This is an example of confirmatory LPA using `tidySEM`

.
The simulated data are based on work by Zegwaard and colleagues, who
sought to establish a typology of caregivers who support a close other
receiving outpatient psychological care. Qualitative research among
experts resulted in a theory postulating the existence of four types of
caregivers (translated from the original Dutch):

**Balanced**

The balanced caregiver experiences relative balance between the costs and benefits of caring for a close other.

**Imbalanced**

The imbalanced caregiver experiences a precarious balance between the costs and benefits of caring for a close other.

**Lonely**

The lonely caregiver experiences a strong sense of isolation.

**Entrapped**

The entrapped caregiver strongly feels a sense of being entangled in responsibilities which are difficult to fulfill.

The goal of this confirmatory study was to validate this hypothesized
class solution in a sample of caregivers. A convenience sample was used,
with no prior sample size justification. To view the data documentation,
run the command `?tidySEM::zegwaard_carecompass`

in the R
console.

To load the data, simply attach the `tidySEM`

package. For
convenience, we assign the variables used for analysis to an object
called `df`

. We first only use the four scales:
`c("burdened", "trapped", "negaffect", "loneliness")`

.

```
# Load required packages
library(tidySEM)
library(ggplot2)
# Load data
<- zegwaard_carecompass[, c("burdened", "trapped", "negaffect",
df "loneliness")]
```

We use `tidySEM::descriptives()`

to describe the data
numerically. Because all scales are continuous, we select only columns
for continuous data to de-clutter the table:

```
<- tidySEM::descriptives(df)
desc <- desc[, c("name", "n", "missing", "unique", "mean", "median",
desc "sd", "min", "max", "skew_2se", "kurt_2se")]
desc
```

The table indicates two potential causes for concern: there is a
small percentage of missingness, and all variables have relatively high
kurtosis. Since there are some missing values, we can conduct an MCAR
test using `mice::mcar(df)`

. According to Hawkins’ test,
there is no evidence to reject the assumptions of multivariate normality
and MCAR, \(\tilde{\chi^2}(6) = 3.78,
\tilde{p} = 0.71\). Missing data will be accounted for using
FIML.

Additionally, we can plot the data. The `ggplot2`

function
`geom_density()`

is useful for continuous data. Visual
inspection confirms the conclusions from the `descriptives()`

table: the data are kurtotic (peaked).

```
<- df
df_plot names(df_plot) <- paste0("Value.", names(df_plot))
<- reshape(df_plot, varying = names(df_plot), direction = "long",
df_plot timevar = "Variable")
ggplot(df_plot, aes(x = Value)) + geom_density() + facet_wrap(~Variable) +
theme_bw()
```

As all variables are continuous, we can use the convenience function
`tidySEM::mx_profiles()`

, which is a wrapper for the generic
function `mx_mixture()`

optimized for continuous indicators.
Its default settings are appropriate for LPA, assuming fixed variances
across classes and zero covariances. Its arguments are `data`

and number of `classes`

. All variables in `data`

are included in the analysis, which is why we first selected the
indicator variables. As this is a confirmatory LCA, we do not follow a
strictly data-driven class enumeration procedure. We will set the
maximum number of classes \(K\) to one
more than the theoretically expected number. We set a seed to ensure
replicable results.

```
set.seed(123)
<- mx_profiles(data = df, classes = 1:5) res
```

This analysis should produce some messages about cluster initialization. These relate to the selection of starting values, which relies on the K-means algorithm and is not robust to missing data. The algorithm automatically switches to hierarchical clustering, no further action is required.

To compare the fit of the theoretical model against other models, we
create a model fit table using `table_fit()`

and retain
relevant columns. We also determine whether any models can be
disqualified.

In this example, all models converge without issues. If, for example,
the two-class solution had not converged, we could use the function
`res[[2]] <- mxTryHard(res[[2]])`

to aid convergence.

Next, we check for local identifiability. The sample size is consistently reported as 513, which means that partially missing cases were indeed included via FIML. The smallest class size occurs in the 5-class model, where the smallest class is assigned 7% of cases, or 38 cases. This model has 28 parameters, approximately 6 per class. We thus have at least five observations per parameter in every class, and do not disqualify the 5-class model.

There are concerns about theoretical interpretability of all solutions, as the entropies and minimum classification probabilities are all low. However, in this confirmatory use case, we address this when interpreting the results.

```
<- table_fit(res) # model fit table
fit c("Name", "LL", "Parameters", "n", "BIC", "Entropy", "prob_min",
fit[, "prob_max", "n_min", "n_max", "np_ratio", "np_local")]
```

the 4-class solution has the lowest BIC, which means it is preferred
over all other solutions including a 1-class solution and a solution
with more classes. Note that a scree plot for the BIC can be plotted by
calling `plot(fit)`

. Following the elbow criterion, a
three-class solution would also be defensible. The function
`ic_weights(tab_compare)`

allows us to compute IC weights; it
indicates that, conditional on the set of models, the 4-class model has
a posterior model probability of nearly 100%.

If we conduct LMR tests, we find that the tests are significant for all pairwise model comparisons, except for the 5-class model:

`lr_lmr(fit)`

We can also use the BLRT test. As it is very computationally expensive, we will use a low number of replications here. In practice, one might use a much higher number (1000+) for published research. Keep in mind that the p-value of the BLRT is subject to Monte Carlo error; if it fluctuates when analyses are replicated or its value is very close to the critical threshold, consider increasing the number of replications.

To accelerate computations, we can use the `future`

package for parallel computing (see `?plan`

to select the
appropriate back-end for your system). To track the function’s progress,
we use the `progressr`

ecosystem, which allows users to
choose how they want to be informed. The example below uses a progress
bar:

```
library(future)
library(progressr)
plan(multisession) # Parallel processing for Windows
handlers("progress") # Progress bar
set.seed(1)
<- BLRT(res, replications = 20) res_blrt
```

In sum, across all class enumeration criteria, there is strong support for a 4-class solution.

In the case of confirmatory LCA, the theory would be refuted by strong evidence against the hypothesized model and number of classes. In the preceding, we only compared the theoretical model against models with different number of classes. Imagine, however, that a Reviewer argues that variance ought to be freely estimated across classes. We could compare our theoretical model against their competing model as follows. Note that we can put two models into a list to compare them.

```
<- mx_profiles(df, classes = 4, variances = "varying")
res_alt <- list(res[[4]], res_alt)
compare table_fit(compare)
```

The alternative model incurs 12 additional parameters for the free variances. Yet, it has a higher BIC, which indicates that this additional complexity does not outweigh the increase in fit.

To interpret the final class solution, we first reorder the 4-class model by class size. This helps prevent label switching.

`<- mx_switch_labels(res[[4]]) res_final `

The 4-class model yielded classes of reasonable size; using
`class_pro`

the largest class comprised 33%, and the smallest
comprised 16% of cases. However, the entropy was low, \(S = .75\), indicating poor class
separability. Furthermore, the posterior classification probability
ranged from \([.81, .92]\), which means
that at least some classes had a high classification error. We produce a
table of the results below.

```
table_results(res_final, columns = c("label", "est", "se", "confint",
"class"))
```

The results are best interpreted by examining a plot of the model and
data, however. Relevant plot functions are
`plot_bivariate()`

, `plot_density()`

, and
`plot_profiles()`

. However, we omit the density plots,
because `plot_bivariate()`

also includes them.

`plot_bivariate(res_final)`

On the diagonal of the bivariate plot are weighted density plots: normal approximations of the density function of observed data, weighed by class probability. On the off-diagonal are plots for each pair of indicators, with the class means indicated by a point, class standard deviations indicated by lines, and covariances indicated by circles. As this model has zero covariances, all circles are round (albeit warped by the different scales of the X and Y axes)

The marginal density plots show that trappedness distinguishes classes rather well. For all other indicators, groups are not always clearly separated in terms of marginal density: class 2 and 3 coalesce on negative affect, 1 and 2 coalesce on loneliness, and 1 and 2 coalesce on burden. Nevertheless, the off-diagonal scatterplots show reasonable bivariate separation for all classes.

We can obtain a more classic profile plot using
`plot_profiles(res_final)`

. This plot conveys less
information than the bivariate plot, but is readily interpretable. Below
is a comparison between the most common type of visualization for LPA,
and the best-practices visualization provided by `tidySEM`

.
Note that the best practices plot includes class means and error bars,
standard deviations, and a ribbon plot of raw data weighted by class
probability to indicate how well the classes describe the observed
distribution. The overlap between the classes is clearly visible in this
figure; this is why the entropy and classification probabilities are
relatively low.

Based on the bivariate plot, we can label class 1 as the
*balanced* type (33%), class 2 as the *imbalanced* type
(29%), class 3 as the *entrapped* type (22%), and class 4 as the
*lonely* type (16%). Note however that the observed classes do
not match the hypothesized pattern of class parameters exactly.

`plot_profiles(res_final)`

We may want to compare the different classes on auxiliary variables
or models. The `BCH()`

function applies three-step analysis,
which compares the classes using a multi-group model, controlling for
classification error. We consider two examples: a single variable, and
an auxiliary model.

For a single (continuous or ordinal) variable, we can call the BCH
function and simply supply the auxiliary variable to the
`data`

argument, omitting the `model`

argument.
Below, we estimate an auxiliary model to compare the sex of patients
between classes:

`<- BCH(res_final, data = zegwaard_carecompass$sexpatient) aux_sex `

To obtain an omnibus likelihood ratio test of the significance of
these sex differences across classes, as well as pairwise comparisons
between classes, use `lr_test(aux_sex)`

. The results indicate
that there are significant sex differences across classes, \(\Delta LL(1) = 8.7, p = .003\). Pairwise
comparisons indicate that class 3 differs significantly from classes 1
and 2. The results can be reported in probability scale using
`table_prob(aux_sex)`

. It appears that the entrapped class
disproportionately cares for female patients.

We can also compare a simple model between classes. Specifically, we will examine whether the distance predicts the frequency of visits differently across classes (treated as continuous).

```
<- zegwaard_carecompass[, c("freqvisit", "distance")]
df_aux $freqvisit <- as.numeric(df_aux$freqvisit)
df_aux<- BCH(res_final, model = "freqvisit ~ distance", data = df_aux) aux_model
```

To obtain an omnibus likelihood ratio test of the difference in
regression coefficients across classes and pairwise comparisons between
classes, use `lr_test(aux_model, compare = "A")`

. The results
indicate that there are no significant sex differences across classes,
\(\Delta LL(3) = 0.98, p = .81\). The
results can be reported using `table_results(aux_model)`

.

This LCA model was developed to help classify care providers in a
clinical context, so that mental healthcare professionals can provide
tailored support to those who take care of their clients. In
`tidySEM`

, it is possible to predict class membership for new
data. Imagine that we administer the care compass questionnaire to a new
individual. We can assign their scale scores to a
`data.frame`

, and supply it to the `predict()`

function via the `newdata`

argument. The result includes the
individual’s most likely class, as well as posterior probabilities for
all classes.

```
<- data.frame(burdened = 2, trapped = 0.5, negaffect = 1.5,
df_new loneliness = 4)
predict(res_final, newdata = df_new)
```