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

Blocking At least one replication is grouped in a homogeneous area C C C Blocking ABD A A B B D D AA A B B B CC C D D D Just replication

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: Y ij =  +  i +  j +  ij –  =mean effect –β i = i th block effect –  j = j th 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 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-SST SSE/(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

ANOVA Sourcedf SSMSF Total  Soils (Block) ** Fertilizer (Trt) ** Error Source(NH 4 ) 2 SO 4 NH 4 NO 3 CO(NH 2 ) 2 Ca(NO 3 ) 2 NaNO 3 Control Mean Standard error of a treatment mean = 0.86CV = 5.6% Standard error of a difference between two treatment means = 1.22

Confidence Interval Estimates

Report of Analysis Differences among sources of nitrogen were highly significant Ammonium sulfate (NH 4 ) 2 SO 4 produced the highest mean yield and CO(NH 2 ) 2 produced the lowest When no nitrogen was added, the yield was only 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

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

Missing Plots If only one plot is missing, you can use the following formula: n Where: B i = sum of remaining observations in the ith block T j = sum of remaining observations in the jth treatment G = grand total of the available observations t, r= number of treatments, blocks, respectively n Total and error df must be reduced by 1 n Used only to obtain a valid ANOVA - No change in Error SS - SS for treatments may be biased upwards Y ij = ( rB i + tT j - 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 Y ij = ( rB i + tT j - G)/[(r-1)(t-1)] ^

Relative Efficiency A way to measure the efficiency of RBD vs CRD RE = [(r-1)MSB + r(t-1)MSE]/(rt-1)MSE n r, t = number of blocks, treatments in the RBD n MSB, MSE = block, error mean squares from the RBD n If RE > 1, RBD was more efficient n (RE - 1)100 = % increase in efficiency n r(RE) = number of replications that would be required in the CRD to obtain the same level of precision Estimated Error for a CRD Observed Error for RBD