# Background

## Assumptions about missingness

There are three assumptions about the process by which data become missing [1].

1. Missing completely at random (MCAR)
2. Missing at random (MAR)
3. Missing not at random (MNAR)

## Probabilistic interpretation

The process by which data become missing is random, and so missing data can be formalised from a probabilistic perspective.

## Mathematical formalism

The following unifies the formalisms in [1], [2], and [3].

- \(D\) : \(n \times p\) data matrix
  - \(D = (D_{obs}, D_{mis})\)
- \(D_{obs}\) : observed values of \(D\)
- \(D_{mis}\) : unobserved values of \(D\)
- \(M\) : \(n \times p\) missingness indicator matrix, where \(M_{ij} = 1\) if \(D_{ij}\) is observed and \(0\) otherwise
- \(j\) : distinct missing data pattern in \(M\)
- \(J\) : total \(j\)
- \(S_j\) : set of cases with missing data pattern \(j\)
- \(m_j\) : number of cases in \(S_j\)
- \(p_j\) : number of observed variables in \(S_j\)
- \(\boldsymbol{\mu}\) : population mean vector
- \(\Sigma\) : population covariance matrix
- \(\hat{\boldsymbol{\mu}}\) : ML estimate of \(\boldsymbol{\mu}\)
- \(\hat{\Sigma}\) : ML estimate of \(\Sigma\)
- \(Q_j\) : \(p \times p_j\) matrix indicating which variables are observed for pattern \(j\)
- \(\hat{\boldsymbol{\mu}}_{obs,j}\) : subset of \(\hat{\boldsymbol{\mu}}\) given by \(\hat{\boldsymbol{\mu}}_{obs,j} \equiv \hat{\boldsymbol{\mu}}Q_j\)
- \(\bar{D}_{obs, j}\) : vector of sample means of observed variables in pattern \(j\).
- \(\hat{\Sigma}_{obs,j}\) : subset of \(\hat{\Sigma}\) given by \(\hat{\Sigma}_{obs,j} \equiv Q_j^{\top}\hat{\Sigma} Q_j\)
- \(\tilde{\Sigma}_{obs, j}\) : accounts for degrees of freedom in \(\hat{\Sigma}_{obs,j}\) given by \(\tilde{\Sigma}_{obs, j} = \frac{n}{n-1}\hat{\Sigma}_{obs,j}\)
- \(d^2\) : statistic used to test MCAR where \(d^2 = \sum_{j=1}^J m_j (\bar{D}_{obs, j} - \hat{\boldsymbol{\mu}}_{obs, j}) \tilde{\Sigma}_{obs, j}^{-1} (\bar{D}_{obs, j} - \hat{\boldsymbol{\mu}}_{obs, j})^{\top}\) =

The definitions of MCAR, MAR, and MNAR are based on the probability distribution of \(M\).

- MCAR
  - \(P(M|D) = P(M)\)
- MAR
  - \(P(M|D) = P(M|D_{obs})\)
- MNAR
  - \(P(M|D) = P(M|D)\)

The above is summarised informally below [1].

Assumption | You can predict \(M\) with:
--- | ---
MCAR | -
MAR | \(D_{obs}\)
MNAR | \(D_{obs}\) and \(D_{mis}\)

## References

[1] King G, Honaker J, Joseph A, Scheve K. Analyzing Incomplete Political Science Data: An Alternative Algorithm for Multiple Imputation. American Political Science Review. 2001 March.

[2] Little RJA. A Test of Missing Completely at Random for Multivariate Data with Missing Values. Journal of the American Statistical Association. 1988;83(404):1198-202.

[3] Rubin DB. Inference and Missing Data. Biometrika. 1976;63(3):581-92.

[4] Joseph G Ibrahim HZ, Tang N. Model Selection Criteria for Missing-Data Problems Using the EM Algorithm. Journal of the American Statistical Asso- ciation. 2008;103(484):1648-58. PMID: 19693282.