Control of Experimental Error

1
Control of Experimental Error

• Blocking -
• A block is a group of homogeneous experimental
  units
• Maximize the variation among blocks in order to
  minimize the variation within blocks
• Reasons for blocking
• To remove block to block variation from the
  experimental error (increase precision)
• Treatment comparisons are more uniform
• Increase the information by allowing the
  researcher to sample a wider range of conditions

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Blocking

• At least one replication is grouped in a
  homogeneous area

3
Criteria for blocking

• Proximity or known patterns of variation in the
  field
• gradients due to fertility, soil type
• animals (experimental units) in a pen (block)
• Time
• planting, harvesting
• Management of experimental tasks
• individuals collecting data
• runs in the laboratory
• Physical characteristics
• height, maturity
• Natural groupings
• branches (experimental units) on a tree (block)

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Randomized Block Design

• Experimental units are first classified into
  groups (or blocks) of plots that are as nearly
  alike as possible
• Linear Model Yij ? ?i ?j ?ij
• ? mean effect
• ßi ith block effect
• ?j jth treatment effect
• ?ij treatment x block interaction, treated as
  error
• Each treatment occurs in each block, the same
  number of times (usually once)
• Also known as the Randomized Complete Block
  Design
• RBD RCB RCBD
• Minimize the variation within blocks - Maximize
  the variation between blocks

5
Pretty doesnt count here
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Randomized Block Design

• Other ways to minimize variation within blocks
• Field operations should be completed in one block
  before moving to another
• If plot management or data collection is handled
  by more than one person, assign each to a
  different block

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Advantages of the RBD

• Can remove site variation from experimental error
  and thus increase precision
• When an operation cannot be completed on all
  plots at one time, can be used to remove
  variation between runs
• By placing blocks under different conditions, it
  can broaden the scope of the trial
• Can accommodate any number of treatments and any
  number of blocks, but each treatment must be
  replicated the same number of times in each block
• Statistical analysis is fairly simple

8
Disadvantages of the RBD

• Missing data can cause some difficulty in the
  analysis
• Assignment of treatments by mistake to the wrong
  block can lead to problems in the analysis
• If there is more than one source of unwanted
  variation, the design is less efficient
• If the plots are uniform, then RBD is less
  efficient than CRD
• As treatment or entry numbers increase, more
  heterogeneous area is introduced and effective
  blocking becomes more difficult. Split plot or
  lattice designs may be better suited.

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Uses of the RBD

• When you have one source of unwanted variation
• Estimates the amount of variation due to the
  blocking factor

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Randomization in an RBD

• Each treatment occurs once in each block
• Assign treatments at random to plots within each
  block
• Use a different randomization for each block

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Analysis of the RBD

• Construct a two-way table of the means and
  deviations for each block and each treatment
  level
• Compute the ANOVA table
• Conduct significance tests
• Calculate means and standard errors
• Compute additional statistics if appropriate
• Confidence intervals
• Comparisons of means
• CV

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The RBD ANOVA
Source df SS MS F Total
rt-1 SSTot Block r-1 SSB
MSB MSB/MSE SSB/(r-1) Treatmen
t t-1 SST MST MST/MSE
SST/(t-1) Error (r-1)(t-1) SSE
MSE SSTot-SSB-SST SSE/(r-1)(t-1)
MSE is the divisor for all F ratios
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Means and Standard Errors
Standard Error of a treatment mean
Confidence interval estimate
Standard Error of a difference
Confidence interval estimate on a difference
t to test difference between two means
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Numerical Example

• Test the effect of different sources of nitrogen
  on the yield of barley
• 5 sources and a control
• Wanted to apply the results over a wide range of
  conditions so the trial was conducted on four
  types of soil
• Soil type is the blocking factor
• Located six plots at random on each of the four
  soil types

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ANOVA
Source df SS MS F Total 23 492.36
Soils (Block) 3 192.56 64.19 21.61 Fertilize
r (Trt) 5 255.28 51.06 17.19 Error 15 44.52
2.97
Standard error of a treatment mean 0.86 CV
5.6 Standard error of a difference between two
treatment means 1.22
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Confidence Interval Estimates
34.41 30.54 29.19 28.86 27.59 23.51 36.25 32.38 3
1.02 30.70 29.42 25.35 38.09 34.21 32.86 32.54 31.
26 27.19
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Report of Analysis

• Differences among sources of nitrogen were highly
  significant
• Ammonium sulfate (NH4)2SO4 produced the highest
  mean yield and CO(NH2)2 produced the lowest
• When no nitrogen was added, the yield was only
  25.35 kg/plot
• Blocking on soil type was effective as evidenced
  by
• large F for Soils (Blocks)
• small coefficient of variation (5.6) for the
  trial

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Is This Experiment Valid?
Full Irrigation
Irrigated Pre-Plant
No Irrigation
19
Missing Plots

• If only one plot is missing, you can use the
  following formula

Yij ( rBi tTj - G)/(r-1)(t-1)

• Where
• Bi sum of remaining observations in the ith
  block
• Tj sum of remaining observations in the jth
  treatment
• G grand total of the available observations
• t, r number of treatments, blocks, respectively

• Total and error df must be reduced by 1
• Used only to obtain a valid ANOVA
• No change in Error SS
• SS for treatments may be biased upwards

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Two or Three Missing Plots

• Estimate all but one of the missing values and
  use the formula
• Use this value and all but one of the remaining
  guessed values and calculate again continue in
  this manner until you have resolved all missing
  plots
• You lose one error degree of freedom for each
  substituted value
• Better approach Let SAS account for missing
  values
• Use a procedure that can accommodate missing
  values (PROC GLM, PROC MIXED)
• Use adjusted means (LSMEANS) rather than MEANS
• degrees of freedom are subtracted automatically
  for each missing observation

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Relative Efficiency

• A way to measure the efficiency of RBD vs CRD

RE (r-1)MSB r(t-1)MSE/(rt-1)MSE

• r, t number of blocks, treatments in the RBD
• MSB, MSE block, error mean squares from the RBD
• If RE gt 1, RBD was more efficient
• (RE - 1)100 increase in efficiency
• r(RE) number of replications that would be
  required in the CRD to obtain the same level of
  precision