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Randomized Block Design - Download as PowerPoint


									Control of Experimental Error
  Blocking -
   – A block is a group of homogeneous experimental
   – 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
 At least one replication is grouped in a
  homogeneous area

      A B D A                 A B D C

      C D B C                 C D B A
      B A D C                 B A D C
      Just replication          Blocking
 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)
 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
   – βi =     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
Pretty doesn’t count here
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
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
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
 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.
Uses of the RBD

 When you have one source of unwanted
 Estimates the amount of variation due to
 the blocking factor
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
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
Source    df           SS                             MS                F

Total     rt-1         SSTot =

                                             
                         i  j Yij  Y
Block     r-1          SSB =                          MSB =            MSB/MSE

                                     
                        t i Yi  Y                   SSB/(r-1)

Treatment t-1          SST =                          MST =            MST/MSE

                                         
                        r j Y j  Y                  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
Means and Standard Errors

Standard Error of a treatment mean               s Y  MSE r

Confidence interval estimate               L   i   Y i  t  MSE r

Standard Error of a difference               s Y  Y   2MSE r
                                                1   2

Confidence interval estimate on a difference
              L   1   2    Y1  Y 2   t  2MSE r

 t to test difference between two means                 t Y1  Y 2
                                                           2MSE r
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
  Source             df           SS          MS             F

  Total              23           492.36

  Soils (Block)      3            192.56     64.19       21.61**

  Fertilizer (Trt)   5            255.28     51.06       17.19**

  Error              15            44.52      2.97

 Source (NH4)2SO4 NH4NO3 CO(NH2)2 Ca(NO3)2 NaNO3 Control
  Mean      36.25         32.38    29.42   31.02     30.70       25.35

Standard error of a treatment mean = 0.86         CV = 5.6%
Standard error of a difference between two treatment means = 1.22
Confidence Interval Estimates
      (NH4)2SO4 NH4NO3   Ca(NO3)2   NaNO3   CO(NO2)2   Control

      34.41    30.54     29.19      28.86    27.59     23.51
      36.25    32.38     31.02      30.70    29.42     25.35
      38.09    34.21     32.86      32.54    31.26     27.19
 Report of Analysis
 Differences among sources of nitrogen were highly
 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
 Blocking on soil type was effective as evidenced by:
   – large F for Soils (Blocks)
   – small coefficient of variation (5.6%) for the trial
Is This Experiment Valid?
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
     Two or Three Missing Plots
            Yij = ( rBi + tTj - G)/[(r-1)(t-1)]
 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
   – Use adjusted means (LSMEANS) rather than MEANS
   – degrees of freedom are subtracted automatically for each missing
Relative Efficiency
 A way to measure the efficiency of RBD vs CRD
      RE = [(r-1)MSB + r(t-1)MSE]/(rt-1)MSE

                 MSECRD      Estimated Error for a CRD
            RE 
                 MSERBD      Observed Error for RBD

     r, t = number of blocks, treatments in the RBD
     MSB, MSE = block, error mean squares from the RBD
     If RE > 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

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