Non-Replicated Experiments in Commercial Dairy Essay (Article)

Introduction

Every trial must have replications or repetitions because the experimental error is estimated in replicated trials. However, dairies commercial farms demonstration trials are impossible to replicate most of the time. This is because of the practical operation at the dairy farm, the manageability of the day-to-day operation, and the cost involved with them. In such situations, it is possible to use statistical experimental design and tools that can be used to analyze the data. This paper will focus on-farm trials and observations, nature and underlying principles of non-replicated tests, and briefly discuss analysis methods.

Importance of Replication

Replication is one of the milestone concepts of experimental design and statistical analysis. It is widely used in animal science research to calculate the experimental error variance against which treatment effects should be compared. It also helps to operational consistency of experimental results (Kuehl, 2000). Replicating treatments in a trial enables the researcher to separate the actual treatment effects from the background noise by absorbing experimental error (Johnson, 2006). Therefore, whenever possible, experimental treatments should be replicated.

However, there are situations where replicating a treatment in different units is not possible, forcing investigators to conduct non-replicated studies. Most of the time, such experiments are frequently used to save costs. According to Machado and Petrie (2006), replication in agricultural science is impractical, expensive, or impossible in certain situations. For example, long-term experiments initiated before the current understanding of statistics, ecological and watershed studies, large field-scale research trials, demonstration plots, geological research, and even unforeseen design mistakes.

Why Non-Replication is Not Used in Dairy Publication

Machado and Petrie (2006) point out that many agricultural researchers consider non-replicated experiments unscientific and unacceptable for publication. Most of the trials published in the animal science journal are from the universities where the farm conducts a scientist trial. Therefore, the structure is designed to use small numbers of animals with enough replication of the experimental united in those studies. A non-replicated study requires special statistical knowledge and a significant number of animals or subsamples to have the power to detect the difference in the treatments. According to Bisgaard (1992), experimental design techniques were initially developed with replications for agricultural and biological research. Although there are early industrial applications of the design of experiments, engineers took a long time to envision that methods used for agricultural and biological studies had relevance to their work. Today, some processes are widely used in industry that can be applied in agricultural, physical, and environmental settings.

Publication of Non-Replicated Data

Due to the lack of replication, a direct estimation of error variance from non-replicated experiments is impossible. This causes problems in the assessment of the significance of estimated effects. However, there are effective methods to overcome this difficulty, such as the one proposed by Miliken and Johnson (1989). Daniel (1959) suggests using normal probability plots for unreplicated two-level factorial experiments. Box and Meyer (1986) and Lenth (1989) provide alternative procedures to normal probability plots.

Unreplicated designs are very useful in industrial experimentation. Applications of these designs have been found in the literature since the 1960s. For example, Michaels (1964) describes unreplicated experimental designs, and Prvan and Street (2002) present an annotated bibliography of application papers on fractional factorial designs with illustrations of unreplicated cases. Ilzarbe et al. (2008) analyzed 77 instances of practical design of experiments applications in engineering published in important scientific journals between 2001 and 2005. Therefore, there are plenty of examples of non-replicated experimental designs that were carefully described and published. The dairy industry needs to follow the same step to take advantage of the sound data produced by several field trials that were never basically because of lack of knowledge or the unwillingness to work with those procedures. Research using experiment design with no replication has a long history. The most famous one is Tukey’s test of additivity, where the degree of freedom (df) is designed for a specific non-additivity structure (Tukey, 1949). Since then, many detailed tests have been deployed, as shown below.

Tests That Can Be Applied to Dairy Field Trials

1. Another Fisher’s contribution, relevant to the analysis of unreplicated data, is the randomization test (Fisher, 1935). For t, the randomization test treats the test statistics as a sample from all the possible sets of results that might have been obtained from a particular set of experimental units (experimental farms if this is a field experiment). Its distribution over the data sets then determines the probability value for each test statistic. The method is useful when the data does not satisfy the distributional assumptions (normality assumption) required for the standard analysis.
2. Daniel (1959) used the idea of detecting outliers in a data set by using a probability plot to identify the outliers. The values that fall off the line correspond to active effects. The method uses the implicit assumption of few functional effects to draw a line through the bulk of small contrasts. However, the method ingeniously avoids the need for estimating σ. He presented an objective graphical method, a standardized probability plot with guardrails, which plots the unsigned contrasts divided by the ordered unsigned contrast corresponding to the order statistic close to 0.683 percentile. Dynamic effects are then identified by the standardized contrasts which exceed their corresponding guardrails. It is the most powerful test when there is only one active effect.
3. Box and Meyer (1986) presented a Bayesian approach based on effect sparsity. They used a scale contaminated model, which assumes that the active effects have a normal distribution. Therefore, contrasts corresponding to the active impacts have distribution to the inert results with a normal distribution. For each product, the marginal posterior probability of being active is computed and declared active if the probability exceeds 0.5. They noted that they could estimate parameters based on ten published analyses of data sets. This provided empirical support for the principle of effect sparsity and motivated their recommendation.
4. Dong (1993) proposed a method based on the trimmed mean of squared contrasts rather than the trimmed median of the unsigned contracts. The method has a small mean squared error which is a good motivation for standardizing the contrasts. He also proposed iteratively calculating the trimmed median of the unsigned contrasts until it stops changing when there are many active effects.
5. Alin and Kurt (2006) have reviewed non-additivity interaction in two-way Anova tables with no replication. They describe some methods for testing non-additivity when there is only one observation. Some of these tests depend on known interaction structure, whereas others do not. However, these methods are straightforward to apply using Microsoft Excel and Statistical packages like R.
6. Payne (2006) has also described new and traditional methods for analyzing unreplicated experiments. One of the methods he recommended was the spatial method. In the spatial methods, the experiment is first analyzed conventionally, treating it as a randomized block design. This design aims to group the units (i.e., dairy farms) into blocks so that the farms in the same block are more similar than those in different blocks. Each treatment occurs an equal number of times in each block (usually once), and the allocation of treatments is randomized, where possible, independently within each block. The analysis estimates and removes between-block differences to estimate treatment effects more precisely. However, the design may constrain which treatments appear on some of the farms to allow reasonable estimates for the parameters in the spatial model. The experimenter has to take account of variation by fitting models to describe how the correlation between each farm and its neighbor’s changes according to their relative locations and the analysis of residual (or restricted) maximum likelihood (Patterson & Thompson, 1971; Gilmour et al., 1995).
7. Wang (2013) has published excellent analysis methods for two-factor unreplicated experiments where one factor is random. His research is motivated by comparing measurement methods. Its foci include parameter estimation, tests of additivity, and prediction of one method given measurements of other methods. Although this proposed test is similar to Mande’s test, his result is a more robust test.
8. Recently, Vivacqua et al. (2015) have published an application of split-plot experiments in time. It extends the experimental design structure by considering an unreplicated factorial plan augmented with one central point and repeated control treatment in a two-time period. The analysis procedure was described in detail unavailable in the literature. Nevertheless, this paper provides an approach to evaluate the significance of meaningful contrasts when the additional treatments evaluated over a two-time period are also unreplicated.

Outliers

One critical point in working with non-replicated design is knowing how to work with outliers. Whether or not an appropriate transformation is used can be more important than the test selected. Slight departures from normality do not affect the distributions of the test statistics too much. Still, outliers impact the sizes of estimates of the effects, so effective detection and elimination are critical to successful process optimization. The active effects as outliers have extensive literature (Barnett & Lewis 1994). Benski (1989) proposed using an outlier test to identify the functional impact to solve this problem. The test is based on a robust estimate of spread, which uses the interquartile range of the contrasts. In particular, it is possible to use interactive graphical methods developed for this problem. However, this topic is out of the scope of this paper and can be discussed in another one.

Conclusion

In conclusion, this paper has reviewed many experiment designs used in other fields that can also be applied in animal science, especially in dairy farms field trails. Given restrictions on time and cost, unreplicated designs will continue to be widely used in the dairy industry, and they could be published if the right statistical tool is used. There are many competing test methods, which are difficult to distinguish among based on performance, and it is easy to come to wrong conclusions if a careful comparison is not made. However, using comparison techniques, the researcher can make the correct decision. The choice of a test is further complicated because no single test performs well over a wide range of dynamic effects. In this regard, we need to use a graphical analysis environment in software designers to provide us with an appropriate choice of tests and an appropriate way to work with outliers. Therefore, it is recommended that the authors take the comprehensive approach of this paper as a guideline for fully describing the statistical method used in his publication, making it replicable and acceptable for publication.

References

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