# Monte Carlo Confidence Intervals with Multiple Imputation

#### Shu Fai Cheung & Sing-Hang Cheung

#### 2025-01-25

# 1 Introduction

This article is a brief illustration of how to use `do_mc()` from the package manymome (Cheung & Cheung, 2024) for a model fitted to multiple imputation datasets to generate Monte Carlo estimates, which can be used by `indirect_effect()` and `cond_indirect_effects()` to form Monte Carlo confidence intervals in the presence of missing data.

For the details of using `do_mc()`, please refer to `vignette("do_mc")`. This article assumes that readers know how to use `do_mc()` and will focus on using it with a model estimated by multiple imputation.

It only supports a model fitted by `semTools::sem.mi()` or `semTools::runMI()`.

# 2 How It Works

When used with multiple imputation, `do_mc()` retrieves the pooled point estimates and variance-covariance matrix of free model parameters and then generates a number of sets of simulated sample estimates using a multivariate normal distribution. Other parameters and implied variances, covariances, and means of variables are then generated from these simulated estimates.

When a \((1 - \alpha)\)% Monte Carlo confidence interval is requested, the \(100(\alpha/2)\)th percentile and the \(100(1 - \alpha/2)\)th percentile are used to form the confidence interval. For a 95% Monte Carlo confidence interval, the 2.5th percentile and 97.5th percentile will be used.

# 3 The Workflow

The following workflow will be demonstrated;

1. Generate datasets using multiple imputation, not covered here (please refer to guides on `mice` or `Amelia`, the two packages supported by `semTools::sem.mi()` and `semTools::runMI()`).

2. Fit the model using `semTools::sem.mi()` or `semTools::runMI()`.

3. Use `do_mc()` to generate the Monte Carlo estimates.

4. Call other functions (e.g, `indirect_effect()` and `cond_indirect_effects()`) to compute the desired effects and form Monte Carlo confidence intervals.

# 4 Demonstration

## 4.1 Multiple Imputation

This data set, with missing data introduced, will be used for illustration.

    library(manymome)
    dat <- data_med
    dat[1, 1] <- dat[2, 3] <- dat[3, 5] <- dat[4, 3] <- dat[5, 2] <- NA
    head(dat)
    #>           x        m        y       c1       c2
    #> 1        NA 17.89644 20.73893 1.426513 6.103290
    #> 2  8.331493 17.92150       NA 2.940388 3.832698
    #> 3 10.327471 17.83178 22.14201 3.012678       NA
    #> 4 11.196969 20.01750       NA 3.120056 4.654931
    #> 5 11.887811       NA 28.47312 4.440018 3.959033
    #> 6  8.198297 16.95198 20.73549 2.495083 3.763712

It has one predictor (`x`), one mediator (`m`), one outcome variable (`y`), and two control variables (`c1` and `c2`).

The following simple mediation model with two control variables (`c1` and `c2`) will be fitted:

plot of chunk do_mc_lavaan_mi_draw_model

In practice, the imputation model needs to be decided and checked (van Buuren, 2018). For the sake of illustration, we just use the default of `mice::mice()` to do the imputation:

    library(mice)
    #>
    #> Attaching package: 'mice'
    #> The following object is masked from 'package:stats':
    #>
    #>     filter
    #> The following objects are masked from 'package:base':
    #>
    #>     cbind, rbind
    set.seed(26245)
    out_mice <- mice(dat, m = 5, printFlag = FALSE)
    dat_mi <- complete(out_mice, action = "all")
    # The first imputed dataset
    head(dat_mi[[1]])
    #>           x        m        y       c1       c2
    #> 1  9.762412 17.89644 20.73893 1.426513 6.103290
    #> 2  8.331493 17.92150 25.68452 2.940388 3.832698
    #> 3 10.327471 17.83178 22.14201 3.012678 3.969419
    #> 4 11.196969 20.01750 24.87107 3.120056 4.654931
    #> 5 11.887811 20.82502 28.47312 4.440018 3.959033
    #> 6  8.198297 16.95198 20.73549 2.495083 3.763712
    # The last imputed dataset
    head(dat_mi[[5]])
    #>           x        m        y       c1       c2
    #> 1  8.301276 17.89644 20.73893 1.426513 6.103290
    #> 2  8.331493 17.92150 22.93143 2.940388 3.832698
    #> 3 10.327471 17.83178 22.14201 3.012678 6.238426
    #> 4 11.196969 20.01750 26.90840 3.120056 4.654931
    #> 5 11.887811 20.82502 28.47312 4.440018 3.959033
    #> 6  8.198297 16.95198 20.73549 2.495083 3.763712

## 4.2 Fit a Model by `semTools::sem.mi()`

We then fit the model by `semTools::sem.mi()`:

    library(semTools)
    #>
    #> ###############################################################################
    #> This is semTools 0.5-6
    #> All users of R (or SEM) are invited to submit functions or ideas for functions.
    #> ###############################################################################
    mod <-
    "
    m ~ x + c1 + c2
    y ~ m + x + c1 + c2
    "
    fit_lavaan <- sem.mi(model = mod,
                         data = dat_mi)
    summary(fit_lavaan)
    #> lavaan.mi object based on 5 imputed data sets.
    #> See class?lavaan.mi help page for available methods.
    #>
    #> Convergence information:
    #> The model converged on 5 imputed data sets
    #>
    #> Rubin's (1987) rules were used to pool point and SE estimates across 5 imputed data sets, and to calculate degrees of freedom for each parameter's t test and CI.
    #>
    #> Parameter Estimates:
    #> Error in if (categorical.flag) {: argument is of length zero

## 4.3 Generate Monte Carlo Estimates

The other steps are identical to those illustrated in `vignette("do_mc")`. It and related functions will use the pooled point estimates and variance-covariance matrix when they detect that the model is fitted by `semTools::sem.mi()` or `semTools::runMI()` (i.e., the fit object is of the class `lavaan.mi`).

We call `do_mc()` on the output of `semTools::sem.mi()` to generate the Monte Carlo estimates of all free parameters and the implied statistics, such as the variances of `m` and `y`, which are not free parameters but are needed to form the confidence interval of the standardized indirect effect.

    mc_out_lavaan <- do_mc(fit = fit_lavaan,
                           R = 10000,
                           seed = 4234)
    #> Stage 1: Simulate estimates
    #> Stage 2: Compute implied statistics

Usually, just three arguments are needed:

- `fit`: The output of `lavaan::sem()`.

- `R`: The number of Monte Carlo replications. Should be at least 10000 in real research.

- `seed`: The seed for the random number generator. To be used by `set.seed()`. It is recommended to set this argument such that the results are reproducible.

Parallel processing is not used. However, the time taken is rarely long because there is no need to refit the model many times.

For the structure of the output, please refer to `vignette("do_mc")`.

## 4.4 Using the Output of `do_mc()` in Other Functions of `manymome`

When calling `indirect_effect()` or `cond_indirect_effects()`, the argument `mc_out` can be assigned the output of `do_mc()`. They will then retrieve the stored simulated estimates to form the Monte Carlo confidence intervals, if requested.

    out_lavaan <- indirect_effect(x = "x",
                                  y = "y",
                                  m = "m",
                                  fit = fit_lavaan,
                                  mc_ci = TRUE,
                                  mc_out = mc_out_lavaan)
    out_lavaan
    #>
    #> == Indirect Effect  ==
    #>
    #>  Path:                 x -> m -> y
    #>  Indirect Effect:      0.656
    #>  95.0% Monte Carlo CI: [0.213 to 1.124]
    #>
    #> Computation Formula:
    #>   (b.m~x)*(b.y~m)
    #>
    #> Computation:
    #>   (0.89141)*(0.73569)
    #>
    #>
    #> Monte Carlo confidence interval with 10000 replications.
    #>
    #> Coefficients of Component Paths:
    #>  Path Coefficient
    #>   m~x       0.891
    #>   y~m       0.736

Reusing the simulated estimates can ensure that all analysis with Monte Carlo confidence intervals are based on the same set of simulated estimates.

# 5 Limitation

Monte Carlo confidence intervals require the variance-covariance matrix of all free parameters. Therefore, only models fitted by `lavaan::sem()` and (since 0.1.9.8) `semTools::sem.mi()` or `semTools::runMI()` are supported. Models fitted by `stats::lm()` do not have a variance-covariance matrix for the regression coefficients from two or more regression models and so are not supported by `do_mc()`.

# 6 Further Information

For further information on `do_mc()`, please refer to its help page.

# 7 Reference

Cheung, S. F., & Cheung, S.-H. (2024). manymome: An R package for computing the indirect effects, conditional effects, and conditional indirect effects, standardized or unstandardized, and their bootstrap confidence intervals, in many (though not all) models. Behavior Research Methods, 56(5), 4862–4882. https://doi.org/10.3758/s13428-023-02224-z

van Buuren, S. (2018). Flexible imputation of missing data (2nd Ed.). CRC Press, Taylor and Francis Group.