Why is MIXED analysis underutilized

The Operations Manual (http://pubs.nrc-cnrc.gc.ca/aicjournals/instruct/operations-manual.pdf) for the Canadian Journal of Plant Science, the Canadian Journal of Soil Science and the Canadian Journal of Animal Science (Revised 2007, page 17) states: ‘‘The GLM procedure of SAS has been widely used for analysis of variance; however, it was designed to analyze data having fixed effects only. Models that have both fixed and random effect should be analyzed using the MIXED procedure of SAS. This is also important in analyzing datasets with repeated observations on the same experimental unit that have heterogeneous variances over time and/or unequal within subject timedependent correlations.’’ Despite these clearly stated guidelines regarding the basic requirements and expectations of the statistical analysis, many submissions to the CJPS have continued the use of GLM when clearly MIXED should be used. A quick inspection of the first two 2008 issues of CJPS reveals that, of the 33 papers with a description of experimental designs and statistical analyses, eight used MIXED, 21 explicitly or implicitly indicated the use of GLM and the remaining four used other software (e.g., SPSS and GenStat). Both fixed and random effects are present in most of these studies judging from their description of experiments, but GLM rather than MIXED has been used in the majority of cases. Such trend of underutilization of MIXED is probably true as well in the previous volumes of CJPS and other agricultural journals. The purpose of this letter is to discuss causes and consequences of such underutilization in crop and agronomic research. Before embarking on such discussion, it is important to briefly review the concept of fixed and random effects. The determination of whether an effect is fixed or random in crop and agronomic studies is not always easy and has been debated in the scientific literature. In crop and agronomic experiments, treatments or combinations of treatments are often chosen intentionally and thus should be fixed effects. Moreover, these experiments are usually carried out at multiple sites and over several years to infer about the treatment performance for future years over a wide region. Such broader inference assumes that site and year effects are random, with sites being a random sample of all possible sites in the region and years being a random set of future years. However, these assumptions are rarely fulfilled in practice since the locations are not always randomly selected, and years may not be representative of future years (Steel et al. 1997). Despite the practical difficulty, both sites and years are generally considered as random for the broader inference. There are several reasons why GLM remains commonly used in the scientific literature for experiments with both fixed and random effects. First, it is often argued that GLM and MIXED give the same results when the data sets are balanced. It is true that obtaining a balanced data set is relatively easier in crop and agronomic experiments than in forestry and animal experiments, where factors such as tree mortality or cost of animals may make it more difficult to achieve the data balance. It is also true that with a balanced data set, the estimated variances of random effects would be identical whether the estimation procedure is the residual (restricted) maximum likelihood (REML) (the default method of MIXED) or TYPE I to Type IV of GLM, provided that these variance estimates are not negative. In this case, GLM would indeed provide the same F-tests of fixed effects if the random effects are specified in the RANDOM statement and the TEST option is added. If the true random effects are small and/or sample sizes are small, the negative variance estimates may be obtained using GLM. However any variance by definition should not be less than zero and negative estimates have no meaning. When this happens, REML (the default in MIXED) sets the negative variance estimates to zero regardless of whether or not the data set is balanced! Such different modes of handling negative variance estimates by GLM vs. MIXED would lead to different F-tests of the same fixed effects. Interestingly, the GLM vs. MIXED difference in F-tests due to the presence of negative variance estimates creates a new issue of which F-test should be used. In other words, should we use an F-test based on negative but unbiased variance components or an F-test based on nonnegative but biased variance components? This remains to be an open question even among statisticians (Littell et al. 2002). Nevertheless,

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