Missing data, effect

The use of constraints and restraints arise from unrealistic bond lengths and angles. It is a reasonable use of statistical analysis to include prior information in a refinement procedure, and geometry is one such restraint/constraint. Poor geometries are often a consequence of missing data down one axis, invariably the result of the missing cone or lack of an epitaxially grown crystal. This lack of data has a profound effect on the refinement process... 

One very simplistic way of handling missing data is to remove those patients with missing data from the analysis in a complete cases analysis or completers analysis. By definition this will be a per-protocol analysis which will omit all patients who do not provide a measure on the primary endpoint and will of course be subject to bias. Such an analysis may well be acceptable in an exploratory setting where we may be looking to get some idea of the treatment effect if every subject were to follow the protocol perfectly, but it would not be acceptable in a confirmatory setting as a primary analysis. 

The analysis of variance lends itself best to balanced factorial designs, whether complete, partially replicated, or otherwise modified. The concept of balance simplifies the calculations tremendously. There are ways of coping with missing data, unequal replication under various conditions, and even some lack of orthogonality in the design, but these methods seem to involve more calculation than the data may deserve. The analysis of variance is a procedure which makes it possible to compare the effects of the variables being studied, first independently of the effects of all other variables, and second in all possible combinations with one another. Sometimes the effect of a variable within a given level of another variable... 

The algorithm used is attributed to J. B. J. Read. For many manipulations on large matrices it is only practical for use with a fairly large computer. The data are arranged in two matrices by sample i and nuclide j one matrix, V, contains the amount of each nuclide in each sample the other matrix, E, contains the variances of these numbers, as estimated from counting statistics, agreement between replicate analyses, and known analytical errors. It is also possible to add an arbitrary term Fik to each variance to account for random effects between samples not considered in the model this is usually done in terms of an additional fractional error. Zeroes are inserted for missing data in cases in which not all nuclides were measured in every sample. 

Some aspects of degree of concern currently can be considered in a quantitative evaluation. For example, EPA considers human and animal data in the process of calculating the RfD, and these data are used as the critical effect when they indicate that developmental effects are the most sensitive endpoints. When a complete database is not available, a database UF is recommended to account for inadequate or missing data. The dose-response nature of the data is considered to an extent in the RfD process, especially when the BMD approach is used to model data and to estimate a low level of response however, there is no approach for including concerns about the slope of the dose-response curve. Because concerns about the slope of the dose-response curve are related to some extent to human exposure estimates, this issue must be considered in risk characterization. (If the MOE is small and the slope of the dose-response curve is very steep, there could be residual uncertainties that must be dealt with to account for the concern that even a small increase in exposure could result in a marked increase in response.) On the other hand, a very shallow slope could be a concern even with a large MOE, because definition of the true biological threshold will be more difficult and an additional factor might be needed to ensure that the RfD is below that threshold. 

A common problem when series of experiments are run, is that for some runs there are missing data. This occurs rather frequently when several responses are measured. Orthogonal designs are balanced designs and each experiment is necessary, and equally important, for the overall picture. Missing data can therefore ruin the analysis and give highly biassed estimates of the effects of the variables. 

---