27  Split-plot Design

27.1 Farming

27.2 CRD

(a) CRD

(b) CRBD

(c) Split-plot design

Abbildung 27.1: Beispiele für verschiedene Designs.

27.3 Example - Soccer

Das folgende hypothetische Beispiel ist adaptiert nach Kowalski et al. (2007)]

game	pitch	twogoals	nplayer	ntouches	eps
G1	large	yes	6p	free	68.5
G1	large	no	5p	free	66.8
G1	large	yes	5p	restricted	58.5
G1	large	no	6p	free	70.8
G1	large	yes	6p	restricted	61.3
G1	large	no	5p	restricted	51.9

27.4 Example - Soccer

Data from full Experiment

27.5 Whole-plot analysis

Average EPS for each game.

27.6 Whole-plot analysis Analysis of variance

game	eps	pitch
G1	62.94	large
G2	57.81	small
G3	62.92	small
G4	64.34	large

mod_whole_plot <- aov(eps ~ pitch, df_bars)
summary(mod_whole_plot)

            Df Sum Sq Mean Sq F value Pr(>F)
pitch        1  10.69  10.685   1.521  0.343
Residuals    2  14.05   7.024               

27.7 Error components split-plot analysis

Error components in a split-plot analysis

27.8 Complete Split-plot analysis

mod_sp <- aov(eps ~ (pitch+nplayer+twogoals+ntouches)^2 + Error(game), df)
summary(mod_sp)

Error: game
          Df Sum Sq Mean Sq F value Pr(>F)
pitch      1  85.48   85.48   1.521  0.343
Residuals  2 112.39   56.20               

Error: Within
                  Df Sum Sq Mean Sq F value Pr(>F)  
nplayer            1  41.18   41.18   4.210 0.0542 .
twogoals           1  45.36   45.36   4.637 0.0443 *
ntouches           1  75.95   75.95   7.765 0.0118 *
pitch:nplayer      1  78.44   78.44   8.019 0.0107 *
pitch:twogoals     1   1.09    1.09   0.111 0.7424  
pitch:ntouches     1  62.44   62.44   6.383 0.0206 *
nplayer:twogoals   1  27.94   27.94   2.856 0.1074  
nplayer:ntouches   1  43.95   43.95   4.492 0.0474 *
twogoals:ntouches  1   2.94    2.94   0.301 0.5899  
Residuals         19 185.86    9.78                 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

27.9 Model

\[\begin{align*} Y_{hij} &= \mu + \alpha_i + \epsilon_{i(h)}^W \\ & + \beta_j + (\alpha\beta)_{ij} + \epsilon_{j(hi)}^S \end{align*}\]

\(\epsilon_{i(h)}^W \sim \mathcal{N}(0,\sigma_{W}^2)\)
\(\epsilon_{jt(hi)}^S \sim \mathcal{N}(0,\sigma_s^2)\)
\(h=1,\ldots,s\)
\(i=1,\ldots,a\)
\(j=1,\ldots,b\)

27.10 A more common example

Hypothetical two-arm RCT

27.11 Data structure

Ausschnitt der Daten

id	time	group	CMJ
S1	Pre	CON	20.5
S2	Pre	CON	20.8
S11	Pre	TRT	20.9
S1	Post	CON	21.5
S11	Post	TRT	24.6
S3	Ret	CON	20.0
S14	Ret	TRT	22.4

27.12 Model

\[\begin{align*} Y_{hij} &= \mu + \theta_h + \alpha_i + \epsilon_{i(h)}^W \\ & + \beta_j + (\alpha\beta)_{ij} + \epsilon_{j(hi)}^S \end{align*}\]

\(\epsilon_{i(h)}^W \sim \mathcal{N}(0,\sigma_{W}^2)\)
\(\epsilon_{jt(hi)}^S \sim \mathcal{N}(0,\sigma_s^2)\)
\(h=1,\ldots,s\)
\(i=1,\ldots,a\)
\(j=1,\ldots,b\)

27.13 Split-plot analysis

mod <- aov(CMJ ~ group*time + Error(id), df_1)
summary(mod)

Error: id
          Df Sum Sq Mean Sq F value  Pr(>F)    
group      1  70.01   70.01   94.31 1.4e-08 ***
Residuals 18  13.36    0.74                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Error: Within
           Df Sum Sq Mean Sq F value   Pr(>F)    
time        2  98.87   49.43   44.58 1.82e-10 ***
group:time  2  42.94   21.47   19.36 1.95e-06 ***
Residuals  36  39.92    1.11                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

27.14 Again, standard analysis is not correct!

mod_falsch <- aov(CMJ ~ id + group*time, df_1)
summary(mod_falsch)

            Df Sum Sq Mean Sq F value   Pr(>F)    
id          19  83.38    4.39   3.957 0.000189 ***
time         2  98.87   49.43  44.579 1.82e-10 ***
group:time   2  42.94   21.47  19.363 1.95e-06 ***
Residuals   36  39.92    1.11                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

27.15 Alternative analysis using mixed models

mod_lmer <- lmer(CMJ ~ time*group + (1|id), df_1)
anova(mod_lmer)

Analysis of Variance Table
           npar Sum Sq Mean Sq F value
time          2 98.866  49.433  50.099
group         1 70.013  70.013  70.957
time:group    2 42.944  21.472  21.761

27.16 Effect size

\[\begin{align*} \hat{\omega}^2_{\text{between}} &= \frac{SS_A -(a-1)MS_{S/A}}{SS_A + SS_{S/A}+MS_{S/A}} \\ \hat{\omega}^2_{\text{within}} &= \frac{(b-1)(MS_b - MS_{B\times S/A})}{SS_B + SS_{B\times S/A}+SS_{S/A}+MS_{S/A}} \\ \hat{\omega}^2_{AB} &= \frac{(a-1)(b-1)(MS_{AB}-MS_{B\times S/A})}{SS_{AB}+SS_{B\times S/A}+MS_{S/A}} \end{align*}\]

\(A=\)group, \(B=\)time, \(AB=\)group:time, \(S/B=\)Error: id, \(B\times S/A=\)Error: within im Beispiel

27.17 Effect size in R

effectsize::omega_squared(mod)

# Effect Size for ANOVA (Type I)

Group  |  Parameter | Omega2 (partial) |       95% CI
-----------------------------------------------------
id     |      group |             0.82 | [0.67, 1.00]
Within |       time |             0.63 | [0.46, 1.00]
Within | group:time |             0.42 | [0.20, 1.00]

- One-sided CIs: upper bound fixed at [1.00].

27.18 Multiple comparisons

mod_em <- emmeans(mod, ~time*group)
pairs(mod_em)

 contrast            estimate    SE   df t.ratio p.value
 Pre CON - Post CON    -1.271 0.471 36.0  -2.699  0.1002
 Pre CON - Ret CON      0.340 0.471 36.0   0.722  0.9780
 Pre CON - Pre TRT      0.226 0.444 52.4   0.509  0.9956
 Pre CON - Post TRT    -4.775 0.444 52.4 -10.750  <.0001
 Pre CON - Ret TRT     -2.863 0.444 52.4  -6.445  <.0001
 Post CON - Ret CON     1.611 0.471 36.0   3.421  0.0181
 Post CON - Pre TRT     1.497 0.444 52.4   3.370  0.0168
 Post CON - Post TRT   -3.504 0.444 52.4  -7.888  <.0001
 Post CON - Ret TRT    -1.592 0.444 52.4  -3.584  0.0092
 Ret CON - Pre TRT     -0.114 0.444 52.4  -0.256  0.9998
 Ret CON - Post TRT    -5.115 0.444 52.4 -11.515  <.0001
 Ret CON - Ret TRT     -3.203 0.444 52.4  -7.211  <.0001
 Pre TRT - Post TRT    -5.002 0.471 36.0 -10.621  <.0001
 Pre TRT - Ret TRT     -3.089 0.471 36.0  -6.560  <.0001
 Post TRT - Ret TRT     1.912 0.471 36.0   4.060  0.0032

P value adjustment: tukey method for comparing a family of 6 estimates 

27.19 Which comparisons are meaninful?!

mod_em2 <- emmeans(mod, ~time|group)
pairs(mod_em2)

group = CON:
 contrast   estimate    SE df t.ratio p.value
 Pre - Post    -1.27 0.471 36  -2.699  0.0277
 Pre - Ret      0.34 0.471 36   0.722  0.7523
 Post - Ret     1.61 0.471 36   3.421  0.0044

group = TRT:
 contrast   estimate    SE df t.ratio p.value
 Pre - Post    -5.00 0.471 36 -10.621  <.0001
 Pre - Ret     -3.09 0.471 36  -6.560  <.0001
 Post - Ret     1.91 0.471 36   4.060  0.0007

P value adjustment: tukey method for comparing a family of 3 estimates 

27.20 Further reading

27.20.1 Allgemein

Kutner u. a. (2005, p.1172), Kowalski, Parker, und Geoffrey Vining (2007), Altman und Krzywinski (2015)

Altman, Naomi, und Martin Krzywinski. 2015. „Points of significance: split plot design“. Nature Methods 12 (3): 165.

Kowalski, Scott M, Peter A Parker, und G Geoffrey Vining. 2007. „Tutorial: Industrial split-plot experiments“. Quality Engineering 19 (1): 1–15.

Kutner, Michael H, Christopher J Nachtsheim, John Neter, und William Li. 2005. Applied Linear Statistical Models. 5. Aufl. McGraw-Hill Irwin New York.