Table Of Content
The model is specified as y ~ drug + Error(cage+rep/litter) or y ~ drug + (1|cage)+(1|rep/litter). As a real-world application, we consider an experiment to characterize and compare multiple antibody assays. Each assay only requires a small amount of patient serum, and in order to provide sufficient sample size for precisely estimating the sensitivity and specificity of each assay, several hundred patient samples were used. For estimating between-plate variability, we would ideally create ten aliquots from several patient sera, and assign one aliquot to each plate. However, the available sera only allowed at most five aliquots of sufficient volume.
ANOVA: Yield versus Batch, Pressure
A Graeco-Latin square is orthogonal between rows, columns, Latin letters and Greek letters. As the treatments were assigned you should have noticed that the treatments have become confounded with the days. Days of the week are not all the same, Monday is not always the best day of the week! Just like any other factor not included in the design you hope it is not important or you would have included it into the experiment in the first place. In this factory you have four machines and four operators to conduct your experiment.
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2.6 Replication Within Blocks
We do not have observations in all combinations of rows, columns, and treatments since the design is based on the Latin square. Together, you can see that going down the columns every pairwise sequence occurs twice, AB, BC, CA, AC, BA, CB going down the columns. The combination of these two Latin squares gives us this additional level of balance in the design, than if we had simply taken the standard Latin square and duplicated it. Here we have used nested terms for both of the block factors representing the fact that the levels of these factors are not the same in each of the replicates. In this case, we have different levels of both the row and the column factors.
Experimental Design
A randomized block design with the following layout was used to compare 4 varieties of rice in 5 blocks. We consider an example which is adapted from Venables and Ripley (2002), the original source isYates (1935) (we will see the full data set in Section 7.3). Atsix different locations (factor block), three plots of land were available.Three varieties of oat (factor variety with levels Golden.rain, Marvellousand Victory) were randomized to them, individually per location. Hence, a block is given by a locationand an experimental unit by a plot of land. In the introductory example, a blockwas given by an individual subject.
Introducing Blocking
For direct comparison with our previous results, we estimate the two interaction contrasts of Table 6.6 in the blocked design. They compare the difference in enzyme levels for D1 (resp. D2) under low and high fat diet to the corresponding difference in the placebo group; estimates and Bonferroni-corrected confidence intervals are shown in Table 7.2. We illustrate the extension using our drug-diet example from Chapter 6 where three drugs were combined with two diets. The six treatment combinations can still be accommodated when blocking by litter, and using four litters of size six results in the experimental layout shown in Figure 7.7 with 24 mice in total. Already including a medium-fat diet would require atypical litter sizes of nine mice, however. The data are shown in Figure 7.1B, where we connect the three observed enzyme levels in each block by a line, akin to an interaction plot.
So far we have discussed experimental designs with fixed factors, that is, the levels of the factors are fixed and constrained to some specific values. In some cases, the levels of the factors are selected at random from a larger population. In this case, the inference made on the significance of the factor can be extended to the whole population but the factor effects are treated as contributions to variance. If the number of times treatments occur together within a block is equal across the design for all pairs of treatments then we call this a balanced incomplete block design (BIBD). In general, we are faced with a situation where the number of treatments is specified, and the block size, or number of experimental units per block (k) is given.
3.2 Defining a Balanced Incomplete Block Design
If this assumption is violated, randomized block ANOVA should not performed. One possible alternative is to treat it like a factorial ANOVA where the independent variables are allowed to interact with each other. An appreciable block-by-treatment interaction means that the differences in enzyme levels between drugs depend on the litter. Treatment contrasts are then litter-specific and this systematic heterogeneity of treatment effects complicates the analysis and precludes a straightforward interpretation of the results.
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This type of experimental design is also used in medical trials where people with similar characteristics are in each block. This may be people who weigh about the same, are of the same sex, same age, or whatever factor is deemed important for that particular experiment. So generally, what you want is for people within each of the blocks to be similar to one another.
This is because the blocking factor is random, and the resulting one-way factor would also be a random factor. The omnibus \(F\)-test for this factor is difficult to interpret, and a contrast analysis would be a futile exercise, since we would compare randomly sampled factor levels among each other. A nuisance factor is a factor that has some effect on the response, but is of no interest to the experimenter; however, the variability it transmits to the response needs to be minimized or explained. We will talk about treatment factors, which we are interested in, and blocking factors, which we are not interested in.
The plot of residuals versus order sometimes indicates a problem with the independence assumption. Here we have four blocks and within each of these blocks is a random assignment of the tips within each block. Back to the hardness testing example, the experimenter may very well want to test the tips across specimens of various hardness levels. To conduct this experiment as a RCBD, we assign all 4 tips to each specimen. Many industrial and human subjects experiments involve blocking, or when they do not, probably should in order to reduce the unexplained variation.
As a results, there will be three parts of the variance in randomized block ANOVA, SS intervention, SS block, and SS error, and together they make up SS total. In doing so, the error variance will be reduced since part of the error variance is now explained by the blocking variable. When the numerator (i.e., error) decreases, the calculated F is going to be larger. We will achieve a smaller P obtained value, and are more likely to reject the null hypothesis. In other words, good blocking variables decreases error, which increases statistical power.
The common use of this design is where you have subjects (human or animal) on which you want to test a set of drugs -- this is a common situation in clinical trials for examining drugs. An assumption that we make when using a Latin square design is that the three factors (treatments, and two nuisance factors) do not interact. If this assumption is violated, the Latin Square design error term will be inflated. The partitioning of the variation of the sum of squares and the corresponding partitioning of the degrees of freedom provides the basis for our orthogonal analysis of variance.
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