ETH p. 1/21. Split Plot Designs. Large and small units Confounding main effects Repeated measures anova
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1 ETH p 1/21 Split Plot Designs Large and small units Confounding main effects Repeated measures anova
2 ETH p 2/21 Study in Dental Medicine Can measurement of electric resistance help in detecting tooth decay? 40 measurements on teeth with and without inflamed gums, with and without special treatment 2 2 factorial with factor A (inflammation) and factor B (special treatment)
3 ETH p 3/21 Correct anova table? Source df MS F A 1 MS A /MS res B 1 MS B /MS res AB 1 MS AB /MS res Residual 36 Total 39 Depends on design structure How many subjects, how many teeth?
4 ETH p 4/21 8 subjects, one tooth each One treatment per person, 5 repeated measurements Stratum Source df F Person A 1 MS A /MS res person B 1 MS B /MS res person AB 1 MS AB /MS res person Residual 4 Total 7 Measurement Residual 32 Total 39
5 ETH p 5/21 5 subjects, 4 teeth each Each person has 2 inflamed and 2 not inflamed teeth Each tooth was measured once with special treatment and once without special treatment Stratum Source df F Person Person 4 Tooth A 1 MS A /MS res tooth Residual 14 Total 15 Measurement B 1 MS B /MS res meas AB 1 MS AB /MS res meas Residual 18 Total 20 Total 39
6 ETH p 6/21 Special properties of this design Replication on three stages: persons, teeth and measurements One factor varies between teeth, the other between measurements main plot= tooth, sub-plot = measurement
7 ETH p 7/21 Split-plot design A first factor needs to be applied to large plots, called main plots Main plots are split into smaller plots, called subplots These are assigned to different levels of a second factor Two different levels for comparing factor levels: effects of the first factor must be examined relative to main plot variation, effects of the second factor must be examined relative to subplot variation
8 ETH p 8/21 Rice experiment 4 irrigation methods I1-I4 on main plots, 3 fertilizer mixtures X,Y, Z on sub-plots, 2 complete replicates Layout: Block I Block II z x y z x y z x x z z y z x x y y y x x y z y z I4 I2 I3 I1 I2 I1 I4 I3 Irrigation is confounded with main plots
9 ETH p 9/21 Model Y ijk = µ + b i + Irr j + ǫ ij + Fert k + (Irr : Fert) jk + δ ijk i = 1,, I; j = 1,, J; k = 1,, n b i : ith block effect Irr j : jth effect of irrigation ǫ ij : main plot error Fert k : kth effect of fertilizer (Irr : Fert) ij : jkth interaction δ ijk : sub-plot error
10 ETH p 10/21 Skeleton Anova Stratum Source df F Blocks Blocks 1 Main plots Irr 3 MS Irr /MS res main Residual 3 Total 7 Sub-plots Fert 2 MS Fert /MS res sub Irr:Fert 6 MS Irr:Fert /MS res sub Residual 8 Total 16 Total 23
11 ETH p 11/21 Data on crop yield (tonnes/hectare) Irrigation Block I I1 I2 I3 I4 Fertilizer x y z Irrigation Block I I1 I2 I3 I4 Fertilizer x y z
12 ETH p 12/21 Graphical display yield 25 yield irrigation fertilizer
13 ETH p 13/21 Anova Table > mod2=aov(yield irrigation*fertilizer+error(block/irrigation)) > summary(mod2) Error: block Df Sum Sq Mean Sq F value Pr(>F) Residuals Error: block:irrigation Df Sum Sq Mean Sq F value Pr(>F) irrigation Residuals Error: Within Df Sum Sq Mean Sq F value Pr(>F) fertilizer ** irrigation:fertilizer Residuals
14 ETH p 14/21 Repeated measures, Longitudinal data Subjects are measured more than once Compare values of an outcome variable before and after a treatment Look at changes over time in an outcome variable Measurements at the same unit are correlated with each other
15 ETH p 15/21 Rheumatoid arthritis Patients with rheumatoid arthritis and normal controls obtained an anti-inflammatory analgesic Serum clonixin levels (in mg/ml) were measured 1/2, 1, 2, 4, 6 and 8 hours after administration of a single dose of three 250 mg tablets of clonixin Time (in hours) Subject 1/
16 Patients with arthritis Time Clonixin levels / ETH p 16/21
17 Controls Time Clonixin levels / ETH p 17/21
18 ETH p 18/21 How to analyze the data? Separate analysis at each time point Summary measure: mean, maximal value, time until maximal value, steepest slope, area under curve (AUC) Repeated measures anova, split-plot approach: subjects=main plots, subject time=subplot Critical assumption is sphericity: same variance of measurements at each time point and the same correlation between measurements taken at different times Correction for deviation from sphericity: Greenhouse-Geisser, Huynh-Feldt
19 ETH p 19/21 Anova table Stratum Source df F Person Treatment 1 MS treat /MS res P erson Residual 22 Total 23 Person Time Time 5 MS time /MS res P erson time Treatment Time 5 Residual 110 Total 120 Total 143
20 ETH p 20/21 R Output > mod1 = aov(clonixin time*group + Error(id),data=rheumalong) > summary(mod1) Error: id Df Sum Sq Mean Sq F value Pr(>F) group Residuals Error: Within Df Sum Sq Mean Sq F value Pr(>F) time <2e-16 *** time:group Residuals
21 ETH p 21/21 Other types of split-plot designs Repeated splitting: a third factor may be applied to sub-subplots Confounding interactions of sub-plot factors in split-plot designs Other designs for main plots, eg Latin squares Strip-plot design Block I Block II
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Chapter 5 Introduction to Factorial Designs 5. Basic Definitions and Principles Stud the effects of two or more factors. Factorial designs Crossed: factors are arranged in a factorial design Main effect: