REM_CFA | R Documentation |

This function uses the robust expectation maximization (REM) algorithm to estimate the parameters of a confirmatory factor analysis model as suggested by Nieser & Cochran (2021).

```
REM_CFA(X, delta = 0.05, model = NA, ctrREM = controlREM())
```

`X` |
data to analyze; should be a data frame or matrix |

`delta` |
hyperparameter between 0 and 1 that captures the researcherâ€™s tolerance of incorrectly down-weighting data from the model (default = 0.05). |

`model` |
string variable that contains each structural equation in a new line where equalities are denoted by the symbol "~". |

`ctrREM` |
control parameters (default: (steps = 25, tol = 1e-6, maxiter = 1e3, min_weights = 1e-30, max_ueps = 0.3, chk_gamma = 0.9, n = 2e4)) |

REM_CFA returns an object of class "REM". The function `summary()`

is used to obtain estimated parameters from the model. An object of class "REM" in Confirmatory Factor Analysis is a list of outputs with four different components: the matched call (call), estimates using traditional expectation maximization (EM_output), estimates using robust expectation maximization (REM_output), and a summary table (summary_table). The list contains the following components:

`call` |
match call |

`model` |
model frame |

`delta` |
hyperparameter between 0 and 1 that captures the researcherâ€™s tolerance of incorrectly down-weighting data from the model |

`k` |
number of factors |

`constraints` |
p x k matrix of zeros and ones denoting the factors (rows) and observed variables (columns) |

`epsilon` |
hyperparameter on the likelihood scale |

`AIC_rem` |
Akaike Information Criterion |

`BIC_rem` |
Bayesian Information Criterion |

`mu` |
item intercepts |

`lambda` |
factor loadings |

`psi` |
unique variances of items |

`gamma` |
average weights |

`weights` |
estimated REM weights |

`ind_lik` |
likelihood value for each individual |

`lik_rem` |
joint log-likelihood evaluated at REM estimates |

`lik` |
joint log-likelihood evaluated at EM estimates |

`summary_table` |
summary of EM and REM estimates, SEs, Z statistics, p-values, and 95% confidence intervals |

Bryan Ortiz-Torres (bortiztorres@wisc.edu); Kenneth Nieser (nieser@stanford.edu)

Nieser, K. J., & Cochran, A. L. (2021). Addressing heterogeneous populations in latent variable settings through robust estimation. Psychological Methods.

`REM_EFA()`

, `summary.REMLA()`

```
# Creating latent model
library(lavaan)
library(GPArotation)
df <- HolzingerSwineford1939
data = df[,-c(1:6)]
model <- "Visual =~ x1 + x2 + x3
Textual =~ x4 + x5 + x6
Speed =~ x7 + x8 + x9"
# Modeling Confirmatory Factor Analysis
model_CFA = REM_CFA(X = data, delta = 0.05, model = model)
summary(model_CFA)
```

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