Package 'lsirm12pl'

Big Five Personality Test

Description

A dataset containing the result of personality test for 50 questions from 1,000 random sampled people.

Usage

data(BFPT)

Format

A matrix with 1,015,341 rows and 50 columns.

Details

A dataset collected in 2016-2018 through an interactive on-line personality test, containing the result of personality test for 50 questions. 1,000 people are random sampled from the original dataset containing 1,015,341 people. The scale is labeled as 1=Disagree, 3=Neutral and 5=Agree.

Source

https://www.kaggle.com/tunguz/big-five-personality-test

---

Diagnostic the result of LSIRM.

Description

diagnostic checks the convergence of MCMC for LSIRM parameters using various diagnostic tools, such as trace plots, posterior density distributions, autocorrelation functions (ACF), and Gelman-Rubin-Brooks plots.

Usage

diagnostic(
  object,
  draw.item = list(beta = c(1), theta = c(1)),
  gelman.diag = FALSE
)

Arguments

object	Object of class lsirm.
draw.item	List; Each key in the list corresponds to a specific parameters such as "beta", "theta", "gamma", "alpha", "sigma", "sigma_sd", and "zw.dist". The values of the list indicate the indices of these parameters. For the key "zw.dist", the value is a matrix with two columns: the first column represents the indices of respondents, and the second column represents the indices of items.
gelman.diag	Logical; If TRUE, the Gelman-Rubin convergence diagnostic will be printed. Default is FALSE.

Value

diagnostic returns plots for checking MCMC convergence for selected parameters.

Examples

# Generate example item response matrix
data     <- matrix(rbinom(500, size = 1, prob = 0.5), ncol=10, nrow=50)

# For 1PL LSIRM
lsirm_result <- lsirm(data ~ lsirm1pl(spikenslab = FALSE, fixed_gamma = FALSE))
diagnostic(lsirm_result)

# For 2PL LSIRM
lsirm_result <- lsirm(data ~ lsirm2pl(spikenslab = FALSE, fixed_gamma = FALSE))
diagnostic(lsirm_result)

---

Goodness-of-fit LSIRM

Description

gof is goodness-of-fit the latent space of fitted LSIRM.

Usage

gof(object, chain.idx = 1)

Arguments

object	Object of class lsirm.
chain.idx	Numeric; Index of MCMC chain. Default is 1.

Value

gof returns the boxplot or AUC plot

Examples

# generate example item response matrix
data     <- matrix(rbinom(500, size = 1, prob = 0.5),ncol=10,nrow=50)
lsirm_result <- lsirm(data ~ lsirm1pl())
gof(lsirm_result)

---

Fit a LSIRM ( Latent Space Item Response Model)

Description

lsirm is used to fit 1PL LSIRM and 2PL LSIRM using Bayesian method as described in Jeon et al. (2021).

Usage

lsirm(formula, ...)

Arguments

formula	The form of formula is lsirm(A ~ <term 1>(<term 2>, <term 3> ...)), where A is binary or continuous item response matrix to be analyzed, <term1> is the model you want to fit and has one of the following values: "lsirm1pl" and "lsirm2pl"., and <term 2>, <term 3>, etc. are each option for the model.
...	Additional arguments for the corresponding function.

Details

The descriptions of options for each model, such as <term 2> and <term 3>, are included in lsirm1pl for 1PL LSIRM and lsirm2pl for 2PL LSIRM.

Value

lsirm returns an object of class list.

See corresponding functions such as lsirm1pl for 1PL LSIRM and lsirm2pl for 2PL LSIRM.

See Also

lsirm1pl for 1PL LSIRM.

lsirm2pl for 2PL LSIRM.

Examples

# generate example item response matrix
data     <- matrix(rbinom(500, size = 1, prob = 0.5),ncol=10,nrow=50)

lsirm_result <- lsirm(data~lsirm1pl())
lsirm_result <- lsirm(data~lsirm2pl())

---

Formula function for LSIRM

Description

lsirm.formula is formula object.

Usage

## S3 method for class 'formula'
lsirm(formula, ...)

Arguments

formula	The form of formula is lsirm(A ~ <term 1>(<term 2>, <term 3> ...)), where A is binary or continuous item response matrix to be analyzed, <term1> is the model you want to fit and has one of the following values: "lsirm1pl" and "lsirm2pl"., and <term 2>, <term 3>, etc., are each option for the model.
...	Additional arguments for the corresponding function.

---

lsirm12pl-package

Description

Analysis of dichotomous and continuous response data using latent factor by both 1PL LSIRM and 2PL LSIRM as described in Jeon et al. (2021) <doi:10.1007/s11336-021-09762-5>. It includes original 1PL LSIRM and 2PL LSIRM provided for binary response data and its extension for continuous response data. Bayesian model selection with spike-and-slab prior and method for dealing data with missing value under missing at random, missing completely at random are also supported. Various diagnostic plots are available to inspect the latent space and summary of estimated parameters.

---

Fit a 1PL LSIRM for binary and continuous item response data

Description

lsirm1pl integrates all functions related to 1PL LSIRM. Various 1PL LSIRM function can be used by setting the spikenslab, fixed_gamma, and missing_data arguments.

This function can be used regardless of the data type, providing a unified approach to model fitting.

Usage

lsirm1pl(
  data,
  spikenslab = FALSE,
  fixed_gamma = FALSE,
  missing_data = NA,
  chains = 1,
  multicore = 1,
  seed = NA,
  ndim = 2,
  niter = 15000,
  nburn = 2500,
  nthin = 5,
  nprint = 500,
  jump_beta = 0.4,
  jump_theta = 1,
  jump_z = 0.5,
  jump_w = 0.5,
  pr_mean_beta = 0,
  pr_sd_beta = 1,
  pr_mean_theta = 0,
  pr_a_theta = 0.001,
  pr_b_theta = 0.001,
  ...
)

Arguments

data	Matrix; a binary or continuous item response matrix for analysis. Each row represents a respondent, and each column contains responses to the corresponding item.
spikenslab	Logical; specifies whether to use a model selection approach. Default is FALSE.
fixed_gamma	Logical; indicates whether to fix gamma at 1. Default is FALSE.
missing_data	Character; the type of missing data assumed. Options are NA, "mar", or "mcar". Default is NA.
chains	Integer; the number of MCMC chains to run. Default is 1.
multicore	Integer; the number of cores to use for parallel execution. Default is 1.
seed	Integer; the seed number for MCMC fitting. Default is NA.
ndim	Integer; the dimension of the latent space. Default is 2.
niter	Integer; the total number of MCMC iterations to run. Default is 15000.
nburn	Integer; the number of initial MCMC iterations to discard as burn-in. Default is 2500.
nthin	Integer; the number of MCMC iterations to thin. Default is 5.
nprint	Integer; the interval at which MCMC samples are displayed during execution. Default is 500.
jump_beta	Numeric; the jumping rule for the beta proposal density. Default is 0.4.
jump_theta	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_z	Numeric; the jumping rule for the z proposal density. Default is 0.5.
jump_w	Numeric; the jumping rule for the w proposal density. Default is 0.5.
pr_mean_beta	Numeric; the mean of the normal prior for beta. Default is 0.
pr_sd_beta	Numeric; the standard deviation of the normal prior for beta. Default is 1.0.
pr_mean_theta	Numeric; the mean of the normal prior for theta. Default is 0.
pr_a_theta	Numeric; the shape parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_b_theta	Numeric; the scale parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
...	Additional arguments for the for various settings. Refer to the functions in the Details.

Details

Additional arguments and return values for each function are documented in the respective function's description.

* For LSIRM with data included missing value are detailed in lsirm1pl_mar and lsirm1pl_mcar.

* For LSIRM using the spike-and-slab model selection approach are detailed in lsirm1pl_ss.

* For continuous version of LSIRM are detailed in lsirm1pl_normal_o.

For 1PL LSIRM with binary item response data, the probability of correct response by respondent $j$ to item $i$ with item effect $\beta_i$, respondent effect $\theta_j$ and the distance between latent position $w_i$ of item $i$ and latent position $z_j$ of respondent $j$ in the shared metric space, with $\gamma$ represents the weight of the distance term:

$logit(P(Y_{j,i} = 1|\theta_j,\beta_i,\gamma,z_j,w_i))=\theta_j+\beta_i-\gamma||z_j-w_i||$

For 1PL LSIRM with continuous item response data, the continuous value of response by respondent $j$ to item $i$ with item effect $\beta_i$, respondent effect $\theta_j$ and the distance between latent position $w_i$ of item $i$ and latent position $z_j$ of respondent $j$ in the shared metric space, with $\gamma$ represents the weight of the distance term:

$Y_{j,i} = \theta_j+\beta_i-\gamma||z_j-w_i|| + e_{j,i}$

where the error $e_{j,i} \sim N(0,\sigma^2)$.

Value

lsirm1pl returns an object of list. The basic return list containing the following components:

data	A data frame or matrix containing the variables used in the model.
bic	A numeric value representing the Bayesian Information Criterion (BIC).
mcmc_inf	Details about the number of MCMC iterations, burn-in periods, and thinning intervals.
map_inf	The log maximum a posteriori (MAP) value and the iteration number at which this MAP value occurs.
beta_estimate	Posterior estimates of the beta parameter.
theta_estimate	Posterior estimates of the theta parameter.
sigma_theta_estimate	Posterior estimates of the standard deviation of theta.
z_estimate	Posterior estimates of the z parameter.
w_estimate	Posterior estimates of the w parameter.
beta	Posterior samples of the beta parameter.
theta	Posterior samples of the theta parameter.
theta_sd	Posterior samples of the standard deviation of theta.
z	Posterior samples of the z parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
w	Posterior samples of the w parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
accept_beta	Acceptance ratio for the beta parameter.
accept_theta	Acceptance ratio for the theta parameter.
accept_z	Acceptance ratio for the z parameter.
accept_w	Acceptance ratio for the w parameter.
...	Additional return values for various settings. Refer to the functions in the Details.

Note

If both spikenslab and fixed_gamma are set TRUE, it returns error because both are related to gamma.

See Also

The LSIRM for 1PL LSIRM for binary item response data as following:

lsirm1pl_o, lsirm1pl_fixed_gamma, lsirm1pl_mar,lsirm1pl_mcar, lsirm1pl_fixed_gamma_mar, lsirm1pl_fixed_gamma_mcar, lsirm1pl_ss, lsirm1pl_mar_ss, and lsirm1pl_mcar_ss

The LSIRM for 1PL LSIRM for continuous item response data as following:

lsirm1pl_normal_o, lsirm1pl_normal_fixed_gamma, lsirm1pl_normal_mar, lsirm1pl_normal_mcar,lsirm1pl_normal_fixed_gamma_mar, lsirm1pl_normal_fixed_gamma_mcar, lsirm1pl_normal_ss, lsirm1pl_normal_mar_ss, lsirm1pl_normal_mcar_ss

Examples

# generate example item response matrix
data     <- matrix(rbinom(500, size = 1, prob = 0.5),ncol=10,nrow=50)
lsirm_result <- lsirm1pl(data)

# The code following can achieve the same result.
lsirm_result <- lsirm(data~lsirm1pl())

---

1PL LSIRM fixing gamma to 1.

Description

lsirm1pl_fixed_gamma is used to fit 1PL LSIRM with gamma fixed to 1. lsirm1pl_fixed_gamma factorizes item response matrix into column-wise item effect, row-wise respondent effect and further embeds interaction effect in a latent space. The resulting latent space provides an interaction map that represents interactions between respondents and items.

Usage

lsirm1pl_fixed_gamma(
  data,
  ndim = 2,
  niter = 15000,
  nburn = 2500,
  nthin = 5,
  nprint = 500,
  jump_beta = 0.4,
  jump_theta = 1,
  jump_z = 0.5,
  jump_w = 0.5,
  pr_mean_beta = 0,
  pr_sd_beta = 1,
  pr_mean_theta = 0,
  pr_a_theta = 0.001,
  pr_b_theta = 0.001,
  verbose = FALSE
)

Arguments

data	Matrix; a binary or continuous item response matrix for analysis. Each row represents a respondent, and each column contains responses to the corresponding item.
ndim	Integer; the dimension of the latent space. Default is 2.
niter	Integer; the total number of MCMC iterations to run. Default is 15000.
nburn	Integer; the number of initial MCMC iterations to discard as burn-in. Default is 2500.
nthin	Integer; the number of MCMC iterations to thin. Default is 5.
nprint	Integer; the interval at which MCMC samples are displayed during execution. Default is 500.
jump_beta	Numeric; the jumping rule for the beta proposal density. Default is 0.4.
jump_theta	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_z	Numeric; the jumping rule for the z proposal density. Default is 0.5.
jump_w	Numeric; the jumping rule for the w proposal density. Default is 0.5.
pr_mean_beta	Numeric; the mean of the normal prior for beta. Default is 0.
pr_sd_beta	Numeric; the standard deviation of the normal prior for beta. Default is 1.0.
pr_mean_theta	Numeric; the mean of the normal prior for theta. Default is 0.
pr_a_theta	Numeric; the shape parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_b_theta	Numeric; the scale parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
verbose	Logical; If TRUE, MCMC samples are printed for each nprint. default value is FALSE

Details

lsirm1pl_fixed_gamma models the probability of correct response by respondent $j$ to item $i$ with item effect $\beta_i$, respondent effect $\theta_j$ and the distance between latent position $w_i$ of item $i$ and latent position $z_j$ of respondent $j$ in the shared metric space:

$logit(P(Y_{j,i} = 1|\theta_j,\beta_i,z_j,w_i))=\theta_j+\beta_i-||z_j-w_i||$

Value

lsirm1pl_fixed_gamma returns an object of list containing the following components:

data	Data frame or matrix containing the variables in the model.
bic	Numeric value with the corresponding BIC.
mcmc_inf	Details about the number of MCMC iterations, burn-in periods, and thinning intervals.
map_inf	The log maximum a posteriori (MAP) value and the iteration number at which this MAP value occurs.
beta_estimate	Posterior estimates of the beta parameter.
theta_estimate	Posterior estimates of the theta parameter.
sigma_theta_estimate	Posterior estimates of the standard deviation of theta.
z_estimate	Posterior estimates of the z parameter.
w_estimate	Posterior estimates of the w parameter.
beta	Posterior samples of the beta parameter.
theta	Posterior samples of the theta parameter.
theta_sd	Posterior samples of the standard deviation of theta.
z	Posterior samples of the z parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
w	Posterior samples of the w parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
accept_beta	Acceptance ratio for the beta parameter.
accept_theta	Acceptance ratio for the theta parameter.
accept_z	Acceptance ratio for the z parameter.
accept_w	Acceptance ratio for the w parameter.

Examples

# generate example item response matrix
data     <- matrix(rbinom(500, size = 1, prob = 0.5),ncol=10,nrow=50)

lsirm_result <- lsirm1pl_fixed_gamma(data)

# The code following can achieve the same result.
lsirm_result <- lsirm(data ~ lsirm1pl(spikenslab = FALSE, fixed_gamma = TRUE))

---

1PL LSIRM fixing gamma to 1 for missing at random data.

Description

lsirm1pl_fixed_gamma_mar is used to fit LSIRM with gamma fixed to 1 in incomplete data assumed to be missing at random. lsirm1pl_fixed_gamma_mar factorizes item response matrix into column-wise item effect, row-wise respondent effect and further embeds interaction effect in a latent space, while considering the missing element under the assumption of missing at random. The resulting latent space provides an interaction map that represents interactions between respondents and items.

Usage

lsirm1pl_fixed_gamma_mar(
  data,
  ndim = 2,
  niter = 15000,
  nburn = 2500,
  nthin = 5,
  nprint = 500,
  jump_beta = 0.4,
  jump_theta = 1,
  jump_z = 0.5,
  jump_w = 0.5,
  pr_mean_beta = 0,
  pr_sd_beta = 1,
  pr_mean_theta = 0,
  pr_a_theta = 0.001,
  pr_b_theta = 0.001,
  missing.val = 99,
  verbose = FALSE
)

Arguments

data	Matrix; a binary or continuous item response matrix for analysis. Each row represents a respondent, and each column contains responses to the corresponding item.
ndim	Integer; the dimension of the latent space. Default is 2.
niter	Integer; the total number of MCMC iterations to run. Default is 15000.
nburn	Integer; the number of initial MCMC iterations to discard as burn-in. Default is 2500.
nthin	Integer; the number of MCMC iterations to thin. Default is 5.
nprint	Integer; the interval at which MCMC samples are displayed during execution. Default is 500.
jump_beta	Numeric; the jumping rule for the beta proposal density. Default is 0.4.
jump_theta	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_z	Numeric; the jumping rule for the z proposal density. Default is 0.5.
jump_w	Numeric; the jumping rule for the w proposal density. Default is 0.5.
pr_mean_beta	Numeric; the mean of the normal prior for beta. Default is 0.
pr_sd_beta	Numeric; the standard deviation of the normal prior for beta. Default is 1.0.
pr_mean_theta	Numeric; the mean of the normal prior for theta. Default is 0.
pr_a_theta	Numeric; the shape parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_b_theta	Numeric; the scale parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
missing.val	Numeric; a number to replace missing values. Default is 99.
verbose	Logical; If TRUE, MCMC samples are printed for each nprint. default value is FALSE.

Details

lsirm1pl_fixed_gamma_mar models the probability of correct response by respondent $j$ to item $i$ with item effect $\beta_i$, respondent effect $\theta_j$ and the distance between latent position $w_i$ of item $i$ and latent position $z_j$ of respondent $j$ in the shared metric space:

$logit(P(Y_{j,i} = 1|\theta_j,\beta_i,z_j,w_i))=\theta_j+\beta_i-||z_j-w_i||$

Under the assumption of missing at random, the model takes the missing element into consideration in the sampling procedure. For the details of missing at random assumption and data augmentation, see References.

Value

lsirm1pl_fixed_gamma_mar returns an object of list containing the following components:

data	Data frame or matrix containing the variables in the model.
missing.val	A number to replace missing values.
bic	Numeric value with the corresponding BIC.
mcmc_inf	Details about the number of MCMC iterations, burn-in periods, and thinning intervals.
map_inf	The log maximum a posteriori (MAP) value and the iteration number at which this MAP value occurs.
beta_estimate	Posterior estimates of the beta parameter.
theta_estimate	Posterior estimates of the theta parameter.
sigma_theta_estimate	Posterior estimates of the standard deviation of theta.
z_estimate	Posterior estimates of the z parameter.
w_estimate	Posterior estimates of the w parameter.
imp_estimate	Probability of imputating a missing value with 1.
beta	Posterior samples of the beta parameter.
theta	Posterior samples of the theta parameter.
theta_sd	Posterior samples of the standard deviation of theta.
z	Posterior samples of the z parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
w	Posterior samples of the w parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
imp	Imputation for missing Values using posterior samples.
accept_beta	Acceptance ratio for the beta parameter.
accept_theta	Acceptance ratio for the theta parameter.
accept_z	Acceptance ratio for the z parameter.
accept_w	Acceptance ratio for the w parameter.

References

Little, R. J., & Rubin, D. B. (2019). Statistical analysis with missing data (Vol. 793). John Wiley & Sons.

Examples

# generate example item response matrix
data     <- matrix(rbinom(500, size = 1, prob = 0.5),ncol=10,nrow=50)

# generate example missing indicator matrix
missing_mat     <- matrix(rbinom(500, size = 1, prob = 0.2),ncol=10,nrow=50)

# make missing value with missing indicator matrix
data[missing_mat==1] <- 99

lsirm_result <- lsirm1pl_fixed_gamma_mar(data)

# The code following can achieve the same result.
lsirm_result <- lsirm(data ~ lsirm1pl(spikenslab = FALSE, fixed_gamma = TRUE,
missing_data = "mar", missing.val = 99))

---

1PL LSIRM fixing gamma to 1 for missing completely at random data.

Description

lsirm1pl_fixed_gamma_mcar is used to fit LSIRM with gamma fixed to 1 in incomplete data assumed to be missing completely at random. lsirm1pl_fixed_gamma_mcar factorizes item response matrix into column-wise item effect, row-wise respondent effect and further embeds interaction effect in a latent space, while ignoring the missing element under the assumption of missing completely at random. The resulting latent space provides an interaction map that represents interactions between respondents and items.

Usage

lsirm1pl_fixed_gamma_mcar(
  data,
  ndim = 2,
  niter = 15000,
  nburn = 2500,
  nthin = 5,
  nprint = 500,
  jump_beta = 0.4,
  jump_theta = 1,
  jump_z = 0.5,
  jump_w = 0.5,
  pr_mean_beta = 0,
  pr_sd_beta = 1,
  pr_mean_theta = 0,
  pr_a_theta = 0.001,
  pr_b_theta = 0.001,
  missing.val = 99,
  verbose = FALSE
)

Arguments

data	Matrix; a binary or continuous item response matrix for analysis. Each row represents a respondent, and each column contains responses to the corresponding item.
ndim	Integer; the dimension of the latent space. Default is 2.
niter	Integer; the total number of MCMC iterations to run. Default is 15000.
nburn	Integer; the number of initial MCMC iterations to discard as burn-in. Default is 2500.
nthin	Integer; the number of MCMC iterations to thin. Default is 5.
nprint	Integer; the interval at which MCMC samples are displayed during execution. Default is 500.
jump_beta	Numeric; the jumping rule for the beta proposal density. Default is 0.4.
jump_theta	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_z	Numeric; the jumping rule for the z proposal density. Default is 0.5.
jump_w	Numeric; the jumping rule for the w proposal density. Default is 0.5.
pr_mean_beta	Numeric; the mean of the normal prior for beta. Default is 0.
pr_sd_beta	Numeric; the standard deviation of the normal prior for beta. Default is 1.0.
pr_mean_theta	Numeric; the mean of the normal prior for theta. Default is 0.
pr_a_theta	Numeric; the shape parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_b_theta	Numeric; the scale parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
missing.val	Numeric; a number to replace missing values. Default is 99.
verbose	Logical; If TRUE, MCMC samples are printed for each nprint. default value is FALSE

Details

lsirm1pl_fixed_gamma_mcar models the probability of correct response by respondent $j$ to item $i$ with item effect $\beta_i$, respondent effect $\theta_j$ and the distance between latent position $w_i$ of item $i$ and latent position $z_j$ of respondent $j$ in the shared metric space:

$logit(P(Y_{j,i} = 1|\theta_j,\beta_i,z_j,w_i))=\theta_j+\beta_i-||z_j-w_i||$

Under the assumption of missing completely at random, the model ignores the missing element in doing inference. For the details of missing completely at random assumption and data augmentation, see References.

Value

lsirm1pl_fixed_gamma_mcar returns an object of list containing the following components:

data	Data frame or matrix containing the variables in the model.
missing.val	A number to replace missing values.
bic	Numeric value with the corresponding BIC.
mcmc_inf	Details about the number of MCMC iterations, burn-in periods, and thinning intervals.
map_inf	The log maximum a posteriori (MAP) value and the iteration number at which this MAP value occurs.
beta_estimate	Posterior estimates of the beta parameter.
theta_estimate	Posterior estimates of the theta parameter.
sigma_theta_estimate	Posterior estimates of the standard deviation of theta.
z_estimate	Posterior estimates of the z parameter.
w_estimate	Posterior estimates of the w parameter.
beta	Posterior samples of the beta parameter.
theta	Posterior samples of the theta parameter.
theta_sd	Posterior samples of the standard deviation of theta.
z	Posterior samples of the z parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
w	Posterior samples of the w parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
accept_beta	Acceptance ratio for the beta parameter.
accept_theta	Acceptance ratio for the theta parameter.
accept_z	Acceptance ratio for the z parameter.
accept_w	Acceptance ratio for the w parameter.

References

Little, R. J., & Rubin, D. B. (2019). Statistical analysis with missing data (Vol. 793). John Wiley & Sons.

Examples

# generate example item response matrix
data     <- matrix(rbinom(500, size = 1, prob = 0.5),ncol=10,nrow=50)

# generate example missing indicator matrix
missing_mat     <- matrix(rbinom(500, size = 1, prob = 0.2),ncol=10,nrow=50)

# make missing value with missing indicator matrix
data[missing_mat==1] <- 99

lsirm_result <- lsirm1pl_fixed_gamma_mcar(data)

# The code following can achieve the same result.
lsirm_result <- lsirm(data ~ lsirm1pl(spikenslab = FALSE, fixed_gamma = TRUE,
missing_data = "mcar", missing.val = 99))

---

1PL LSIRM for missing at random data.

Description

lsirm1pl_mar is used to fit 1PL LSIRM in incomplete data assumed to be missing at random. lsirm1pl_mar factorizes item response matrix into column-wise item effect, row-wise respondent effect and further embeds interaction effect in a latent space, while considering the missing element under the assumption of missing at random. The resulting latent space provides an interaction map that represents interactions between respondents and items.

Usage

lsirm1pl_mar(
  data,
  ndim = 2,
  niter = 15000,
  nburn = 2500,
  nthin = 5,
  nprint = 500,
  jump_beta = 0.4,
  jump_theta = 1,
  jump_gamma = 0.025,
  jump_z = 0.5,
  jump_w = 0.5,
  pr_mean_beta = 0,
  pr_sd_beta = 1,
  pr_mean_theta = 0,
  pr_mean_gamma = 0.5,
  pr_sd_gamma = 1,
  pr_a_theta = 0.001,
  pr_b_theta = 0.001,
  missing.val = 99,
  verbose = FALSE
)

Arguments

data	Matrix; a binary or continuous item response matrix for analysis. Each row represents a respondent, and each column contains responses to the corresponding item.
ndim	Integer; the dimension of the latent space. Default is 2.
niter	Integer; the total number of MCMC iterations to run. Default is 15000.
nburn	Integer; the number of initial MCMC iterations to discard as burn-in. Default is 2500.
nthin	Integer; the number of MCMC iterations to thin. Default is 5.
nprint	Integer; the interval at which MCMC samples are displayed during execution. Default is 500.
jump_beta	Numeric; the jumping rule for the beta proposal density. Default is 0.4.
jump_theta	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_gamma	Numeric; the jumping rule for the gamma proposal density. Default is 0.025
jump_z	Numeric; the jumping rule for the z proposal density. Default is 0.5.
jump_w	Numeric; the jumping rule for the w proposal density. Default is 0.5.
pr_mean_beta	Numeric; the mean of the normal prior for beta. Default is 0.
pr_sd_beta	Numeric; the standard deviation of the normal prior for beta. Default is 1.0.
pr_mean_theta	Numeric; the mean of the normal prior for theta. Default is 0.
pr_mean_gamma	Numeric; mean of log normal prior for gamma. Default is 0.5.
pr_sd_gamma	Numeric; standard deviation of log normal prior for gamma. Default is 1.0.
pr_a_theta	Numeric; the shape parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_b_theta	Numeric; the scale parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
missing.val	Numeric; a number to replace missing values. Default is 99.
verbose	Logical; If TRUE, MCMC samples are printed for each nprint. default value is FALSE.

Details

lsirm1pl_mar models the probability of correct response by respondent $j$ to item $i$ with item effect $\beta_i$, respondent effect $\theta_j$ and the distance between latent position $w_i$ of item $i$ and latent position $z_j$ of respondent $j$ in the shared metric space, with $\gamma$ represents the weight of the distance term:

$logit(P(Y_{j,i} = 1|\theta_j,\beta_i,\gamma,z_j,w_i))=\theta_j+\beta_i-\gamma||z_j-w_i||$

Under the assumption of missing at random, the model takes the missing element into consideration in the sampling procedure. For the details of missing at random assumption and data augmentation, see References.

Value

lsirm1pl_mar returns an object of list containing the following components:

data	Data frame or matrix containing the variables in the model.
missing.val	A number to replace missing values.
bic	Numeric value with the corresponding BIC.
mcmc_inf	Details about the number of MCMC iterations, burn-in periods, and thinning intervals.
map_inf	The log maximum a posteriori (MAP) value and the iteration number at which this MAP value occurs.
beta_estimate	Posterior estimates of the beta parameter.
theta_estimate	Posterior estimates of the theta parameter.
sigma_theta_estimate	Posterior estimates of the standard deviation of theta.
gamma_estimate	posterior estimates of gamma parameter.
z_estimate	Posterior estimates of the z parameter.
w_estimate	Posterior estimates of the w parameter.
imp_estimate	Probability of imputating a missing value with 1.
beta	Posterior samples of the beta parameter.
theta	Posterior samples of the theta parameter.
gamma	Posterior samples of the gamma parameter.
theta_sd	Posterior samples of the standard deviation of theta.
z	Posterior samples of the z parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
w	Posterior samples of the w parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
imp	Imputation for missing Values using posterior samples.
accept_beta	Acceptance ratio for the beta parameter.
accept_theta	Acceptance ratio for the theta parameter.
accept_z	Acceptance ratio for the z parameter.
accept_w	Acceptance ratio for the w parameter.
accept_gamma	Acceptance ratio for the gamma parameter.

References

Little, R. J., & Rubin, D. B. (2019). Statistical analysis with missing data (Vol. 793). John Wiley & Sons.

Examples

# generate example item response matrix
data     <- matrix(rbinom(500, size = 1, prob = 0.5),ncol=10,nrow=50)

# generate example missing indicator matrix
missing_mat     <- matrix(rbinom(500, size = 1, prob = 0.2),ncol=10,nrow=50)

# make missing value with missing indicator matrix
data[missing_mat==1] <- 99

lsirm_result <- lsirm1pl_mar(data)

# The code following can achieve the same result.
lsirm_result <- lsirm(data ~ lsirm1pl(spikenslab = FALSE, fixed_gamma = FALSE,
missing_data ='mar', missing.val = 99))

---

1PL LSIRM with model selection approach for missing at random data.

Description

lsirm1pl_mar_ss is used to fit 1PL LSIRM with model selection approach based on spike-and-slab priors in incomplete data assumed to be missing at random. lsirm1pl_mar_ss factorizes item response matrix into column-wise item effect, row-wise respondent effect and further embeds interaction effect in a latent space, while considering the missing element under the assumption of missing at random. The resulting latent space provides an interaction map that represents interactions between respondents and items.

Usage

lsirm1pl_mar_ss(
  data,
  ndim = 2,
  niter = 15000,
  nburn = 2500,
  nthin = 5,
  nprint = 500,
  jump_beta = 0.4,
  jump_theta = 1,
  jump_gamma = 1,
  jump_z = 0.5,
  jump_w = 0.5,
  pr_mean_beta = 0,
  pr_sd_beta = 1,
  pr_mean_theta = 0,
  pr_spike_mean = -3,
  pr_spike_sd = 1,
  pr_slab_mean = 0.5,
  pr_slab_sd = 1,
  pr_a_theta = 0.001,
  pr_b_theta = 0.001,
  pr_xi_a = 1,
  pr_xi_b = 1,
  missing.val = 99,
  verbose = FALSE
)

Arguments

data	Matrix; a binary or continuous item response matrix for analysis. Each row represents a respondent, and each column contains responses to the corresponding item.
ndim	Integer; the dimension of the latent space. Default is 2.
niter	Integer; the total number of MCMC iterations to run. Default is 15000.
nburn	Integer; the number of initial MCMC iterations to discard as burn-in. Default is 2500.
nthin	Integer; the number of MCMC iterations to thin. Default is 5.
nprint	Integer; the interval at which MCMC samples are displayed during execution. Default is 500.
jump_beta	Numeric; the jumping rule for the beta proposal density. Default is 0.4.
jump_theta	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_gamma	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_z	Numeric; the jumping rule for the z proposal density. Default is 0.5.
jump_w	Numeric; the jumping rule for the w proposal density. Default is 0.5.
pr_mean_beta	Numeric; the mean of the normal prior for beta. Default is 0.
pr_sd_beta	Numeric; the standard deviation of the normal prior for beta. Default is 1.0.
pr_mean_theta	Numeric; the mean of the normal prior for theta. Default is 0.
pr_spike_mean	Numeric; the mean of spike prior for log gamma. Default is -3.
pr_spike_sd	Numeric; the standard deviation of spike prior for log gamma. Default is 1.
pr_slab_mean	Numeric; the mean of spike prior for log gamma. Default is 0.5.
pr_slab_sd	Numeric; the standard deviation of spike prior for log gamma. Default is is 1.
pr_a_theta	Numeric; the shape parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_b_theta	Numeric; the scale parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_xi_a	Numeric; the first shape parameter of beta prior for latent variable xi. Default is 1.
pr_xi_b	Numeric; the second shape parameter of beta prior for latent variable xi. Default is 1.
missing.val	Numeric; a number to replace missing values. Default is 99.
verbose	Logical; If TRUE, MCMC samples are printed for each nprint. Default is FALSE.

Details

lsirm1pl_mar_ss models the probability of correct response by respondent $j$ to item $i$ with item effect $\beta_i$, respondent effect $\theta_j$ and the distance between latent position $w_i$ of item $i$ and latent position $z_j$ of respondent $j$ in the shared metric space, with $\gamma$ represents the weight of the distance term:

$logit(P(Y_{j,i} = 1 |\theta_j,\beta_i,\gamma,z_j,w_i))=\theta_j+\beta_i-\gamma||z_j-w_i||$

Under the assumption of missing at random, the model takes the missing element into consideration in the sampling procedure. For the details of missing at random assumption and data augmentation, see References. lsirm1pl_mar_ss model include model selection approach based on spike-and-slab priors for log gamma. For detail of spike-and-slab priors, see References.

Value

lsirm1pl_mar_ss returns an object of list containing the following components:

data	Data frame or matrix containing the variables in the model.
missing.val	A number to replace missing values.
bic	Numeric value with the corresponding BIC.
mcmc_inf	Details about the number of MCMC iterations, burn-in periods, and thinning intervals.
map_inf	The log maximum a posteriori (MAP) value and the iteration number at which this MAP value occurs.
beta_estimate	Posterior estimates of the beta parameter.
theta_estimate	Posterior estimates of the theta parameter.
sigma_theta_estimate	Posterior estimates of the standard deviation of theta.
gamma_estimate	posterior estimates of gamma parameter.
z_estimate	Posterior estimates of the z parameter.
w_estimate	Posterior estimates of the w parameter.
beta	Posterior samples of the beta parameter.
theta	Posterior samples of the theta parameter.
gamma	Posterior samples of the gamma parameter.
theta_sd	Posterior samples of the standard deviation of theta.
z	Posterior samples of the z parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
w	Posterior samples of the w parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
pi	Posterior samples of phi which is indicator of spike and slab prior. If phi is 1, log gamma follows the slab prior, otherwise follows the spike prior.
imp	Imputation for missing Values using posterior samples.
accept_beta	Acceptance ratio for the beta parameter.
accept_theta	Acceptance ratio for the theta parameter.
accept_z	Acceptance ratio for the z parameter.
accept_w	Acceptance ratio for the w parameter.
accept_gamma	Acceptance ratio for the gamma parameter.
pi_estimate	Posterior estimation of phi. inclusion probability of gamma. if estimation of phi is less than 0.5, choose Rasch model with gamma = 0, otherwise latent space model with gamma > 0.
imp_estimate	Probability of imputating a missing value with 1.

References

Little, R. J., & Rubin, D. B. (2019). Statistical analysis with missing data (Vol. 793). John Wiley & Sons. Ishwaran, H., & Rao, J. S. (2005). Spike and slab variable selection: frequentist and Bayesian strategies. The Annals of Statistics, 33(2), 730-773.

Examples

# generate example item response matrix
data     <- matrix(rbinom(500, size = 1, prob = 0.5),ncol=10,nrow=50)

# generate example missing indicator matrix
missing_mat     <- matrix(rbinom(500, size = 1, prob = 0.2),ncol=10,nrow=50)

# make missing value with missing indicator matrix
data[missing_mat==1] <- 99

lsirm_result <- lsirm1pl_mar_ss(data)

# The code following can achieve the same result.
lsirm_result <- lsirm(data ~ lsirm1pl(spikenslab = TRUE, fixed_gamma = FALSE,
missing_data = 'mar', missing = 99))

---

1PL LSIRM for missing completely at random data.

Description

lsirm1pl_mcar is used to fit 1PL LSIRM in incomplete data assumed to be missing completely at random. lsirm1pl_mcar factorizes item response matrix into column-wise item effect, row-wise respondent effect and further embeds interaction effect in a latent space, while ignoring the missing element under the assumption of missing completely at random. The resulting latent space provides an interaction map that represents interactions between respondents and items.

Usage

lsirm1pl_mcar(
  data,
  ndim = 2,
  niter = 15000,
  nburn = 2500,
  nthin = 5,
  nprint = 500,
  jump_beta = 0.4,
  jump_theta = 1,
  jump_gamma = 0.025,
  jump_z = 0.5,
  jump_w = 0.5,
  pr_mean_beta = 0,
  pr_sd_beta = 1,
  pr_mean_theta = 0,
  pr_mean_gamma = 0.5,
  pr_sd_gamma = 1,
  pr_a_theta = 0.001,
  pr_b_theta = 0.001,
  missing.val = 99,
  verbose = FALSE
)

Arguments

data	Matrix; a binary or continuous item response matrix for analysis. Each row represents a respondent, and each column contains responses to the corresponding item.
ndim	Integer; the dimension of the latent space. Default is 2.
niter	Integer; the total number of MCMC iterations to run. Default is 15000.
nburn	Integer; the number of initial MCMC iterations to discard as burn-in. Default is 2500.
nthin	Integer; the number of MCMC iterations to thin. Default is 5.
nprint	Integer; the interval at which MCMC samples are displayed during execution. Default is 500.
jump_beta	Numeric; the jumping rule for the beta proposal density. Default is 0.4.
jump_theta	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_gamma	Numeric; the jumping rule for the gamma proposal density. Default is 0.025.
jump_z	Numeric; the jumping rule for the z proposal density. Default is 0.5.
jump_w	Numeric; the jumping rule for the w proposal density. Default is 0.5.
pr_mean_beta	Numeric; the mean of the normal prior for beta. Default is 0.
pr_sd_beta	Numeric; the standard deviation of the normal prior for beta. Default is 1.0.
pr_mean_theta	Numeric; the mean of the normal prior for theta. Default is 0.
pr_mean_gamma	Numeric; mean of log normal prior for gamma. Default is 0.5.
pr_sd_gamma	Numeric; standard deviation of log normal prior for gamma. Default is 1.0.
pr_a_theta	Numeric; the shape parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_b_theta	Numeric; the scale parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
missing.val	Numeric; A number to replace missing values. Default is 99.
verbose	Logical; If TRUE, MCMC samples are printed for each nprint. Default is FALSE.

Details

lsirm1pl_mcar models the probability of correct response by respondent $j$ to item $i$ with item effect $\beta_i$, respondent effect $\theta_j$ and the distance between latent position $w_i$ of item $i$ and latent position $z_j$ of respondent $j$ in the shared metric space, with $\gamma$ represents the weight of the distance term:

$logit(P(Y_{j,i} = 1|\theta_j,\beta_i,\gamma,z_j,w_i))=\theta_j+\beta_i-\gamma||z_j-w_i||$

Under the assumption of missing completely at random, the model ignores the missing element in doing inference. For the details of missing completely at random assumption and data augmentation, see References.

Value

lsirm1pl_mcar returns an object of list containing the following components:

data	A data frame or matrix containing the variables used in the model.
missing.val	A number to replace missing values.
bic	Numeric value with the corresponding BIC.
mcmc_inf	Details about the number of MCMC iterations, burn-in periods, and thinning intervals.
map_inf	The log maximum a posteriori (MAP) value and the iteration number at which this MAP value occurs.
beta_estimate	Posterior estimates of the beta parameter.
theta_estimate	Posterior estimates of the theta parameter.
sigma_theta_estimate	Posterior estimates of the standard deviation of theta.
gamma_estimate	Posterior estimates of gamma parameter.
z_estimate	Posterior estimates of the z parameter.
w_estimate	Posterior estimates of the w parameter.
beta	Posterior samples of the beta parameter.
theta	Posterior samples of the theta parameter.
theta_sd	Posterior samples of the standard deviation of theta.
gamma	Posterior samples of the gamma parameter.
z	Posterior samples of the z parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
w	Posterior samples of the w parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
accept_beta	Acceptance ratio for the beta parameter.
accept_theta	Acceptance ratio for the theta parameter.
accept_z	Acceptance ratio for the z parameter.
accept_w	Acceptance ratio for the w parameter.
accept_gamma	Acceptance ratio for the gamma parameter.

References

Little, R. J., & Rubin, D. B. (2019). Statistical analysis with missing data (Vol. 793). John Wiley & Sons.

Examples

# generate example item response matrix
data     <- matrix(rbinom(500, size = 1, prob = 0.5),ncol=10,nrow=50)

# generate example missing indicator matrix
missing_mat     <- matrix(rbinom(500, size = 1, prob = 0.2),ncol=10,nrow=50)

# make missing value with missing indicator matrix
data[missing_mat==1] <- 99

lsirm_result <- lsirm1pl_mcar(data)

# The code following can achieve the same result.
lsirm_result <- lsirm(data ~ lsirm1pl(spikenslab = FALSE, fixed_gamma = FALSE,
missing_data ='mcar', missing.val = 99))

---

1PL LSIRM with model selection approach for missing completely at random data.

Description

lsirm1pl_mcar_ss is used to fit 1PL LSIRM with model selection approach based on spike-and-slab priors in incomplete data assumed to be missing completely at random. lsirm1pl_mcar_ss factorizes item response matrix into column-wise item effect, row-wise respondent effect and further embeds interaction effect in a latent space, while ignoring the missing element under the assumption of missing completely at random. The resulting latent space provides an interaction map that represents interactions between respondents and items.

Usage

lsirm1pl_mcar_ss(
  data,
  ndim = 2,
  niter = 15000,
  nburn = 2500,
  nthin = 5,
  nprint = 500,
  jump_beta = 0.4,
  jump_theta = 1,
  jump_gamma = 1,
  jump_z = 0.5,
  jump_w = 0.5,
  pr_mean_beta = 0,
  pr_sd_beta = 1,
  pr_mean_theta = 0,
  pr_spike_mean = -3,
  pr_spike_sd = 1,
  pr_slab_mean = 0.5,
  pr_slab_sd = 1,
  pr_a_theta = 0.001,
  pr_b_theta = 0.001,
  pr_xi_a = 1,
  pr_xi_b = 1,
  missing.val = 99,
  verbose = FALSE
)

Arguments

data	Matrix; a binary or continuous item response matrix for analysis. Each row represents a respondent, and each column contains responses to the corresponding item.
ndim	Integer; the dimension of the latent space. Default is 2.
niter	Integer; the total number of MCMC iterations to run. Default is 15000.
nburn	Integer; the number of initial MCMC iterations to discard as burn-in. Default is 2500.
nthin	Integer; the number of MCMC iterations to thin. Default is 5.
nprint	Integer; the interval at which MCMC samples are displayed during execution. Default is 500.
jump_beta	Numeric; the jumping rule for the beta proposal density. Default is 0.4.
jump_theta	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_gamma	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_z	Numeric; the jumping rule for the z proposal density. Default is 0.5.
jump_w	Numeric; the jumping rule for the w proposal density. Default is 0.5.
pr_mean_beta	Numeric; the mean of the normal prior for beta. Default is 0.
pr_sd_beta	Numeric; the standard deviation of the normal prior for beta. Default is 1.0.
pr_mean_theta	Numeric; the mean of the normal prior for theta. Default is 0.
pr_spike_mean	Numeric; the mean of spike prior for log gamma. Default is -3.
pr_spike_sd	Numeric; the standard deviation of spike prior for log gamma. Default is 1.
pr_slab_mean	Numeric; the mean of spike prior for log gamma. Default is 0.5.
pr_slab_sd	Numeric; the standard deviation of spike prior for log gamma. Default is 1.
pr_a_theta	Numeric; the shape parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_b_theta	Numeric; the scale parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_xi_a	Numeric; the first shape parameter of beta prior for latent variable xi. Default is 1.
pr_xi_b	Numeric; the second shape parameter of beta prior for latent variable xi. Default is 1.
missing.val	Numeric; a number to replace missing values. Default is 99.
verbose	Logical; If TRUE, MCMC samples are printed for each nprint. Default is FALSE.

Details

lsirm1pl_mcar_ss models the probability of correct response by respondent $j$ to item $i$ with item effect $\beta_i$, respondent effect $\theta_j$ and the distance between latent position $w_i$ of item $i$ and latent position $z_j$ of respondent $j$ in the shared metric space, with $\gamma$ represents the weight of the distance term:

$logit(P(Y_{j,i} = 1 |\theta_j,\beta_i,\gamma,z_j,w_i))=\theta_j+\beta_i-\gamma||z_j-w_i||$

Under the assumption of missing completely at random, the model ignores the missing element in doing inference. For the details of missing completely at random assumption and data augmentation, see References. lsirm1pl_mcar_ss model include model selection approach based on spike-and-slab priors for log gamma. For detail of spike-and-slab priors, see References.

Value

lsirm1pl_mcar_ss returns an object of list containing the following components:

data	Data frame or matrix containing the variables in the model.
missing.val	A number to replace missing values.
bic	Numeric value with the corresponding BIC.
mcmc_inf	Details about the number of MCMC iterations, burn-in periods, and thinning intervals.
map_inf	The log maximum a posteriori (MAP) value and the iteration number at which this MAP value occurs.
beta_estimate	Posterior estimates of the beta parameter.
theta_estimate	Posterior estimates of the theta parameter.
sigma_theta_estimate	Posterior estimates of the standard deviation of theta.
gamma_estimate	posterior estimates of gamma parameter.
z_estimate	Posterior estimates of the z parameter.
w_estimate	Posterior estimates of the w parameter.
beta	Posterior samples of the beta parameter.
theta	Posterior samples of the theta parameter.
gamma	Posterior samples of the gamma parameter.
theta_sd	Posterior samples of the standard deviation of theta.
z	Posterior samples of the z parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
w	Posterior samples of the w parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
pi	Posterior samples of phi which is indicator of spike and slab prior. If phi is 1, log gamma follows the slab prior, otherwise follows the spike prior.
accept_beta	Acceptance ratio for the beta parameter.
accept_theta	Acceptance ratio for the theta parameter.
accept_z	Acceptance ratio for the z parameter.
accept_w	Acceptance ratio for the w parameter.
accept_gamma	Acceptance ratio for the gamma parameter.

References

Little, R. J., & Rubin, D. B. (2019). Statistical analysis with missing data (Vol. 793). John Wiley & Sons. Ishwaran, H., & Rao, J. S. (2005). Spike and slab variable selection: frequentist and Bayesian strategies. The Annals of Statistics, 33(2), 730-773.

Examples

# generate example item response matrix
data     <- matrix(rbinom(500, size = 1, prob = 0.5),ncol=10,nrow=50)

# generate example missing indicator matrix
missing_mat     <- matrix(rbinom(500, size = 1, prob = 0.2),ncol=10,nrow=50)

# make missing value with missing indicator matrix
data[missing_mat==1] <- 99

lsirm_result <- lsirm1pl_mcar_ss(data)

# The code following can achieve the same result.
lsirm_result <- lsirm(data ~ lsirm1pl(spikenslab = TRUE, fixed_gamma = FALSE,
missing_data ='mcar', missing.val = 99))

---

1PL LSIRM fixing gamma to 1 with normal likelihood

Description

lsirm1pl_normal_fixed_gamma is used to fit 1PL LSIRM for continuous variable with gamma fixed to 1. lsirm1pl_normal_fixed_gamma factorizes continuous item response matrix into column-wise item effect, row-wise respondent effect and further embeds interaction effect in a latent space, while ignoring the missing element under the assumption of missing at random. The resulting latent space provides an interaction map that represents interactions between respondents and items.

Usage

lsirm1pl_normal_fixed_gamma(
  data,
  ndim = 2,
  niter = 15000,
  nburn = 2500,
  nthin = 5,
  nprint = 500,
  jump_beta = 0.4,
  jump_theta = 1,
  jump_z = 0.5,
  jump_w = 0.5,
  pr_mean_beta = 0,
  pr_sd_beta = 1,
  pr_mean_theta = 0,
  pr_a_theta = 0.001,
  pr_b_theta = 0.001,
  pr_a_eps = 0.001,
  pr_b_eps = 0.001,
  verbose = FALSE
)

Arguments

data	Matrix; a binary or continuous item response matrix for analysis. Each row represents a respondent, and each column contains responses to the corresponding item.
ndim	Integer; the dimension of the latent space. Default is 2.
niter	Integer; the total number of MCMC iterations to run. Default is 15000.
nburn	Integer; the number of initial MCMC iterations to discard as burn-in. Default is 2500.
nthin	Integer; the number of MCMC iterations to thin. Default is 5.
nprint	Integer; the interval at which MCMC samples are displayed during execution. Default is 500.
jump_beta	Numeric; the jumping rule for the beta proposal density. Default is 0.4.
jump_theta	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_z	Numeric; the jumping rule for the z proposal density. Default is 0.5.
jump_w	Numeric; the jumping rule for the w proposal density. Default is 0.5.
pr_mean_beta	Numeric; the mean of the normal prior for beta. Default is 0.
pr_sd_beta	Numeric; the standard deviation of the normal prior for beta. Default is 1.0.
pr_mean_theta	Numeric; the mean of the normal prior for theta. Default is 0.
pr_a_theta	Numeric; the shape parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_b_theta	Numeric; the scale parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_a_eps	Numeric; the shape parameter of inverse gamma prior for variance of data likelihood. Default is 0.001.
pr_b_eps	Numeric; the scale parameter of inverse gamma prior for variance of data likelihood Default is 0.001.
verbose	Logical; If TRUE, MCMC samples are printed for each nprint. default value is FALSE.

Details

lsirm1pl_normal_fixed_gamma models the continuous value of response by respondent $j$ to item $i$ with item effect $\beta_i$, respondent effect $\theta_j$ and the distance between latent position $w_i$ of item $i$ and latent position $z_j$ of respondent $j$ in the shared metric space:

$Y_{j,i} = \theta_j+\beta_i-||z_j-w_i|| + e_{j,i}$

where the error $e_{j,i} \sim N(0,\sigma^2)$.

Value

lsirm1pl_normal_fixed_gamma returns an object of list containing the following components:

data	A data frame or matrix containing the variables used in the model.
bic	A numeric value representing the Bayesian Information Criterion (BIC).
mcmc_inf	Details about the number of MCMC iterations, burn-in periods, and thinning intervals.
map_inf	The log maximum a posteriori (MAP) value and the iteration number at which this MAP value occurs.
beta_estimate	Posterior estimates of the beta parameter.
theta_estimate	Posterior estimates of the theta parameter.
sigma_theta_estimate	Posterior estimates of the standard deviation of theta.
z_estimate	Posterior estimates of the z parameter.
w_estimate	Posterior estimates of the w parameter.
beta	Posterior samples of the beta parameter.
theta	Posterior samples of the theta parameter.
theta_sd	Posterior samples of the standard deviation of theta.
z	Posterior samples of the z parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
w	Posterior samples of the w parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
accept_beta	Acceptance ratio for the beta parameter.
accept_theta	Acceptance ratio for the theta parameter.
accept_z	Acceptance ratio for the z parameter.
accept_w	Acceptance ratio for the w parameter.
sigma_estimate	Posterior estimates of the standard deviation.
sigma	Posterior samples of the standard deviation.

Examples

# generate example (continuous) item response matrix
data     <- matrix(rnorm(500, mean = 0, sd = 1),ncol=10,nrow=50)

# generate example missing indicator matrix
missing_mat     <- matrix(rbinom(500, size = 1, prob = 0.2),ncol=10,nrow=50)

# make missing value with missing indicator matrix
data[missing_mat==1] <- 99

lsirm_result <- lsirm1pl_normal_fixed_gamma(data)

# The code following can achieve the same result.
lsirm_result <- lsirm(data ~ lsirm1pl(spikenslab = FALSE, fixed_gamma = TRUE))

---

1PL LSIRM fixing gamma to 1 with normal likelihood for missing at random data.

Description

lsirm1pl_normal_fixed_gamma_mar is used to fit 1PL LSIRM for continuous variable with gamma fixed to 1 in incomplete data assumed to be missing at random.

lsirm1pl_normal_fixed_gamma_mar factorizes continuous item response matrix into column-wise item effect, row-wise respondent effect and further embeds interaction effect in a latent space, while considering the missing element under the assumption of missing at random. The resulting latent space provides an interaction map that represents interactions between respondents and items.

Usage

lsirm1pl_normal_fixed_gamma_mar(
  data,
  ndim = 2,
  niter = 15000,
  nburn = 2500,
  nthin = 5,
  nprint = 500,
  jump_beta = 0.4,
  jump_theta = 1,
  jump_z = 0.5,
  jump_w = 0.5,
  pr_mean_beta = 0,
  pr_sd_beta = 1,
  pr_mean_theta = 0,
  pr_a_theta = 0.001,
  pr_b_theta = 0.001,
  pr_a_eps = 0.001,
  pr_b_eps = 0.001,
  missing.val = 99,
  verbose = FALSE
)

Arguments

data	Matrix; a binary or continuous item response matrix for analysis. Each row represents a respondent, and each column contains responses to the corresponding item.
ndim	Integer; the dimension of the latent space. Default is 2.
niter	Integer; the total number of MCMC iterations to run. Default is 15000.
nburn	Integer; the number of initial MCMC iterations to discard as burn-in. Default is 2500.
nthin	Integer; the number of MCMC iterations to thin. Default is 5.
nprint	Integer; the interval at which MCMC samples are displayed during execution. Default is 500.
jump_beta	Numeric; the jumping rule for the beta proposal density. Default is 0.4.
jump_theta	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_z	Numeric; the jumping rule for the z proposal density. Default is 0.5.
jump_w	Numeric; the jumping rule for the w proposal density. Default is 0.5.
pr_mean_beta	Numeric; the mean of the normal prior for beta. Default is 0.
pr_sd_beta	Numeric; the standard deviation of the normal prior for beta. Default is 1.0.
pr_mean_theta	Numeric; the mean of the normal prior for theta. Default is 0.
pr_a_theta	Numeric; the shape parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_b_theta	Numeric; the scale parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_a_eps	Numeric; the shape parameter of inverse gamma prior for variance of data likelihood. Default is 0.001.
pr_b_eps	Numeric; the scale parameter of inverse gamma prior for variance of data likelihood Default is 0.001.
missing.val	Numeric; a number to replace missing values. Default is 99.
verbose	Logical; If TRUE, MCMC samples are printed for each nprint. Default is FALSE.

Details

lsirm1pl_normal_fixed_gamma_mar models the continuous value of response by respondent $j$ to item $i$ with item effect $\beta_i$, respondent effect $\theta_j$ and the distance between latent position $w_i$ of item $i$ and latent position $z_j$ of respondent $j$ in the shared metric space:

$Y_j,i = \theta_j+\beta_i-||z_j-w_i|| + e_{ji}$

where the error $e_ji ~ N(0,\sigma^2)$ Under the assumption of missing at random, the model takes the missing element into consideration in the sampling procedure. For the details of missing at random assumption and data augmentation, see References.

Value

lsirm1pl_normal_fixed_gamma_mar returns an object of list containing the following components:

data	Data frame or matrix containing the variables in the model.
missing.val	A number to replace missing values.
bic	Numeric value with the corresponding BIC.
mcmc_inf	Details about the number of MCMC iterations, burn-in periods, and thinning intervals.
map_inf	The log maximum a posteriori (MAP) value and the iteration number at which this MAP value occurs.
beta_estimate	Posterior estimates of the beta parameter.
theta_estimate	Posterior estimates of the theta parameter.
sigma_theta_estimate	Posterior estimates of the standard deviation of theta.
z_estimate	Posterior estimates of the z parameter.
w_estimate	Posterior estimates of the w parameter.
imp_estimate	Probability of imputating a missing value with 1.
beta	Posterior samples of the beta parameter.
theta	Posterior samples of the theta parameter.
theta_sd	Posterior samples of the standard deviation of theta.
z	Posterior samples of the z parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
w	Posterior samples of the w parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
imp	Imputation for missing Values using posterior samples.
accept_beta	Acceptance ratio for the beta parameter.
accept_theta	Acceptance ratio for the theta parameter.
accept_z	Acceptance ratio for the z parameter.
accept_w	Acceptance ratio for the w parameter.
sigma_estimate	Posterior estimates of the standard deviation.
sigma	Posterior samples of the standard deviation.

Examples

# generate example (continuous) item response matrix
data     <- matrix(rnorm(500, mean = 0, sd = 1),ncol=10,nrow=50)

# generate example missing indicator matrix
missing_mat     <- matrix(rbinom(500, size = 1, prob = 0.2),ncol=10,nrow=50)

# make missing value with missing indicator matrix
data[missing_mat==1] <- 99

lsirm_result <- lsirm1pl_normal_fixed_gamma_mar(data)

# The code following can achieve the same result.
lsirm_result <- lsirm(data ~ lsirm1pl(spikenslab = FALSE, fixed_gamma = TRUE,
missing_data = "mar", missing.val = 99))

---

1PL LSIRM fixing gamma to 1 with normal likelihood for missing completely at random data.

Description

lsirm1pl_normal_fixed_gamma_mcar is used to fit 1PL LSIRM for continuous variable with gamma fixed to 1 in incomplete data assumed to be missing completely at random.

lsirm1pl_normal_fixed_gamma_mcar factorizes continuous item response matrix into column-wise item effect, row-wise respondent effect and further embeds interaction effect in a latent space, while ignoring the missing element under the assumption of missing completely at random. The resulting latent space provides an interaction map that represents interactions between respondents and items.

Usage

lsirm1pl_normal_fixed_gamma_mcar(
  data,
  ndim = 2,
  niter = 15000,
  nburn = 2500,
  nthin = 5,
  nprint = 500,
  jump_beta = 0.4,
  jump_theta = 1,
  jump_z = 0.5,
  jump_w = 0.5,
  pr_mean_beta = 0,
  pr_sd_beta = 1,
  pr_mean_theta = 0,
  pr_a_theta = 0.001,
  pr_b_theta = 0.001,
  pr_a_eps = 0.001,
  pr_b_eps = 0.001,
  missing.val = 99,
  verbose = FALSE
)

Arguments

data	Matrix; a binary or continuous item response matrix for analysis. Each row represents a respondent, and each column contains responses to the corresponding item.
ndim	Integer; the dimension of the latent space. Default is 2.
niter	Integer; the total number of MCMC iterations to run. Default is 15000.
nburn	Integer; the number of initial MCMC iterations to discard as burn-in. Default is 2500.
nthin	Integer; the number of MCMC iterations to thin. Default is 5.
nprint	Integer; the interval at which MCMC samples are displayed during execution. Default is 500.
jump_beta	Numeric; the jumping rule for the beta proposal density. Default is 0.4.
jump_theta	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_z	Numeric; the jumping rule for the z proposal density. Default is 0.5.
jump_w	Numeric; the jumping rule for the w proposal density. Default is 0.5.
pr_mean_beta	Numeric; the mean of the normal prior for beta. Default is 0.
pr_sd_beta	Numeric; the standard deviation of the normal prior for beta. Default is 1.0.
pr_mean_theta	Numeric; the mean of the normal prior for theta. Default is 0.
pr_a_theta	Numeric; the shape parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_b_theta	Numeric; the scale parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_a_eps	Numeric; the shape parameter of inverse gamma prior for variance of data likelihood. Default is 0.001.
pr_b_eps	Numeric; the scale parameter of inverse gamma prior for variance of data likelihood. Default is 0.001.
missing.val	Numeric; a number to replace missing values. Default is 99.
verbose	Logical; If TRUE, MCMC samples are printed for each nprint. Default is FALSE.

Details

lsirm1pl_normal_fixed_gamma_mcar models the continuous value of response by respondent $j$ to item $i$ with item effect $\beta_i$, respondent effect $\theta_j$ and the distance between latent position $w_i$ of item $i$ and latent position $z_j$ of respondent $j$ in the shared metric space:

$Y_{j,i} = \theta_j+\beta_i-||z_j-w_i|| + e_{j,i}$

where the error $e_{j,i} \sim N(0,\sigma^2)$ Under the assumption of missing completely at random, the model ignores the missing element in doing inference. For the details of missing completely at random assumption and data augmentation, see References.

Value

lsirm1pl_normal_fixed_gamma_mcar returns an object of list containing the following components:

data	Data frame or matrix containing the variables in the model.
missing.val	A number to replace missing values.
bic	Numeric value with the corresponding BIC.
mcmc_inf	Details about the number of MCMC iterations, burn-in periods, and thinning intervals.
map_inf	The log maximum a posteriori (MAP) value and the iteration number at which this MAP value occurs.
beta_estimate	Posterior estimates of the beta parameter.
theta_estimate	Posterior estimates of the theta parameter.
sigma_theta_estimate	Posterior estimates of the standard deviation of theta.
z_estimate	Posterior estimates of the z parameter.
w_estimate	Posterior estimates of the w parameter.
beta	Posterior samples of the beta parameter.
theta	Posterior samples of the theta parameter.
theta_sd	Posterior samples of the standard deviation of theta.
z	Posterior samples of the z parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
w	Posterior samples of the w parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
accept_beta	Acceptance ratio for the beta parameter.
accept_theta	Acceptance ratio for the theta parameter.
accept_z	Acceptance ratio for the z parameter.
accept_w	Acceptance ratio for the w parameter.
sigma_estimate	Posterior estimates of the standard deviation.
sigma	Posterior samples of the standard deviation.

Examples

# generate example (continuous) item response matrix
data     <- matrix(rnorm(500, mean = 0, sd = 1),ncol=10,nrow=50)

# generate example missing indicator matrix
missing_mat     <- matrix(rbinom(500, size = 1, prob = 0.2),ncol=10,nrow=50)

# make missing value with missing indicator matrix
data[missing_mat==1] <- 99

lsirm_result <- lsirm1pl_normal_fixed_gamma_mcar(data)

# The code following can achieve the same result.
lsirm_result <- lsirm(data ~ lsirm1pl(spikenslab = FALSE, fixed_gamma = TRUE,
missing_data = "mcar", missing.val = 99))

---

1PL LSIRM with normal likelihood for missing at random data.

Description

lsirm1pl_normal_mar is used to fit LSIRM for continuous variable with 1pl in incomplete data assumed to be missing at random. lsirm1pl_normal_mar factorizes continuous item response matrix into column-wise item effect, row-wise respondent effect and further embeds interaction effect in a latent space, while considering the missing element under the assumption of missing at random. The resulting latent space provides an interaction map that represents interactions between respondents and items.

Usage

lsirm1pl_normal_mar(
  data,
  ndim = 2,
  niter = 15000,
  nburn = 2500,
  nthin = 5,
  nprint = 500,
  jump_beta = 0.4,
  jump_theta = 1,
  jump_gamma = 1,
  jump_z = 0.5,
  jump_w = 0.5,
  pr_mean_beta = 0,
  pr_sd_beta = 1,
  pr_mean_theta = 0,
  pr_mean_gamma = 0.5,
  pr_sd_gamma = 1,
  pr_a_theta = 0.001,
  pr_b_theta = 0.001,
  pr_a_eps = 0.001,
  pr_b_eps = 0.001,
  missing.val = 99,
  verbose = FALSE
)

Arguments

data	Matrix; a binary or continuous item response matrix for analysis. Each row represents a respondent, and each column contains responses to the corresponding item.
ndim	Integer; the dimension of the latent space. Default is 2.
niter	Integer; the total number of MCMC iterations to run. Default is 15000.
nburn	Integer; the number of initial MCMC iterations to discard as burn-in. Default is 2500.
nthin	Integer; the number of MCMC iterations to thin. Default is 5.
nprint	Integer; the interval at which MCMC samples are displayed during execution. Default is 500.
jump_beta	Numeric; the jumping rule for the beta proposal density. Default is 0.4.
jump_theta	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_gamma	Numeric; the jumping rule for the gamma proposal density. Default is 0.025
jump_z	Numeric; the jumping rule for the z proposal density. Default is 0.5.
jump_w	Numeric; the jumping rule for the w proposal density. Default is 0.5.
pr_mean_beta	Numeric; the mean of the normal prior for beta. Default is 0.
pr_sd_beta	Numeric; the standard deviation of the normal prior for beta. Default is 1.0.
pr_mean_theta	Numeric; the mean of the normal prior for theta. Default is 0.
pr_mean_gamma	Numeric; mean of log normal prior for gamma. Default is 0.5.
pr_sd_gamma	Numeric; standard deviation of log normal prior for gamma. Default is 1.0.
pr_a_theta	Numeric; the shape parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_b_theta	Numeric; the scale parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_a_eps	Numeric; the shape parameter of inverse gamma prior for variance of data likelihood. Default is 0.001.
pr_b_eps	Numeric; the scale parameter of inverse gamma prior for variance of data likelihood. Default is 0.001.
missing.val	Numeric; a number to replace missing values. Default is 99.
verbose	Logical; If TRUE, MCMC samples are printed for each nprint. Default is FALSE.

Details

lsirm1pl_normal_mar models the continuous value of response by respondent $j$ to item $i$ with item effect $\beta_i$, respondent effect $\theta_j$ and the distance between latent position $w_i$ of item $i$ and latent position $z_j$ of respondent $j$ in the shared metric space, with $\gamma$ represents the weight of the distance term:

$Y_j,i = \theta_j+\beta_i-\gamma||z_j-w_i|| + e_{ji}$

where the error $e_{ji} \sim N(0,\sigma^2)$ Under the assumption of missing at random, the model takes the missing element into consideration in the sampling procedure. For the details of missing at random assumption and data augmentation, see References.

Value

lsirm1pl_normal_mar returns an object of list containing the following components:

data	Data frame or matrix containing the variables in the model.
missing.val	A number to replace missing values.
bic	Numeric value with the corresponding BIC.
mcmc_inf	Details about the number of MCMC iterations, burn-in periods, and thinning intervals.
map_inf	The log maximum a posteriori (MAP) value and the iteration number at which this MAP value occurs.
beta_estimate	Posterior estimates of the beta parameter.
theta_estimate	Posterior estimates of the theta parameter.
sigma_theta_estimate	Posterior estimates of the standard deviation of theta.
gamma_estimate	posterior estimates of gamma parameter.
z_estimate	Posterior estimates of the z parameter.
w_estimate	Posterior estimates of the w parameter.
imp_estimate	Probability of imputating a missing value with 1.
beta	Posterior samples of the beta parameter.
theta	Posterior samples of the theta parameter.
gamma	Posterior samples of the gamma parameter.
theta_sd	Posterior samples of the standard deviation of theta.
z	Posterior samples of the z parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
w	Posterior samples of the w parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
imp	Imputation for missing Values using posterior samples.
accept_beta	Acceptance ratio for the beta parameter.
accept_theta	Acceptance ratio for the theta parameter.
accept_z	Acceptance ratio for the z parameter.
accept_w	Acceptance ratio for the w parameter.
accept_gamma	Acceptance ratio for the gamma parameter.
sigma_estimate	Posterior estimates of the standard deviation.
sigma	Posterior samples of the standard deviation.

References

Little, R. J., & Rubin, D. B. (2019). Statistical analysis with missing data (Vol. 793). John Wiley & Sons.

Examples

# generate example (continuous) item response matrix
data     <- matrix(rnorm(500, mean = 0, sd = 1),ncol=10,nrow=50)

# generate example missing indicator matrix
missing_mat     <- matrix(rbinom(500, size = 1, prob = 0.2),ncol=10,nrow=50)

# make missing value with missing indicator matrix
data[missing_mat==1] <- 99

lsirm_result <- lsirm1pl_normal_mar(data)

# The code following can achieve the same result.
lsirm_result <- lsirm(data ~ lsirm1pl(spikenslab = FALSE, fixed_gamma = FALSE,
missing_data ='mar', missing.val = 99))

---

1PL LSIRM with normal likelihood and model selection approach for missing at random data.

Description

lsirm1pl_normal_mar_ss is used to fit 1PL LSIRM with model selection approach based on spike-and-slab priors for continuous variable with 1pl in incomplete data assumed to be missing at random. lsirm1pl_normal_mar_ss factorizes continuous item response matrix into column-wise item effect, row-wise respondent effect and further embeds interaction effect in a latent space, while considering the missing element under the assumption of missing at random. The resulting latent space provides an interaction map that represents interactions between respondents and items.

Usage

lsirm1pl_normal_mar_ss(
  data,
  ndim = 2,
  niter = 15000,
  nburn = 2500,
  nthin = 5,
  nprint = 500,
  jump_beta = 0.4,
  jump_theta = 1,
  jump_gamma = 1,
  jump_z = 0.5,
  jump_w = 0.5,
  pr_mean_beta = 0,
  pr_sd_beta = 1,
  pr_mean_theta = 0,
  pr_spike_mean = -3,
  pr_spike_sd = 1,
  pr_slab_mean = 0.5,
  pr_slab_sd = 1,
  pr_a_theta = 0.001,
  pr_b_theta = 0.001,
  pr_a_eps = 0.001,
  pr_b_eps = 0.001,
  pr_xi_a = 0.001,
  pr_xi_b = 0.001,
  missing.val = 99,
  verbose = FALSE
)

Arguments

data	Matrix; a binary or continuous item response matrix for analysis. Each row represents a respondent, and each column contains responses to the corresponding item.
ndim	Integer; the dimension of the latent space. Default is 2.
niter	Integer; the total number of MCMC iterations to run. Default is 15000.
nburn	Integer; the number of initial MCMC iterations to discard as burn-in. Default is 2500.
nthin	Integer; the number of MCMC iterations to thin. Default is 5.
nprint	Integer; the interval at which MCMC samples are displayed during execution. Default is 500.
jump_beta	Numeric; the jumping rule for the beta proposal density. Default is 0.4.
jump_theta	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_gamma	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_z	Numeric; the jumping rule for the z proposal density. Default is 0.5.
jump_w	Numeric; the jumping rule for the w proposal density. Default is 0.5.
pr_mean_beta	Numeric; the mean of the normal prior for beta. Default is 0.
pr_sd_beta	Numeric; the standard deviation of the normal prior for beta. Default is 1.0.
pr_mean_theta	Numeric; the mean of the normal prior for theta. Default is 0.
pr_spike_mean	Numeric; the mean of spike prior for log gamma. Default is -3.
pr_spike_sd	Numeric; the standard deviation of spike prior for log gamma. Default is 1.
pr_slab_mean	Numeric; the mean of spike prior for log gamma. Default is 0.5.
pr_slab_sd	Numeric; the standard deviation of spike prior for log gamma. Default is is 1.
pr_a_theta	Numeric; the shape parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_b_theta	Numeric; the scale parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_a_eps	Numeric; the shape parameter of inverse gamma prior for variance of data likelihood. Default is 0.001.
pr_b_eps	Numeric; the scale parameter of inverse gamma prior for variance of data likelihood Default is 0.001.
pr_xi_a	Numeric; the first shape parameter of beta prior for latent variable xi. Default is 1.
pr_xi_b	Numeric; the second shape parameter of beta prior for latent variable xi. Default is 1.
missing.val	Numeric; a number to replace missing values. Default is 99.
verbose	Logical; If TRUE, MCMC samples are printed for each nprint. Default is FALSE.

Details

lsirm1pl_normal_mar_ss models the continuous value of response by respondent $j$ to item $i$ with item effect $\beta_i$, respondent effect $\theta_j$ and the distance between latent position $w_i$ of item $i$ and latent position $z_j$ of respondent $j$ in the shared metric space, with $\gamma$ represents the weight of the distance term:

$Y_{j,i} = \theta_j+\beta_i-\gamma||z_j-w_i|| + e_{j,i}$

where the error $e_{j,i} \sim N(0,\sigma^2)$ Under the assumption of missing at random, the model takes the missing element into consideration in the sampling procedure. For the details of missing at random assumption and data augmentation, see References. lsirm1pl_normal_mar_ss model include model selection approach based on spike-and-slab priors for log gamma. For detail of spike-and-slab priors, see References.

Value

lsirm1pl_normal_mar_ss returns an object of list containing the following components:

data	Data frame or matrix containing the variables in the model.
missing.val	A number to replace missing values.
bic	Numeric value with the corresponding BIC.
mcmc_inf	Details about the number of MCMC iterations, burn-in periods, and thinning intervals.
map_inf	The log maximum a posteriori (MAP) value and the iteration number at which this MAP value occurs.
beta_estimate	Posterior estimates of the beta parameter.
theta_estimate	Posterior estimates of the theta parameter.
sigma_theta_estimate	Posterior estimates of the standard deviation of theta.
gamma_estimate	posterior estimates of gamma parameter.
z_estimate	Posterior estimates of the z parameter.
w_estimate	Posterior estimates of the w parameter.
beta	Posterior samples of the beta parameter.
theta	Posterior samples of the theta parameter.
gamma	Posterior samples of the gamma parameter.
theta_sd	Posterior samples of the standard deviation of theta.
z	Posterior samples of the z parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
w	Posterior samples of the w parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
pi	Posterior samples of phi which is indicator of spike and slab prior. If phi is 1, log gamma follows the slab prior, otherwise follows the spike prior.
imp	Imputation for missing Values using posterior samples.
accept_beta	Acceptance ratio for the beta parameter.
accept_theta	Acceptance ratio for the theta parameter.
accept_z	Acceptance ratio for the z parameter.
accept_w	Acceptance ratio for the w parameter.
accept_gamma	Acceptance ratio for the gamma parameter.
pi_estimate	Posterior estimation of phi. inclusion probability of gamma. if estimation of phi is less than 0.5, choose Rasch model with gamma = 0, otherwise latent space model with gamma > 0.
imp_estimate	Probability of imputating a missing value with 1.
sigma_estimate	Posterior estimates of the standard deviation.
sigma	Posterior samples of the standard deviation.

References

Little, R. J., & Rubin, D. B. (2019). Statistical analysis with missing data (Vol. 793). John Wiley & Sons. Ishwaran, H., & Rao, J. S. (2005). Spike and slab variable selection: frequentist and Bayesian strategies. The Annals of Statistics, 33(2), 730-773.

Examples

# generate example (continuous) item response matrix
data     <- matrix(rnorm(500, mean = 0, sd = 1),ncol=10,nrow=50)

# generate example missing indicator matrix
missing_mat     <- matrix(rbinom(500, size = 1, prob = 0.2),ncol=10,nrow=50)

# make missing value with missing indicator matrix
data[missing_mat==1] <- 99

lsirm_result <- lsirm1pl_normal_mar_ss(data)

# The code following can achieve the same result.
lsirm_result <- lsirm(data ~ lsirm1pl(spikenslab = TRUE, fixed_gamma = FALSE,
missing_data = 'mar', missing = 99))

---

1PL LSIRM with normal likelihood for missing completely at random data.

Description

lsirm1pl_normal_mcar is used to fit LSIRM with 1pl in incomplete data assumed to be missing completely at random. lsirm1pl_normal_mcar factorizes continuous item response matrix into column-wise item effect, row-wise respondent effect and further embeds interaction effect in a latent space, while ignoring the missing element under the assumption of missing completely at random. The resulting latent space provides an interaction map that represents interactions between respondents and items.

Usage

lsirm1pl_normal_mcar(
  data,
  ndim = 2,
  niter = 15000,
  nburn = 2500,
  nthin = 5,
  nprint = 500,
  jump_beta = 0.4,
  jump_theta = 1,
  jump_gamma = 1,
  jump_z = 0.5,
  jump_w = 0.5,
  pr_mean_beta = 0,
  pr_sd_beta = 1,
  pr_mean_theta = 0,
  pr_mean_gamma = 0.5,
  pr_sd_gamma = 1,
  pr_a_theta = 0.001,
  pr_b_theta = 0.001,
  pr_a_eps = 0.001,
  pr_b_eps = 0.001,
  missing.val = 99,
  verbose = FALSE
)

Arguments

data	Matrix; a binary or continuous item response matrix for analysis. Each row represents a respondent, and each column contains responses to the corresponding item.
ndim	Integer; the dimension of the latent space. Default is 2.
niter	Integer; the total number of MCMC iterations to run. Default is 15000.
nburn	Integer; the number of initial MCMC iterations to discard as burn-in. Default is 2500.
nthin	Integer; the number of MCMC iterations to thin. Default is 5.
nprint	Integer; the interval at which MCMC samples are displayed during execution. Default is 500.
jump_beta	Numeric; the jumping rule for the beta proposal density. Default is 0.4.
jump_theta	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_gamma	Numeric; the jumping rule for the gamma proposal density. Default is 0.025
jump_z	Numeric; the jumping rule for the z proposal density. Default is 0.5.
jump_w	Numeric; the jumping rule for the w proposal density. Default is 0.5.
pr_mean_beta	Numeric; the mean of the normal prior for beta. Default is 0.
pr_sd_beta	Numeric; the standard deviation of the normal prior for beta. Default is 1.0.
pr_mean_theta	Numeric; the mean of the normal prior for theta. Default is 0.
pr_mean_gamma	Numeric; mean of log normal prior for gamma. Default is 0.5.
pr_sd_gamma	Numeric; standard deviation of log normal prior for gamma. Default is 1.0.
pr_a_theta	Numeric; the shape parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_b_theta	Numeric; the scale parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_a_eps	Numeric; the shape parameter of inverse gamma prior for variance of data likelihood. Default is 0.001.
pr_b_eps	Numeric; the scale parameter of inverse gamma prior for variance of data likelihood. Default is 0.001.
missing.val	Numeric; a number to replace missing values. Default is 99.
verbose	Logical; If TRUE, MCMC samples are printed for each nprint. Default is FALSE.

Details

lsirm1pl_normal_mcar models the continuous value of response by respondent $j$ to item $i$ with item effect $\beta_i$, respondent effect $\theta_j$ and the distance between latent position $w_i$ of item $i$ and latent position $z_j$ of respondent $j$ in the shared metric space, with $\gamma$ represents the weight of the distance term:

$Y_{j,i} = \theta_j+\beta_i-\gamma||z_j-w_i|| + e_{j,i}$

where the error $e_{j,i} \sim N(0,\sigma^2)$ Under the assumption of missing completely at random, the model ignores the missing element in doing inference. For the details of missing at random assumption and data augmentation, see References.

Value

lsirm1pl_normal_mcar returns an object of list containing the following components:

data	A data frame or matrix containing the variables used in the model.
missing.val	A number to replace missing values.
bic	Numeric value with the corresponding BIC.
mcmc_inf	Details about the number of MCMC iterations, burn-in periods, and thinning intervals.
map_inf	The log maximum a posteriori (MAP) value and the iteration number at which this MAP value occurs.
beta_estimate	Posterior estimates of the beta parameter.
theta_estimate	Posterior estimates of the theta parameter.
sigma_theta_estimate	Posterior estimates of the standard deviation of theta.
gamma_estimate	Posterior estimates of gamma parameter.
z_estimate	Posterior estimates of the z parameter.
w_estimate	Posterior estimates of the w parameter.
beta	Posterior samples of the beta parameter.
theta	Posterior samples of the theta parameter.
theta_sd	Posterior samples of the standard deviation of theta.
gamma	Posterior samples of the gamma parameter.
z	Posterior samples of the z parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
w	Posterior samples of the w parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
accept_beta	Acceptance ratio for the beta parameter.
accept_theta	Acceptance ratio for the theta parameter.
accept_z	Acceptance ratio for the z parameter.
accept_w	Acceptance ratio for the w parameter.
accept_gamma	Acceptance ratio for the gamma parameter.
sigma_estimate	Posterior estimates of the standard deviation.
sigma	Posterior samples of the standard deviation.

References

Little, R. J., & Rubin, D. B. (2019). Statistical analysis with missing data (Vol. 793). John Wiley & Sons.

Examples

# generate example (continuous) item response matrix
data     <- matrix(rnorm(500, mean = 0, sd = 1),ncol=10,nrow=50)

# generate example missing indicator matrix
missing_mat     <- matrix(rbinom(500, size = 1, prob = 0.2),ncol=10,nrow=50)

# make missing value with missing indicator matrix
data[missing_mat==1] <- 99

lsirm_result <- lsirm1pl_normal_mcar(data)

# The code following can achieve the same result.
lsirm_result <- lsirm(data ~ lsirm1pl(spikenslab = FALSE, fixed_gamma = FALSE,
missing_data ='mcar', missing.val = 99))

---

1PL LSIRM with normal likelihood and model selection approach for missing completely at random data.

Description

lsirm1pl_normal_mcar_ss is used to fit LSIRM with model selection approach based on spike-and-slab priors for continuous variable with 1pl in incomplete data assumed to be missing completely at random. lsirm1pl_normal_mcar_ss factorizes continuous item response matrix into column-wise item effect, row-wise respondent effect and further embeds interaction effect in a latent space, while ignoring the missing element under the assumption of missing completely at random. The resulting latent space provides an interaction map that represents interactions between respondents and items.

Usage

lsirm1pl_normal_mcar_ss(
  data,
  ndim = 2,
  niter = 15000,
  nburn = 2500,
  nthin = 5,
  nprint = 500,
  jump_beta = 0.4,
  jump_theta = 1,
  jump_gamma = 1,
  jump_z = 0.5,
  jump_w = 0.5,
  pr_mean_beta = 0,
  pr_sd_beta = 1,
  pr_mean_theta = 0,
  pr_spike_mean = -3,
  pr_spike_sd = 1,
  pr_slab_mean = 0.5,
  pr_slab_sd = 1,
  pr_a_theta = 0.001,
  pr_b_theta = 0.001,
  pr_a_eps = 0.001,
  pr_b_eps = 0.001,
  pr_xi_a = 0.001,
  pr_xi_b = 0.001,
  missing.val = 99,
  verbose = FALSE
)

Arguments

data	Matrix; a binary or continuous item response matrix for analysis. Each row represents a respondent, and each column contains responses to the corresponding item.
ndim	Integer; the dimension of the latent space. Default is 2.
niter	Integer; the total number of MCMC iterations to run. Default is 15000.
nburn	Integer; the number of initial MCMC iterations to discard as burn-in. Default is 2500.
nthin	Integer; the number of MCMC iterations to thin. Default is 5.
nprint	Integer; the interval at which MCMC samples are displayed during execution. Default is 500.
jump_beta	Numeric; the jumping rule for the beta proposal density. Default is 0.4.
jump_theta	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_gamma	Numeric; the jumping rule for the theta proposal density. Default is 1.0.
jump_z	Numeric; the jumping rule for the z proposal density. Default is 0.5.
jump_w	Numeric; the jumping rule for the w proposal density. Default is 0.5.
pr_mean_beta	Numeric; the mean of the normal prior for beta. Default is 0.
pr_sd_beta	Numeric; the standard deviation of the normal prior for beta. Default is 1.0.
pr_mean_theta	Numeric; the mean of the normal prior for theta. Default is 0.
pr_spike_mean	Numeric; the mean of spike prior for log gamma. Default is -3.
pr_spike_sd	Numeric; the standard deviation of spike prior for log gamma. Default is 1.
pr_slab_mean	Numeric; the mean of spike prior for log gamma. Default is 0.5.
pr_slab_sd	Numeric; the standard deviation of spike prior for log gamma. Default is is 1.
pr_a_theta	Numeric; the shape parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_b_theta	Numeric; the scale parameter of the inverse gamma prior for the variance of theta. Default is 0.001.
pr_a_eps	Numeric; the shape parameter of inverse gamma prior for variance of data likelihood. Default is 0.001.
pr_b_eps	Numeric; the scale parameter of inverse gamma prior for variance of data likelihood. Default is 0.001.
pr_xi_a	Numeric; the first shape parameter of beta prior for latent variable xi. Default is 1.
pr_xi_b	Numeric; the second shape parameter of beta prior for latent variable xi. Default is 1.
missing.val	Numeric; a number to replace missing values. Default is 99.
verbose	Logical; If TRUE, MCMC samples are printed for each nprint. Default is FALSE.

Details

lsirm1pl_normal_mcar_ss models the continuous value of response by respondent $j$ to item $i$ with item effect $\beta_i$, respondent effect $\theta_j$ and the distance between latent position $w_i$ of item $i$ and latent position $z_j$ of respondent $j$ in the shared metric space, with $\gamma$ represents the weight of the distance term:

$Y_{j,i} = \theta_j+\beta_i-\gamma||z_j-w_i|| + e_{j,i}$

where the error $e_{j,i} \sim N(0,\sigma^2)$. Under the assumption of missing completely at random, the model ignores the missing element in doing inference. For the details of missing at random assumption and data augmentation, see References. lsirm1pl_normal_mcar_ss model include model selection approach based on spike-and-slab priors for log gamma. For detail of spike-and-slab priors, see References.

Value

lsirm1pl_normal_mcar_ss returns an object of list containing the following components:

data	Data frame or matrix containing the variables in the model.
missing.val	A number to replace missing values.
bic	Numeric value with the corresponding BIC.
mcmc_inf	Details about the number of MCMC iterations, burn-in periods, and thinning intervals.
map_inf	The log maximum a posteriori (MAP) value and the iteration number at which this MAP value occurs.
beta_estimate	Posterior estimates of the beta parameter.
theta_estimate	Posterior estimates of the theta parameter.
sigma_theta_estimate	Posterior estimates of the standard deviation of theta.
gamma_estimate	posterior estimates of gamma parameter.
z_estimate	Posterior estimates of the z parameter.
w_estimate	Posterior estimates of the w parameter.
beta	Posterior samples of the beta parameter.
theta	Posterior samples of the theta parameter.
gamma	Posterior samples of the gamma parameter.
theta_sd	Posterior samples of the standard deviation of theta.
z	Posterior samples of the z parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
w	Posterior samples of the w parameter, represented as a 3-dimensional matrix where the last axis denotes the dimension of the latent space.
pi	Posterior samples of phi which is indicator of spike and slab prior. If phi is 1, log gamma follows the slab prior, otherwise follows the spike prior.
accept_beta	Acceptance ratio for the beta parameter.
accept_theta	Acceptance ratio for the theta parameter.
accept_z	Acceptance ratio for the z parameter.
accept_w	Acceptance ratio for the w parameter.
accept_gamma	Acceptance ratio for the gamma parameter.
sigma_estimate	Posterior estimates of the standard deviation.
sigma	Posterior samples of the standard deviation.

References

Little, R. J., & Rubin, D. B. (2019). Statistical analysis with missing data (Vol. 793). John Wiley & Sons. Ishwaran, H., & Rao, J. S. (2005). Spike and slab variable selection: frequentist and Bayesian strategies. The Annals of Statistics, 33(2), 730-773.

Examples

# generate example (continuous) item response matrix
data     <- matrix(rnorm(500, mean = 0, sd = 1),ncol=10,nrow=50)

# generate example missing indicator matrix
missing_mat     <- matrix(rbinom(500, size = 1, prob = 0.2),ncol=10,nrow=50)

# make missing value with missing indicator matrix
data[missing_mat==1] <- 99

lsirm_result <- lsirm1pl_normal_mcar_ss(data)

# The code following can achieve the same result.
lsirm_result <- lsirm(data ~ lsirm1pl(spikenslab = TRUE, fixed_gamma = FALSE,
missing_data ='mcar', missing.val = 99))

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