bomegas {Bayesrel} | R Documentation |

## Estimate Bayesian reliability estimates for multidimensional scales following common factor models

### Description

When supplying a multidimensional data set the function estimates the reliability of the set by means of omega_total and the general factor saturation of the set by means of omega_hierarchical. The data may follow multiple multi-factorial factor models: a) the second-order factor model (crossloadings possible), b) the bi-factor model (no crossloadings), c) the correlated factor model (crossloadings possible, only omega_t) Error-covariances are not estimable.

The prior distributions of omega_t and omega_h are computed from the prior distributions of the respective factor model parameters. Specifically, normal distributions for the factor loadings and factor scores; an inverse gamma distribution for the manifest and latent residuals; an inverse Wishart distribution for the covariance matrix of the latent variables (correlated model). A Gibbs sampler iteratively draws samples from the conditional posterior distributions of the factor model parameters. The posterior distributions of omega_t and omega_h are computed from the posterior samples of the factor model parameters.

The output contains the posterior distributions of omega_t and omega_h (only for second-order and bi-factor model), their mean, and credible intervals. If desired, one may also find the posterior implied covariance matrices, and the posterior factor model parameters in the output.

### Usage

```
bomegas(
data,
n.factors = NULL,
model = NULL,
model.type = "second-order",
n.iter = 2000,
n.burnin = 200,
n.chains = 3,
thin = 1,
interval = 0.95,
missing = "impute",
a0 = NA,
b0 = NA,
l0 = NA,
A0 = NA,
c0 = NA,
d0 = NA,
beta0 = NA,
B0 = NA,
p0 = NA,
R0 = NA,
param.out = FALSE,
callback = function() {
},
disableMcmcCheck = FALSE
)
```

### Arguments

`data` |
A matrix or data.frame containing multivariate observations, rows = observations, columns = variables/items |

`n.factors` |
A number specifying the number of group factors that the items load on |

`model` |
A string that by default NULL (=balanced) distributes the items evenly among the number of group factors. This only works if the items are a multiple of the number of group factors and the items are already grouped in the data set, meaning, e.g., items 1-5 load on one factor, 6-10 on another, and so on. A model file can be specified in lavaan syntax style (f1=~.+.+.) to relate the items to the group factors. The items' names need to equal the column names in the data set, aka the variable names |

`model.type` |
A string indicating the factor model estimated, by default this is the "second-order" model. Another option is the "bi-factor" model; and eventually "correlated" for the correlated factor model |

`n.iter` |
A number for the iterations of the Gibbs Sampler |

`n.burnin` |
A number for the burnin in the Gibbs Sampler |

`n.chains` |
A number for the chains to run for the MCMC sampling |

`thin` |
A number for the thinning of the MCMC samples |

`interval` |
A number specifying the credible interval, the interval is the highest posterior density interval (HPD) |

`missing` |
A string denoting the missing data handling, can be "impute" or "listwise". With impute the missing data will be estimated during the MCMC sampling as further unknown parameters |

`a0` |
A number for the shape of the prior inverse gamma distribution for the manifest residual variances, when left as NA, the default value is set to 2 |

`b0` |
A number for the scale of the prior inverse gamma distribution for the manifest residual variances, when left as NA, the default value is set to 1 |

`l0` |
A number for the mean of the prior normal distribution for the manifest loadings, when left as NA, the default value is set to 0; can be a single value or a loading matrix |

`A0` |
A number for scaling the variance of the prior normal distribution for the manifest loadings, when left as NA, the default value is set to 1 |

`c0` |
A number for the shape of the prior inverse gamma distribution for the latent residual variances, when left as NA, the default value is set to 2; necessary only for the second-order model |

`d0` |
A number for the scale of the prior inverse gamma distribution for the latent residual variances, when left as NA, the default value is set to 1, necessary only for the second-order model |

`beta0` |
A number for the mean of the prior normal distribution for the g-factor loadings, when left as NA, the default value is set to 0, can be a single value or a vector |

`B0` |
A number for scaling the variance of the prior normal distribution for the g-factor loadings, when left as NA, the default value is set to 2.5 |

`p0` |
A number for the shape of the prior inverse gamma distribution for the variance of the g-factor, when left as NA, the default value is set to q^2-q when q are the number of group factors for the second-order and bi-factor model |

`R0` |
A number for the scale of the prior inverse gamma distribution for the variance of the g-factor, when left as NA, the default value is set to the number of items for the second-order and bi-factor model |

`param.out` |
A logical indicating if loadings and residual variances should be attached to the result, by default FALSE because it saves memory |

`callback` |
An empty function for implementing a progressbar call from a higher program (e.g., JASP) |

`disableMcmcCheck` |
A logical disabling the MCMC settings check for running tests and the likes |

### Value

The posterior means and the highest posterior density intervals for omega_t and omega_h (for the second-order and bi-factor model)

### References

Lee S (2007).
*Structural equation modeling: A Bayesian approach*.
John Wiley & Sons.

### Examples

```
# specify the model syntax relating the group factors to the items:
model <- "f1 =~ U17_r + U22_r + U29_r + U34_r
f2 =~ U4 + U14 + U19 + U27
f3 =~ U6 + U16 + U28 + U48
f4 =~ U23_r + U31_r + U36_r + U46_r
f5 =~ U10_r + U20_r + U35_r + U52_r"
# specifying the model can be omitted if the model structure is simple and balanced,
# meaning the an equal number of items load on each factor, and the items relate to the factors
# in the same order as they appear in the data.
# Note that the iterations are set very low for smoother running examples, you should use
# at least the defaults:
res <- bomegas(upps, n.factors = 5, model = NULL, n.iter = 200, n.burnin = 50,
n.chains = 2, missing = "listwise", model.type = "second-order")
```

*Bayesrel*version 0.7.7 Index]