6.2. Timing of the statistical analysis plan
6.3. Decision criteria
6.4. Statistical analysis plan structure
There is a considerable body of literature explaining statistical methods for observational studies but very little addressing the statistical analysis plan. A clear guide to general principles and the need for a plan is given in Design of Observational Studies (P.R. Rosenbaum, Springer Series in Statistics, 2010. Chapter 18), which also gives useful advice on how to test complex hypotheses in a way that minimizes the chances of drawing incorrect conclusions.
Planning analyses for randomised clinical trials is covered in a number of publications. These often give checklists of the component parts of an analysis plan and much of this applies equally to non-randomised designs. A good reference in this respect is the International Conference on Harmonisation of Technical Requirements for Registration of Pharmaceuticals for Human Use (ICH). ICH E9 ‘Statistical Principles for Clinical Trials’. While specific guidance on the statistical analysis plan for epidemiological studies is sparse, the following principles will apply to most of the studies.
Pre-specification of statistical and epidemiological analyses can be challenging for data that are not collected specifically to answer the study questions. This is often the case in observational studies, where secondary data are used. However, thoughtful specification of the way missing values will be handled or the use of a small part of the data as a pilot set to guide analysis can be useful techniques to overcome such problems. A feature common to most studies is that some not pre-specified analyses will be performed in response to observations in the data to help interpretation of results. It is important to distinguish between such data-driven analyses and the pre-specified findings. Post-hoc modifications to the analysis strategy should be noted and explained. The statistical analysis plan provides a confirmation of this process.
A particular concern in retrospective studies is that decisions about the analysis should be made blinded to any knowledge of the results. This should be a consideration in the study design, particularly when feasibility studies are to be performed to inform the design phase. Feasibility studies should be independent of the main study results.
The study protocol will have specified the questions to be addressed in the study and will contain a generic description of the study type and the statistical techniques. However, the statistical analysis plan is likely to be the document in which the statistics to be calculated and tabular and graphical presentations are fully described. Since the decision criteria for the study are specified in terms of the observed values of these detailed statistics, it is worth formulating the statistical analysis plan at an early stage and, in particular, before any informal inspection of aspects of the data or results that might influence opinions regarding the study hypotheses. Ideally the statistical analysis plan will be developed as soon as the protocol is finalised.
If decisions are to be made based on the results of the study, a section of the statistical analysis plan should explain the different outcomes that might be selected for each decision, which statistics influence the decision making process and which values of the statistics will be considered to support each outcome. Often the statistical analysis will employ standard routines incorporated in statistical packages that have outputs seen as implicit decision criteria – for instance p values or confidence intervals. However, different applications of the study may require lower or higher strength of evidence – for instance policy recommendations regarding drug licensing may require a lower chance of false positive decisions than the classical one when deciding whether further investigation is needed for a product safety issue. Hence consideration of decision making criteria with explicit reference to the type of decision to be made is beneficial.
The statistical and epidemiological analysis plan is usually structured to reflect the protocol and will address, where relevant, the following points:
Missing data, occur when no data value is stored for the variable in the current observation. Missing data are a common occurrence and can have a significant effect on the conclusions that can be drawn from the data. There are different patterns of missing data: completely at random, at random or not at random.
The book Statistical analysis with missing data (Little RJA, Rubin DB. 2nd ed.,Wiley 2002) describes many aspects of the handling of missing data. The section ‘Handling of missing values’ in Rothman’s Modern Epidemiology, 3rd ed. (K. Rothman, S. Greenland, T. Lash. Lippincott Williams & Wilkins, 2008) is a summary of the state of the art, focused on practical issues for epidemiologists. Ways of dealing with such data include complete subject analysis (subjects with missing values are deleted from the analyses) and imputation methods (missing data are predicted based on the observed values and the pattern of missingness). A method commonly used in epidemiology is to create a category of the variable, or an indicator, for the missing values. This practice can be invalid even if the data are missing completely at random and should be avoided (see Indicator and Stratification Methods for Missing Explanatory Variables in Multiple Linear Regression. J Am Stat Assoc 1996;91(433):222–230).
A concise review of methods to handle missing data is also provided in the section ‘Missing data’ of the Encyclopedia of Epidemiologic Methods (Gail MH, Benichou J, Editors. Wiley 2000). Identifying the pattern of missing data is important as some methods for handling missing data assume a defined pattern of missingness. Biased results may be obtained if it is incorrectly assumed that data are missing at random. In general, it is desirable to show that conclusions drawn from the data are not sensitive to the particular strategy used to handle missing values. To investigate this, it may be helpful to repeat the analysis with a variety of approaches.
Other useful references on handling of missing data include the books Multiple Imputation for Nonresponse in Surveys (Rubin DB, Wiley, 2004) and Analysis of Incomplete Multivariate Data (Schafer JL, Chapman & Hall/CRC, 1997), and the articles Using the outcome for imputation of missing predictor values was preferred (J Clin Epi 2006;59(10):1092-101), Recovery of information from multiple imputation: a simulation study (Emerg Themes Epidemiol 2012;9(1):3) and Evaluation of two-fold fully conditional specification multiple imputation for longitudinal electronic health record data (Stat Med. 2014;33(21):3725-37).