The semfindr
package contains functions for doing structural equation modeling (SEM)
diagnostics, such as identifying influential cases and computing various
diagnostic measures. This document illustrates how to use
`semfindr`

to do casewise sensitivity analysis: Assessing the
influence of a case on parameter estimates and model fit measures.

It supports two approaches: the leave-one-out approach presented by
Pek and MacCallum (2011), and the approximate approach that approximates
the influence of a case without refitting a model. It can generate some
plots based on similar plots available in the `car`

package
by Fox and Weisberg (2019) for casewise sensitivity analysis.

Under this approach, for a case of concern, the model is fitted again without this case, and then results such as parameter estimates are compared. This approach is exact but can be time consuming because the model needs to be fitted again for each case under consideration.

To remove the need to refit the model many times whenever a case
influence statistic is requested, `semfindr`

adopts this
workflow:

Decide cases to examine. All cases will be examined, by default.

For each selected case, remove it and refit the model.

Store the results.

Any case influence statistics can then computed without the need to repeat Step 2.

Users can do as much diagnostic analysis as they want without repeating the time consuming refitting step. Step 2 can also be conducted without the need to decide in advance the influence statistics to compute. Some statistics, such as generalized Cook’s distance, is a function of the parameters selected, and the parameters to examine may depend on the results of other statistics and so may change during the analysis.

The following sections illustrates how to use the major functions.

The sample dataset is `pa_dat`

, provided in the package,
with variables `iv1`

, `iv2`

, `m1`

, and
`dv`

, and 100 cases. For convenience, we assign
`pa_dat`

to a new symbol, `dat`

.

```
library(semfindr)
dat <- pa_dat
head(dat)
#> m1 dv iv1 iv2
#> 1 0.32067106 1.4587148 0.2055776 -0.42187811
#> 2 0.15360231 -0.3809220 0.1853543 0.15229953
#> 3 0.35136439 -0.4886773 0.9151424 1.16670950
#> 4 -0.56529330 -0.9766142 0.2884440 0.04563409
#> 5 -1.60657017 -1.0948066 -0.5756171 -0.18184854
#> 6 0.03143301 0.5859886 0.1420111 0.06286986
```

Assume that the target model under examination is a path model with two predictors, one mediator, and one outcome variable:

We fit the model by `lavaan::sem()`

:

We refit the model 100 times, each time with one case removed:

This example takes about 4 to 8 seconds. For larger samples or more
complicated models, `lavaan_rerun()`

supports parallel
processing by setting `parallel`

to `TRUE`

.

`lavaan_rerun()`

also supports selecting cases using the
Mahalanobis distance on all variables in the model or on the residuals
of outcome variables. See the help page of `lavaan_rerun()`

or `vignette("selecting_cases", package = "semfindr")`

for
details.

If this process is slow, users can save the results by
`base::saveRDS()`

such that users can load it for sensitivity
analysis later, without the need to repeat these steps in each R
session.

One intuitive way to assess case influence is to compute the changes in parameter estimates if a case is included, with the changes standardized by their standard errors (Pek & MacCallum, 2011, Equation 7):

```
fit_est_change <- est_change(fit_rerun)
fit_est_change
#>
#> -- Standardized Case Influence on Parameter Estimates --
#>
#> m1~iv1 m1~iv2 dv~m1 m1~~m1 dv~~dv gcd
#> 16 0.052 -0.038 -0.237 -0.004 0.624 0.450
#> 43 -0.403 -0.263 -0.135 0.223 0.120 0.302
#> 65 0.152 0.191 0.363 0.076 0.161 0.221
#> 85 -0.174 0.216 -0.119 0.335 -0.052 0.208
#> 51 0.421 -0.057 0.094 0.089 -0.044 0.200
#> 34 -0.314 -0.192 -0.109 0.189 0.030 0.178
#> 32 -0.247 0.195 -0.191 0.193 0.001 0.175
#> 18 -0.273 0.035 0.101 0.260 -0.046 0.156
#> 20 -0.239 0.204 -0.141 0.183 -0.032 0.156
#> 100 -0.001 -0.225 -0.069 0.305 -0.056 0.152
#>
#> Note:
#> - Changes are standardized raw changes if a case is included.
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by generalized Cook's distance.
```

The output is a matrix-like object of the class “est_change”, with a
print method (`print.est_change()`

). By default, the cases
are sorted in descending order based on generalized Cook’s distance
(`gcd`

, described below), and only the first 10 cases are
printed.

The standardized change is a measure of *influence* if a case
is *included*. If the standardized change of a parameter for a
case is positive, then including this case increases the estimate of
this parameter.

For example, the standardized change of the path from
`iv1`

to `m1`

is 0.024 for the first case. The
estimates of this path with and without the first case are 0.215 and
0.212, respectively. The estimate of this path is larger when this case
is included than when this case is excluded. (Recall that 0.024 is the
change standardized by the standard error of the estimate).

`est_change()`

also computes the generalized Cook’s
distance (Cook, 1977; Pek & MacCallum, 2011, Equation 6),
*gCD* (labelled in lowercase in the output as `gcd`

),
using the parameters examined. *gCD* is analogous to Cook’s
distance in multiple regression. It measures the overall influence in
the parameters if a case is included.

```
#> m1~iv1 m1~iv2 dv~m1 m1~~m1 dv~~dv gcd
#> 16 0.052 -0.038 -0.237 -0.004 0.624 0.450
#> 43 -0.403 -0.263 -0.135 0.223 0.120 0.302
#> 65 0.152 0.191 0.363 0.076 0.161 0.221
#> 85 -0.174 0.216 -0.119 0.335 -0.052 0.208
#> 51 0.421 -0.057 0.094 0.089 -0.044 0.200
```

Pek and MacCallum recommended computing generalized Cook’s distance
for subset of parameters that researchers would like to assess case
influence. This can be done by specifying the parameters to be included.
For example, we may compute the changes and the *gCD* only for
path coefficients, using the argument `parameters`

:

```
fit_est_change_paths_only <- est_change(fit_rerun,
parameters = c("m1 ~ iv1",
"m1 ~ iv2",
"dv ~ m1"))
fit_est_change_paths_only
#>
#> -- Standardized Case Influence on Parameter Estimates --
#>
#> m1~iv1 m1~iv2 dv~m1 gcd
#> 43 -0.403 -0.263 -0.135 0.238
#> 51 0.421 -0.057 0.094 0.190
#> 65 0.152 0.191 0.363 0.189
#> 34 -0.314 -0.192 -0.109 0.142
#> 32 -0.247 0.195 -0.191 0.138
#> 20 -0.239 0.204 -0.141 0.121
#> 85 -0.174 0.216 -0.119 0.093
#> 11 0.010 0.149 -0.257 0.088
#> 18 -0.273 0.035 0.101 0.087
#> 13 0.274 0.059 -0.068 0.082
#>
#> Note:
#> - Changes are standardized raw changes if a case is included.
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by generalized Cook's distance.
```

If all paths are to be included, the following call will also work:

Although the 16^{th} case has the largest *gCD* based
on all parameters, the 43^{th} case has the largest *gCD*
based on regression paths only. Therefore, when examining *gCD*,
it is better to compute it only for parameters that are theoretically
important.

See the help page of `est_change()`

for further
information.

The standardized changes in parameter may not be easy to interpret.
If the original units are interpretable, users can compute the
*raw* changes, that is, the changes in parameter estimates if a
case is included, not standardized by their standard errors. This can be
done by `est_change_raw()`

:

```
fit_est_change_raw <- est_change_raw(fit_rerun)
fit_est_change_raw
#>
#> -- Case Influence on Parameter Estimates --
#>
#> id m1~iv1 id m1~iv2 id dv~m1 id m1~~m1 id dv~~dv
#> 1 51 0.046 43 -0.026 65 0.039 61 0.043 16 0.108
#> 2 43 -0.043 94 0.024 11 -0.027 85 0.041 9 0.051
#> 3 34 -0.033 100 -0.022 16 -0.024 100 0.038 76 0.050
#> 4 13 0.030 85 0.021 32 -0.021 18 0.032 25 0.050
#> 5 18 -0.029 20 0.020 99 0.020 42 0.029 91 0.043
#> 6 32 -0.026 32 0.019 79 0.019 43 0.028 17 0.039
#> 7 20 -0.025 65 0.019 93 0.018 32 0.024 65 0.030
#> 8 75 0.021 34 -0.019 22 0.017 34 0.024 26 0.029
#> 9 42 -0.020 64 -0.017 61 -0.017 20 0.023 62 0.027
#> 10 68 0.020 52 0.016 25 -0.015 40 0.023 90 0.024
#>
#> Note:
#> - Changes are raw changes if a case is included.
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by the absolute changes for each variable.
```

The output is a matrix-like object of the class “est_change”, with a
print method (`print.est_change()`

). If the output was
generated `est_change_raw()`

, by default, each column of
parameter is sorted in the descending order of the absolute value, with
case IDs inserted.

For example, the change of the path from `iv1`

to
`m1`

is -0.043 for the 43^{rd} case. The estimate of
this path with and without the 43^{rd} cases are 0.215 and
0.258, respectively. The estimate with the 43^{rd} case included
is smaller than the estimate with the 43^{rd} case excluded. The
raw changes is 0.215 - 0.258 or -0.043.

If desired, `est_change_raw()`

can also compute the
changes in parameters in the *standardized solution*, by setting
`standardized`

to `TRUE`

:

```
fit_est_change_raw_std <- est_change_raw(fit_rerun,
standardized = TRUE)
fit_est_change_raw_std
#>
#> -- Case Influence on Standardized Parameter Estimates --
#>
#> id m1~iv1 id m1~iv2 id dv~m1 id m1~~m1 id dv~~dv id iv1~~iv1 id iv1~~iv2
#> 1 51 0.042 100 -0.023 16 -0.033 43 0.026 16 0.030 1 0 87 -0.041
#> 2 43 -0.032 43 -0.021 65 0.029 94 -0.022 65 -0.024 2 0 60 -0.035
#> 3 13 0.028 94 0.020 25 -0.018 100 0.022 25 0.017 3 0 45 0.029
#> 4 34 -0.026 99 0.019 11 -0.018 34 0.021 11 0.016 4 0 91 -0.027
#> 5 18 -0.025 34 -0.017 99 0.016 99 -0.019 99 -0.013 5 0 27 0.025
#> 6 32 -0.023 87 0.016 93 0.015 52 -0.016 9 0.013 6 0 43 0.024
#> 7 20 -0.023 52 0.015 9 -0.014 65 -0.016 93 -0.013 7 0 57 -0.024
#> 8 68 0.021 40 -0.013 22 0.014 27 -0.014 43 0.012 8 0 50 0.024
#> 9 85 -0.020 20 0.012 43 -0.013 40 0.013 22 -0.012 9 0 69 0.022
#> 10 42 -0.019 61 -0.012 79 0.013 18 0.013 79 -0.011 10 0 71 0.020
#> id iv2~~iv2
#> 1 1 0
#> 2 2 0
#> 3 3 0
#> 4 4 0
#> 5 5 0
#> 6 6 0
#> 7 7 0
#> 8 8 0
#> 9 9 0
#> 10 10 0
#>
#> Note:
#> - Changes are raw changes if a case is included.
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by the absolute changes for each variable.
```

Note that the variances of `iv1`

and `iv2`

are
necessarily equal to one in the standardized solution and so the raw
changes are equal to zero for all cases.

For example, these are standardized solutions of the full sample and the sample with the 43rd case removed:

```
standardizedSolution(fit, se = FALSE)[1, ]
#> lhs op rhs est.std
#> 1 m1 ~ iv1 0.178
standardizedSolution(sem(mod, dat[-43, ]), se = FALSE)[1, ]
#> lhs op rhs est.std
#> 1 m1 ~ iv1 0.21
```

The change of the standardized estimate of the path from
`iv1`

to `m1`

is -0.032 for the 43^{rd}
case. The standardized estimates of this path with and without the
43^{rd} cases are 0.178 and 0.21, respectively. The estimate of
the standardized coefficient from `iv1`

to `m1`

is
smaller than the estimate with the 43^{rd} case removed. The raw
changes of standardized estimate is 0.178 - 0.21 or -0.032.

`est_change_raw()`

also supports computing the changes for
selected parameters:

```
fit_est_change_raw_std_paths <- est_change_raw(fit_rerun,
standardized = TRUE,
parameters = c("m1 ~ iv1",
"m1 ~ iv2",
"dv ~ m1"))
fit_est_change_raw_std_paths
#>
#> -- Case Influence on Standardized Parameter Estimates --
#>
#> id dv~m1 id m1~iv1 id m1~iv2
#> 1 16 -0.033 51 0.042 100 -0.023
#> 2 65 0.029 43 -0.032 43 -0.021
#> 3 25 -0.018 13 0.028 94 0.020
#> 4 11 -0.018 34 -0.026 99 0.019
#> 5 99 0.016 18 -0.025 34 -0.017
#> 6 93 0.015 32 -0.023 87 0.016
#> 7 9 -0.014 20 -0.023 52 0.015
#> 8 22 0.014 68 0.021 40 -0.013
#> 9 43 -0.013 85 -0.020 20 0.012
#> 10 79 0.013 42 -0.019 61 -0.012
#>
#> Note:
#> - Changes are raw changes if a case is included.
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by the absolute changes for each variable.
```

If all parameters of the same operators are to be included, e.g.,
`"~"`

for all regression paths, this form will also work:

See the help page of `est_change_raw()`

for further
information.

One commonly used measure for identifying outliers is Mahalanobis
distance (Mahalanobis, 1936; Pek & MacCallum, 2011, Equation 9).
`mahalanobis_rerun()`

can be used to compute the Mahalanobis
distance of each case on all the variables used in the target model:

```
fit_md <- mahalanobis_rerun(fit_rerun)
fit_md
#>
#> -- Mahalanobis Distance --
#>
#> md
#> 16 11.530
#> 99 11.312
#> 87 11.091
#> 43 10.181
#> 51 9.869
#> 13 8.476
#> 91 8.078
#> 71 7.757
#> 17 7.555
#> 68 7.472
#>
#> Note:
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by Mahalanobis distance in decreasing order.
```

The output is a matrix-like object of the class “md_semfindr”, with a
print method (`print.md_semfindr()`

). By default, cases are
sorted in descending order of Mahalanobis distance.

Note that a case with a large Mahalanobis distance is not necessarily an influential case (Pek & MacCallum, 2011). Therefore, influence measures should be examined to avoid overlooking cases that are not extreme but are influential.

See the help page of `mahalanobis_rerun()`

for further
information.

Another intuitive measure of influence is the difference in a measure
of model fit between the analysis with a case included and that with the
case excluded. This can be done by `fit_measures_change()`

,
which simply gets any fit measures supported by
`lavaan::fitMeasures()`

from the results from
`lavaan_rerun`

:

```
fit_mc <- fit_measures_change(fit_rerun,
fit_measures = c("chisq", "cfi", "tli", "rmsea"))
fit_mc
#>
#> -- Case Influence on Fit Measures --
#>
#> chisq cfi tli rmsea
#> 1 0.154 -0.002 -0.005 0.002
#> 2 -0.019 0.001 0.003 -0.001
#> 3 -0.417 0.008 0.021 -0.007
#> 4 -0.154 0.004 0.009 -0.003
#> 5 0.097 0.000 0.001 0.001
#> 6 0.116 -0.001 -0.003 0.001
#> 7 -0.631 0.014 0.034 -0.011
#> 8 0.120 0.002 0.005 0.001
#> 9 0.524 -0.012 -0.030 0.008
#> 10 0.697 -0.013 -0.033 0.011
#>
#> Note:
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
```

The output is a matrix-like object of the class
“fit_measures_change”, with a print method
(`print.fit_measures_change()`

). By default, only the first
10 cases are printed.

To sort cases by a specific measure, set `sort_by`

to the
column name to be used for sorting cases. By default, cases are sorted
in descending order of the *absolute* value of the selected
column.

```
print(fit_mc, sort_by = "chisq")
#>
#> -- Case Influence on Fit Measures --
#>
#> chisq cfi tli rmsea
#> 91 1.760 -0.034 -0.085 0.031
#> 17 -1.591 0.027 0.066 -0.025
#> 25 1.580 -0.031 -0.079 0.028
#> 16 -1.533 0.019 0.048 -0.024
#> 87 -1.381 0.030 0.074 -0.022
#> 43 1.306 -0.030 -0.075 0.022
#> 90 0.930 -0.016 -0.039 0.015
#> 97 -0.919 0.017 0.042 -0.015
#> 13 -0.909 0.020 0.050 -0.015
#> 62 0.863 -0.015 -0.038 0.014
#>
#> Note:
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by chisq in decreasing order on absolute values.
```

The value is computed by \(M_\textrm{full sample} - M_\textrm{one case removed}\). Therefore, if the value for a case is positive, the measure is higher when this case is included than when this case is excluded. If the value is negative, the measure is smaller when this case is included than when this case is excluded. This convention is selected such that the interpretation is consistent with that for changes in parameter estimates.

For example, the change in CFI for the 43rd case is -0.03. Therefore, including the 43rd Case yields a CFI smaller than when this case is exclude, and the difference is 0.03.

The argument `fit_measures`

is passed to
`lavaan::fitMeasures()`

to specify the measures to be
computed. The default values are
`c("chisq", "cfi", "tli", "rmsea")`

. Therefore, this argument
can be omitted if they are the desired measures of fit:

See the help page of `fit_measures_change()`

for further
information.

We can also use `influence_stat()`

to compute the previous
measures. It calls `fit_measures_change()`

,
`est_change()`

, and `mahalanobis_rerun()`

and then
merges their results into one object:

```
fit_influence <- influence_stat(fit_rerun)
fit_influence
#>
#> -- Standardized Case Influence on Parameter Estimates --
#>
#> m1~iv1 m1~iv2 dv~m1 m1~~m1 dv~~dv gcd
#> 16 0.052 -0.038 -0.237 -0.004 0.624 0.450
#> 43 -0.403 -0.263 -0.135 0.223 0.120 0.302
#> 65 0.152 0.191 0.363 0.076 0.161 0.221
#> 85 -0.174 0.216 -0.119 0.335 -0.052 0.208
#> 51 0.421 -0.057 0.094 0.089 -0.044 0.200
#> 34 -0.314 -0.192 -0.109 0.189 0.030 0.178
#> 32 -0.247 0.195 -0.191 0.193 0.001 0.175
#> 18 -0.273 0.035 0.101 0.260 -0.046 0.156
#> 20 -0.239 0.204 -0.141 0.183 -0.032 0.156
#> 100 -0.001 -0.225 -0.069 0.305 -0.056 0.152
#>
#> Note:
#> - Changes are standardized raw changes if a case is included.
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by generalized Cook's distance.
#>
#> -- Case Influence on Fit Measures --
#>
#> chisq cfi rmsea tli
#> 1 0.154 -0.002 0.002 -0.005
#> 2 -0.019 0.001 -0.001 0.003
#> 3 -0.417 0.008 -0.007 0.021
#> 4 -0.154 0.004 -0.003 0.009
#> 5 0.097 0.000 0.001 0.001
#> 6 0.116 -0.001 0.001 -0.003
#> 7 -0.631 0.014 -0.011 0.034
#> 8 0.120 0.002 0.001 0.005
#> 9 0.524 -0.012 0.008 -0.030
#> 10 0.697 -0.013 0.011 -0.033
#>
#> Note:
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#>
#> -- Mahalanobis Distance --
#>
#> md
#> 16 11.530
#> 99 11.312
#> 87 11.091
#> 43 10.181
#> 51 9.869
#> 13 8.476
#> 91 8.078
#> 71 7.757
#> 17 7.555
#> 68 7.472
#>
#> Note:
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by Mahalanobis distance in decreasing order.
```

The output is a matrix-like object of the class “influence_stat”,
with a print method (`print.influence_stat()`

). If printed,
it will print the results using the methods described above.

One major use of `influence_stat()`

is to provide
information for the diagnostic plots introduced below.

`semfindr`

provides several functions to generate
diagnostic plots. All these functions accept an output of
`influence_stat()`

and returns a `ggplot2`

plot,
which can be further customized if desired by other `ggplot2`

functions.

To visualize the *gCDs* of cases, we can plot an index plot
with cases on the horizontal axis and the *gCD* on the vertical
axis using `gcd_plot()`

:

The plot shows that, compared to other cases, the 16th case has the
largest *gCD* (based on all free parameters).

`largest_gcd`

controls the number of cases with the
largest `gcd`

to be labelled. The default is 1.

More options of `gcd_plot()`

can be found on its help
page.

An index plot can be computed on the Mahalanobis distance given by
`influence_stat()`

:

This plot illustrates that, although the 87th and 99th cases are also
large on Mahalanobis distance, they are not influential cases when
assessed by *gCD*.

`largest_m`

is used to control how many cases with high
Mahalanobis distance on all the variables in the fitted model will be
labelled. The default is 1.

More options for `md_plot()`

can be found on its help
page.

To examine how *gCD* relates to a selected measure of model
fit (`gof`

), `gcd_gof_plot()`

can be used:

`largest_gcd`

determines the number of cases with largest
`gcd`

to be labelled, and `largest_fit_measure`

determines the number of cases with largest *absolute* change in
the selected measure of model fit to be labelled. The default is 1 for
both arguments.

More options of `gcd_gof_plot()`

can be found on its help
page.

The function `gcd_gof_md_plot()`

can be used to plot a
bubble plot of a selected measure of model fit against Mahalanobis
distance, with the bubble size determined by generalized Cook’s
distance. This plot is similar to the plot by
`car::influencePlot()`

for regression models.

```
gcd_gof_md_plot(fit_influence,
fit_measure = "rmsea",
largest_gcd = 3,
largest_fit_measure = 3,
largest_md = 3,
circle_size = 15)
```

`circle_size`

controls the size of the largest bubble.
Increase this number when the size difference is too small between
bubbles. `largest_gcd`

, `largest_fit_measure`

, and
`largest_md`

controls the number of cases with highest
absolute values one the these measures to be labelled. Their default
values are 1.

More options of `gcd_gof_md_plot()`

can be found from its
help page.

The function `est_change_plot()`

can be used to plot an
index plot of standardized or raw changes using the output of
`est_change()`

or `est_change_raw()`

.

For example, using the output generated by `est_change()`

above, it can generate an index plot for each parameter:

`largest_change`

controls the number of cases with the
largest change to be labelled. The default is 1. The cases to be
labelled is determined separately for each parameter.

The function also supports plotting the changes only for selected
parameters, using `parameters`

:

It can also plot the raw changes. For example:

Last, the output of `influence_stat()`

can also be used.
The case influence will be extracted from the object. For example, the
following call, using `fit_influence`

instead of
`fit_est_change_raw`

, will generate the same plot.

More options of `est_change_plot()`

can be found on its
help page.

The function `est_change_gcd_plot()`

can be used to plot,
for each selected parameter, casewise standardized changes using the
output of `est_change()`

against *gCD*.

For example, using the output generated by `est_change()`

above, it can generate an index plot for each parameter:

`largest_gcd`

controls the number of cases with the
largest *gCD* to be labelled. The default is 1.

The function also supports plotting the changes only for selected
parameters, using `parameters`

:

It does not support plotting the raw changes against *gCD*
because *gCD* is not computed by
`est_change_raw()`

.

Last, the output of `influence_stat()`

can also be used.
The case influence will be extracted from the object. For example, the
following call, using `fit_influence`

instead of
`fit_est_change`

, will generate the same plot.

More options of `est_change_gcd_plot()`

can be found on
its help page.

The leave-one-out approach is exact because the model is fitted
twice, with and without the target case. However, this can be time
consuming for some models and datasets. The `semfindr`

package also supports the approximate approach that uses casewise scores
(from `lavaan::lavScores()`

) and casewise likelihood to
approximate the influence of a case *without* refitting a model.
This approach is not exact but is much faster than the leave-one-out
approach because the model is not fitted again.

This approach can be used together with the leave-one-out approach, using the approximate approach to identify potentially influential cases and then use the leave-one-out approach to compute the exact influence.

Most the functions for the leave-one-out approach has their
approximate approach counterparts. Therefore, only their usage will be
illustrated here. Please refer to the previous section on the meanings
of the influence statistics. The major difference is, all functions for
the approximate approach use the output of `lavaan`

directly.
There is no need to use `lavaan_rerun()`

.

For the technical details on the approximate approach, please refer
to the vignette *Approximate Case Influence Using Scores and Casewise
Likelihood*
(`vignette("casewise_scores", package = "semfindr")`

).

The function `est_change_approx()`

can be used to compute
the approximate standardized change. The first argument is the output of
`lavaan`

:

```
fit_est_change_approx <- est_change_approx(fit)
fit_est_change_approx
#>
#> -- Approximate Standardized Case Influence on Parameter Estimates --
#>
#> m1~iv1 m1~iv2 dv~m1 m1~~m1 dv~~dv gcd_approx
#> 16 0.052 -0.038 -0.228 -0.006 0.572 0.372
#> 43 -0.387 -0.249 -0.135 0.201 0.116 0.270
#> 65 0.150 0.189 0.355 0.071 0.148 0.203
#> 85 -0.170 0.211 -0.118 0.315 -0.054 0.187
#> 51 0.405 -0.052 0.094 0.075 -0.046 0.179
#> 34 -0.306 -0.186 -0.110 0.176 0.028 0.163
#> 32 -0.241 0.190 -0.189 0.181 -0.002 0.161
#> 20 -0.234 0.199 -0.140 0.172 -0.034 0.144
#> 18 -0.269 0.035 0.101 0.246 -0.048 0.143
#> 100 -0.001 -0.221 -0.069 0.290 -0.058 0.137
#>
#> Note:
#> - Changes are approximate standardized raw changes if a case is included.
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by approximate generalized Cook's distance.
```

The output is a matrix-like object of the class “est_change”, with a
print method (`print.est_change()`

). By default, the cases
are sorted in descending order based on approximate generalized Cook’s
distance (`gcd_approx`

, described below), and only the first
10 cases are printed.

The column `gcd_approx`

indicates that the *gCD* is
only an approximate value.

Like `est_change()`

, it also supports computing the
approximate *gCD* based on selected parameters. For example, the
following computes the *gCD* based on regression coefficients
only:

```
fit_est_change_approx_paths <- est_change_approx(fit,
parameters = "~")
fit_est_change_approx_paths
#>
#> -- Approximate Standardized Case Influence on Parameter Estimates --
#>
#> m1~iv1 m1~iv2 dv~m1 gcd_approx
#> 43 -0.387 -0.249 -0.135 0.217
#> 65 0.150 0.189 0.355 0.177
#> 51 0.405 -0.052 0.094 0.172
#> 34 -0.306 -0.186 -0.110 0.132
#> 32 -0.241 0.190 -0.189 0.130
#> 20 -0.234 0.199 -0.140 0.114
#> 85 -0.170 0.211 -0.118 0.087
#> 11 0.010 0.149 -0.254 0.084
#> 18 -0.269 0.035 0.101 0.082
#> 13 0.267 0.056 -0.068 0.076
#>
#> Note:
#> - Changes are approximate standardized raw changes if a case is included.
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by approximate generalized Cook's distance.
```

See the help page of `est_change_approx()`

for further
information.

The function `est_change_raw_approx()`

computes the
approximate raw changes of parameter estimates, not standardized by
their standard errors. The first argument is the output of
`lavaan`

:

```
fit_est_change_raw_approx <- est_change_raw_approx(fit)
fit_est_change_raw_approx
#>
#> -- Approximate Case Influence on Parameter Estimates --
#>
#> id m1~iv1 id m1~iv2 id dv~m1 id m1~~m1 id dv~~dv
#> 1 51 0.042 43 -0.025 65 0.037 61 0.042 16 0.106
#> 2 43 -0.040 94 0.023 11 -0.027 85 0.040 9 0.050
#> 3 34 -0.032 100 -0.022 16 -0.024 100 0.037 76 0.049
#> 4 18 -0.028 85 0.021 32 -0.020 18 0.031 25 0.049
#> 5 13 0.028 20 0.020 99 0.020 42 0.028 91 0.043
#> 6 32 -0.025 32 0.019 79 0.018 43 0.025 17 0.039
#> 7 20 -0.024 65 0.019 93 0.018 32 0.023 26 0.028
#> 8 75 0.021 34 -0.018 22 0.017 34 0.022 65 0.027
#> 9 42 -0.020 64 -0.016 61 -0.016 40 0.022 62 0.027
#> 10 68 0.018 52 0.016 25 -0.015 20 0.022 90 0.024
#>
#> Note:
#> - Changes are approximate raw changes if a case is included.
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by the absolute changes for each variable.
```

The output is a matrix-like object of the class “est_change”, with a
print method (`print.est_change()`

). If the output was
generated `est_change_raw_approx()`

, by default, each column
of parameter is sorted in the descending order of the absolute value,
with case IDs inserted.

Unlike `est_change_raw()`

,
`est_change_raw_approx()`

does not support raw changes in the
standardized solution.

See the help page of `est_change_raw_approx()`

for further
information.

The function `mahalanobis_rerun()`

actually does not need
the leave-one-out approach. Therefore, it can also be used in the
approximate approach by setting the first argument to the output of
`lavaan`

:

```
fit_md <- mahalanobis_rerun(fit)
fit_md
#>
#> -- Mahalanobis Distance --
#>
#> md
#> 16 11.530
#> 99 11.312
#> 87 11.091
#> 43 10.181
#> 51 9.869
#> 13 8.476
#> 91 8.078
#> 71 7.757
#> 17 7.555
#> 68 7.472
#>
#> Note:
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by Mahalanobis distance in decreasing order.
```

The results are the same whether the output of `lavaan`

or
`lavaan_reun()`

is used.

The function `fit_measures_change_approx()`

computes the
approximate changes in selected fit measures. The first argument is the
output of `lavaan`

:

```
fit_mc_approx <- fit_measures_change_approx(fit,
fit_measures = c("chisq", "cfi", "tli", "rmsea"))
fit_mc_approx
#>
#> -- Approximate Case Influence on Fit Measures --
#>
#> chisq cfi tli rmsea
#> 1 0.160 -0.002 -0.005 0.002
#> 2 -0.019 0.001 0.003 -0.001
#> 3 -0.389 0.008 0.019 -0.007
#> 4 -0.151 0.004 0.009 -0.003
#> 5 0.097 0.000 0.001 0.001
#> 6 0.116 -0.001 -0.003 0.001
#> 7 -0.596 0.013 0.032 -0.010
#> 8 0.119 0.002 0.005 0.001
#> 9 0.543 -0.012 -0.031 0.008
#> 10 0.703 -0.013 -0.033 0.011
#>
#> Note:
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
```

The output is a matrix-like object of the class
“fit_measures_change”, with a print method
(`print.fit_measures_change()`

). By default, only the first
10 cases are printed.

To sort cases by a specific measure, set `sort_by`

to the
column name to be used for sorting cases. By default, cases are sorted
in descending order of the *absolute* value of the selected
column.

```
print(fit_mc_approx, sort_by = "chisq")
#>
#> -- Approximate Case Influence on Fit Measures --
#>
#> chisq cfi tli rmsea
#> 91 1.846 -0.035 -0.089 0.033
#> 25 1.621 -0.032 -0.080 0.029
#> 43 1.392 -0.031 -0.078 0.024
#> 17 -1.389 0.023 0.058 -0.022
#> 16 -1.283 0.016 0.039 -0.021
#> 87 -1.146 0.026 0.064 -0.019
#> 90 0.944 -0.016 -0.040 0.016
#> 34 0.876 -0.021 -0.052 0.014
#> 62 0.874 -0.015 -0.038 0.014
#> 97 -0.855 0.016 0.039 -0.014
#>
#> Note:
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by chisq in decreasing order on absolute values.
```

These measures are the default values. Therefore, if only these four measures are needed, the following will also work:

See the help page of `fit_measures_change_approx()`

for
further information.

The all-in-one function `influence_stat()`

can be used to
compute approximate influence statistics by calling
`fit_measures_change_approx()`

and
`est_change_approx()`

. This can be done simply by using the
output of `lavaan`

as the first argument:

```
fit_influence_approx <- influence_stat(fit)
fit_influence_approx
#>
#> -- Approximate Standardized Case Influence on Parameter Estimates --
#>
#> m1~iv1 m1~iv2 dv~m1 m1~~m1 dv~~dv gcd_approx
#> 16 0.052 -0.038 -0.228 -0.006 0.572 0.372
#> 43 -0.387 -0.249 -0.135 0.201 0.116 0.270
#> 65 0.150 0.189 0.355 0.071 0.148 0.203
#> 85 -0.170 0.211 -0.118 0.315 -0.054 0.187
#> 51 0.405 -0.052 0.094 0.075 -0.046 0.179
#> 34 -0.306 -0.186 -0.110 0.176 0.028 0.163
#> 32 -0.241 0.190 -0.189 0.181 -0.002 0.161
#> 20 -0.234 0.199 -0.140 0.172 -0.034 0.144
#> 18 -0.269 0.035 0.101 0.246 -0.048 0.143
#> 100 -0.001 -0.221 -0.069 0.290 -0.058 0.137
#>
#> Note:
#> - Changes are approximate standardized raw changes if a case is included.
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by approximate generalized Cook's distance.
#>
#> -- Approximate Case Influence on Fit Measures --
#>
#> chisq cfi rmsea tli
#> 1 0.160 -0.002 0.002 -0.005
#> 2 -0.019 0.001 -0.001 0.003
#> 3 -0.389 0.008 -0.007 0.019
#> 4 -0.151 0.004 -0.003 0.009
#> 5 0.097 0.000 0.001 0.001
#> 6 0.116 -0.001 0.001 -0.003
#> 7 -0.596 0.013 -0.010 0.032
#> 8 0.119 0.002 0.001 0.005
#> 9 0.543 -0.012 0.008 -0.031
#> 10 0.703 -0.013 0.011 -0.033
#>
#> Note:
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#>
#> -- Mahalanobis Distance --
#>
#> md
#> 16 11.530
#> 99 11.312
#> 87 11.091
#> 43 10.181
#> 51 9.869
#> 13 8.476
#> 91 8.078
#> 71 7.757
#> 17 7.555
#> 68 7.472
#>
#> Note:
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by Mahalanobis distance in decreasing order.
```

The output is a matrix-like object of the class “influence_stat”,
with a print method (`print.influence_stat()`

). If printed,
it will print the results using the methods described above.

See the help page of `influence_stat()`

for further
information.

All the diagnostic plot functions can also be used to visualize case
influence statistics based of the approximate approach. The method used
will be determined by the output of `influence_stat()`

,
`est_change_approx()`

, and
`est_change_raw_approx()`

and users use them in exactly the
same way as for the leave-one-out approach. Therefore, only sample code
is presented below, using the output of `influence_stat()`

,
`est_change_approx()`

, and
`est_change_raw_approx()`

based on the approximate approach
generated in the previous section.

Note that all the plots noted in the titles and axis labels that the statistics are approximate values.

This plot is the same for both the leave-one-out approach and the approximate approach.

Like the leave-one-out approach, the output of
`influence_stat()`

can also be used. For example, replacing
`fit_est_change_raw_approx`

by
`fit_influence_approx`

will generate the same plot:

Note `largest_gcd`

controls the number of cases with the
largest *approximated* *gCD* to be labelled. The default
is 1.

Like the leave-one-out approach, the output of
`influence_stat()`

can also be used. For example, replacing
`fit_est_change_approx`

by `fit_influence_approx`

will generate the same plot:

The examples above use row numbers to identify cases. If users have
meaningful case IDs, they can be used to label case ( see
`vignette("user_id", package = "semfindr")`

). If users want
to refit the model only with selected cases removed one-by-one,
`lavaan_rerun()`

supports various methods to examine only
selected cases (see
`vignette("selecting_cases", package = "semfindr")`

).

Last, all the plot functions return `ggplot`

graph
objects. Users can further modify them to suit their needs. They also
have `*_aes`

arguments that can be used to customize the plot
generated. Please see their help pages on how to use these
arguments.

Cook, R. D. (1977). Detection of influential observation in linear
regression. *Technometrics, 19*(1), 15-18.

Fox J., & Weisberg, S. (2019). *An R companion to applied
regression* (3rd Edition). Sage. https://socialsciences.mcmaster.ca/jfox/Books/Companion/.

Mahalanobis, P. C. (1936). On the generalized distance in statistics.
*Proceedings of the National Institute of Science of India*, 2,
49-55.

Pek, J., & MacCallum, R. (2011). Sensitivity analysis in
structural equation models: Cases and their influence. *Multivariate
Behavioral Research, 46*(2), 202-228. https://doi.org/10.1080/00273171.2011.561068