Description Usage Arguments Value Note Author(s) References See Also Examples

This function returns a (*1-α*)% confidence interval (CI) for a well–defined nonlinear function of the coefficients in single–level and multilevel structural equation models. The `ci`

function uses the Monte Carlo (`type="MC"`

) and the asymptotic normal theory (`type="asymp"`

) with the multivariate delta standard error (Asymptotic–Delta) method (Sobel, 1982) to compute a CI. In addition, for each of the methods, when a user specifies `plot=TRUE`

and `plotCI=TRUE`

, a plot of the sampling distribution of the quantity of interest in the `quant`

argument and an overlaid plot of the CI will be produced. When `type="all"`

and `plot=TRUE`

, two overlaid plots of the sampling distributions corresponding to each method will be produced; when `plotCI=TRUE`

, then the overlaid plots of the CIs for both methods will be displayed as well.

1 2 3 |

`mu` |
(1) a vector of means (e.g., coefficient estimates) for the normal random variables. A user can assign a name to each mean value, e.g., |

`Sigma` |
either a covariance matrix or a vector that stacks all the columns of the lower triangle variance–covariance matrix one underneath the other. |

`quant` |
quantity of interest, which is a nonlinear/linear function of the model parameters. Argument |

`alpha` |
significance level for the CI. The default value is .05. |

`type` |
method used to compute a CI. It takes on the values |

`plot` |
when |

`plotCI` |
when |

`n.mc` |
Monte Carlo sample size. The default sample size is 1e+6. |

`H0` |
False. If |

`mu0` |
a vector of means (e.g., coefficient estimates) for the normal random variables that satisft the null hypothesis |

`Sigma0` |
either a covariance matrix or a vector that stacks all the columns of the lower triangle variance–covariance matrix one underneath the other. If it is not provided, then |

`...` |
additional arguments. |

When `type`

is `"MC"`

or `"asymp"`

, `ci`

returns a list that contains:

`(` |
a vector of lower and upper confidence limits, |

`Estimate` |
a point estimate of the quantity of interest, |

`SE` |
standard error of the quantity of interest, |

`MC Error` |
When |

When `type="all"`

, `ci`

returns a list of two objects, each of which a list that contains the results produced by each method as described above.

The web applications for this function is available at http://amp.gatech.edu/MonteCarlo.

Davood Tofighi [email protected] and David P. MacKinnon [email protected]

Tofighi, D., and MacKinnon, D. P. (2016). Monte Carlo confidence intervals for complex functions of indirect effects. *Structural Equation Modeling: A Multidisciplinary Journal*, **23**, 194-205. http://doi.org/10.1080/10705511.2015.1057284

1 2 3 4 5 6 7 8 9 10 11 12 | ```
ci(mu=c(b1=1,b2=.7,b3=.6, b4= .45), Sigma=c(.05,0,0,0,.05,0,0,.03,0,.03),
quant=~b1*b2*b3*b4, type="all", plot=TRUE, plotCI=TRUE)
#An Example of Conservative Null Sampling Distribution
ci(c(b1=.3,b2=.4,b3=.3), c(.01,0,0,.01,0,.02),
quant=~b1*b2*b3, type="mc", plot=TRUE, plotCI=TRUE,
H0=TRUE, mu0=c(b1=.3,b2=.4,b3=0) )
#An Example of Less Conservative Null Sampling Distribution
ci(c(b1=.3,b2=.4,b3=.3), c(.01,0,0,.01,0,.02),
quant=~b1*b2*b3, type="mc", plot=TRUE, plotCI=TRUE,
H0=TRUE, mu0=c(b1=0,b2=.4,b3=0.1) )
``` |

```
Loading required package: MASS
Loading required package: lavaan
This is lavaan 0.6-3
lavaan is BETA software! Please report any bugs.
Loading required package: e1071
$MC
$MC[[1]]
2.5 % 97.5 %
0.02327788 0.50148971
$MC$Estimate
[1] 0.189122
$MC$SE
[1] 0.1253406
$MC$MCError
[1] 1.253406e-07
$MC$p
[1] 0.011528
attr(,"quant")
~b1 * b2 * b3 * b4
$Asymptotic
$Asymptotic$`97.5% CI`
[1] -0.04040623 0.41840623
$Asymptotic$Estimate
[1] 0.189
$Asymptotic$SE
[,1]
[1,] 0.1170461
attr(,"quant")
~b1 * b2 * b3 * b4
[[1]]
[[1]][[1]]
2.5 % 97.5 %
0.001570451 0.093490055
[[1]]$Estimate
[1] 0.03601567
[[1]]$SE
[1] 0.02396109
[[1]]$MCError
[1] 2.396109e-08
[[1]]$p
[1] 0.03636
attr(,"quant")
~b1 * b2 * b3
[[2]]
[[2]]$CI
2.5 % 97.5 %
-0.002164623 0.074296014
[[2]]$Estimate
[1] 1.902105e-05
[[2]]$SE
[1] 0.01840343
[[2]]$p
[1] 0.06073
[[2]][[5]]
b1 b2 b3
0.3 0.4 0.0
[[1]]
[[1]][[1]]
2.5 % 97.5 %
0.001484511 0.093643023
[[1]]$Estimate
[1] 0.0360057
[[1]]$SE
[1] 0.02398737
[[1]]$MCError
[1] 2.398737e-08
[[1]]$p
[1] 0.036824
attr(,"quant")
~b1 * b2 * b3
[[2]]
[[2]]$CI
2.5 % 97.5 %
0.02054618 0.05156672
[[2]]$Estimate
[1] 1.200391e-05
[[2]]$SE
[1] 0.007146034
[[2]]$p
[1] 0.002568
[[2]][[5]]
b1 b2 b3
0.0 0.4 0.1
```

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