# Unit 6- Split Plot ANOVA

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Published on March 5, 2014

Author: lveselka

Source: slideshare.net

Split Plot Analysis of Variance Designs PSYCHOLOGY 3800, LAB 002

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

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

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

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: Example Research question: Does a participant’s level of amusement after watching different types of ‘80s action movies change depending on their level of sleep deprivation? Variables of Interest: Independent variables (i.e. factors): (A) between-subjects: (B) within-subjects: sleep group (2 levels) ’80s action movie condition (3 levels) Dependent variable: self-reported level of amusement (5-point Likert scale, high scores reflecting greater level of amusement)

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

Split Plot ANOVA: Example Split Plot Design Sleep status Action movie from 1980s Ghostbusters The Terminator Indiana Jones Sleep deprived Tons of sleep Participants get assigned to a group, and experience all levels of second factor in that group.

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)… Sleep status Action movie from 1980s Ghostbusters The Terminator Indiana Jones Sleep deprived 4.105 3.975 2.425 3.502 Tons of sleep 3.540 3.555 2.970 3.355 3.823 3.765 2.698

Split Plot ANOVA: Effects main effect of sleep (if significant, know that these means differ significantly) Sleep status Action movie from 1980s Ghostbusters The Terminator Indiana Jones Sleep deprived 4.105 3.975 2.425 3.502 Tons of sleep 3.540 3.555 2.970 3.355 3.823 3.765 2.698

Split Plot ANOVA: Effects main effect of movies (if significant, know that at least two of these means differ significantly) Sleep status Action movie from 1980s Ghostbusters The Terminator Indiana Jones Sleep deprived 4.105 3.975 2.425 3.502 Tons of sleep 3.540 3.555 2.970 3.355 3.823 3.765 2.698

Split Plot ANOVA: Effects interaction effect (if significant, know that cell means differ significantly) Sleep status Action movie from 1980s Ghostbusters The Terminator Indiana Jones Sleep deprived 4.105 3.975 2.425 Tons of sleep 3.540 3.555 2.970 From here, we must decide on an approach to interpreting interaction effect. (same as in two-way ANOVA)

Split Plot ANOVA: Effects Interpreting the Interaction: Option #1 o  simple main effects of movie at each level of sleep Sleep status Action movie from 1980s Ghostbusters The Terminator Indiana Jones 4.105 3.975 2.425 3.540 3.555 2.970 Sleep deprived Tons of sleep Sleep deprived: G vs. T G vs. I T vs. I 6 comparisons Tons of Sleep: G vs. T G vs. I T vs. I

Split Plot ANOVA: Effects Interpreting the Interaction: Option #2 o  simple main effects of sleep at each level of movie Sleep status Action movie from 1980s Ghostbusters Indiana Jones 4.105 3.975 2.425 3.540 Sleep deprived The Terminator 3.555 2.970 Tons of sleep Ghostbusters: sleep deprived vs. tons of sleep Terminator: sleep deprived vs. tons of sleep Indiana Jones: sleep deprived vs. tons of sleep 3 comparisons

Split Plot ANOVA: Assumptions 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: Example Analysis in SPSS

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

Split Plot ANOVA: SPSS Analysis 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 )

Split Plot ANOVA: SPSS Analysis 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)

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

Split Plot ANOVA: SPSS Analysis Plots Menu request both types of plots to help you decide in which way you would like to frame/interpret the interaction

Split Plot ANOVA: SPSS Analysis Once all selections have been made, click “OK” to run the analyses.

Split Plot ANOVA: Example Output for Overall Effects

Split Plot ANOVA: SPSS Output Descriptive Statistics: Values for the Interaction •  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 = Recall that we can do the SE calculations quickly in Excel: http://uwo3800g.tumblr.com/post/78007754575/calculating-standard-error-in-excel € s n

Split Plot ANOVA: SPSS Output 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 = € s n

Split Plot ANOVA: SPSS Output 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)

Split Plot ANOVA: SPSS Output Descriptive Statistics: Obtaining Data for the Between-Subjects Factor 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)

Split Plot ANOVA: SPSS Output Descriptive Statistics: Obtaining Data for the Between-Subjects Factor •  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

Split Plot ANOVA: SPSS Output Test of Assumptions: Mauchly’s Test 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)

Split Plot ANOVA: SPSS Output Test of Assumptions: Levene’s Test Ghostbusters: Terminator: Indiana Jones: Levene F(1, 38) = 1.048, ns Levene F(1, 38) = 0.427, ns 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.

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

Split Plot ANOVA: SPSS Output Omnibus Test: Within-Subjects Effects (Movie) 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)

Split Plot ANOVA: SPSS Output 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

Split Plot ANOVA: SPSS Output 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 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 Ghostbusters vs. Indiana Jones Terminator vs. Indiana Jones # of levels in factor q(3, 76) = 0.394, ns q(3, 76) = 8.827, p < .01 q(3, 76) = 8.433, p < .01 sphericity assumed dferror *critical values found using online calculator: http://vassarstats.net/tabs.html critical values: .05 = 3.39 .01 = 4.25

Split Plot ANOVA: Post Hoc Analyses 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) 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 *critical values found using online calculator: http://vassarstats.net/tabs.html

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 Mean Amusement Rating 5 4 3 2 1 0 Sleep deprived Lots of Sleep Sleep Status

Split Plot ANOVA: Post Hoc Analyses Interaction 1: Simple Main Effects of Movie at Each Level of Sleep Step 1: Run a MANOVA using the syntax option in SPSS levels of within-subjects factor (movie) name of within-subjects factor (number of levels) *syntax has been given to you for your assignment between-subjects factor (sleep) with coding comparing means of within-subject factor (movie) at first level of sleep-status (sleep-deprived)

Split Plot ANOVA: Post Hoc Analyses Interaction 1: Simple Main Effects of Movie at Each Level of Sleep Reading the very bottom table of the MANOVA output… movies at sleep-deprived: F(2, 76) = 27.13, p < .001 movies at tons-of-sleep: F(2, 76) = 3.45, p < .05 at least two movie means differ significantly at each sleep level  proceed with Tukey’s HSD to pinpoint differences

Split Plot ANOVA: Post Hoc Analyses Interaction 1: Simple Main Effects of Movie at Each Level of Sleep Step 2: follow up with separate Tukey’s HSD analyses of condition means using the POSTHOC program (sphericity assumed values, no pooled error term) 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 withinsubjects factor dferror •  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

Split Plot ANOVA: Post Hoc Analyses Interaction 2 : Simple Main Effects of Sleep at Each Level of Movie Mean Amusement Rating i.e. simple main effects of between-subjects factor at each level of within-subjects factor 5 4 3 2 1 0 Ghostbusters Terminator Indiana Jones 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.

Split Plot ANOVA: Post Hoc Analyses Interaction 2 : Simple Main Effects of Sleep at Each Level of Movie Step 1: run a one-way ANOVA in SPSS for all variables Analyze  Compare Means  One-Way ANOVA 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.

Split Plot ANOVA: Post Hoc Analyses Interaction 2 : Simple Main Effects of Sleep at Each Level of Movie •  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)

Split Plot ANOVA: Post Hoc Analyses Interaction 2 : Simple Main Effects of Sleep at Each Level of Movie Step 2: Calculate pooled error terms for the analysis using the POSTHOC program •  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)

Split Plot ANOVA: Post Hoc Analyses Interaction 2 : Simple Main Effects of Sleep at Each Level of Movie 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)

€ Split Plot ANOVA: Post Hoc Analyses Interaction 2 : Simple Main Effects of Sleep at Each Level of Movie Step 3: Calculate F-obtained values for each comparison by hand Calculating the F-obtained: MSBG F= MSerror where… Reporting the F-obtained: F(dfBG, dferror) = calculated value where… MSBG = between-groups MS value of interest (one-way ANOVA output) dfBG = between-groups df value of interest (one-way ANOVA output) MSerror = pooled MS error value (from POSTHOC) dferror = pooled df error value, rounded up (from POSTHOC)

Split Plot ANOVA: Post Hoc Analyses Interaction 2 : Simple Main Effects of Sleep at Each Level of Movie Step 3: Calculate F-obtained values for each comparison by hand Numerator values… FGhostbusters = € Denominator value… (as outputted by POSTHOC program) € € FTerminator = MSBG 3.192 = = 3.324 MSerror .9603333 MSBG 1.764 = = 1.837 MSerror .9603333 FIndiana Jones = MSBG 2.970 = = 3.093 MSerror .9603333

Split Plot ANOVA: Post Hoc Analyses Interaction 2 : Simple Main Effects of Sleep at Each Level of Movie 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) 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). •  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)

Split Plot ANOVA: Post Hoc Analyses Interaction 2 : Simple Main Effects of Sleep at Each Level of Movie 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 q(df1, df2) = obtained value # levels in between-subjects factor dferror (pooled, rounded up) 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)

Assignment #6

Assignment: What to Submit •  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 Report for Question #1 Example: Method 1 of Interpreting Interaction (point form) 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…

Assignment: What to Report for Question #1 Example: Method 2 of Interpreting Interaction (point form) 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… Don’t forget to address part B (in full sentences)!

Assignment: What to Report in Results Section •  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

Helpful Hints •  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 withinsubjects 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 betweensubjects factor

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

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