Also known as feature extraction or dimension reduction in machine learning, the goal of variable reduction is to reduce the number of predictors by deriving a new set of variables intended to be informative and non-redundant from a set of measured data. This method can be used to simplify models, which can benefit model interpretation, shorten fitting time, and improve generalization (by reducing overfitting).

### Quick and Exploratory Method

Let’s start by fitting a multiple linear regression model with the attitude dataset, available is base R, to predict the overall rating by employees of their organization with the remaining variables (handling of employee complaints, special privileges, opportunity of learning, raises, a feedback considered too critical and opportunity of advancement).

data("attitude")
model <- lm(rating ~ ., data = attitude)
parameters(model)
#> Parameter   | Coefficient |    SE |          95% CI | t(23) |      p
#> --------------------------------------------------------------------
#> (Intercept) |       10.79 | 11.59 | [-13.19, 34.76] |  0.93 | 0.362
#> complaints  |        0.61 |  0.16 | [  0.28,  0.95] |  3.81 | < .001
#> privileges  |       -0.07 |  0.14 | [ -0.35,  0.21] | -0.54 | 0.596
#> learning    |        0.32 |  0.17 | [ -0.03,  0.67] |  1.90 | 0.070
#> raises      |        0.08 |  0.22 | [ -0.38,  0.54] |  0.37 | 0.715
#> critical    |        0.04 |  0.15 | [ -0.27,  0.34] |  0.26 | 0.796
#> advance     |       -0.22 |  0.18 | [ -0.59,  0.15] | -1.22 | 0.236

We can explore a reduction of the number of parameters with the reduce_parameters() function.

newmodel <- reduce_parameters(model)
parameters(newmodel)
#> Parameter                                                              | Coefficient |   SE |         95% CI | t(27) |      p
#> -----------------------------------------------------------------------------------------------------------------------------
#> (Intercept)                                                            |       64.63 | 1.57 | [61.41, 67.85] | 41.19 | < .001
#> raises 0 88/learning 0 82/complaints 0 78/privileges 0 70/advance 0 68 |        4.62 | 0.90 | [ 2.78,  6.46] |  5.16 | < .001
#> critical -0 80                                                         |        3.41 | 1.59 | [ 0.14,  6.67] |  2.14 | 0.041

This output hints at the fact that the model could be represented via two “latent” dimensions, one correlated with all the positive things that a company has to offer, and the other one related to the amount of negative critiques received by the employees. These two dimensions have a positive and negative relationship with the company rating, respectively.

What does reduce_parameters() exactly do?

This function performs a reduction in the parameter space (the number of variables). It starts by creating a new set of variables, based on the chosen method (the default method is “PCA”, but other are available via the method argument, such as “cMDS”, “DRR” or “ICA”). Then, it names this new dimensions using the original variables that correlate the most with it. For instance, in the example above a variable named raises_0.88/learning_0.82/complaints_0.78/privileges_0.70/advance_0.68 means that the respective variables (raises, learning, complaints, privileges, advance) correlate maximally (with coefficients of .88, .82, .78, .70, .68, respectively) with this dimension.

reduce_parameters(model, method = "cMDS") %>%
parameters()
#> Parameter                                                 | Coefficient |   SE |         95% CI | t(26) |      p
#> ----------------------------------------------------------------------------------------------------------------
#> (Intercept)                                               |       64.63 | 1.41 | [61.73, 67.53] | 45.80 | < .001
#> raises 0 85/complaints 0 84/learning 0 83/privileges 0 74 |        0.43 | 0.07 | [ 0.28,  0.57] |  6.14 | < .001
#> advance -0 60                                             |        0.32 | 0.13 | [ 0.04,  0.59] |  2.36 | 0.026
#> critical -0 65                                            |       -0.24 | 0.15 | [-0.56,  0.07] | -1.61 | 0.120

A different method (Classical Multidimensional Scaling - cMDS) suggests that negative critiques do not have a significant impact on the rating, and that the lack of opportunities of career advancement is a separate dimension with an importance on its own.

Although reduce_parameters() function can be useful in exploratory data analysis, it’s best to perform the dimension reduction step in a separate and dedicated stage, as this is a very important process in the data analysis workflow.

### Principal Component Analysis (PCA)

PCA is a widely used procedure that lies in-between dimension reduction and structural modeling. Indeed, one of the ways of reducing the number of predictors is to extract a new set of uncorrelated variables that will represent variance of your initial dataset. But how the original variables relate between themselves can also be a question on its own.

We can apply the principal_components() function to do the the predictors of the model:

pca <- principal_components(insight::get_predictors(model), n = "auto")
pca
#>
#> Variable   | PC1  |  PC2  | Complexity
#> --------------------------------------
#> complaints | 0.78 | 0.31  |    1.31
#> privileges | 0.70 | 0.31  |    1.37
#> learning   | 0.82 | 0.22  |    1.14
#> raises     | 0.88 | -0.12 |    1.03
#> critical   | 0.40 | -0.80 |    1.47
#> advance    | 0.68 | -0.32 |    1.43
#>
#> The 2 principal components accounted for 69.59% of the total variance of the original data (PC1 = 52.82%, PC2 = 16.77%).

The principal_components() function automatically selected one component (if the number of components is not specified, this function uses n_factors() to estimate the optimal number to keep) and returned the loadings, i.e., the relationship with all of the original variables.

As we can see here, it seems that our new component captured the essence (more than half of the total variance present in the original dataset) of all our other variables together. We can extract the values of this component for each of our observation using the predict() method and add in the response variable of our initial dataset.

newdata <- predict(pca)
newdata$rating <- attitude$rating

We can know update the model with this new component:

update(model, rating ~ PC1, data = newdata) %>%
parameters()
#> Parameter   | Coefficient |   SE |         95% CI | t(28) |      p
#> ------------------------------------------------------------------
#> (Intercept) |       64.63 | 1.67 | [61.22, 68.05] | 38.78 | < .001
#> PC1         |        4.62 | 0.95 | [ 2.67,  6.57] |  4.86 | < .001

#### Using the psych package for PCA

You can also use different packages for models, such as psych (Revelle 2018) or FactoMineR for PCA or Exploratory Factor Analysis (EFA), as it allows for more flexibility and control when running such procedures.

The functions from this package are fully supported by parameters through the model_parameters() function. For instance, we can redo the above analysis using the psych package as follows:

library(psych)

# Fit the PCA
pca <- model_parameters(psych::principal(attitude, nfactors = 1))
pca
#>
#> Variable   | PC1  | Complexity | Uniqueness
#> -------------------------------------------
#> rating     | 0.80 |    1.00    |    0.37
#> complaints | 0.85 |    1.00    |    0.28
#> privileges | 0.68 |    1.00    |    0.53
#> learning   | 0.83 |    1.00    |    0.32
#> raises     | 0.86 |    1.00    |    0.26
#> critical   | 0.36 |    1.00    |    0.87
#> advance    | 0.58 |    1.00    |    0.66
#>
#> The unique principal component (varimax rotation) accounted for 53.09% of the total variance of the original data.

Note: By default, psych::principal() uses a varimax rotation to extract rotated components, possibly leading to discrepancies in the results.

Finally, refit the model:

df <- cbind(attitude, predict(pca))

update(model, rating ~ PC1, data = df) %>%
model_parameters()

## References

Revelle, William. 2018. Psych: Procedures for Psychological, Psychometric, and Personality Research. Evanston, Illinois: Northwestern University. https://CRAN.R-project.org/package=psych.