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orms of missingness take different types with different impacts on the validity of conclusions from research missing completely at random missing at random and missing not at random missing data can be handled similarly as censored data types edit understanding the reasons why data are missing is important for handling the remaining data correctly if values are missing completely at random the data sample is likely still representative of the population but if the values are missing systematically analysis may be biased for example in a study of the relation between iq and income if participants with an above average iq tend to skip the question what is your salary analyses that do not take into account this missing at random mar pattern see below may falsely fail to find a positive association between iq and salary because of these problems methodologists routinely advise researchers to design studies to minimize the occurrence of missing values 2 graphical models can be used to describe the missing data mechanism in detail 3 4 the graph shows the probability distributions of the estimations of the expected intensity of depression in the population the number of cases is 60 let the true population be a standardised normal distribution and the non response probability be a logistic function of the intensity of depression the conclusion is the more data is missing mnar the more biased are the estimations we underestimate the intensity of depression in the population missing completely at random edit values in a data set are missing completely at random mcar if the events that lead to any particular data item being missing are independent both of observable variables and of unobservable parameters of interest and occur entirely at random 5 when data are mcar the analysis performed on the data is unbiased however data are rarely mcar in the case of mcar the missingness of data is unrelated to any study variable thus the participants with completely observed data are in effect a random sample of all the participants assigned a particular intervention with mcar the random assignment of treatments is assumed to be preserved but that is usually an unrealistically strong assumption in practice 6 missing at random edit missing at random mar occurs when the missingness is not random but where missingness can be fully accounted for by variables where there is complete information 7 since mar is an assumption that is impossible to verify statistically we must rely on its substantive reasonableness 8 an example is that males are less likely to fill in a depression survey but this has nothing to do with their level of depression after accounting for maleness depending on the analysis method these data can still induce parameter bias in analyses due to the contingent emptiness of cells male very high depression may have zero entries however if the parameter is estimated with full information maximum likelihood mar will provide asymptotically unbiased estimates citation needed missing not at random edit missing not at random mnar also known as nonignorable nonresponse is data that is neither mar nor mcar i e the value of the variable that s missing is related to the reason it s missing 5 to extend the previous example this would occur if men failed to fill in a depression survey because of their level of depression samuelson and spirer 1992 discussed how missing and or distorted data about demographics law enforcement and health could be indicators of patterns of human rights violations they gave several fairly well documented examples 9 structured missingness edit missing data can also arise in subtle ways that are not well accounted for in classical theory an increasingly encountered problem arises in which data may not be mar but missing values exhibit an association or structure either explicitly or implicitly such missingness has been described as structured missingness 10 structured missingness commonly arises when combining information from multiple studies each of which may vary in its design and measurement set and therefore only contain a subset of variables from the union of measurement modalities in these situations missing values may relate to the various sampling methodologies used to collect the data or reflect characteristics of the wider population of interest and so may impart useful information for instance in a health context structured missingness has been observed as a consequence of linking clinical genomic and imaging data 10 the presence of structured missingness may be a hindrance to make effective use of data at scale including through both classical statistical and current machine learning methods for example there might be bias inherent in the reasons why some data might be missing in patterns which might have implications in predictive fairness for machine learning models furthermore established methods for dealing with missing data such as imputation do not usually take into account the structure of the missing data and so development of new formulations is needed to deal with structured missingness appropriately or effectively finally characterising structured missingness within the classical framework of mcar mar and mnar is a work in progress 11 planned missingness edit missing data can also be a deliberate part of study design specifically planned missingness is a research design strategy employed in survey research in which data are intentionally left uncollected from individual respondents typically by administering randomly sampled subsets of items to each participant to reduce burden while preserving the ability to estimate parameters for the full item set across the sample 12 13 techniques of dealing with missing data edit missing data reduces the representativeness of the sample and can therefore distort inferences about the population generally speaking there are three main approaches to handle missing data 1 imputation where values are filled in the place of missing data 2 omission where samples with invalid data are discarded from further analysis and 3 analysis by directly applying methods unaffected by the missing values one systematic review addressing the prevention and handling of missing data for patient centered outcomes research identified 10 standards as necessary for the prevention and handling of missing data these include standards for study design study conduct analysis and reporting 14 in some practical application the experimenters can control the level of missingness and prevent missing values before gathering the data for example in computer questionnaires it is often not possible to skip a question a question has to be answered otherwise one cannot continue to the next so missing values due to the participant are eliminated by this type of questionnaire though this method may not be permitted by an ethics board overseeing the research in survey research it is common to make multiple efforts to contact each individual in the sample often sending letters to attempt to persuade those who have decided not to participate to change their minds 15 161 187 however such techniques can either help or hurt in terms of reducing the negative inferential effects of missing data because the kind of people who are willing to be persuaded to participate after initially refusing or not being home are likely to be significantly different from the kinds of people who will still refuse or remain unreachable after additional effort 15 188 198 in situations where missing values are likely to occur the researcher is often advised on planning to use methods of data analysis methods that are robust to missingness an analysis is robust when we are confident that mild to moderate violations of the technique s key assumptions will produce little or no bias or distortion in the conclusions drawn about the population imputation edit main article imputation statistics some data analysis techniques are not robust to missingness and require to fill in or impute the missing data rubin 1987 argued that repeating imputation even a few times five or less enormously improves the quality of estimation 2 for many practical purposes two or three imputations capture most of the relative efficiency that could be captured with a larger number of imputations however a too small number of imputations can lead to a substantial loss of statistical power and some scholars now recommend 20 to 100 or more 16 any multiply imputed data analysis must be repeated for each of the imputed data sets and in some cases the relevant statistics must be combined in a relatively complicated way 2 multiple imputation is not conducted in specific disciplines as there is a lack of training or misconceptions about them 17 methods such as listwise deletion have been used to impute data but it has been found to introduce additional bias 18 there is a beginner guide that provides a step by step instruction how to impute data 19 the expectation maximization algorithm is an approach in which values of the statistics which would be computed if a complete dataset were available are estimated imputed taking into account the pattern of missing data in this approach values for individual missing data items are not usually imputed interpolation edit main article interpolation in the mathematical field of numerical analysis interpolation is a method of constructing new data points within the range of a discrete set of known data points in the comparison of two paired samples with missing data a test statistic that uses all available data without the need for imputation is the partially overlapping samples t test 20 this is valid under normality and assuming mcar partial deletion edit methods which involve reducing the data available to a dataset having no missing values include listwise deletion casewise deletion pairwise deletion full analysis edit methods which take full account of all information available without the distortion resulting from using imputed values as if they were actually observed generative approaches the expectation maximization algorithm full information maximum likelihood estimation discriminative approaches max margin classification of data with absent features 21 22 partial identification methods may also be used 23 model based techniques edit model based techniques often using graphs offer additional tools for testing missing data types mcar mar mnar and for estimating parameters under missing data conditions for example a test for refuting mar mcar reads as follows for any three variables x y and z where z is fully observed and x and y partially observed the data should satisfy x r y r x z displaystyle x perp perp r_ y r_ x z in words the observed portion of x should be independent on the missingness status of y conditional on every value of z failure to satisfy this condition indicates that the problem belongs to the mnar category 24 remark these tests are necessary for variable based mar which is a slight variation of event based mar 25 26 27 when data falls into mnar category techniques are available for consistently estimating parameters when certain conditions hold in the model 3 for example if y explains the reason for missingness in x and y itself has missing values the joint probability distribution of x and y can still be estimated if the missingness of y is random the estimand in this case will be p x y p x y p y p x y r x 0 r y 0 p y r y 0 displaystyle begin aligned p x y p x y p y p x y r_ x 0 r_ y 0 p y r_ y 0 end aligned where r x 0 displaystyle r_ x 0 and r y 0 displaystyle r_ y 0 denote the observed portions of their respective variables different model structures may yield different estimands and different procedures of estimation whenever consistent estimation is possible the preceding estimand calls for first estimating p x y displaystyle p x y from complete data and multiplying it by p y displaystyle p y estimated from cases in which y is observed regardless of the status of x moreover in order to obtain a consistent estimate it is crucial that the first term be p x y displaystyle p x y as opposed to p y x displaystyle p y x in many cases model based techniques permit the model structure to undergo refutation tests 27 any model which implies the independence between a partially observed variable x and the missingness indicator of another variable y i e r y displaystyle r_ y conditional on r x displaystyle r_ x can be submitted to the following refutation test x r y r x 0 displaystyle x perp perp r_ y r_ x 0 finally the estimands that emerge from these techniques are derived in closed form and do not require iterative procedures such as expectation maximization that are susceptible to local optima 28 a special class of problems appears when the probability of the missingness depends on time for example in the trauma databases the probability to lose data about the trauma outcome depends on the day after trauma in these cases various non stationary markov chain models are applied 29 see also edit censoring condition in which the value of a measurement or observation is only partially known expectation maximization algorithm iterative method for finding maximum likelihood estimates in statistical models imputation process of replacing missing data with substituted values indicator variable numeric stand ins in regression analysis pages displaying short descriptions of redirect targets inverse probability weighting statistical technique latent variable concept in statistics pages displaying short descriptions of redirect targets matrix completion filling in missing entries of a matrix references edit messner sf 1992 exploring the consequences of erratic data reporting for cross national research on homicide journal of quantitative criminology 8 2 155 173 doi 10 1007 bf01066742 s2cid 133325281 1 2 3 4 hand david j adèr herman j mellenbergh gideon j 2008 advising on research methods a consultant s companion huizen netherlands johannes van kessel pp 305 332 isbn 978 90 79418 01 5 1 2 mohan karthika pearl judea tian jin 2013 graphical models for inference with missing data advances in neural information processing systems 26 pp 1277 1285 karvanen juha 2015 study design in causal models scandinavian journal of statistics 42 2 361 377 arxiv 1211 2958 doi 10 1111 sjos 12110 s2cid 53642701 1 2 polit df beck ct 2012 nursing research generating and assessing evidence for nursing practice 9th ed philadelphia usa wolters klower health lippincott williams wilkins deng 2012 10 05 on biostatistics and clinical trials archived from the original on 15 march 2016 retrieved 13 may 2016 home archived from the original on 2015 09 10 retrieved 2015 08 01 little roderick j a rubin donald b 2002 statistical analysis with missing data 2nd ed wiley bibcode 2002samd book l samuelson douglas a spirer herbert f 1992 12 31 chapter 3 use of incomplete and distorted data in inference about human rights violations human rights and sta...
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