Splitplot and Repeated measures designs

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Split-plot andRepeated measuresdesigns

• Kingshuk Roy Choudhury
• Statistics Dept., UCC

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Topics

• Split Plot Experiments Design and Analysis
• Repeated Measures
• a. Growth Curve Analysis

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What is split-plot ?

• In agricultural experiments, certain factors are
  assigned to geographical areas called whole plots
  (e.g. individual farms), while other factors are
  applied to smaller sub-plots or split plots (e.g.
  fields within each farm).
• e.g. One factor in our experiment is insecticide,
  which has to be sprayed on entire farms (whole
  plots) by airplanes. Different varieties of
  seeds could be used in individual fields within a
  farm. With appropriate randomisation, this
  constitutes a split-plot design.
• Four factors in the study insecticide, farms
  (whole plots), varieties, fields (split-plots).

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Split-plot example
Insecticide A
Farm 1
Farm 2
Fields
Insecticide B
Farm 4
Farm 3

• Whole plot factor insecticide ( 2 levels)
  
• Split-plot factor seed (2 levels)
• Randomisation occurs only within each farm

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Randomised block design
Insecticide A
Farm 1
Farm 2
Insecticide B
Farm 4
Farm 3

• Balance does not exist within farms
• Overall balance exists
• Farm effect can influence comparison

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The split-unit principle

• In split-plot designs, we sacrifice precision in
  estimating the whole-plot factor in order to be
  better able to measure the sub-plot (or
  split-plot) factor.
• E.g. in previous expt., the sub-plot factor is
  seed type and this effect is estimated very
  precisely because the seed factor is balanced
  within each farm.Thus differences between farms
  would not affect the seed type comparison.
• On the other hand, the main plot factor is
  insecticide type and this comparison may be
  affected because Insecticide A is applied to
  farms 1 and 2, whereas Insecticide B is applied
  to farms 3 and 4, which may have different
  growing conditions than 1 and 2.

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Other split plot examples

• Comparing milking machines. Each machine produces
  substantial amount of milk. Other comparisons,
  such as different methods of pasteurising or
  cooling, can be conducted with smaller amounts of
  milk per treatment.
• Comparing harvester combines. Each combine
  harvests a large batch of corn. Threshing
  machines can be compared on smaller amounts taken
  from the same combine.
• In industrial production, certain processes
  produce goods in large batches, whereas other
  processes can work on smaller amounts. Batch
  effect whole plot factor

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Guayule experiment

• Experiment to determine effect of treating
  guayule (rubber plant) seeds prior to planting.
• 8 varieties of seeds and 4 different treatments
  on seeds. Each variety of seed is treated in a
  batch of 100.
• Seeds are planted in a flat (planting tray). Only
  one variety per flat, each trt is equi-present in
  each flat (400 seeds in all). Three replications
  of the experiment.
• Total no. of flats
• Factors in the experiment
• Whole plots
• Whole plot factor
• Sub-plot factor

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Guayule experiment visualised
Variety 1
Variety 4
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Guayule data
Yield or outcome of experiment is no. of seeds
germinated from each batch
variety trt reps plants flats 1
V1 T1 1 66 1.V1 2 V2
T1 1 77 1.V2 3 V3 T1 1
51 1.V3 . 8 V8 T1
1 49 1.V8 9 V1 T2 1 12
1.V1 10 V2 T2 1 26 1.V2 11
V3 T2 1 8 1.V3 .. 33
V1 T1 2 63 2.V1 34 V2
T1 2 47 2.V2 35 V3 T1 2
81 2.V3 ...
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Summarising Guayule data
Rounded cell means
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Split-plot ANOVA
Whole plot analysis (based on flat totals)
Split-plot analysis (based on data within flats)
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Points to note in split-plot ANOVA

• Analysis of whole plot and sub-plot factors is
  separate
• This is because sub-plot measurements
  (treatment averages) are correlated within the
  whole plots (flats).
• Comparison of two different factor combinations
  (e.g. V1 vs. V4 or V1 T3 vs. V2 T4) can be done
  by means of a t-test. Numerator of t-test is just
  difference in averages. Denominator of t-test is
  standard error.
• Calculation of standard errors of estimated
  effects is complicated due to the correlation
  between sub-plot measurements. Formulae for
  standard errors can be found in texts (e.g. Mead
  et. al. ) and uses ANOVA results.