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Topic 6. Randomized Complete Block Design (RCBD) [ST&D Chapter 9 sections 9.1 to 9.7 (except 9.6) and Chapter 15: section 15.8]
6. 1. Variability in the completely randomized design (CRD)
In the CRD it is assumed that all experimental units are uniform. This is not always true in practice, and it is necessary to develop methods to deal with such variability. When comparing two methods of fertilization, if one region of the field has much greater natural fertility than the others, a treatment effect might be incorrectly ascribed to the treatment applied to that part of the field, leading to a Type I error. For this reason, when conducting a CRD, it is always advocated to include as much of the native variability of the experiment as possible within each experimental unit (e.u.), making each e.u. as representative of the whole experiment, and the whole experiment as uniform, as possible. In actual field studies, plots are designed to be long and narrow to achieve this objective. But if the e.u. are more variable, experimental error (MSE) is larger, F (MST/MSE) is smaller, and the experiment is less sensitive. And if the experiment is replicated in a variety of situations to increase its scope the variability increases even further. This additional variability needs to be removed from the analysis so that the actual effects of the treatment can be detected. This is the purpose of blocking.
**6. 2. Randomized complete block design (RCBD)
- Definition** The RCBD assumes that a population of experimental units can be divided into a number of relatively homogeneous subpopulations or blocks. The treatments are then randomly assigned to experimental units such that each treatment occurs equally often (usually once) in each block –i.e. each block contains all treatments. Blocks usually represent levels of naturally occurring differences or sources of variation that are unrelated to the treatments. In the analysis, the variation among blocks can be partitioned out of the experimental error (MSE), thereby reducing this quantity and increasing the power of the test. 6. 2. 2. Example: Consider a field trial comparing three cultivars (A, B, and C) of sugar beet with four replications (in this case, the field is divided into 12 plots; each plot is a replication / e.u.). Suppose the native level of soil nitrogen at the field site varies from high at the north end to low at the south end. In such situation, yield is expected to vary from one end of the field to the other, regardless of cultivar differences. This violates the assumption that the error terms are randomly distributed since the residuals will tend to be positive at the north end of the field and negative at the south end.
North end of field (^) Hi N 1 2 3
4 5 6
7 8 9
10 11 12 South end of field (^) Low N
One strategy to minimize the impact of this variability in native soil fertility on the analysis of treatment effects is to divide the field into four east-west blocks of three plots each. Block North end of field (^) Hi N
1 1 2 3
2 1 2 3
3 1 2 3
4 1 2 3 South end of field (^) Low N
Because these blocks run perpendicular to the nitrogen gradient, the soil within each of these blocks will be relatively uniform. This is the basic idea of the randomized complete block design. Remember that in the completely randomized design (CRD) , each e.u. in the experiment has an equal chance of being assigned any treatment level (i.e. a single randomization is performed for the entire experiment). This is not the case in an RCBD. In the randomized complete block design (RCBD) , each e.u. in a given block has the same chance of being chosen for each treatment (i.e. a separate randomization is performed for each block). Within each block, a fixed number (often 1) of e.u.'s will be assigned to each treatment level. The term "complete" refers to the fact that all treatment levels are represented in each block (and, by symmetry, that all blocks are represented in each treatment level).
After the four separate randomizations, one for each block, the field could look like this:
Block North end of field (^) Hi N
1 B^ A^ C
2 A B C
3 A C B
4 A C B South end of field (^) Low N
6. 2. 3. The linear model
In the case of a single replication per block-treatment combination (like the example above), the underlying linear model that explains each observation is:
Yij = + i + j + ij.
Here, as before, i represents the effect of treatment i (i= 1,…, t) such that the average of
each treatment level is Ti i. In a similar way, j represents the effect of Block j (j =
1, ..., r). , such that the average of each block is Bj j. As always, ij are the
residuals, the deviations of each observation from their expected values. The model in dot notation:
Power 1 – β
distribution of the difference between two means (RCBD and CRD).
α/
MSE (^) RCBD =MSE (^) CRD
df MSE : CRD > RCBD Critical F : RCBD > CRD
Rejection threshold RCBD
Rejection threshold CRD
When there are no significant differences among blocks power CRD > power RCBD
Situation 1: No differences among blocks (i.e. no block effects)
If the RCBD design were applied to an experiment in which the blocks were really no different from one another (i.e. there were no significant block effect), the MSE for the CRD and the RCBD will estimate the same error variance. However the degrees of freedom of the MSE for the RCBD will be smaller than those for the CRD and therefore the critical F value will be larger. The larger critical value in the RCBD moves the threshold of rejection further from the mean than in the CRD. This change in the rejection threshold affects the Type II error ( ) and the power of the test (1- ). Under this scenario, the probability of accepting a false null hypothesis ( ) will be smaller in the CRD than in the RCBD. In other words, the CRD would in this situation be more powerful.
Situation 2: Significant difference among blocks
Now suppose that there really are substantial differences among blocks as well as among treatments (H 0 is false). In a CRD, this variation due to differences among blocks would remain in the error. This larger MSE would make the F statistic (MST/MSE) for the CRD smaller (less significant) than the F statistic for the RCBD.
Under this scenario, the RCBD would still have a larger critical F value in the table because of the lost degrees of freedom; but this may be more than compensated by the smaller MSE. If the effect of the reduced MSE (increased F) outweighs the effect of the larger critical value
(rejection threshold further from 0), the net result will be a smaller , and thus a larger
power (1-) in the RCBD relative to the CRD.
Obviously, one should only use the RCBD when the variation explained by the blocks more than offsets the penalty associated with having fewer error degrees of freedom. So how can one determine when an RCBD is appropriate? This question is answered using the concept of efficiency (introduced in Section 1.4.4.6. and further developed in section
- 3.). 6. 2. 5. Example (from Little and Hills book)
This experiment was conducted to investigate the effect of estrogen on weight gain in sheep.
The four treatments in the experiment are a factorial combinations of two separate factors: Gender of sheep (male and female) and amount of estrogen (S0 and S3). Although this experiment could be analyzed as a factorial, in this example we are treating the four treatments and four levels of a single factor (gender-estrogen combination).
Sheep from four different ranches were involved in the experiment. Anticipating that differences in herd management may affect the results, the researchers blocked by ranch. The completeness of an RCBD demanded, therefore, that each ranch volunteer four sheep to the experiment, two males and two females, providing one replication of each treatment level from each ranch.
Table 6.1 RCBD. Effect of estrogen on weight gain in sheep (lbs).
Block (=ranch) Treatment Treatment I II III IV Total Mean F-S0 47 52 62 51 212 53 M-S0 50 54 67 57 228 57 F-S3 57 53 69 57 236 59 M-S3 54 65 74 59 252 63 Block Total 208 224 272 224 928 Block Mean 52 56 68 56 58
Table 6.2 RCBD ANOVA
Source of Variation df SS MS F^1 Totals 15 854 Blocks 3 576 192.00 24.69** Treatments^3 208 69.33^ 8.91** Error 9 70 7. (^1) F (3, 9, 0.05) = 3.
Table 6.3 CRD ANOVA
Recall that the F statistic = MST/MSE. The experimental design primarily affects the MSE since the degrees of freedom for treatments is always (t – 1) and the variation due to treatments is independent of (i.e. orthogonal to) the variation due to blocks and the experimental error. The information per replication in a given design is:
2
I
Therefore, the relative efficiency of one design another is
2 1
2 2
2 2
2 1 2
1 1 : (^21)
I
I
RE
In reality, we never know the true experimental error ( ^2 ); we only have an estimate of
it (MSE). To pay for this lack of knowledge, a correction factor is introduced into the expressions for information (I) and relative efficiency (RE) (Cochram and Cox, 1957). The following formulas include this correction factor and give an estimate of the relative amount of information provided by two designs:
df MSE
df I MSE
MSE^1
2 1 1
1 2 2
2 2
2
1 1
1
2
1 1 : (^2) ( 1 )( 3 )
df df MSE
df df MSE
df MSE
df
df MSE
df
I
I
RE
MSE MSE
MSE MSE
MSE
MSE
MSE
MSE
where MSEi is the mean square error from experimental design i. If this ratio is greater than 1, it means that Design 1 provides more information per replication and is therefore more efficient than Design 2. If RE1:2 = 2, for example, each replication in Design 1 provides twice as much information as each replication in Design 2. Design 1 is twice as efficient.
The main problem with the approach is how to estimate MSE for the alternative design. Suppose an experiment is conducted as an RCBD. The MSE for this design is simply given by the analysis (MSE (^) RCBD). But now we wish to ask the question: What would have been the value of the MSE if the experiment had been conducted as a CRD? In fact, it was not conducted as a CRD. The treatments were not randomized according to a CRD. Because of this, one cannot just re-analyze the data as though it were a CRD and use the MSE from the analysis as a valid estimate of MSECRD.
MSECRD can be estimated , however, by the following formula (ST&D 222):
B T e
B RCBD T e RCBD CRD (^) df df df
df MSB df df MSE MS E
where MSB and MSE are the block and error mean squares in the original design (RCBD), and dfB , dfT, and dfe are the block, treatment, and error degrees of freedom in the original design. To obtain this formula, the total SS of the two designs are assumed equal. This equation is then expanded such that the SS are rewritten in terms of the underlying variance components of the expected MS. Simplification of the terms generates the above estimate (for a complete derivation, see Sokal & Rohlf 1995, Biometry 838-839).
From the sheep experiment, MSERCBD = 7.78 and MSBRCBD = 192.0. Therefore:
ˆ ( )^3 (^192 ) (^39 )^7.^78
B T e
B RCBD T e RCBD CRD (^) df df df
df MSB df df MSE MSE
and…
2 1
1 2 : (^)
MSE MSE RCBD
MSE MSE CRD RCBD CRD df df MSE
df df MSE RE
Interpretation: It takes 5.51 replications in the CRD to produce the same amount of information as one replication in the RCBD. Or, the RCBD is 5.51 time more efficient than the CRD in this case. It was a very good idea to block by ranch.
6. 4. Assumptions of the model
The model for the RCBD with a single replication per block-treatment combination is:
Yij = + i + j + ij
Thus, when we use ij as estimates of the true experimental error, we are assuming that
i*j = 0.
This assumption of no interaction in a two-way ANOVA is referred to as the assumption of additivity of the main effects. If this assumption is violated, all F-tests will be very inefficient and possibly misleading, particularly if the interaction effect is very large.
Example : A significant interaction term will result if the effect of the two factors A and B on the response variable Y is multiplicative rather than additive. This is one form of non-additivity.
Table 6.4. Additive and multiplicative effects
Factor A
Factor B (^) 1= +1 2= +2 3= +
1= +
2 3 4 Additive effects 1 2 3 Multiplicative effects 0 0.30 0.48 Log of multiplicative effects
2= +
6 7 8 Additive effects 5 10 15 Multiplicative effects 0.70 1.00 1.18 Log of multiplicative effects
In Table 6.4 additive and multiplicative treatment effects are shown in a hypothetical
two-way ANOVA. Let us assume that the population mean is =0. Then the mean of the e.u.’s subjected to level 1 of factor A and level 1 of factor B should be 2 (1+1) by the conventional additive model. Similarly, the expected subgroup mean subjected to level 3 of factor A and level 2 of factor B is 8, since the respective contributions to the mean are 3 and 5. If the process is multiplicative rather than additive, however, as occurs in a variety of physicochemical and biological phenomena, the expected values are quite different. For treatment A 3 B 2 , the expected value is 15, the product of 3 and 5.
If multiplicative data of this sort are analyzed by a conventional ANOVA, the interaction SS will be large due to the non-additivity of the treatment effects. If this SS is embedded in the SSE, as in the case of an RCBD with one e.u. per block-treatment combination, the estimate of the experimental error will be artificially large, thereby making all F tests artificially insensitive.
In the case of multiplicative effects, there is a simple remedy. Transforming the variable by taking the log of each mean will restore additivity. The third line in each cell gives the logarithm of the expected value, assuming multiplicative relations. After the transformation, the increments are strictly additive again ( 1 =0, 2 =0.30, 3 =0.48, 1 =0,
1 =0.70). This is a good illustration of how transformations of scale can be used to meet the assumptions of analysis of variance.
6. 4. 1. Tukey’s test for non-additivity
John Tukey devised a very clever method of testing for significant non-additive effects (i.e. interactions) in datasets that lack the degrees of freedom necessary to include such effects (i.e. interactions) directly in the model. Here's the logic behind the test:
To begin, recall that under our linear model, each observation is characterized as:
yij i j ij
Therefore, the predicted value of each individual is given by:
predij i j
In looking at these two equations, the first thing to notice is that, if we had no error in our experiment
(i.e. if ij 0 ), the observed data would exactly
match its predicted values and a correlation plot of the two would yield a perfect line with slope = 1:
Now let's introduce some error. If the errors in the experiment are in fact random and independent (criteria of the ANOVA and something achieved
by proper randomization from the outset), then ij
will be a random variable that causes no systematic deviation from this linear relationship, as indicated in the next plot:
As this plot shows, while random error may decrease the overall strength of correlation, it will not systematically compromise its underlying linear nature.
But what happens when you have an interaction (e.g. Block * Treatment) but lack the degrees of freedom necessary to include it in the linear model (e.g. when you have only 1 replication per block*treatment combination)? In this case, the df and the variation assigned to the interaction are relegated to the error term. Under such circumstances, you can think of the error term as now containing two separate components:
ij RANDOMij B*T Interaction Effects
While the first component is random and will not affect the underlying linear correlation seen above, the second component is non-random and will cause
Observed vs. Predicted Values (RCBD, no error)
10
12
14
16
18
20
10 12 14 16 18 20 Predicted
Observed…
…
Observed vs. Predicted (RCBD, with error)
10
12
14
16
18
20
10 12 14 16 18 20 Predicted
Observed..
Observed vs. Predicted (RCBD, with error and BT)*
10
12
14
16
18
20
10 12 14 16 18 20 Predicted
Observed..
The Tukey test is NS (p = 0.5395 > 0.05); therefore, we fail to reject the null hypothesis of additivity and we are justified in using the MSE as a valid estimate of the true experimental error.
Please note: This test is necessary ONLY when there is one observation per block- treatment combination. If there are two or more replications per block-treatment combination, the block*treatment interaction can be tested directly in an exploratory model.
6. 4. 2. Diagnostic checking of the Model
Discrepancies of many different kinds between the tentative model and the observed data can be detected by studying residuals. These residuals, the third value in each cell in the table below, are the quantities remaining after the systematic contributions associated with the assumed model (in this case treatments and blocks) are removed (See ST&D 213-214).
Table 6.5 Yield of penicillin from four different protocols (A – D). Blocks are different stocks of an important reagent. The numbers below each observation (O) are the predicted values (P = Grand Mean + Treatment effect + Block effect) and the residuals (R).
Block
Treatment Block
Mean
Block
A B C D Effect
Stock 1 O: 89 P: 90
R: -
O: 88 P: 91 R: -
O: 97 P: 95 R: 2
O: 94 P: 92 R: 2
Stock 2 O: 84 P: 81 R: 3
O: 77 P: 82 R: -
O: 92 P: 86 R: 6
O: 79 P: 83 R: -
Stock 3 O: 81 P: 83 R: -
O: 87 P: 84 R: 3
O: 87 P: 88 R: -
O: 85 P: 85 R: 0
Stock 4 O: 87 P: 86 R: 1
O: 92 P: 87 R: 5
O: 89 P: 91 R: -
O: 84 P: 88 R: -
Stock 5 O: 79 P: 80 R: -
O: 81 P: 81 R: 0
O: 80 P: 85 R: -
O: 88 P: 82 R: 6
Treatment mean 84 85 89 86 Mean=
Treatment effect -2 -1 3 0
SAS program This experiment, organized as an RCBD, has one replication per block-treatment combination. Therefore, in addition to testing for normality and variance homogeneity of residuals, one must also test for non-additivity of block and treatment effects. The SAS code for such an analysis:
Data Penicillin; Do Block = 1 to 4 ; Do Trtmt = 1 to 4 ; Input Yield @@; Output; End; End; Cards; 89 88 97 94 84 77 92 79 81 87 87 85 87 92 89 84 79 81 80 88 ; Proc GLM Data = Penicillin; Class Block Trtmt; Model Yield = Block Trtmt; Output out = PenPR p = Pred r = Res; Proc Univariate Data = PenPR normal; * Testing for normality of residuals; Var Res; Proc GLM Data = Penicillin; * Testing for variance homogeneity (1 way ANOVA); Class Trtmt; Model Yield = Trtmt; Means Trtmt / hovtest = Levene; Proc Plot Data = PenPR; * Generating a plot of predicted vs. residual values; Plot ResPred = Trtmt; Proc GLM Data = PenPR; * Testing for nonadditivity; Class Block Trtmt; Model Yield = Block Trtmt PredPred; Run ; Quit ;
The OUT option after MODEL creates a new data set, named ‘PenPR’ in this example, that includes the original variables plus the predicted (‘Pred’) and residual (‘Res’) variables.
This dataset meets all assumptions: normality, variance homogeneity, and additivity:
Tests for Normality
Test --Statistic--- -----p Value------ Shapiro-Wilk W 0.967406 Pr < W 0.6994 NS
Levene's Test for Homogeneity of Yield Variance ANOVA of Squared Deviations from Group Means
Sum of Mean Source DF Squares Square F Value Pr > F Trtmt 3 922.2 307.4 0.39 0.7620 NS Error 16 12618.8 788.
Tukey's 1 df test for nonadditivity
Source DF Type III SS Mean Square F Value Pr > F
Block 3 28.29215818 9.43071939 0.28 0. Trtmt 3 28.26814524 9.42271508 0.28 0. Pred*Pred 1 26.46875000 26.46875000 0.79 0.3901 NS
6.5 Example of nesting within an RCBD
The following dataset was created assuming that each sheep was weighed two separate times at the end of the experiment (i.e. two measurements (subsamples) were taken on each experimental unit). The subsample values were created such that their averages yield the values of each experimental unit in the original dataset:
Data Lambs; Input Sex_Est $ Ranch Animal Gain @@; Cards; f0 1 1 46 f0 2 1 51 f0 3 1 61 f0 4 1 50 m0 1 1 49 m0 2 1 53 m0 3 1 66 m0 4 1 56 f3 1 1 56 f3 2 1 52 f3 3 1 68 f3 4 1 56 m3 1 1 53 m3 2 1 64 m3 3 1 73 m3 4 1 58
f0 1 1 48 f0 2 1 53 f0 3 1 63 f0 4 1 52 m0 1 1 51 m0 2 1 55 m0 3 1 68 m0 4 1 58 f3 1 1 58 f3 2 1 54 f3 3 1 70 f3 4 1 58 m3 1 1 55 m3 2 1 66 m3 3 1 75 m3 4 1 60 ; Proc GLM Data = Lambs Order = Data; Class Ranch Sex_Est Animal; Model Gain = Ranch Sex_Est Animal(RanchSex_Est); Random Animal(RanchSex_Est); Test h = Sex_Est e = Animal(Ranch*Sex_Est);
In nested models, specify the correct error term in all contrasts and mean comparisons; Contrast 'sex' Sex_Est 1 - 1 1 - 1 / e = Animal(RanchSex_Est); Contrast 'estrogen' Sex_Est 1 1 - 1 - 1 / e = Animal(RanchSex_Est); Contrast 'interaction' Sex_Est 1 - 1 - 1 1 / e = Animal(RanchSex_Est); Means Sex_Est / Tukey e = Animal(Ranch*Sex_Est);
*Below is a Tukey test with the incorrect error for comparison; Means Sex_Est / Tukey;
Proc Varcomp Method = Type1; Class Ranch Sex_Est Animal; Model Gain = Ranch Sex_Est Animal(Ranch*Sex_Est); Run ; Quit ;
Now that there are subsamples, the experimental unit (Animal) must appear in the model. As with all experimental units, the label "Animal" is simply an ID. To specify a particular animal in the experiment, the Ranch (i.e. block) and the level of Sex_Est (i.e. treatment) must be specified. Hence the syntax:
Animal(Ranch*Sex_Est)
The ID "Animal" only has meaning within specific Block-Treatment combinations. Animals are random samples from larger populations; hence Animal(Ranch*Sex_Est) is declared as a random variable.
As in the CRD, it is important to note that the error term to test differences among treatments is the variance among experimental units (MSEE). Thus
Animal(Ranch*Sex_Est) must be declared as the error term for each hypothesis tested regarding treatment means, whether the overall F test or subsequent contrasts or mean comparisons. If the error term is not specified, SAS will test every hypothesis using the variance among subsamples (MSSE).
Output (compare to output of the original experiment with no subsamples)
Overall ANOVA
Sum of Source DF Squares Mean Square F Value Pr > F
Model 15 1708.000000 113.866667 56.93 <. Error 16 32.000000 2. Corrected Total 31 1740.
Partitioning of model effects ( automatic wrong F tests for treatment and block )
Source DF Type III SS Mean Square F Value Pr > F
Ranch 3 1152.000000 384.000000 192.00 <. Sex_Est 3 416.000000 138.666667 69.33 <. Animal(Ranch*Sex_Es) 9 140.000000 15.555556 7.78 0.
Correct F test for treatment
Tests of Hypotheses Using the Type III MS for Animal(RanchSex_Es) as an Error Term*
Source DF Type III SS Mean Square F Value Pr > F Sex_Est 3 416.0000000 138.6666667 8.91 0.0046 **
Note that 8.91 is exactly the same as in the previous analysis (Topic 6.2.5, Table 6.2) using the average of the two subsamples)
Correct contrasts Contrast DF Contrast SS Mean Square F Value Pr > F
sex 1 128.0000000 128.0000000 8.23 0.0185 * estrogen 1 288.0000000 288.0000000 18.51 0.0020 ** interaction 1 0.0000000 0.0000000 0.00 1.0000 NS
Correct Tukey means separation Minimum Significant Difference 6. Tukey Grouping Mean N Sex_Est A 63.000 8 m B A 59.000 8 f B A 57.000 8 m B 53.000 8 f Incorrect Tukey means separation Minimum Significant Difference 2. Tukey Grouping Mean N Sex_Est A 63.0000 8 m B 59.0000 8 f B 57.0000 8 m C 53.0000 8 f
Type 1 Analysis of Variance
Source Expected Mean Square Ranch Var(Error) + 2 Var(Animal(RanchSex_Es)) + 8 Var(Ranch) Sex_Est Var(Error) + 2 Var(Animal(RanchSex_Es)) + 8 Var(Sex_Est) Animal(RanchSex_Es) Var(Error) + 2 Var(Animal(RanchSex_Es)) Error Var(Error)
model gain= sex_est; means sex_est/ HOVTEST = LEVENE ; run ; quit ;
Output LAMB 2 Reps ANOVA Dependent Variable: gain Exploratory Source DF SS MS F Value Pr > F Model 15 1700.5 113.4 59.47 <. Error 16 30.5 1. Corrected Total 31 1731. Source DF SS MS F Value Pr > F block 3 1132.1 377.4 198.0 <. sex_est 3 426.1 142.0 74.5 <. blocksex_est* 9 142.3 15.8 8.3 0.
Final Source DF SS MS F Value Pr > F Model 6 1558.2 259.7 37.6 <. Error 25 172.8 6. Corrected Total 31 1731. Source DF SS MS F Value Pr > F block 3 1132.1 377.4 54.6 <. sex_est 3 426.1 142.0 20.6 <. Still significant in spite of large blocksex_est interaction*
Contrast DF SS MS F Value Pr > F sex 1 132.0 132.0 19.10 0. estrogen 1 294.0 294.0 42.54 <. interaction 1 0.03 0.03 0.00 0.
Tukey's Studentized Range (HSD) Test for gain Minimum Significant Diff. 1. Grouping Mean N sex_est A 63.000 8 m B 59.000 8 f B 57.000 8 m C 52.875 8 f
Tests for Normality Test --Statistic--- -----p Value------ Shapiro-Wilk W 0.978499 Pr < W 0.7549 OK
Levene's Test for Homogeneity of Variance
Sum of Mean Source DF Squares Square F Value Pr > F sex_est 3 3181.5 1060.5 0.54 0.6567 OK Error 28 54660.5 1952.
