Chris Brien

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Design and analysis of experiments with a
laboratory phase subsequent to an initial phase

• Chris Brien
• University of South Australia
• Bronwyn Harch
• Ray Correll
• CSIRO Mathematical and Information Sciences

Chris.brien_at_unisa.edu.au
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Outline

• Designing two-phase experiments
• A biodiversity example
• When first-phase factors do not divide lab
  factors
• Trend adjustment in the biodiversity example
• Taking trend into account in design
• Duplicates

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Notation

• Factor relationships
• AB factors A and B are crossed
• A/B factor B is nested within A
• Generalized factor
• A?B is the ab-level factor formed from the
  combinations of A with a levels and B with b
  levels
• Symbolic mixed model
• Fixed terms random terms (AB Blocks/Runs)
• AB A B A?B
• A/B/C A A?B A?B?C
• Sources in ANOVA table
• AB a source for the interaction of A and B
• BA a source for the effects of B nested within A

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1. Designing two-phase experiments

• Two-phase experiments as introduced by McIntyre
  (1955)
• Consider special case of second phase a
  laboratory phase

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General considerations

• Need to randomize laboratory phase so involve two
  randomizations
• 1st-phase treatments to 1st-phase, unrandomized
  factors
• latter to unrandomized, laboratory factors
• Often have a sequence of analyses to be performed
  and how should one group these over time.
• Fundamental difference between 1st and 2nd
  randomizations
• 1st has randomized factors crossed and nested
• 2nd has two sets of factors and all combinations
  of the two sets are not observable within sets
  are crossed or nested ? tendency to ignore 1st
  phase, unrandomized factors.
• Categories of designs
• Lab phase factors purely hierarchical or involve
  crossed rows and columns
• Two-phase randomizations are composed or
  randomized-inclusive (Brien Bailey, 2006)
  related to whether 1st-phase, unrandomized
  factors divide laboratory unrandomized factors
• Treatments added in laboratory phase or not
• Lab duplicates included or not

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1a) A Biodiversity example
(Harch et al., 1997)

• Effect of tillage treatments on bacterial and
  fungal diversity
• Two-phase experiment field and laboratory phase
• Field phase
• 2 tillage treatments assigned to plots using RCBD
  with 4 blocks
• 2 soil samples taken at each of 2 depths
• 2 ? 4 ? 2 ? 2 32 samples

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Laboratory phase

• Then analysed soil samples in the lab using Gas
  Chromatography - Fatty Acid Methyl Ester
  (GC-FAME) analysis
• 2 preprocessing methods randomized to 2 samples
  in each Plot?Depth
• All samples analysed twice necessary?
• once on days 1 2 again on day 3

• In each Int2, 16 samples analyzed

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Processing order within Int1?Int2

• Logical as similar to order obtained from field
• But confounding with systematic laboratory
  effects
• Preprocessing method effects
• Depth effects
• Depths assigned to lowest level - sensible?

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Towards an analysis
2 Samples in B, P, D 4 Blocks 2 Plots in B 2
Depths
32 samples

• 64 analyses divided up hierarchically by 6 x
  2-level factors Int1Int6 of size 32, , 2
  analyses, respectively.

• Dashed arrows indicate systematic assignment

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Analysis of example for lab variability

• Variability for
• Int4 gtInt5 gt Int6
• 8 gt 4 gt2 Analyses
• Int1, Int2, Int3 small (lt Int4)

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Alternative blocking for the biodiversity example

• Want to assign the 32 samples to 64 analyses
• Consider with the experimenter
• Uninteresting effects Blocks
• Large effects Depth?
• Some treatments best changed infrequently
  Methods?
• Period over which analyses effectively
  homogeneous 16 analyses? 4 analyses? 2 analyses?

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Alternative blocking for the biodiversity example

• For now, divide 64 analyses into 2 Occasions
  Int1, 4 Times Int2?Int3, 8 Analyses
  Int4?Int5?Int6

• Blocks of 8 would be best as 2 Plots x 2 Depths x
  2 Methods, but Blocks Times too variable.
• Best if pairs of analyses in a block.
• Also Times are similar ? could take 4 Times x 2
  Analyses.
• Many other possibilities e.g. blocks of size 4
  with Depths randomized to pairs of blocks.

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Proposed laboratory design

• Organise 64 analyses into blocks of 8

• Randomization of field units ignores treats
• Two composed randomizations (Brien and Bailey,
  2006)
• Field treats to samples to analyses
• Two independent randomizations (Brien and Bailey,
  2006)
• Field and lab treats to samples
• Experiment with
• hierarchical lab phase, composed randomizations,
  duplicates and treatments added at laboratory
  phase.

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Decomposition table for proposed design

• Important for design shows confounding and
  apportionment of variability

Each of the 15 lines is a separate subspace in
the final decomp-osition Note Residual df
determined by field phase

• Randomization-based mixed model (Brien Bailey,
  2006)
• TillMethDep ((Blk/Plot)Dep)/Sample Dep
  Occ/Int/Anl

• Or TillMethDep ((Blk/Till)Dep)/Meth Dep
  Occ/Int/Anl

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1b) When first-phase factors do not divide lab
factors

• Need to use a nonorthogonal design and two
  randomized-inclusive randomizations (Brien and
  Bailey, 2006)
• Willow experiment (Peacock et al, 2003)
• Beetle damage inhibiting rust on willows?
• Glasshouse and lab phases
• Example here same problem but different details
• Will be an experiment with
• hierarchical lab phase, randomized-inclusive
  randomizations, no duplicates and no treatments
  added at laboratory phase

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Willow experiment (contd)

• Glasshouse 60 locations each with a plant
• 12 damages to assign to locations.
• Only 6 locations per bench
• Damages does not divide no. locations or benches
  so IBD
• Use RIBD with v 12, k 6, E 0.893, bound
  0.898.
• Randomize between Reps, Benches within Reps and
  Locations within Benches.

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Willow experiment (contd)

• Lab phase disk/plant put onto 20 plates, 3 disks
  /plate
• Plates divided into 5 groups for processing on an
  Occasion
• Locations does not divide Cells
• divide 6 Locations into 2 sets of 3 cannot do
  this ignoring Damages
• RIBD related to 1st-phase (v 12, k 3, r 5, E
  0.698, bound 0.721)
• In fact got this design using CycDesgN (Whittaker
  et al, 2002) and combined pairs of blocks to get
  1st-phase.
• To include Locations, read numbers as Locations
  with these Damages.
• Renumber Locations to L1 and L2 to identify those
  assigned same Plate.

• Sometimes better design if allow for lab phase in
  designing 1st

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Decomposition table for proposed design

• Each of the 6 lines is a separate subspace in the
  final decomposition.
• Note Residual df for Locations from 1st phase is
  39 and has been reduced to 29 in lab phase.
• xs are strata variances or portions of EMSq
  from cells and ?s from locations.
• Four estimable variance functions xO hR, xOP
  hRB, xOP hRBL, xOPC hRBL, although 2nd may be
  difficult.
• Randomization-based mixed model (Brien Bailey,
  2006) that corresponds to estimable quantities
• Damages Rep/Benches/L1 Occasions?Plates?Cells
  .
• Must have Locations in the form of L1 in this
  model - i.e. cannot ignore unrandomized factors
  from 1st phase.

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Willow experiment (contd)

• Glasshouse 50 locations each with a plant
• 10 Damages assigned to 5 blocks using RCBD
• Disks taken high, middle low leafs

• Lab phase 6 disks assigned put onto 25 plates, 2
  locations x 3 leaves per plate
• Plates divided into 5 groups for processing on an
  Occasion.
• Locations assigned using a resolvable PBIB(2) for
  v 10, k 2 and r 5.
• cannot assign Locations, ignoring Damages
• use L1 to explicitly identify which Damage
  assigned to a Location - connects them

• Cannot ignore Locations as need it in the
  analysis.

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Willow experiment ANOVA

• Mixed model (a model of convenience)
• DamagesLeafPosn Occasions/Plates
  Blocks?(LocationsLeafPosn)

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2. Trend in the biodiversity example

• Trend can be a problem in laboratory phase. Is it
  here?
• Plot of Lab-only residuals in run order for 8
  Analyses within Times

• Linear trend that varies evident
• Proposed design (4 x 2) is appropriate (? trend
  low Times variability) ? smallest Analysis
  Residual

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Trend adjustment for example

• REML analysis with vector of 18 for each
  Occasion
• Significant different linear trends (p lt 0.001)
• Effect on fixed effects

• Trend adjustment reduced
• Tillage effect from -0.99 to -0.07
• PlotBlock component from 13.25 to 0.001.
• Low PlotBlock df makes this dubious.

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4. Taking trend into account in design

• Cox (1958, section 14.2) discusses trend
  elimination
• concludes that, where the estimation of trend not
  required, use of blocking preferred to trend
  adjustment
• Yeh, Bradley and Notz (1985) combine blocking for
  trend and adjustment provide trend-free and
  nearly trend-free designs with blocks
• allow for common quadratic trends within blocks
• minimize the effects of adjustment
• Look at design of laboratory phase
• for field phase with RCBD, b 3, v 18
• 3 Occasions in lab phase to which 3 Blocks
  randomized
• allow for different linear cubic Trends within
  each Occasion

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Different designs for blocks of 18 analyses

• RCBD for this no. treats relatively efficient
  when adjusting for trend
• Blocks assigned to 3 Occasions 6 Analyses
  (blocking perpendicular to trend?)
• Use when Occasions variability low e.g.
  recalibration
• Nearly Trend-Free (using Yeh, Bradley and Notz ,
  1985) worse than RCBD for different trends
• optimal for common linear trend.
• Still to investigate designs that protect against
  different trends.

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Comparing RCBD with RIBDs for k 6,9

• Use Relative Efficiencies
• av. pairwise variance of RCBD to RIBD for sets
  of generated data
• Generate using random model
• Y Occasion IntervalOccasion
• AnalysesOccasions?Interval
• PlotsBlocks

• Expect efficiency
• k 6 gt k 9
• RIBD gt RCBD
• provided
• gBP not dominant and
• gOI is non-zero.
• How much?
• gBP lt 10
• gOI 0.5 (very little extra required, but after
  trend adjustment)

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Resolvable design with cols latinized rows
using CycDesgN (Whittaker et al, 2002), Intra E
0.49

• Expect LRCD gt RIBD if gIA ? 0 and gBP not
  dominant
• How much?
• If gIA gt 1 irrespective of gOI. Again only small
  gs.
• Expect LRCD gt RCD if Occasions different.
• REs ? as gBP ?
• (LRCD/RCD lt 2 if gBP 2.5).

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4. Duplicates

• Commonly used, but only need in two-phase
  experiments if Lab variation large compared to
  field.
• Possibilities
• Separated analyze all then reanalyze all in
  different random order
• Nested some analyzed then these reanalyzed in
  a different random order
• Crossed some analyzed then these reanalyzed in
  same order
• Consecutive duplicate immediately follows first
  analysis
• Randomized some analysed everything randomized
• From ANOVAs and REs to randomized, when adjusting
  for different cubic trends, conclude
• Separated duplicates superior, with nested
  duplicates 2nd best little gain in efficiency if
  gOI 0.5 and gBP is considerable
• Crossed and consecutive duplicates perform poorly
  with RE lt 1 often

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5. Summary for lab phase design

• Two-phase initial expt lab phase
• Leads to ? 2 randomizations composed or
  r-inclusive related to whether 1st phase,
  unrandomized factors divide laboratory,
  unrandomized factors
• Use of pseudofactors with r-inclusive does not
  ignore field terms and makes explicit what has
  occurred
• Adding treatments in lab phase leads to more
  randomizations
• Cannot improve on field design but can make worse
• Important to have some idea of likely laboratory
  variation
• Will there be recalibration or the like?
• Are consistent differences between and/or across
  Occasions likely?
• How does the magnitude of the field and
  laboratory variation compare?
• Are trends probable common vs different linear
  vs cubic?
• Will laboratory duplicates be necessary and how
  will they be arranged?
• If yes, separated duplicates best but other
  arrangements may be OK.
• RCBD will suffice if
• field variation gtgt lab variation, in which case
  duplicates unnecessary.
• after adjustment for trend, no extra laboratory
  variation, except Occasions
• can block across occasions when no Occasion
  differences
• If Intervals differences, RIBD better than RCBD -
  not much needed.
• LRCD better than RIBD provided, after trend
  adjustment, moderate consistent differences
  between Analyses across Occasions.

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References

• Brien, C.J., and Bailey, R.A. (2006) Multiple
  randomizations (with discussion). J. Roy.
  Statist. Soc., Ser. B, 68, 571609.
• Cox, D.R. (1958) Planning of Experiments. New
  York, Wiley.
• John, J.A. and Williams, E.R. (1995) Cyclic and
  Computer Generated Designs. Chapman Hall,
  London.
• Harch, B.E., Correll, R.L., Meech, W., Kirkby,
  C.A. and Pankhurst, C.E. (1997) Using the Gini
  coefficient with BIOLOG substrate utilisation
  data to provide an alternative quantitative
  measure for comparing bacterial soil communities.
  Journal of Microbial Methods, 30, 91101.
• McIntyre, G. (1955) Design and analysis of two
  phase experiments. Biometrics, 11, p.32434.
• Peacock, L., Hunter, P., Yap, M. and Arnold, G.
  (2003) Indirect interactions between rust
  (Melampsora epitea) and leaf beetle (Phratora
  vulgatissima) damage on Salix. Phytoparasitica,
  31, 22635.
• Whitaker, D., Williams, E.R. and John, J.A.
  (2002) CycDesigN A Package for the Computer
  Generation of Experimental Designs. (Version 2.0)
  CSIRO, Canberra, Australia. http//www.ffp.csiro.a
  u/software
• Yeh, C.-M., Bradely, R.A. and Notz, W.I. (1985)
  Nearly Trend-Free Block Designs. J. Amer.
  Statist. Assoc., 392, 98592.

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Web address for link to Multitiered experiments
site
http//chris.brien.name/multitier