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Missing data are unavoidable in epidemiological and clinical research but their potential to undermine the validity of research results has often been overlooked in the medical literature.1 This is partly because statistical methods that can tackle problems arising from missing data have, until recently, not been readily accessible to medical researchers. However, multiple imputation—a relatively flexible, general purpose approach to dealing with missing data—is now available in standard statistical software,2 3 4 5 making it possible to handle missing data semiroutinely. Results based on this computationally intensive method are increasingly reported, but it needs to be applied carefully to avoid misleading conclusions.
In this article, we review the reasons why missing data may lead to bias and loss of information in epidemiological and clinical research. We discuss the circumstances in which multiple imputation may help by reducing bias or increasing precision, as well as describing potential pitfalls in its application. Finally, we describe the recent use and reporting of analyses using multiple imputation in general medical journals, and suggest guidelines for the conduct and reporting of such analyses.
Researchers usually address missing data by including in the analysis only complete cases —those individuals who have no missing data in any of the variables required for that analysis. However, results of such analyses can be biased. Furthermore, the cumulative effect of missing data in several variables often leads to exclusion of a substantial proportion of the original sample, which in turn causes a substantial loss of precision and power.
The risk of bias due to missing data depends on the reasons why data are missing. Reasons for missing data are commonly classified as: missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR) (box 1).6 This nomenclature is widely used, even though the phrases convey little about their technical meaning and practical implications, which can be subtle. When it is plausible that data are missing at random, but not completely at random, analyses based on complete cases may be biased. Such biases can be overcome using methods such as multiple imputation that allow individuals with incomplete data to be included in analyses. Unfortunately, it is not possible to distinguish between missing at random and missing not at random using observed data. Therefore, biases caused by data that are missing not at random can be addressed only by sensitivity analyses examining the effect of different assumptions about the missing data mechanism.
A variety of ad hoc approaches are commonly used to deal with missing data. These include replacing missing values with values imputed from the observed data (for example, the mean of the observed values), using a missing category indicator,7 and replacing missing values with the last measured value (last value carried forward).8 None of these approaches is statistically valid in general, and they can lead to serious bias. Single imputation of missing values usually causes standard errors to be too small, since it fails to account for the fact that we are uncertain about the missing values.
When there are missing outcome data in a randomised controlled trial, a common sensitivity analysis is to explore “best” and “worst” case scenarios by replacing missing values with “good” outcomes in one group and “bad” outcomes in the other group. This can be useful if there are only a few missing values of a binary outcome, but because imputing all missing values to good or bad is a strong assumption the sensitivity analyses can give a very wide range of estimates of the intervention effect, even if there are only a moderate number of missing outcomes. When outcomes are quantitative (numerical) such sensitivity analyses are not possible because there is no obvious good or bad outcome.
There are circumstances in which analyses of complete cases will not lead to bias. When missing data occur only in an outcome variable that is measured once in each individual, then such analyses will not be biased, provided that all variables associated with the outcome being missing can be included as covariates (under a missing at random assumption). Missing data in predictor variables also do not cause bias in analyses of complete cases if the reasons for the missing data are unrelated to the outcome.9 10 In these circumstances, specialist methods to address missing data may lessen the loss of precision and power resulting from exclusion of individuals with incomplete predictor variables but are not required in order to avoid bias.
If we assume data are missing at random (box 1), then unbiased and statistically more powerful analyses (compared with analyses based on complete cases) can generally be done by including individuals with incomplete data. Sometimes this is possible by building a more general model incorporating information on partially observed variables—for example, using random effects models to incorporate information on partially observed variables from intermediate time points11 12 or by using bayesian methods to incorporate partially observed variables into a full statistical model from which the analysis of interest can be derived.13 Other approaches include weighting the analysis to allow for the missing data,14 15 and maximum likelihood estimation that simultaneously models the reasons for missing data and the associations of interest in the substantive analysis.13 Here, we focus on multiple imputation, which is a popular alternative to these approaches.
Multiple imputation is a general approach to the problem of missing data that is available in several commonly used statistical packages. It aims to allow for the uncertainty about the missing data by creating several different plausible imputed data sets and appropriately combining results obtained from each of them.
The first stage is to create multiple copies of the dataset, with the missing values replaced by imputed values. These are sampled from their predictive distribution based on the observed data—thus multiple imputation is based on a bayesian approach. The imputation procedure must fully account for all uncertainty in predicting the missing values by injecting appropriate variability into the multiple imputed values; we can never know the true values of the missing data.
The second stage is to use standard statistical methods to fit the model of interest to each of the imputed datasets. Estimated associations in each of the imputed datasets will differ because of the variation introduced in the imputation of the missing values, and they are only useful when averaged together to give overall estimated associations. Standard errors are calculated using Rubin’s rules,16 which take account of the variability in results between the imputed datasets, reflecting the uncertainty associated with the missing values. Valid inferences are obtained because we are averaging over the distribution of the missing data given the observed data.
Consider, for example, a study investigating the association of systolic blood pressure with the risk of subsequent coronary heart disease, in which data on systolic blood pressure are missing for some people. The probability that systolic blood pressure is missing is likely to decrease with age (doctors are more likely to measure it in older people), increasing body mass index, and history of smoking (doctors are more likely to measure it in people with heart disease risk factors or comorbidities). If we assume that data are missing at random and that we have systolic blood pressure data on a representative sample of individuals within strata of age, smoking, body mass index, and coronary heart disease, then we can use multiple imputation to estimate the overall association between systolic blood pressure and coronary heart disease.
Multiple imputation has potential to improve the validity of medical research. However, the multiple imputation procedure requires the user to model the distribution of each variable with missing values, in terms of the observed data. The validity of results from multiple imputation depends on such modelling being done carefully and appropriately. Multiple imputation should not be regarded as a routine technique to be applied at the push of a button—whenever possible specialist statistical help should be obtained.
A recent BMJ article reported the development of the QRISK tool for cardiovascular risk prediction, based on a large general practice research database.17 The researchers correctly identified a difficulty with missing data in their database and used multiple imputation to handle the missing data in their analysis. In their published prediction model, however, cardiovascular risk was found to be unrelated to cholesterol (coded as the ratio of total to high density lipoprotein cholesterol), which was surprising.18 The authors have subsequently clarified that when they restricted their analysis to individuals with complete information (no missing data) there was a clear association between cholesterol and cardiovascular risk. Furthermore, a similar result was obtained after using a revised, improved, imputation procedure.19 It is thus important to be aware of problems that can occur in multiple imputation analyses, which we discuss below.
Often an analysis explores the association between one or more predictors and an outcome but some of the predictors have missing values. In this case, the outcome carries information about the missing values of the predictors and this information must be used.20 For example, consider a survival model relating systolic blood pressure to time to coronary heart disease, fitted to data that have some missing values of systolic blood pressure. When missing systolic blood pressure values are imputed, individuals who develop coronary heart disease should have larger values, on average, than those who remain disease free. Failure to include the coronary heart disease outcome and time to this outcome when imputing the missing systolic blood pressure values would falsely weaken the association between systolic blood pressure and coronary heart disease.
Many multiple imputation procedures assume that data are normally distributed, so including non-normally distributed variables may introduce bias. For example, if a biochemical factor had a highly skewed distribution but was implicitly assumed to be normally distributed, then imputation procedures could produce some implausibly low or even negative values. A pragmatic approach here is to transform such variables to approximate normality before imputation and then transform the imputed values back to the original scale. Different problems arise when data are missing in binary or categorical variables. Some procedures21 may handle these types of missing data better than others,13 and this area requires further research.22 23
“Missing at random” is an assumption that justifies the analysis, not a property of the data. For example, the missing at random assumption may be reasonable if a variable that is predictive of missing data in a covariate of interest is included in the imputation model, but not if the variable is omitted from the model. Multiple imputation analyses will avoid bias only if enough variables predictive of missing values are included in the imputation model. For example, if individuals with high socioeconomic status are both more likely to have their systolic blood pressure measured and less likely to have high systolic blood pressure then, unless socioeconomic status is included in the model used when imputing systolic blood pressure, multiple imputation will underestimate mean systolic blood pressure and may wrongly estimate the association between systolic blood pressure and coronary heart disease.
It is sensible to include a wide range of variables in imputation models, including all variables in the substantive analysis, plus, as far as computationally feasible, all variables predictive of the missing values themselves and all variables influencing the process causing the missing data, even if they are not of interest in the substantive analysis.24 Failure to do so may mean that the missing at random assumption is not plausible and that the results of the substantive analysis are biased.
Some data are inherently missing not at random because it is not possible to account for systematic differences between the missing values and the observed values using the observed data. In such cases multiple imputation may give misleading results. For example, consider a study investigating predictors of depression. If individuals are more likely to miss appointments because they are depressed on the day of the appointment, then it may be impossible to make the missing at random assumption plausible, even if a large number of variables is included in the imputation model. When data are missing not at random, bias in analyses based on multiple imputation may be as big as or bigger than the bias in analyses of complete cases. Unfortunately, it is impossible to determine from the data how large a problem this may be. The onus rests on the data analyst to consider all the possible reasons for missing data and assess the likelihood of missing not at random being a serious concern.
Where complete cases and multiple imputation analyses give different results, the analyst should attempt to understand why, and this should be reported in publications.
Multiple imputation is computationally intensive and involves approximations. Some algorithms need to be run repeatedly in order to yield adequate results, and the required run length increases when more data are missing. Unforeseen difficulties may arise when the algorithms are run in settings different from those in which they were developed—for example, with high proportions of missing data, very large numbers of variables, or small numbers of observations. These points are discussed more fully elsewhere.25
The imputation models that were used in the original and revised versions of the QRISK cardiovascular risk prediction tool discussed above have been clarified.26 The main reasons for the unexpected finding of a null association between cholesterol level and cardiovascular risk were omission of the cardiovascular disease outcome when imputing missing cholesterol values and calculation of the ratio of cholesterol to HDL based on imputed cholesterol and HDL values, which led to extreme values of the ratio being included in estimations. The impact of these pitfalls was increased by the high proportion of missing data (70% of HDL cholesterol values were missing).
Multiple imputation usually involves much more complicated statistical modelling than the single regression analyses commonly reported in medical research papers. However, constraints on the length of medical research papers mean that the details of the imputation procedures are often reported briefly, or not at all. Peer reviewers’ lack of familiarity with multiple imputation may make it difficult for them to ask appropriate questions about the methods employed.
To examine recent use and reporting of multiple imputation, we searched four major general medical journals (New England Journal of Medicine, Lancet, BMJ, and JAMA) from 2002 to 2007 for articles reporting original research findings in which multiple imputation had been used. Articles were located by using search facilities on each journal’s website to search for the phrase “multiple imputation” in the full text of all articles published during the specified period. We found 59 articles, and the reported use of multiple imputation roughly doubled over the six years.
The tabletable summarises the results of our survey. Various methods for multiple imputation were used, with the specific method often reported only vaguely (for instance with a book reference). Thirty six papers reported at least some information on the amount of missing data, but only seven fully or partially reported comparisons of distributions of key variables in individuals with and without missing data. The number of imputation based datasets was reported in 22 papers. Results of both imputed and complete cases analyses were fully reported in only seven papers, with one reporting sensitivity analyses. It was thus rarely possible to assess the impact of allowing for missing data. The variables used in imputation models were rarely listed, and the plausibility of the missing at random assumption was rarely assessed or discussed.
In the era of online supplements to research papers, it is feasible and reasonable for authors to provide sufficient detail of imputation analyses to facilitate peer review, without distracting from the substantive research question. Box 2 lists the information that should be provided, either as supplements or within the main paper. This extends guidance provided as part of the STROBE initiative to strengthen the reporting of observational studies,27 and complements suggestions for reporting of analyses using multiple imputation in the epidemiological literature.28
Box 3 relates the suggested guidelines to the use of multiple imputation in a published paper that examined the cost effectiveness of chemotherapy with that of standard palliative care in patients with advanced non-small cell lung cancer.
Burton et al32 used data from a randomised controlled trial to compare the cost effectiveness of chemotherapy with that of standard palliative care in patients with advanced non-small cell lung cancer. Costs were obtained for a subset of 115 patients but were complete for only 82 patients.
They gave the extent and distribution of missing data in table 1 of their paper. Patient and tumour characteristics were stated to be comparable in those with complete and incomplete data, but the effect of treatment on survival was stated to differ. The authors used the multiple imputation procedure in SAS statistical software (PROC MI) to impute the missing data. Variables included in the imputation models were listed. Five imputed datasets were created. A total run length of 12500 iterations was used with imputations made after every 2500th imputation. Log and logit transformations were used to deal with non-normality, and a two stage procedure was used to deal with variables with a high proportion of zero values (semicontinuous distributions). Complete data were transformed back to their original scales before analysis.
The complete case analysis resulted in a higher mean cost for chemotherapy compared with palliative care (£2804 (€3285; $4580), 95% confidence interval £1236 to £4290) than did the analyses using multiple imputation (£2384, 95% CI £833 to £3954). The complete case analyses implied that chemotherapy was not cost effective (mean net monetary benefit −£3346), but the multiple imputation analyses implied that it was cost effective (mean net monetary benefit £1186), although confidence intervals were wide.
In the discussion, the authors noted the multiple imputation analysis “assumes that the incomplete cost data are missing at random such that the missingness of the cost components are associated only with the observed data, either the observed covariates or effectiveness.” They did not, however, discuss whether the missing at random assumption was plausible or conduct sensitivity analyses investigating the robustness of the findings to assumed missing not at random mechanisms.
We are enthusiastic about the potential for multiple imputation and other methods14 to improve the validity of medical research results and to reduce the waste of resources caused by missing data. The cost of multiple imputation analyses is small compared with the cost of collecting the data. It would be a pity if the avoidable pitfalls of multiple imputation slowed progress towards the wider use of these methods. It is no longer excusable for missing values and the reason they arose to be swept under the carpet, nor for potentially misleading and inefficient analyses of complete cases to be considered adequate. We hope that the pitfalls and guidelines discussed here will contribute to the appropriate use and reporting of methods to deal with missing data.
We thank Lucinda Billingham for checking our description of the article described in box 3.
Contributors: JACS, IRW, JBC, and JRC wrote the first draft of the paper. MS conducted the review of the use of multiple imputation in medical journals and analysed the data. All authors contributed to the final draft and subsequent redrafts of the paper. JACS, IRW, and JRC will act as guarantors
Funding: Funded by UK Medical Research Council grant G0600599. IRW was supported by MRC grant U.1052.00.006 and JBC by NHMRC (Australia) grant 334336.
Competing interests: None declared.
Provenance and peer review: Not commissioned; externally peer reviewed.
Cite this as: BMJ 2009;338:b2393