Statistical Design and Analysis of Experiments

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The response is defined by the vector of yields of maize and beans measured by weight Kg per square meter. Sole crops show a zero in one of the yields. The DEA model should be run with 56 DMUs 14 treatments and 4 blocks , but there were missing data in two of them treatment 7 - block 2, and treatment 9 - block 4. The analysis of variance of the experiment based on efficiency responses is shown in Table 2.

The statistical analysis is carried out assuming the data generating process of the block design with DEA efficiency measurements as the observations on the dependent variable. Treatment contrasts of interest were also included in the analysis. Efficiency seems to be the same for all variety combinations of maize and bean used in intercropping rows d and e , Table 2. Overall, there is no significant gain in efficiency of sole plots over intercropping row a , Table 2. Intercropped beans however are significantly more efficient than sole beans row b , Table 2.

An interesting aspect of this intercropping experiment is that it shows a smaller risk for intercropped plots. Efficiency variation of intercropped plots is dominated by efficiency variation in sole plots. This remark is evident from Figure 1 where treatments refer to intercropping and treatments refer to sole crops treatments show less dispersion than treatments Efficiency responses belong to the interval 0.

In this context, one may perform an analysis of variance considering ranks as the response variable. The approach is non-parametric. We notice that this analysis leads to similar conclusions. The second experiment we consider is the intercropping of groundnut and maize in a randomized block design laid out as a split plot. The underlying principle of the split plot is this: the levels of a factor A treatment A are applied to the experimental plots arranged as in the randomized block design. Each experimental plot is then divided into subplots to which the levels of a second factor B treatment B are applied.

EMIS Statistical Design and Analysis of Experiments

In this context each experimental plot becomes a block for factor B. The randomization is carried out in two stages. First the levels of A are randomized over the whole experimental plots. Then the levels of B are randomized over the subplots. The data generating process for the split plot assuming the randomized block design is as follows. As before, the design matrix is singular but all parametric functions of interest are estimable. Covariates may also be superimposed in the context of the design layout. Table 3 shows production and efficiency data for an intercropping experiment involving maize and groundnuts.

The experiment was carried out in the substation of Morwa, in Samuru, Nigeria. It is described in Carvalho The whole experimental plot treatment levels are defined by factor A with 5 levels, which are repeated in 4 blocks. The subplot treatment, factor B, comprises two levels of nitrogen treated as qualitative. We denote by the plant densities and by the nitrogen levels. Figure 2 shows clearly an increase in efficiency due to nitrogen level 2 independently of the plant density treatments Tx2 have higher response that treatments Tx1.

It is also evident the dominance of plant density 4. A non-parametric analysis of variance is shown in Table 4. The response considered for this analysis is rank of efficiency. The ranks are assumed to follow the data generating process of the split plot with the randomized block design.

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Table 5 provides information on the average efficiency rank per treatment means over efficiency ranks for the intercropping of maize and groundnut. The two error component are involved, the whole plot error and the subplot error. The variance of the difference is given by. The sampling distribution associated with this unbiased estimate is not chi-square but a linear combination of chi-square distributions. There is not an exact least significant difference for the comparison of two A means. Using the method of moments an approximate value can be computed.

The procedure is similar to the one used when the means of two populations with distinct variances are compared.

Factorial Designs 1: Introduction

We see that the visual impression of Figure 2 is confirmed. Responses associated with nitrogen level 2 are dominant. Treatment combination 42 has the highest efficiency. We see that level 2 significantly dominates level 1. We emphasize at this point that a similar analysis using non-transformed data leads to the same conclusions.

It leads to the data in Table 6 and the analysis shown in Table 7 when the responses are ranks. At this level of significance the least significant difference for A comparisons is 7. The A means are The best density is level 4, which is superior significantly to densities 1, 2, and 5 and marginally to density 3.

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The least significant difference for comparison of nitrogen levels is 3. Level 2 is superior as before. The assumptions behind the statistical analysis carried in both examples of the previous section are strong and can be questioned. Typically, errors are uncorrelated and normally distributed. DEA efficiency responses cannot be normally distributed. The use of DEA efficiency measures as a response variable in regression models has been questioned recently in the econometric literature. Besides normality, another issue in this discussion is that the analysis is carried out in two stages and in the first stage a correlation between experimental units is induced by the computation and the definition of the response variable and the potential association of covariates with the error term.

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In this context, one loses the distributional properties of the least squares estimators. For designed experiments however, the analysis of variance is known to be robust against many departures from the linear models defining the data generating process. Including non-normality and correlation among the experimental plots responses. On the other hand, covariates are not considered random in a designed experiment.

The random allocation of treatments to the experimental plots is the key feature that validates the statistical analysis besides allowing a method to verify if the significance levels induced by the normal theory are correct or not. Fisher extensively promoted this experimental process. Based on the robustness of the statistical analysis of designed experiments, we should stress once again that the issues of correlation and non-normality are of secondary importance in many applications.

For example, with unitary inputs and a single response, the measure of efficiency proposed here amounts simply to divide the values of the response by its sample maximum. Since t and F statistics are invariant by scale transformations, the statistical results are unaffected by the two-stage process. The situation with multiple outputs is more complex.

Our experience is that the correlation is not sufficiently strong to invalidate the analysis. In a previous study Souza et al. A way to overcome the discussion on whether or not the classical analysis of variance is valid is to derive the distribution, induced by the randomization of treatments to the experimental plots, of the test statistics of concern, and compare them with the distribution generated under normal theory, i.

Consider the F statistics used to test equality of treatment effects. Each treatment assignment generates a value for F. The collection of these F values defines the distribution of F under randomization. Typically the number of possible treatment allocations is large and one works with a random sample from this population. Here we use 10, random allocations, which will generate 10, F values. We note that in these simulations the DEA responses are fixed in each experimental plot.

Only the treatment allocations change. For the split plot arranged in r blocks with a levels of factor A applied to main plots and b levels of factor B applied to the subplots, there will be a! Here we also use 10, random allocations. We used SAS v9. Table 8 shows the quantiles of the two distributions involved. The quantiles of the two distributions are close. The analyses with normal theory and with randomization are coincident. The empirical p-value does not point to any significant discrepancy. Table 9 shows the quantiles of the distributions of these test statistics under normality and under randomization.

Although the approximations for the split plot are not as good as for the previous randomized block design, the distributions reasonably agree. Empirical p values are 0.

Statistical Design and Analysis of Industrial Experiments

We propose the use of DEA output oriented efficiency measurements, under constant returns to scale, assuming a unitary input, as the response of each experimental plot in a designed experiment whenever the output vector is defined by non-negative yields. The technique consists in applying the normal theory or non-parametric rank methods to the efficiency measurements.

This proposed approach, with the validation test, has never been used before in intercropping experiments.