This vignette has been stripped down to comply with CRAN package size policies. To view the complete vignette, including graphics, see the package website.

This is an example of exploratory LCA with continuous indicators, or finite Gaussian mixture modeling, using tidySEM. The present example uses data collected by Alkema as part of a study on ocean microplastics. To view its documentation, run the command ?tidySEM::alkema_microplastics in the R console. The original analyses are available at https://github.com/cjvanlissa/lise_microplastics; in this vignette, we take a different approach to the analysis to showcase other possibilities.

Loading the Data

To load the data, simply attach the tidySEM package. For convenience, we assign the variables used for analysis to an object called df. As explained in the paper, the classes are quite different for lines, films, and fragments. For this reason, we here only use data from fragments. The indicators are fragments’ length and width.

# Load required packages
library(tidySEM) 
library(ggplot2)
# Load data
df_analyze <- alkema_microplastics[alkema_microplastics$category == "Fragment", ]
df <- df_analyze[ ,c("length", "width")]

Examining the Data

As per the best practices, the first step in LCA is examining the observed data. We use tidySEM::descriptives() to describe the data numerically. Because all items are categorical, we remove columns for continuous data to de-clutter the table:

desc <- tidySEM::descriptives(df)
desc <- desc[, c("name", "type", "n", "missing", "unique", 
"mean", "median", "sd", "min", "max", "skew_2se", "kurt_2se")]
knitr::kable(desc, caption = "Descriptive statistics")
Descriptive statistics
name type n missing unique mean median sd min max skew_2se kurt_2se
length numeric 5605 0 2086 2.9 2.4 1.9 1.0 69.2 137 2116
width numeric 5605 0 2079 2.0 1.6 1.1 0.2 6.8 22 37

Additionally, we can plot the data. The ggplot2 function geom_density() is useful for continuous data:

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

The data are correctly coded as numeric. There are no missing values; if any variables had missing values, we would report an MCAR test with mice::mcar(), and explain that missing data are accounted for using FIML. Note that the data are extremely right-skewed and kurtotic, as also evident from the plot. With this in mind, it can be useful to transform and rescale the data. We will use a log transformation.

df_plot$Value <- log(df_plot$Value)
ggplot(df_plot, aes(x = Value)) +
  geom_density() +
  facet_wrap(~Variable)+
  theme_bw()

The log transformation addresses all aforementioned concerns regarding skew and kurtosis. Let’s reshape the data to wide format and examine a scatterplot:

df <- reshape(df_plot, direction = "wide", v.names = "Value")[, -1]
names(df) <- gsub("Value.", "", names(df), fixed = TRUE)
ggplot(df, aes(x = length, y = width)) +
  geom_point(alpha = .1) +
  theme_bw()

Conducting Latent Profile Analysis

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 an exploratory LCA, we will conduct a rather extensive search across model specifications and number of classes. We will set the maximum number of classes \(K\) to four; depending on the results, we can always choose to increase it later. We set a seed to ensure replicable results. As the analysis takes a long time to compute, it is prudent to save the results to disk immediately, so as not to lose them. For this, we use the function saveRDS(). We can later use res <- readRDS("res_gmm.RData") to load the analysis from the file.

set.seed(123)
res <- mx_profiles(data = df,
                   classes = 1:4,
                   variances = c("equal", "varying"),
                   covariances = c("zero", "equal",
                                   "varying"),
                   expand_grid = TRUE)
saveRDS(res, "res_gmm.RData")

Class Enumeration

To compare the fit of the estimated models, we create a model fit table using table_fit() and retain relevant columns. We also determine whether any models can be disqualified.

fit <- table_fit(res)

There were no indications of convergence problems during estimation. Next, we check for local identifiability. The sample size is 5605. We can calculate the ratio of observations to parameters and append it to the fit table as follows:

fit$par_ratio <- (5605*fit$n_min) / (fit$Parameters/fit$Classes)

As can be seen from the fit table below, the lowest ratio of observations to parameters is 18, which is no cause for concern. However, these classes comprise a very small percentage of the total sample size.

There are, however, concerns about the interpretability of all solutions, as many of the entropies and minimum classification probabilities are low. Only a few models have acceptable entropies around .86 and minimum classification probabilities around .94. Note that the BIC and the entropy are strongly correlated. If we omit the 1-class models, for which entropy is technically not defined, we see that cor(fit$BIC[!fit$Classes == 1], fit$Entropy[!fit$Classes == 1]) returns 0.85. This strong correlation indicates that an increase in fit comes with a decrease in class separability. This illustrates why entropy should not be treated as a model fit criterion.

fit[ , c("Name", "LL", "Parameters", "par_ratio",
         "BIC", "Entropy",
         "prob_min", "prob_max", 
         "n_min", "n_max",
         "lmr_p")]
Model fit table
Name LL Parameters par_ratio BIC Entropy prob_min prob_max n_min n_max lmr_p
equal var 1 -8107 4 1401 16249 1.00 1.00 1.00 1.00 1.00 NA
equal var 2 -5211 7 564 10483 0.87 0.94 0.97 0.35 0.65 0
equal var 3 -4138 10 342 8363 0.83 0.88 0.95 0.20 0.46 0
equal var 4 -3500 13 246 7113 0.83 0.89 0.93 0.14 0.35 0
free var 1 -8107 4 1401 16249 1.00 1.00 1.00 1.00 1.00 NA
free var 2 -5138 9 501 10353 0.85 0.94 0.97 0.40 0.60 0
free var 3 -4005 14 374 8131 0.83 0.89 0.95 0.31 0.35 0
free var 4 -3331 19 245 6826 0.84 0.88 0.93 0.21 0.33 0
equal var, equal cov 1 -3389 5 1121 6820 1.00 1.00 1.00 1.00 1.00 NA
equal var, equal cov 2 -3082 8 432 6234 0.72 0.86 0.95 0.31 0.69 0
equal var, equal cov 3 -3030 11 256 6155 0.67 0.71 0.93 0.17 0.56 0
equal var, equal cov 4 -3021 14 220 6162 0.61 0.68 0.83 0.14 0.34 0
free var, equal cov 1 -3389 5 1121 6820 1.00 1.00 1.00 1.00 1.00 NA
free var, equal cov 2 -2545 10 97 5176 0.64 0.51 0.98 0.09 0.91 0
free var, equal cov 3 -2257 15 72 4643 0.68 0.54 0.94 0.06 0.64 0
free var, equal cov 4 -2069 20 26 4310 0.63 0.52 0.90 0.02 0.50 0
equal var, free cov 1 -3389 5 1121 6820 1.00 1.00 1.00 1.00 1.00 NA
equal var, free cov 2 -2552 9 108 5181 0.65 0.52 0.98 0.09 0.91 0
equal var, free cov 3 -2359 13 84 4831 0.68 0.56 0.93 0.07 0.65 0
equal var, free cov 4 -2174 17 28 4494 0.63 0.60 0.88 0.02 0.50 0
free var, free cov 1 -3389 5 1121 6820 1.00 1.00 1.00 1.00 1.00 NA
free var, free cov 2 -2575 11 407 5245 0.56 0.81 0.91 0.40 0.60 0
free var, free cov 3 -2111 17 36 4370 0.65 0.51 0.88 0.04 0.56 0
free var, free cov 4 -2024 23 15 4247 0.62 0.50 0.91 0.02 0.48 0

Next, we plot a scree plot for the BIC by calling plot(fit):

plot(fit) + theme(axis.text.x = element_text(angle = 90, vjust = 0.5, hjust=1))

It is not immediately clear which solution to prefer. Looking at the blocks of 1-4 class models for each model specification, it appears that the BIC keeps decreasing with the addition of more classes. Across the blocks, the BIC keeps decreasing with increasingly complex model specifications. Similarly, all LMR tests are significant. The function ic_weights(fit) allows us to compute IC weights for all models in the set; it prefers the most complex model with a posterior model probability of nearly 100%. However, the smallest class in this model contains only 85 cases, about 1.6% of the sample. We can wonder whether such a small class has a meaningful interpretation.

The analysis thus indicates support for increasingly complex models, and those complex models are ever less interpretable and relevant, as indicated by low entropy and class size, respectively. This suggests a potential risk of overfitting. We may instead choose the most parsimoneous model that fits the data well. To aid in this decision, we plot the BIC and the Entropy, omitting the 1-class solutions because there is clear evidence that more classes are needed:

df_plot <- fit
df_plot <- df_plot[!df_plot$Classes == 1, ]
ggplot(df_plot, aes(x = BIC, y = Entropy, label = Name)) +
  geom_point() +
  geom_label() +
  theme_bw()

It appears that the 2-class model with equal variances and covariances has an above-average fit, and relatively high entropy. We thus proceed with this model.

Interpreting the Final Class Solution

For convenience, we assign the final model to a separate object:

res_final <- res[["equal var, equal cov 2"]]

The 4-class model yielded classes of reasonable size; the largest class comprised 68%, and the smallest comprised 32% of cases. The entropy was relatively low, \(S = .72\), indicating poor class separability. Furthermore, the posterior classification probability ranged from \([.86, .95]\), which means that classification error was non-negligible. We produce a table of the results below.

table_results(res_final, columns = c("label", "est", "se", "confint", "class"))
Two-class model results
label est se confint class
mix2.weights[1,2] 0.46 0.02 [0.42, 0.50] NA
Variances.length 0.11 0.00 [0.10, 0.11] class1
Covariances.length.WITH.width 0.08 0.00 [0.07, 0.08] class1
Variances.width 0.10 0.00 [0.09, 0.10] class1
Means.length 0.68 0.01 [0.66, 0.69] class1
Means.width 0.30 0.01 [0.29, 0.32] class1
Means.length 1.51 0.01 [1.49, 1.53] class2
Means.width 1.12 0.01 [1.10, 1.15] class2

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.

The bivariate and marginal plots show that the classes are not clearly separable, as also evident from the low entropy. At the same time however, it is clear that the distributions are non-normal, and the second class accounts for some of this non-normality. The first class (68%) accounts for smaller fragments, and the second class (32%) accounts for some of the right-skew in fragments’ length and width. We can simply label class 1 as small fragments, and class 2 as larger fragments.

Auxiliary Analyses

Finally, 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. For example, we can test whether polymer type differs between the two classes:

df_pt <- mx_dummies(df_analyze$poly_type)
aux_pt <- BCH(res_final, model = "poly_typeOther | t1
                                  poly_typePE | t1
                                  poly_typePP | t1", data = df_pt)
aux_pt <- mxTryHardOrdinal(aux_pt)

To obtain an omnibus likelihood ratio test of the significance of the differences in polymer type across classes, use lr_test(aux_pt). The results indicate that there are significant differences in polymer types across classes, \(\Delta LL(3) = 14.08, p = .003\). The results can be reported in probability scale using table_prob(aux_pt). To test differences for specific polymer types, we can use Wald tests:

wald_test(aux_pt, "class1.Thresholds[1,1] = class2.Thresholds[1,1];class1.Thresholds[1,2] = class2.Thresholds[1,2];class1.Thresholds[1,3] = class2.Thresholds[1,3]")

The results indicate that there is no significant difference in the prevalence of “Other” polymer types across classes. However, PE is significantly more prevalent in class 2, and PP is significantly more prevalent in class 1.