---
title: "Introduction to tidyLPA"
author: "Joshua M. Rosenberg"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Introduction to tidyLPA}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
  % \VignetteDepends{dplyr}
---
  
```{r, echo = FALSE}
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
```

Latent Profile Analysis (LPA) is a statistical modeling approach for estimating distinct profiles, or groups, of variables. In the social sciences and in educational research, these profiles could represent, for example, how different youth experience dimensions of being engaged (i.e., cognitively, behaviorally, and affectively) at the same time. 

tidyLPA provides the functionality to carry out LPA in R. In particular, tidyLPA provides functionality to specify different models that determine whether and how different parameters (i.e., means, variances, and covariances) are estimated and to specify (and compare solutions for) the number of profiles to estimate.

This introduction to tidyLPA vignette is an overview of LPA and the tidyLPA package. This vignette covers the following topics:

1. Background on Latent Profile Analysis
1. Description of the goals of tidyLPA
1. Software approach to carrying out LPA: Interface to mclust (and to MPlus)
1. An example
1. More information on model specification
1. Other functionality
1. Conclusion

# Background on Latent Profile Analysis (LPA)

Latent Profile Analysis (LPA) is a statistical modeling approach for estimating distinct profiles of variables. In the social sciences and in educational research, these profiles could represent, for example, how different youth experience dimensions of being engaged (i.e., cognitively, behaviorally, and affectively) at the same time. Note that LPA works best with continuous variables (and, in some cases, ordinal variables), but is not appropriate for dichotomous (binary) variables.

Many analysts have carried out LPA using a latent variable modeling approach. From this approach, different parameters - means, variances, and covariances - are freely estimated across profiles, fixed to be the same across profiles, or constrained to be zero. The MPlus software is commonly used to estimate these models (see [here](https://www.statmodel.com/examples/mixture.shtml)) using the expectation-maximization (EM) algorithm to obtain the maximum likelihood estimates for the parameters. 

Different *models* (or how or whether parameters are estimated) can be specified and estimated. While MPlus is widely-used (and powerful), it is costly, closed-source, and can be difficult to use, particularly with respect to interpreting or using the output of specified models as part of a reproducible workflow. 

# Description of the goals of tidyLPA

The goal of tidyLPA is to make it easy to carry out LPA using R. In particular, tidyLPA provides an interface to the powerful and widely-used [mclust](https://www.stat.washington.edu/mclust/) package for Gaussian Mixture Modeling. This means that tidyLPA does not contain code to carry out LPA directly, but rather provides "wrappers" to mclust functions that make them easier to use. The primary contributions of tidyLPA are to:

1. Provide functionality to specify models that are common to LPA
2. Make it easier to use the output in subsequent analysis through a ["tidy" interface](https://CRAN.R-project.org/package=tidyverse/vignettes/manifesto.html), in that:
- input and output are both a `data.frame` (specifically its modified version, a `tibble`) that can be used to create plots or can be used in subsequent analyses
- uses the "pipe" operator, `%>%` to compose functions
- being designed and documented to be easy to use, especially for beginners (but also to provide options for finer-grained choices for estimating the model and for viewing more specific forms of the LPA output)

## Software approach to carrying out LPA: Interface to mclust (and to MPlus)

In the open-source R software, there is not yet a tool to easily carry out LPA, though there are many tools that one could use to. For example, the [R version of OpenMx](https://openmx.ssri.psu.edu/) can be used for this purpose (and to specify almost any model possible to specify within a latent variable modeling approach). However, while OpenMx is very flexible, it can also be challenging to use. 

Other tools in R allow for estimating Gaussian mixture models, or models of multivariate Gaussian (or normal) distributions. In this framework, the term "mixture component" has a similar meaning to a profile. While much more constraining than the latent variable modeling framework, the approach is often similar or the same: the EM algorithm is used to (aim to) obtain the maximum likelihood estimates for the parameters being estimated. Like in the latent variable modeling framework, different models can be specified. 

In addition to following the same general approach, using tools that are designed for Gaussian mixture modeling have other benefits, some efficiency-related (see [RMixMod](https://cran.r-project.org/package=Rmixmod), which uses compiled C++ code) and others in terms of ease-of-use (i.e., the plot methods built-in to RMixMod, mclust, and other tools). However, they also have some drawbacks, in that it can be difficult to translate between the model specifications, which are often described in terms of the geometric properties of the multivariate distributions being estimated (i.e., "spherical, equal volume"), rather than in terms of whether and how the means, variances, and covariances are estimated. They also may use different default settings (than those encountered in MPlus) in terms of the EM algorithm, which can make comparing results across tools challenging.

This package focuses on models that are commonly specified as part of LPA. Because MPlus is so widely-used, it can be helpful to compare output from other software to MPlus. The functions in tidyLPA that use mclust have been benchmarked to MPlus for a series of simple models (with small datasets and for models with small numbers of profiles. This [becnhmarking vignette](https://data-edu.github.io/tidyLPA/articles/benchmarking-mclust-and-mplus.html) contains information on how mclust and Mplus compare. As long as you have purchased MPlus (and installed MplusAutomation), this vignette can be used to replicate all of the results for the benchmark. Note that most of the output is identical, though there are some differences in the hundredths decimal places for some. Because of differences in settings for the EM algorithm and particularly for the start values (random starts for MPlus and starting values from hierarchical clustering for mclust), differences may be expected for more complex data and models. An important direction for the development of tidyLPA (the functions that use mclust) is to continue to understand when and why the output differs from MPlus output. Note that tidyLPA also provides functions to interface to MPlus, though these are not the focus of the package, as they require MPlus to be purchased and installed in order to be used. 

# Example

## Installation

You can install tidyLPA from CRAN with:

```{r, eval = FALSE}
install.packages("tidyLPA")
```

You can also install the development version of tidyLPA from GitHub with:

```{r gh-installation, eval = FALSE}
install.packages("devtools")
devtools::install_github("data-edu/tidyLPA")
```

## Mclust

Here is a brief example using the built-in `pisaUSA15` data set and variables for broad interest, enjoyment, and self-efficacy. Note that we first type the name of the data frame, followed by the unquoted names of the variables used to create the profiles. We also specify the number of profiles and the model. See `?estimate_profiles` for more details.

In these examples, we pass the results of one function to the next by *piping* (using the `%>%` operator, loaded from the `dplyr` package). We pass the data to a function that selects relevant variables, and then to `estimate_profiles`:

```{r, message = F}
library(tidyLPA)
library(dplyr)
```

```{r}
pisaUSA15[1:100, ] %>%
    select(broad_interest, enjoyment, self_efficacy) %>%
    single_imputation() %>%
    estimate_profiles(3)
```

## Mplus

We can use Mplus simply by changing the package argument for `estimate_profiles()` (not run):

```{r, eval = FALSE}
pisaUSA15[1:100, ] %>%
    select(broad_interest, enjoyment, self_efficacy) %>%
    single_imputation() %>%
    estimate_profiles(3, package = "MplusAutomation")
```

A simple summary of the analysis is printed to the console (and its posterior probability). The resulting object can be further passed down a pipeline to other functions, such as `plot`, `compare_solutions`, `get_data`, `get_fit`, etc. This is the "tidy" part, in that the function can be embedded in a tidy analysis pipeline.

If you have Mplus installed, you can call the version of this function that uses MPlus in the same way, by adding the argument `package = "MplusAutomation`.

We can plot the profiles by piping the output to `plot_profiles()`:

```{r, eval = TRUE}
pisaUSA15[1:100, ] %>%
    select(broad_interest, enjoyment, self_efficacy) %>%
    single_imputation() %>%
    scale() %>%
    estimate_profiles(3) %>% 
    plot_profiles()
```

## Comparing a wide range of solutions

The function `compare_solutions()` compares the fit of several estimated models, with varying numbers of profiles and model specifications:

```{r, eval = TRUE}
pisaUSA15[1:100, ] %>%
    select(broad_interest, enjoyment, self_efficacy) %>%
    single_imputation() %>%
    estimate_profiles(1:3, 
                      variances = c("equal", "varying"),
                      covariances = c("zero", "varying")) %>%
    compare_solutions(statistics = c("AIC", "BIC"))
```

## Passing additional arguments

Additional arguments can be passed as follows.

For MPlus (here, the additional argument is `ANALYSIS`; see:

```{r, eval = FALSE}
pisaUSA15[1:100, ] %>%
    select(broad_interest, enjoyment, self_efficacy) %>%
    single_imputation() %>% 
    estimate_profiles(3, 
                      package = "mplus", 
                      ANALYSIS = "starts = 100, 20;")
```

For mclust (here, the additional argument is to `prior`; see `?mclust::mclust` for other options):

```{r, eval = FALSE}
pisaUSA15[1:100, ] %>%
    select(broad_interest, enjoyment, self_efficacy) %>%
    single_imputation() %>% 
    estimate_profiles(3,
                      prior = priorControl())
```

# More information on model specifications

## Model specification

In addition to the number of profiles (specified with the `n_profiles` argument), the model can be specified in terms of whether and how the variable variances and covariances are estimated.

The models are specified by passing arguments to the `variance` and `covariance` arguments. The possible values for these arguments are:

- `variances`: "equal" and "varying"
- `covariances`: "varying", "equal", and "zero"

If no values are specified for these, then the variances are constrained to be equal across classes, and covariances are fixed to 0 (conditional independence of the indicators).

These arguments allow for four models to be specified:

* Equal variances and covariances fixed to 0 (Model 1)
* Varying variances and covariances fixed to 0 (Model 2)
* Equal variances and equal covariances (Model 3)
* Varying variances and varying covariances (Model 6)

Two additional models (Models 4 and 5) can be fit using MPlus. More information on the models can be found in the [vignette](https://data-edu.github.io/tidyLPA/articles/Introduction_to_tidyLPA.html).

Here is an example of specifying a model with varying variances and covariances (Model 6):

```{r, eval = TRUE}
pisaUSA15[1:100, ] %>%
    select(broad_interest, enjoyment, self_efficacy) %>%
    estimate_profiles(3, 
                      variances = "varying",
                      covariances = "varying")
```

In general, the approach to choosing the model is similar to choosing the number of profiles, requiring **deciding on the basis of evidence from multiple sources**, including information criteria, statistical tests, and concerns of interpretability and parsimony. The article by [Pastor and colleagues (2007)](https://www.sciencedirect.com/science/article/pii/S0361476X06000543) has helpful information on the model specifications. Here, the six models that are possible to specify in LPA are described in terms of how the variables used to create the profiles are estimated. 

Note that *p* represents different profiles and each parameterization is represented by a 4 x 4 covariance matrix and therefore would represent the parameterization for a four-profile solution. In all of the models, the means are estimated freely in the different profiles. Imagine that each row and column represents a different variable, i.e., the first row (and column) represents broad interest, the second enjoyment, the third self-efficacy, and the fourth another variable, i.e., future goals and plans.

### 1. Equal variances, and covariances fixed to 0 (model 1)

In this model, which corresponds to the mclust model wit the name "EEI", the variances are estimated to be equal across profiles, indicated by the absence of a p subscript for any of the diagonal elements of the matrix. The covariances are constrained to be zero, as indicated by the 0's between every combination of the variables. 

It is specified with `variances = "equal"` and `covariances = "zero"`. 

This model is highly constrained but also parsimonious: the profiles are estimated in such a way that the variables' variances are identical for each of the profiles, and the relationships between the variables are not estimated. In this way, less degrees of freedom are taken used to explain the observations that make up the data. However, estimating more parameters--as in the other models--may better explain the data, justifying the addition in complexity that their addition involves (and their reduction in degrees of freedom). This model is sometimes referred to as a *class-invariant* parameterization.

$$
\left[ \begin{matrix} { \sigma  }_{ 1 }^{ 2 } & 0 & 0 & 0 \\ 0 & { \sigma  }_{ 2 }^{ 2 } & 0 & 0 \\ 0 & 0 & { \sigma  }_{ 3 }^{ 2 } & 0 \\ 0 & 0 & 0 & { \sigma  }_{ 4 }^{ 2 } \end{matrix} \right] 
$$

### 2. Varying variances and covariances fixed to 0 (model 2)

This model corresponds to the mclust model "VVI" and allows for the variances to be freely estimated across profiles. The covariances are constrained to zero. 

It is specified with `variances = "varying"` and `covariances = "zero"`. 

Thus, it is more flexible (and less parsimonious) than model 1, but in terms of the covariances, is more constrained than model 2. This model is sometimes referred to as a *class-varying diagonal* parameterization.

$$ 
\left[ \begin{matrix} { \sigma  }_{ 1p }^{ 2 } & 0 & 0 & 0 \\ 0 & { \sigma  }_{ 2p }^{ 2 } & 0 & 0 \\ 0 & 0 & { \sigma  }_{ 3p }^{ 2 } & 0 \\ 0 & 0 & 0 & { \sigma  }_{ 4p }^{ 2 } \end{matrix} \right] 
$$


### 3. Equal variances and equal covariances (model 3)

This model corresponds to the mclust model "EEE". In this model, the variances are still constrained to be the same across the profiles, although now the covariances are estimated (but like the variances, are constrained to be the same across profiles). 

It is specified with `variances = "equal"` and `covariances = "equal"`. 

Thus, this model is the first to estimate the covariance (or correlations) of the variables used to create the profiles, thus adding more information that can be used to better understand the characteristics of the profiles (and, potentially, better explain the data). This model is sometimes referred to as a *class-invariant unrestricted* parameterization.

$$
\left[ \begin{matrix} { \sigma  }_{ 1 }^{ 2 } & { \sigma  }_{ 21 } & { \sigma  }_{ 31 } & { \sigma  }_{ 41 } \\ { \sigma  }_{ 12 } & { \sigma  }_{ 2 }^{ 2 } & { \sigma  }_{ 23 } & { \sigma  }_{ 24 } \\ { \sigma  }_{ 13 } & { \sigma  }_{ 12 } & { \sigma  }_{ 3 }^{ 2 } & { \sigma  }_{ 33 } \\ { \sigma  }_{ 14 } & { \sigma  }_{ 12 } & { \sigma  }_{ 12 } & { \sigma  }_{ 4 }^{ 2 } \end{matrix} \right] 
$$


### 4. Varying means, varying variances, and equal covariances (model 4)

This model, which specifies for the variances to be freely estimated across the profiles and for the covariances to be estimated to be equal across profiles, extends model 3. 

It is specified with `variances = "varying"` and `covariances = "equal"`. 

Unfortunately, this model cannot be specified with mclust, though it can be with MPlus; this model *can* be used with the functions to interface to MPlus described below.

$$
\left[ \begin{matrix} { \sigma  }_{ 1p }^{ 2 } & { \sigma  }_{ 21 } & { \sigma  }_{ 31 } & { \sigma  }_{ 41 } \\ { \sigma  }_{ 12 } & { \sigma  }_{ 2p }^{ 2 } & { \sigma  }_{ 23 } & { \sigma  }_{ 24 } \\ { \sigma  }_{ 13 } & { \sigma  }_{ 12 } & { \sigma  }_{ 3p }^{ 2 } & { \sigma  }_{ 33 } \\ { \sigma  }_{ 14 } & { \sigma  }_{ 12 } & { \sigma  }_{ 12 } & { \sigma  }_{ 4p }^{ 2 } \end{matrix} \right] 
$$

### 5. Varying means, equal variances, and varying covariances (model 5)

This model specifies the variances to be equal across the profiles, but allows the covariances to be freely estimated across the profiles. 

It is specified with `variances = "equal"` and `covariances = "varying"`. 

Like model 4, this model cannot be specified with mclust, though it can be with MPlus. Again, this model *can* be used with the functions to interface to MPlus described below.

$$
\left[ \begin{matrix} { \sigma  }_{ 1 }^{ 2 } & { \sigma  }_{ 21p } & { \sigma  }_{ 31p } & { \sigma  }_{ 41p } \\ { \sigma  }_{ 12p } & { \sigma  }_{ 2 }^{ 2 } & { \sigma  }_{ 23p } & { \sigma  }_{ 24p } \\ { \sigma  }_{ 13p } & { \sigma  }_{ 12p } & { \sigma  }_{ 3 }^{ 2 } & { \sigma  }_{ 33p } \\ { \sigma  }_{ 14p } & { \sigma  }_{ 12p } & { \sigma  }_{ 12p } & { \sigma  }_{ 4 }^{ 2 } \end{matrix} \right] \quad 
$$

### 6. Varying variances and varying covariances (model 6)

This model corresponds to the mclust model "VVV". It allows the variances and the covariances to be freely estimated across profiles. 

It is specified with `variances = "varying"` and `covariances = "varying"`. 

Thus, it is the most complex model, with the potential to allow for understanding many aspects of the variables that are used to estimate the profiles and how they are related. However, it is less parsimonious than all of the other models, and the added parameters should be considered in light of how preferred this model is relative to those with more simple specifications. This model is sometimes referred to as a *class-varying unrestricted* parameterization.

$$
\left[ \begin{matrix} { \sigma  }_{ 1p }^{ 2 } & { \sigma  }_{ 21p } & { \sigma  }_{ 31p } & { \sigma  }_{ 41p } \\ { \sigma  }_{ 12p } & { \sigma  }_{ 2p }^{ 2 } & { \sigma  }_{ 23p } & { \sigma  }_{ 24p } \\ { \sigma  }_{ 13p } & { \sigma  }_{ 12p } & { \sigma  }_{ 3p }^{ 2 } & { \sigma  }_{ 33p } \\ { \sigma  }_{ 14p } & { \sigma  }_{ 12p } & { \sigma  }_{ 12p } & { \sigma  }_{ 4p }^{ 2 } \end{matrix} \right] 
$$

## Other functionality 

### Getting estimates

There is a lot of output that is possible to obtain from the `estimate_profiles()` function - much more than a tidy data frame, which is the default. The easiest way to access it is by using the `get_estimates()` function.

```{r, eval = TRUE}
m3 <- pisaUSA15[1:100, ] %>%
    select(broad_interest, enjoyment, self_efficacy) %>%
    estimate_profiles(3)

get_estimates(m3)
```

Other options include how the raw data is processed. 

We can center or scale the data before estimating the profiles with the `scale()` or `poms()` functions:

```{r, eval = TRUE}
pisaUSA15[1:100, ] %>%
    select(broad_interest, enjoyment, self_efficacy) %>%
    scale() %>%
    estimate_profiles(4) %>%
    plot_profiles()

pisaUSA15[1:100, ] %>%
    select(broad_interest, enjoyment, self_efficacy) %>%
    poms() %>%
    estimate_profiles(4) %>%
    plot_profiles()
```

## Getting data

Since we often wish to use the estimated profiles in subsequent analyses, we may want the original `data.frame`, with variables that are predictors or outcomes of the profiles, included. Here, we created profiles with just two of the three variables, to demonstrate how the third variable is still returned in the output. We can return this `data.frame`, and not just one with the variables used to create the profiles and the profile assignments (and posterior probabilities), using the function `get_data()`:

```{r, eval = TRUE}
get_data(m3)
```

We note that *if more than one model is fit at once, then the data is returned not in the (wide) format above, but in long form*, e.g.:

```{r}
m4 <- pisaUSA15[1:100, ] %>%
    select(broad_interest, enjoyment, self_efficacy) %>%
    single_imputation() %>% 
    estimate_profiles(c(3, 4))

get_data(m4)
```

## Getting fit statistics 

If we wish to work with the fit statistics more programatically, we can use the `get_fit()` function:

```{r}
get_fit(m4)
```

Descriptions of these fit indices follows:

* LogLik: Log-likelihood of the data, given the model.  
* AIC: Aikake information criterion; based on -2 log-likelihood, and penalized by number of parameters.  
* AWE: Approximate weight of evidence; combines information on model fit and on classification errors (Celeux et al., 1997).  
* BIC: Bayesian information criterion; based on -2 log-likelihood, and penalized by number of parameters adjusted by sample size.
* CAIC: Consistent Aikake information criterion; based on -2 log-likelihood, and penalized by number of parameters adjusted by sample size.  
* CLC: Classification Likelihood Criterion; based on -2 log-likelihood, and penalized by the entropy (Biernacki, 1997).  
* KIC: Kullback information criterion; based on -2 log-likelihood, and penalized by 3 times the number of parameters -1 (Cavanaugh, 1999).  
* SABIC: Sample size-adjusted Bayesian information criterion (Sclove, 1987).  
* ICL: Integrated completed likelihood (Biernacki, Celeux, & Govaert, 2000).  
* Entropy: A measure of classification uncertainty, reverse-coded so that 1 reflects complete certainty of classification, and 0 complete uncertainty (see Celeux & Soromenho, 1996).  
* Prob. Min.: Minimum of the diagonal of the average latent class probabilities for most likely class membership, by assigned class. The minimum should be as high as possible, reflecting greater classification certainty (cases are assigned to classes they have a high probability of belonging to; see Jung & Wickrama, 2008).  
* Prob. Max.: Maximum of the diagonal of the average latent class probabilities for most likely class membership, by assigned class. The maximum should also be as high as possible, reflecting greater classification certainty (cases are assigned to classes they have a high probability of belonging to).  
* N Min.: Proportion of the sample assigned to the smallest class (based on most likely class membership).  
* N Max.: Proportion of the sample assigned to the largest class (based on most likely class membership).  
* BLRT: bootstrapped likelihood test.  
* BLRT p-value: p-value for the bootstrapped likelihood ratio test.  

**Notes**

This is related to [prcr](https://github.com/jrosen48/prcr), for use of two-step cluster analysis to carry out person-oriented analyses.

To contribute, file issues via GitHub [here](https://github.com/data-edu/tidyLPA/issues) or get in touch [via email](mailto:jrosen@msu.edu) or [Twitter](https://twitter.com/jrosenberg6432).

**References**

Pastor, D. A., Barron, K. E., Miller, B. J., & Davis, S. L. (2007). A latent profile analysis of college students’ achievement goal orientation. *Contemporary Educational Psychology, 32*(1), 8-47. (https://www.sciencedirect.com/science/article/pii/S0361476X06000543

**Helpful resources**

* [Hennig et al's (2015)](https://www.amazon.com/Handbook-Cluster-Analysis-Handbooks-Statistical/dp/1466551887/ref=sr_1_1?ie=UTF8&qid=1517773186&sr=8-1&keywords=cluster+analysis+handbook) handbook for an overview of mixture models, of which LPA is often considered an instance of.

* [Collins and Lanza (2013)](https://books.Google.com/books?hl=en&lr=&id=gPJQWKsgh3YC&oi=fnd&pg=PT12&dq=collins+lanza&ots=_0L9qnxxun&sig=Vx9RhJgIv0zbttIgvYLxaUQwtFI#v=onepage&q=collins%20lanza&f=false) for a book on the related approach (for use with dichotomous, rather than continuous variables used to create the profiles) Latent Class Analysis (LCA)

**How to cite tidyLPA**

> Rosenberg, J. M., Beymer, P. N., Anderson, D. J., Van Lissa, C. J., & Schmidt, J. A. (2018). tidyLPA: An R Package to Easily Carry Out Latent Profile Analysis (LPA) Using Open-Source or Commercial Software. *Journal of Open Source Software, 3*(30), 978, https://doi.org/10.21105/joss.00978

You can also cite the most latest version with the following citation:

> Rosenberg, J. M., van Lissa, C. J.,  Beymer, P. N., Anderson, D. J., Schell, M. J. & Schmidt, J. A. (2019). tidyLPA: Easily carry out Latent Profile Analysis (LPA) using open-source or commercial software [R package]. https://data-edu.github.io/tidyLPA/

## Acknowledgments

This material is based upon work supported by the National Science Foundation under *Grant No.: DRL#1661064*. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not reflect the views of the National Science Foundation.