

€

FTerminator =MSBGMSerror

=1.764.9603333

=1.837

€

FIndiana Jones =MSBGMSerror

=2.970

.9603333= 3.093


Step 3: determine the level of significance for your obtained F-value


Ghostbusters sleep deprived vs. sleep affluent F(1, 94) = 3.324, ns (exact p = .0715)

Terminator sleep deprived vs. sleep affluent F(1, 94) = 1.837, ns (exact p = .1786)

Indiana Jones sleep deprived vs. sleep affluent F(1, 94) = 3.093, ns (exact p = .0819)

•  to determine p-value: http://vassarstats.net/tabs.html (“F to p” calculator) enter your F-obtained value and your df values, click “Calculate” will output exact p-value for each F-value (note: anything above .05 is non-significant)

With no significant effects at any of movie levels, we do not have to report any q-obtained values (this is where our analysis of SMEs ends).


Step 4: for any levels at which F-obtained is significant, report q-statistics


•  the q-obtained values that you would need for this step will be found in your POSTHOC output generated at Step 2 (obtaining a pooled error term)

•  to determine whether specific q-obtained values are significant, use the online calculator: http://vassarstats.net/tabs.html

•  you will get far more q-values than you will need to report in your output, so select out only the comparisons in which you are interested

Recall: if there are only two means being compared at each level, then this step is not needed, because a significant F-value tells as that at least two of our means differ

significantly (and with only two means, we know which two differ)

q(df1, df2) = obtained value

# levels in between-subjects factor dferror (pooled, rounded up)

Assignment #6

•  3-page report in APA-style

•  two main sections: 1)  response to question #1 (part A in point-form and single-spaced, part B in

sentence form and double-spaced) 2)  formal APA-style results section describing overall results (double-spaced)

•  all output and hand calculations (by hand or done in Excel) 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

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 sleep affluent participants   MG = 3.54, MT = 3.56, MI = 2.97   etc…

Example: Method 1 of Interpreting Interaction (point form)

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, 94) = 3.32, ns (sleep deprived = sleep affluent)   q-values not needed due to non-significant F-obtained

•  simple main effect of movies for Terminator   MDEPRIVED = 3.98, MLOTS = 3.56   etc…

Example: Method 2 of Interpreting Interaction (point form)

Don’t forget to address part B (in full sentences)!

Assignment: What to Report for Question #1

•  introductory paragraph o  general overview of study o  provide design being used (split plot analysis of variance) o  identify IVs (and levels) specify which is between-subjects, within-subjects o  identify DV and scoring

•  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 (one identified in #1b) o  report descriptive statistics for the conditions being compared o  provide interpretation and caution regarding interpretation of main effects

•  main effect for within-subjects variable o  descriptive values o  report F-statistics (with df and p-value), effect size, power o  if significant, post hoc tests (q-values) o  provide interpretation

•  main effect for between-subjects variable o  report F-statistics (with df and p-value), effect size, power o  if significant, post hoc tests (q-values) o  provide interpretation

•  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

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

unit 6- split plot anova

Education

independent o split

repeated anova o

o simple main effects

interaction o

subjects factor o

sleep group

effects main effect

o simple main effects