| \(Y\) |
\(n \times p\) matrix of partially observed sample data |
2.2.3 |
| \(R\) |
\(n \times p\) matrix, 0-1 response indicator of \(Y\) |
2.2.3 |
| \(X\) |
\(n \times q\) matrix of predictors, used for various purposes |
2.2.3 |
| \(Y_\mathrm{obs}\) |
observed sample data, values of \(Y\) with \(R=1\) |
2.2.3 |
| \(Y_\mathrm{mis}\) |
unobserved sample data, values of \(Y\) with \(R=0\) |
2.2.3 |
| \(n\) |
sample size |
2.2.3 |
| \(m\) |
number of multiple imputations |
2.2.3 |
|
|
|
| \(\psi\) |
parameters of the missing data model that relates \(Y\) to \(R\) |
2.2.4 |
| \(\theta\) |
parameters of the scientifically interesting model for the full data \(Y\) |
2.2.5 |
| \(Q\) |
\(k \times 1\) vector with \(k\) scientific estimands |
2.3.1 |
| \(\hat Q\) |
\(k \times 1\) vector, estimate of \(Q\) calculated from a hypothetically complete sample |
|
| \(U\) |
\(k \times k\) matrix, within-imputation variance due to sampling |
2.3.1 |
|
|
|
| \(\ell\) |
imputation number, where \(\ell = 1, \dots, m\) |
2.3.2 |
| \(Y_\ell\) |
\(\ell^\mathrm{th}\) imputed dataset, where \(\ell = 1, \dots, m\) |
2.3.2 |
| \(\bar Q\) |
\(k \times 1\) vector, estimate of \(Q\) calculated from the incompletely observed sample |
2.3.2 |
| \(\bar U\) |
\(k \times k\), estimate of \(U\) from the incomplete data |
2.3.2 |
| \(B\) |
\(k \times k\), between-imputation variance due to nonresponse |
2.3.2 |
| \(T\) |
total variance of \((Q-\bar Q)\), \(k \times k\) matrix |
2.3.2 |
|
|
|
| \(\lambda\) |
proportion of the variance attributable to the missing data for a scalar parameter |
2.3.5 |
| \(\gamma\) |
fraction of information missing due to nonresponse |
2.3.5 |
| \(r\) |
relative increase of variance due to nonresponse for a scalar parameter |
2.3.5 |
| \(\bar\lambda\) |
\(\lambda\) for multivariate \(Q\) |
2.3.5 |
| \(\bar r\) |
\(r\) for multivariate \(Q\) |
2.3.5 |
|
|
|
| \(\nu_\mathrm{old}\) |
old degrees of freedom |
2.3.6 |
| \(\nu\) |
adjusted degrees of freedom |
2.3.6 |
|
|
|
| \(y\) |
univariate \(Y\) |
3.2.1 |
| \(y_\mathrm{obs}\) |
vector with \(n_1\) observed data values in \(y\) |
3.2.1 |
| \(y_\mathrm{mis}\) |
vector with \(n_0\) missing data values in \(y\) |
3.2.1 |
| \(\dot y\) |
vector \(n_0\) imputed values in \(y\) |
3.2.1 |
| \(X_\mathrm{obs}\) |
subset of \(n_1\) rows of \(X\) for which \(y\) is observed |
3.2.1 |
| \(X_\mathrm{mis}\) |
subset of \(n_0\) rows of \(X\) for which \(y\) is missing |
3.2.1 |
| \(\hat\beta\) |
estimate of regression weight \(\beta\) |
3.2 |
| \(\dot\beta\) |
simulated regression weight for \(\beta\) |
3.2.1 |
| \(\hat\sigma^2\) |
estimate of residual variance \(\sigma^2\) |
3.2.1 |
| \(\dot\sigma^2\) |
simulated residual variance for \(\sigma^2\) |
3.2.1 |
| \(\kappa\) |
ridge parameter |
3.2.2 |
| \(\eta\) |
distance parameter in predictive mean matching |
3.4.2 |
| \(\hat y_i\) |
vector of \(n_1\) predicted values given \(X_\mathrm{obs}\) |
3.4.2 |
| \(\hat y_j\) |
vector of \(n_0\) predicted values given \(X_\mathrm{mis}\) |
3.4.2 |
| \(\delta\) |
shift parameter in nonignorable models |
3.8.1 |
|
|
|
| \(Y_j\) |
\(j^\mathrm{th}\) column of \(Y\) |
4.1.1 |
| \(Y_{-j}\) |
all columns of \(Y\) except \(Y_j\) |
4.1.1 |
| \(I_{jk}\) |
proportion of usable cases for imputing \(Y_j\) from \(Y_k\) |
4.1.2 |
| \(O_{jk}\) |
proportion of observed cases in \(Y_j\) to impute \(Y_k\) |
4.1.2 |
| \(I_j\) |
influx statistic to impute \(Y_j\) from \(Y_{-j}\) |
4.1.3 |
| \(O_j\) |
outflux statistic to impute \(Y_{-j}\) from \(Y_j\) |
4.1.3 |
|
|
|
| \(\phi\) |
parameters of the imputation model that models the distribution of \(Y\) |
4.3.1 |
| \(M\) |
number of iterations |
4.5 |
|
|
|
| \(D_1\) |
test statistic of Wald test |
5.3.1 |
| \(r_1\) |
\(r\) for Wald test |
5.3.1 |
| \(\nu_1\) |
\(\nu\) for Wald test |
5.3.1 |
| \(D_2\) |
test statistic for \(\chi^2\)-test |
5.3.2 |
| \(r_2\) |
\(r\) for \(\chi^2\)-test |
5.3.2 |
| \(\nu_2\) |
\(\nu\) for \(\chi^2\)-test |
5.3.2 |
| \(D_3\) |
test statistic for likelihood ratio test |
5.3.3 |
| \(r_3\) |
\(r\) for likelihood ratio test |
5.3.3 |
| \(\nu_3\) |
\(\nu\) for likelihood ratio test |
5.3.3 |
|
|
|
| \(C\) |
number of classes |
7.2 |
| \(c\) |
class index, \(c=1,\dots,C\) |
7.2 |
| \(y_c\) |
outcome vector for cluster \(c\) |
7.2 |
| \(X_c\) |
design matrix, fixed effects |
7.2 |
| \(Z_c\) |
design matrix, random effects |
7.2 |
| \(\Omega\) |
covariance matrix, random effects |
7.2 |
|
|
|
| \(Y_i(1)\) |
outcome of unit \(i\) under treatment |
8.1 |
| \(Y_i(0)\) |
outcome of unit \(i\) under control |
8.1 |
| \(\tau_i\) |
individual causal effect |
8.1 |
| \(\tau\) |
average causal effect |
8.1 |
| \(\dot\tau_{i\ell}\) |
simulated \(\tau_i\) in \(\ell^\mathrm{th}\) imputed data set |
8.3 |
| \(\hat\tau_{i}\) |
estimate of \(\tau_i\) |
8.3 |
| \(\hat\sigma_i^2\) |
variance estimate of \(\hat\tau_{i}\) |
8.3 |
| \(\dot\tau_{\ell}\) |
simulated within-replication average causal effect |
8.4.3 |
| \(\dot\sigma_{\ell}^2\) |
variance of \(\dot\tau_{\ell}\) |
8.4.3 |