In statistics, **missing data**, or **missing values**, occur when no data value is stored for the variable in an observation. Missing data are a common occurrence and can have a significant effect on the conclusions that can be drawn from the data.

- Types
- Missing completely at random
- Missing at random
- Missing not at random
- Techniques of dealing with missing data
- Imputation
- Partial deletion
- Full analysis
- Model-based techniques
- See also
- References
- Further reading
- External links
- Background
- Software

Missing data can occur because of nonresponse: no information is provided for one or more items or for a whole unit ("subject"). Some items are more likely to generate a nonresponse than others: for example items about private subjects such as income. Attrition is a type of missingness that can occur in longitudinal studies—for instance studying development where a measurement is repeated after a certain period of time. Missingness occurs when participants drop out before the test ends and one or more measurements are missing.

Data often are missing in research in economics, sociology, and political science because governments or private entities choose not to, or fail to, report critical statistics,^{ [1] } or because the information is not available. Sometimes missing values are caused by the researcher—for example, when data collection is done improperly or mistakes are made in data entry.^{ [2] }

These forms 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.

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] }

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 (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 (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.

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.^{ [9] }

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.^{ [10] }^{: 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.^{ [10] }^{: 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.

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 (5 or less) enormously improves the quality of estimation.^{ [2] } For many practical purposes, 2 or 3 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.^{ [11] } 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] }

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.

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.^{ [12] } This is valid under normality and assuming MCAR

Methods which involve reducing the data available to a dataset having no missing values include:

- Listwise deletion/casewise deletion
- Pairwise deletion

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
^{ [13] }^{ [14] }

- Max-margin classification of data with absent features

Partial identification methods may also be used.^{ [15] }

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: .

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.^{ [16] }

(Remark: These tests are necessary for variable-based MAR which is a slight variation of event-based MAR.^{ [17] }^{ [18] }^{ [19] })

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:

where and 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 from complete data and multiplying it by 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 as opposed to .

In many cases model based techniques permit the model structure to undergo refutation tests.^{ [19] } Any model which implies the independence between a partially observed variable *X* and the missingness indicator of another variable *Y* (i.e. ), conditional on can be submitted to the following refutation test: .

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.^{ [20] }

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. ^{ [21] }

**Analysis of variance** (**ANOVA**) is a collection of statistical models and their associated estimation procedures used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the *t*-test beyond two means.

**Statistical inference** is the process of using data analysis to infer properties of an underlying distribution of probability. Inferential statistical analysis infers properties of a population, for example by **testing hypotheses** and deriving estimates. It is assumed that the observed data set is sampled from a larger population.

In statistics, the **logistic model** is used to model the probability of a certain class or event existing such as pass/fail, win/lose, alive/dead or healthy/sick. This can be extended to model several classes of events such as determining whether an image contains a cat, dog, lion, etc. Each object being detected in the image would be assigned a probability between 0 and 1, with a sum of one.

**Factor analysis** is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called **factors**. For example, it is possible that variations in six observed variables mainly reflect the variations in two unobserved (underlying) variables. Factor analysis searches for such joint variations in response to unobserved latent variables. The observed variables are modelled as linear combinations of the potential factors, plus "error" terms.

In statistical modeling, **regression analysis** is a set of statistical processes for estimating the relationships between a dependent variable and one or more independent variables. The most common form of regression analysis is linear regression, in which one finds the line that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line that minimizes the sum of squared differences between the true data and that line. For specific mathematical reasons, this allows the researcher to estimate the conditional expectation of the dependent variable when the independent variables take on a given set of values. Less common forms of regression use slightly different procedures to estimate alternative location parameters or estimate the conditional expectation across a broader collection of non-linear models.

In statistics, **imputation** is the process of replacing missing data with substituted values. When substituting for a data point, it is known as "**unit imputation**"; when substituting for a component of a data point, it is known as "**item imputation**". There are three main problems that missing data causes: missing data can introduce a substantial amount of bias, make the handling and analysis of the data more arduous, and create reductions in efficiency. Because missing data can create problems for analyzing data, imputation is seen as a way to avoid pitfalls involved with listwise deletion of cases that have missing values. That is to say, when one or more values are missing for a case, most statistical packages default to discarding any case that has a missing value, which may introduce bias or affect the representativeness of the results. Imputation preserves all cases by replacing missing data with an estimated value based on other available information. Once all missing values have been imputed, the data set can then be analysed using standard techniques for complete data. There have been many theories embraced by scientists to account for missing data but the majority of them introduce bias. A few of the well known attempts to deal with missing data include: hot deck and cold deck imputation; listwise and pairwise deletion; mean imputation; non-negative matrix factorization; regression imputation; last observation carried forward; stochastic imputation; and multiple imputation.

**Random forests** or **random decision forests** are an ensemble learning method for classification, regression and other tasks that operates by constructing a multitude of decision trees at training time. For classification tasks, the output of the random forest is the class selected by most trees. For regression tasks, the mean or average prediction of the individual trees is returned. Random decision forests correct for decision trees' habit of overfitting to their training set. Random forests generally outperform decision trees, but their accuracy is lower than gradient boosted trees. However, data characteristics can affect their performance.

In statistics, econometrics, epidemiology and related disciplines, the method of **instrumental variables** (**IV**) is used to estimate causal relationships when controlled experiments are not feasible or when a treatment is not successfully delivered to every unit in a randomized experiment. Intuitively, IVs are used when an explanatory variable of interest is correlated with the error term, in which case ordinary least squares and ANOVA give biased results. A valid instrument induces changes in the explanatory variable but has no independent effect on the dependent variable, allowing a researcher to uncover the causal effect of the explanatory variable on the dependent variable.

The following is a glossary of terms used in the mathematical sciences statistics and probability.

In statistics, the number of **degrees of freedom** is the number of values in the final calculation of a statistic that are free to vary.

**Robust statistics** is statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, such as estimating location, scale, and regression parameters. One motivation is to produce statistical methods that are not unduly affected by outliers. Another motivation is to provide methods with good performance when there are small departures from parametric distribution. For example, robust methods work well for mixtures of two normal distributions with different standard-deviations; under this model, non-robust methods like a t-test work poorly.

In statistics, a **confounder** is a variable that influences both the dependent variable and independent variable, causing a spurious association. Confounding is a causal concept, and as such, cannot be described in terms of correlations or associations.

In statistics, a **fixed effects model** is a statistical model in which the model parameters are fixed or non-random quantities. This is in contrast to random effects models and mixed models in which all or some of the model parameters are random variables. In many applications including econometrics and biostatistics a fixed effects model refers to a regression model in which the group means are fixed (non-random) as opposed to a random effects model in which the group means are a random sample from a population. Generally, data can be grouped according to several observed factors. The group means could be modeled as fixed or random effects for each grouping. In a fixed effects model each group mean is a group-specific fixed quantity.

**Multilevel models** are statistical models of parameters that vary at more than one level. An example could be a model of student performance that contains measures for individual students as well as measures for classrooms within which the students are grouped. These models can be seen as generalizations of linear models, although they can also extend to non-linear models. These models became much more popular after sufficient computing power and software became available.

In statistics, **ignorability** is a feature of an experiment design whereby the method of data collection do not depend on the missing data. A missing data mechanism such as a treatment assignment or survey sampling strategy is "ignorable" if the missing data matrix, which indicates which variables are observed or missing, is independent of the missing data conditional on the observed data.

**Bootstrapping** is any test or metric that uses random sampling with replacement, and falls under the broader class of resampling methods. Bootstrapping assigns measures of accuracy to sample estimates. This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods.

In statistics, **censoring** is a condition in which the value of a measurement or observation is only partially known.

In the statistical analysis of observational data, **propensity score matching** (**PSM**) is a statistical matching technique that attempts to estimate the effect of a treatment, policy, or other intervention by accounting for the covariates that predict receiving the treatment. PSM attempts to reduce the bias due to confounding variables that could be found in an estimate of the treatment effect obtained from simply comparing outcomes among units that received the treatment versus those that did not. Paul R. Rosenbaum and Donald Rubin introduced the technique in 1983.

The **analysis of clinical trials** involves many related topics including:

**Roderick Joseph Alexander Little** is an academic statistician, whose main research contributions lie in the statistical analysis of data with missing values and the analysis of complex sample survey data. Little is Richard D. Remington Distinguished University Professor of Biostatistics in the Department of Biostatistics at the University of Michigan, where he also holds academic appointments in the Department of Statistics and the Institute for Social Research.

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- Missing values-envision
- psychwiki.com: Missing Values, Identifying Missing Values, and Dealing with Missing Values
- missingdata.org.uk, Department of Medical Statistics, London School of Hygiene & Tropical Medicine
- Spatial and temporal Trend Analysis of Long Term rainfall records in data-poor catchments with missing data, a case study of Lower Shire floodplain in Malawi for the Period 1953–2010. https://www.hydrol-earth-syst-sci-discuss.net/hess-2017-601/hess-2017-601.pdf
- R-miss-tastic, A unified platform for missing values methods and workflows.

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