# Symbol Description

Symbol Description Section
$$Y$$ $$n \times p$$ 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