### Note

An in-development `R`

package to carry out LPA and this analysis is available here: https://github.com/jrosen48/tidymixmod.

# Introduction

Along with starting to use MPlus, I’ve become (more) interested in trying to find out how to carry out Latent Profile Analysis (LPA) in R, focused on two options: OpenMx and MCLUST. The two are very different: OpenMx is an option for general latent variable modeling (i.e., it can be used to specify a wide range of latent variable models, from Confirmatory Factor Analysis to a Growth Mixture Model), while MCLUST is (primarily) for clustering (and classification) using mixture models.This post is a first attempt about carrying out LPA using MCLUST in R. MCLUST does model-based clustering, as or as the authors describe it, Gaussian Mixture Modeling, since LPA can be seen as one way to do mixture modeling through general latent variable modeling software. Another post will focus on OpenMx (and MPlus), both of which do that.

# LPA models

When carrying out LPA, we can fit an unconstrained model, in which not only the measured variables’ means, but also their variance and covariances, are different, or freely-estimated, across profiles, or mixture components. However, we also sometimes estimate commonly-specified models that are more constrained as part of a model-building process.

Namely, we are commonly interested in those in which the residual variances are constrained (to be the same across components) or freely-estimated (to be different across components) and the covariances are fixed (to be zero, i.e. the covariance matrix is diagonal, in that only the cells on the diagonal, or the variances, are estimated) or freely-estimated.

Let’s get started.

# Test (example) code using the “iris” data

Here is a first pass, using the iris data, which is built into R.

First, let’s explore fit indices for each of the three models with between one and six mixture components. Like MCLUST does in its built-in function, we can plot these values of the Bayesian Information Criteria (BIC).

```
library(tidyverse, warn.conflicts = FALSE)
library(mclust)
library(hrbrthemes)
data(iris)
df <- select(iris, -Species)
explore_model_fit <- function(df, n_profiles_range = 1:9, model_names = c("EII", "VVI", "EEE", "VVV")) {
x <- mclustBIC(df, G = n_profiles_range, modelNames = model_names)
y <- x %>%
as.data.frame.matrix() %>%
rownames_to_column("n_profiles") %>%
rename(`Constrained variance, fixed covariance` = EII,
`Freed variance, fixed covariance` = VVI,
`Constrained variance, constrained covariance` = EEE,
`Freed variance, freed covariance` = VVV)
y
}
fit_output <- explore_model_fit(df, n_profiles_range = 1:6)
to_plot <- fit_output %>%
gather(`Covariance matrix structure`, val, -n_profiles) %>%
mutate(`Covariance matrix structure` = as.factor(`Covariance matrix structure`),
val = abs(val)) # this is to make the BIC values positive (to align with more common formula / interpretation of BIC)
library(forcats)
to_plot$`Covariance matrix structure` <- fct_relevel(to_plot$`Covariance matrix structure`,
"Constrained variance, fixed covariance",
"Freed variance, fixed covariance",
"Constrained variance, constrained covariance",
"Freed variance, freed covariance")
ggplot(to_plot, aes(x = n_profiles, y = val, color = `Covariance matrix structure`, group = `Covariance matrix structure`)) +
geom_line() +
geom_point() +
ylab("BIC (smaller value is better)") +
theme_ipsum_rc()
```

From red to purple, the models become less constrained (more free). It appears that a *two* or *three* profile (mixture component) model with freely-estimated residual variances and covariances, or a *four* profile model with constrained residual covariances and variances, fit best (based on interpreting the BIC).

Given this, we can fit (and inspect) a model, say, the three profile model with freely-estimated residual variance and covariances.

```
create_profiles_mclust <- function(df,
n_profiles,
variance_structure = "freed",
covariance_structure = "freed"){
if (variance_structure == "constrained" & covariance_structure == "fixed") {
model_name <- "EEI"
} else if (variance_structure == "freed" & covariance_structure == "fixed") {
model_name <- "VVI"
} else if (variance_structure == "constrained" & covariance_structure == "constrained") {
model_name <- "EEE"
} else if (variance_structure == "freed" & covariance_structure == "freed") {
model_name <- "VVV"
} else if (variance_structure == "fixed") {
stop("variance_structure cannot equal 'fixed' using this function; change this to 'constrained' or 'freed' or try one of the models from mclust::Mclust()")
}
x <- Mclust(df, G = n_profiles, modelNames = model_name)
print(summary(x))
dff <- bind_cols(df, classification = x$classification)
proc_df <- dff %>%
mutate_at(vars(-classification), scale) %>%
group_by(classification) %>%
summarize_all(funs(mean)) %>%
mutate(classification = paste0("Profile ", 1:n_profiles)) %>%
mutate_at(vars(-classification), function(x) round(x, 3)) %>%
rename(profile = classification)
return(proc_df)
}
m3 <- create_profiles_mclust(df, 3, variance_structure = "freed", covariance_structure = "freed")
```

```
## ----------------------------------------------------
## Gaussian finite mixture model fitted by EM algorithm
## ----------------------------------------------------
##
## Mclust VVV (ellipsoidal, varying volume, shape, and orientation) model with 3 components:
##
## log.likelihood n df BIC ICL
## -180.186 150 44 -580.8399 -584.0537
##
## Clustering table:
## 1 2 3
## 50 45 55
```

We can then plot the mean values for the variables used to estimate the model for each of the two profiles. Of course, there are other models that we may want to inspect with different covariance matrix structures or profile numbers.

```
m3 %>%
gather(key, val, -profile) %>%
ggplot(aes(x = profile, y = val, fill = key, group = key)) +
geom_col(position = "dodge") +
ylab("Z-score") +
xlab("") +
scale_fill_discrete("") +
theme_ipsum_rc()
```

One big question: Are the residual covariance structures correctly specified? There are a lot of possible specifications (see help file here). I think they are right, based on their definitions, inspecting their covariance matrices, and inspecting their plots. But they might not be.

Please send me a message if you have any comments.

A shout-out to my colleagues You-Kyung Lee and Kristy Robinson for helping me begin to understand LPA.