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## PowerPoint Slideshow about 'Control of Experimental Error' - Leo

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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

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

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 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.

Uses of the RBD

- When you have one source of unwanted variation
- 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

The RBD ANOVA

Source df SS MS F

Total rt-1 SSTot =

Block r-1 SSB = MSB = MSB/MSE

SSB/(r-1)

Treatment t-1 SST = MST = MST/MSE

SST/(t-1)

Error (r-1)(t-1) SSE = MSE =

SSTot-SSB-SSTSSE/(r-1)(t-1)

MSE is the divisor for all F ratios

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

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 (NH4)2SO4 NH4NO3 CO(NH2)2 Ca(NO3)2 NaNO3 Control

Mean 36.25 32.38 29.42 31.02 30.70 25.35

ANOVASource 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

Standard error of a treatment mean = 0.86 CV = 5.6%

Standard error of a difference between two treatment means = 1.22

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 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

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

Yij = ( rBi + tTj - G)/[(r-1)(t-1)]

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

Relative Efficiency

- A way to measure the efficiency of RBD vs CRD

RE = [(r-1)MSB + r(t-1)MSE]/(rt-1)MSE

Estimated Error for a CRD

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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