README.Rmd

---
output: github_document
editor_options: 
  markdown: 
    wrap: 80
---

<!-- README.md is generated from README.Rmd. Please edit that file -->

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse   = TRUE,
  comment    = "#>",
  fig.path   = "man/figures/README-",
  dpi        = 300,
  fig.width  = 10,
  fig.height = 7,
  dev        = "svg"
)

set.seed(1234)

# preload libraries here to avoid messages
library(mifa)
library(psych)
library(ggplot2)
library(tidyr)

ggplot2::theme_set(theme_minimal(base_size = 16))
```

# mifa - multiple imputation for factor analysis
<img src='man/figures/logo.png' align="right" height="139" />

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`mifa` is an R package that implements multiple imputation of covariance
matrices to allow to perform factor analysis on incomplete data. It works as
follows:

1.  Impute missing values multiple times using *Multivariate Imputation with
    Chained Equations* (MICE) from the [mice](https://amices.org/mice/) package.

2.  Combine the covariance matrices of the imputed data sets into a single
    covariance matrix using Rubin's rules<sup>^[Rubin D. B. Multiple imputation
    for nonresponse in surveys (2004). John Wiley & Sons.]</sup>

3.  Use the combined covariance matrix for exploratory factor analysis.

`mifa` also provides two types of confidence intervals for the variance
explained by different numbers of principal components: 
Fieller confidence intervals (parametric) for larger samples<sup>^[Fieller, E. 
C. (1954). Some problems in interval estimation. Journal of the Royal 
Statistical Society. Series B (Methodological): 175-185.]</sup> 
and bootstrapped confidence intervals (nonparametric) for smaller 
samples.<sup>^[Shao, J. & Sitter, R. R. (1996). Bootstrap for imputed survey 
data. Journal of the American Statistical Association 91.435 (1996): 1278-1288. doi: [10.1080/01621459.1996.10476997](https://dx.doi.org/10.1080/01621459.1996.10476997)]</sup>

**For more information about the method, see:**

Nassiri, V., Lovik, A., Molenberghs, G., Verbeke, G. (2018). On using multiple
imputation for exploratory factor analysis of incomplete data. *Behavior
Research Methods* 50, 501–517. doi:  [10.3758/s13428-017-1013-4](https://doi.org/10.3758/s13428-017-1013-4)

*Note:* The paper was accompanied by an implementation in R, and this package
emerged from it. The repository appears to have been abandoned by the authors,
but you can still find it [here](https://github.com/vahidnassiri/mifa).

## Installation

Install from CRAN with:

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

Or install the development version from [Github](https://github.com/teebusch/mifa) with:

``` r
# install.packages("devtools")
devtools::install_github("teebusch/mifa")
```

## Usage

### Example Data

For this example we use the `bfi` data set from the `psych` package. It contains
2,800 subjects' answers to 25 personality self-report items and 3 demographic
variables (sex, education, and age). Each of the 25 personality questions is
meant to tap into one of the "Big 5" personality factors, as indicated by their
names: **O**penness, **C**onscientiousness, **A**greeableness, ,
**E**xtraversion, **N**euroticism. There are missing responses for most items.
Instead of dropping the incomplete cases from the analysis, we will use `mifa`
to impute them, and then perform a factor analysis on the imputed covariance
matrix.

### Imputing the Covariance Matrix

First, we use `mifa()` to impute the covariance matrix and get an idea how many
factors we should use. We use the `cov_vars` argument to tell `mifa` to use
`gender`, `education`, and `age` for the imputations, but exclude them from the
covariance matrix:

```{r run-mifa, messages=FALSE}
library(mifa)
library(psych)

mi <- mifa(
  data      = bfi, 
  cov_vars  = -c(gender, education, age),
  n_pc      = 2:8, 
  ci        = "fieller", 
  print     = FALSE
)

mi
```

### Factor Analysis

It looks like the first 5 principal components explain more than half of the
variance in the responses, so we perform a factor analysis with 5 factors, using
the `fa()` function from the `psych` package. We can get the imputed covariance
matrix of our data from `mi$cov_combined`. From there on, it's business as
usual.

```{r message=FALSE}
fit <- fa(mi$cov_combined, n.obs = nrow(bfi), nfactors = 5)
```

The factor diagram shows that the five factors correspond nicely to the 5 types
of questions:

```{r fa-diagram, fig.height = 8}
fa.diagram(fit)
```

We can add the factor scores to the original data, in order to explore group
differences. Because we need complete data to calculate factor scores, we first
impute a single data set with mice:

```{r}
data_imp <- mice::complete(mice::mice(bfi, 1, print = FALSE))

fct_scores <- data.frame(factor.scores(data_imp[, 1:25], fit)$scores)

data_imp <- data.frame(
  Gender        = factor(data_imp$gender),
  Extraversion  = fct_scores$MR1,
  Neuroticism   = fct_scores$MR2,
  Conscientious = fct_scores$MR3,
  Openness      = fct_scores$MR4,
  Agreeableness = fct_scores$MR5
)

levels(data_imp$Gender) <- c("Male", "Female")
```

Then we can visualize the group differences:

```{r fa-group-comparison}
library(ggplot2)
library(tidyr)

data_imp2 <- tidyr::pivot_longer(data_imp, -Gender, "factor")

ggplot(data_imp2) +
  geom_density(aes(value, linetype = Gender)) +
  facet_wrap(~ factor, nrow = 2) +
  theme(legend.position = c(.9, .1))
```

## Further Reading