This article is a brief illustration of how
`indirect_effect()`

can be used to estimate the indirect
effects among latent variables and form bootstrap confidence intervals
for these effects.

This is the sample dataset used for illustration:

```
library(manymome)
<- data_sem
dat print(round(head(dat), 1))
#> x01 x02 x03 x04 x05 x06 x07 x08 x09 x10 x11 x12 x13 x14
#> 1 0.2 -1.1 -0.2 2.7 -0.1 2.2 1.7 0.2 -1.9 -2.1 -0.9 -0.5 -0.6 -1.5
#> 2 0.1 0.3 0.9 -0.5 -1.0 0.8 -0.3 -0.9 0.8 -1.5 -0.7 1.0 -0.8 -1.4
#> 3 -0.1 -0.1 2.4 -0.2 -0.1 -1.6 -1.9 -2.4 -0.3 -0.2 0.5 -0.7 0.2 -0.7
#> 4 -0.5 1.1 -0.7 -0.5 1.0 -0.6 -0.9 -1.5 -1.5 -0.5 -2.2 -1.6 -0.8 -0.2
#> 5 -0.8 -0.3 -0.9 -0.2 0.3 -2.1 -0.5 -1.8 -0.1 -2.7 -0.4 -1.2 0.3 -2.6
#> 6 0.9 0.3 0.3 1.0 -1.8 -0.4 0.8 -0.4 -1.1 -2.1 0.4 0.0 -0.3 2.2
```

This dataset has 14 variables, which are indicators of four latent
factors: `f1`

, `f2`

, `f3`

, and
`f4`

.

Suppose this is the model to be fitted:

This model can be fitted by `lavaan::sem()`

:

```
<-
mod "
f1 =~ x01 + x02 + x03
f2 =~ x04 + x05 + x06 + x07
f3 =~ x08 + x09 + x10
f4 =~ x11 + x12 + x13 + x14
f3 ~ f1 + f2
f4 ~ f1 + f3
"
<- sem(model = mod,
fit_med data = dat)
```

These are the estimates of the paths between the latent variables:

```
<- parameterEstimates(fit_med)
est $op == "~", ]
est[est#> lhs op rhs est se z pvalue ci.lower ci.upper
#> 15 f3 ~ f1 0.243 0.120 2.018 0.044 0.007 0.479
#> 16 f3 ~ f2 0.326 0.102 3.186 0.001 0.125 0.526
#> 17 f4 ~ f1 0.447 0.125 3.592 0.000 0.203 0.692
#> 18 f4 ~ f3 0.402 0.090 4.445 0.000 0.225 0.579
```

Suppose that for the free parameters, we would like to use ML to form the confidence intervals. For indirect effects, we want to use bootstrapping.

Although bootstrap estimates can be generated and stored the first
time we call `indirect_effect()`

, we illustrate using
`do_boot()`

to generate the bootstrap estimates to be used by
`indirect_effect()`

:

```
<- do_boot(fit_med,
boot_out_med R = 100,
seed = 98171,
ncores = 1)
```

Please see `vignette("do_boot")`

or the help page of
`do_boot()`

on how to use this function. In real research,
`R`

, the number of bootstrap samples, should be set to 2000
or even 5000. The argument `ncores`

can usually be omitted
unless users want to manually control the number of CPU cores to be used
in parallel processing.

Even though path coefficients are not labelled, we can still use
`indirect_effect()`

to estimate the indirect effect and form
its bootstrap confidence interval for any path in the model. By reusing
the generated bootstrap estimates, there is no need to repeat the
resampling and estimation.

Suppose we want to estimate the indirect effect from `f1`

to `f4`

through `f3`

:

```
<- indirect_effect(x = "f1",
out_f1f3f4 y = "f4",
m = "f3",
fit = fit_med,
boot_ci = TRUE,
boot_out = boot_out_med)
out_f1f3f4#>
#> == Indirect Effect ==
#>
#> Path: f1 -> f3 -> f4
#> Indirect Effect 0.098
#> 95.0% Bootstrap CI: [-0.007 to 0.216]
#>
#> Computation Formula:
#> (b.f3~f1)*(b.f4~f3)
#> Computation:
#> (0.24307)*(0.40186)
#>
#> Percentile confidence interval formed by nonparametric bootstrapping
#> with 100 bootstrap samples.
#>
#> Coefficients of Component Paths:
#> Path Coefficient
#> f3~f1 0.243
#> f4~f3 0.402
```

The indirect effect is 0.098, with 95% confidence interval [-0.007, 0.216].

Similarly, we can estimate the indirect effect from `f2`

to `f4`

through `f3`

:

```
<- indirect_effect(x = "f2",
out_f2f3f4 y = "f4",
m = "f3",
fit = fit_med,
boot_ci = TRUE,
boot_out = boot_out_med)
out_f2f3f4#>
#> == Indirect Effect ==
#>
#> Path: f2 -> f3 -> f4
#> Indirect Effect 0.131
#> 95.0% Bootstrap CI: [0.049 to 0.254]
#>
#> Computation Formula:
#> (b.f3~f2)*(b.f4~f3)
#> Computation:
#> (0.32561)*(0.40186)
#>
#> Percentile confidence interval formed by nonparametric bootstrapping
#> with 100 bootstrap samples.
#>
#> Coefficients of Component Paths:
#> Path Coefficient
#> f3~f2 0.326
#> f4~f3 0.402
```

The indirect effect is 0.131, with 95% confidence interval [0.049, 0.254].

The standardized indirect effect from `f1`

to
`f4`

through `f3`

can be estimated by setting
`standardized_x`

and `standardized_y`

to
`TRUE:

```
<- indirect_effect(x = "f1",
std_f1f3f4 y = "f4",
m = "f3",
fit = fit_med,
boot_ci = TRUE,
boot_out = boot_out_med,
standardized_x = TRUE,
standardized_y = TRUE)
std_f1f3f4#>
#> == Indirect Effect ==
#>
#> Path: f1 -> f3 -> f4
#> Indirect Effect 0.073
#> 95.0% Bootstrap CI: [-0.005 to 0.157]
#>
#> Computation Formula:
#> (b.f3~f1)*(b.f4~f3)*sd_f1/sd_f4
#> Computation:
#> (0.24307)*(0.40186)*(0.87470)/(1.17421)
#>
#> Percentile confidence interval formed by nonparametric bootstrapping
#> with 100 bootstrap samples.
#>
#> Coefficients of Component Paths:
#> Path Coefficient
#> f3~f1 0.243
#> f4~f3 0.402
#>
#> NOTE: The effects of the component paths are from the model, not standardized.
```

The standardized indirect effect is 0.073, with 95% confidence interval [-0.005, 0.157].

Similarly, we can estimate the standardized indirect effect from
`f2`

to `f4`

through `f3`

:

```
<- indirect_effect(x = "f2",
std_f2f3f4 y = "f4",
m = "f3",
fit = fit_med,
boot_ci = TRUE,
boot_out = boot_out_med,
standardized_x = TRUE,
standardized_y = TRUE)
std_f2f3f4#>
#> == Indirect Effect ==
#>
#> Path: f2 -> f3 -> f4
#> Indirect Effect 0.116
#> 95.0% Bootstrap CI: [0.044 to 0.204]
#>
#> Computation Formula:
#> (b.f3~f2)*(b.f4~f3)*sd_f2/sd_f4
#> Computation:
#> (0.32561)*(0.40186)*(1.03782)/(1.17421)
#>
#> Percentile confidence interval formed by nonparametric bootstrapping
#> with 100 bootstrap samples.
#>
#> Coefficients of Component Paths:
#> Path Coefficient
#> f3~f2 0.326
#> f4~f3 0.402
#>
#> NOTE: The effects of the component paths are from the model, not standardized.
```

The standardized indirect effect is 0.116, with 95% confidence interval [0.044, 0.204].

Note that, unlike the confidence intervals in
`lavaan::standardizedSolution()`

, the confidence intervals
formed by `indirect_effect()`

are the bootstrap confidence
intervals formed based on the bootstrap estimates, rather than intervals
based on the delta method.

Note that the results of `indirect_effect()`

can be added
using `+`

.

For example, to find the total effect of `f1`

on
`f4`

, we also need to compute the direct effect from
`f1`

to `f4`

. Although it is already available in
the `lavaan`

output, we still use
`indirect_effect()`

to compute it so that it can be added to
the indirect effect computed above with bootstrap confidence
interval:

```
<- indirect_effect(x = "f1",
out_f1f4 y = "f4",
fit = fit_med,
boot_ci = TRUE,
boot_out = boot_out_med)
out_f1f4#>
#> == Effect ==
#>
#> Path: f1 -> f4
#> Effect 0.447
#> 95.0% Bootstrap CI: [0.203 to 0.753]
#>
#> Computation Formula:
#> (b.f4~f1)
#> Computation:
#> (0.44749)
#>
#> Percentile confidence interval formed by nonparametric bootstrapping
#> with 100 bootstrap samples.
```

We can now compute the total effect:

```
<- out_f1f3f4 + out_f1f4
out_f1_total
out_f1_total#>
#> == Indirect Effect ==
#>
#> Path: f1 -> f3 -> f4
#> Path: f1 -> f4
#> Function of Effects: 0.545
#> 95.0% Bootstrap CI: [0.318 to 0.858]
#>
#> Computation of the Function of Effects:
#> (f1->f3->f4)
#> +(f1->f4)
#>
#>
#> Percentile confidence interval formed by nonparametric bootstrapping
#> with 100 bootstrap samples.
```

The total effect of `f1`

on `f4`

is 0.545, with
95% confidence interval [0.318, 0.858].

Subtraction can also be conducted using `-`

. For example,
we can compute the difference between the indirect effect of
`f1`

on `f4`

and the direct effect of
`f1`

on `f4`

:

```
<- out_f1f4 - out_f1f3f4
out_f1_diff
out_f1_diff#>
#> == Indirect Effect ==
#>
#> Path: f1 -> f4
#> Path: f1 -> f3 -> f4
#> Function of Effects: 0.350
#> 95.0% Bootstrap CI: [-0.027 to 0.700]
#>
#> Computation of the Function of Effects:
#> (f1->f4)
#> -(f1->f3->f4)
#>
#>
#> Percentile confidence interval formed by nonparametric bootstrapping
#> with 100 bootstrap samples.
```

The difference in effects is 0.350, with 95% confidence interval [-0.027, 0.700].

If there are several indirect paths in a model, the function
`all_indirect_paths()`

can be used to automatically identify
all indirect paths (a path with at least one mediator) in a model:

```
<- all_indirect_paths(fit = fit_med)
all_paths
all_paths#> Call:
#> all_indirect_paths(fit = fit_med)
#> Path(s):
#> path
#> 1 f1 -> f3 -> f4
#> 2 f2 -> f3 -> f4
```

The output is a `all_paths`

-class object. It can be used
in `many_indirect_effects()`

```
<- many_indirect_effects(paths = all_paths,
out_all fit = fit_med,
boot_ci = TRUE,
boot_out = boot_out_med)
```

The first argument, `paths`

, is the output of
`all_indirect_paths()`

. The other arguments will be passed to
`indirect_effect()`

.

The output is an `indirect_list`

-class object, which is a
list of the outputs of `indirect_effects()`

. If printed, a
summary of the indirect effects will be printed:

```
out_all#>
#> == Indirect Effect(s) ==
#> ind CI.lo CI.hi Sig
#> f1 -> f3 -> f4 0.098 -0.007 0.216
#> f2 -> f3 -> f4 0.131 0.049 0.254 Sig
#>
#> - [CI.lo to CI.hi] are 95.0% percentile confidence intervals by
#> nonparametric bootstrapping with 100 samples.
#> - The 'ind' column shows the indirect effects.
#>
```

The output of `many_indirect_effects()`

is a named list,
names being the path name as appeared in the output. Individual indirect
effects can be extracted using either the indices or the path names

An example using index:

```
<- out_all[[1]]
out1
out1#>
#> == Indirect Effect ==
#>
#> Path: f1 -> f3 -> f4
#> Indirect Effect 0.098
#> 95.0% Bootstrap CI: [-0.007 to 0.216]
#>
#> Computation Formula:
#> (b.f3~f1)*(b.f4~f3)
#> Computation:
#> (0.24307)*(0.40186)
#>
#> Percentile confidence interval formed by nonparametric bootstrapping
#> with 100 bootstrap samples.
#>
#> Coefficients of Component Paths:
#> Path Coefficient
#> f3~f1 0.243
#> f4~f3 0.402
```

An example using path name (though not recommended because the name is usually long):

```
<- out_all[["f2 -> f3 -> f4"]]
out2
out2#>
#> == Indirect Effect ==
#>
#> Path: f2 -> f3 -> f4
#> Indirect Effect 0.131
#> 95.0% Bootstrap CI: [0.049 to 0.254]
#>
#> Computation Formula:
#> (b.f3~f2)*(b.f4~f3)
#> Computation:
#> (0.32561)*(0.40186)
#>
#> Percentile confidence interval formed by nonparametric bootstrapping
#> with 100 bootstrap samples.
#>
#> Coefficients of Component Paths:
#> Path Coefficient
#> f3~f2 0.326
#> f4~f3 0.402
```

The extracted element can be used just like the outputs of
`indirect_effect()`

in previous section.

Users can customize the search. For example, if a model has control
variables, they can be excluded in the search for indirect paths. Users
can also limit the search to paths that start from or end at selected
variables. See the help page of `all_indirect_paths()`

and
`many_indirect_effects()`

for the arguments available.

Not demonstrated in this document, total indirect effect can be
computed by `total_indirect_effect()`

from the output of
`many_indirect_effects()`

. Please refer to
`vignette("med_lm")`

for an example and the help page of
`total_indirect_effect()`

.

For further information on `do_boot()`

and
`indirect_effect()`

, please refer to their help pages, or
`vignette("manymome")`

and
`vignette("do_boot")`

.