Also known as feature selection in machine learning, the goal of variable selection is to identify a subset of predictors to simplify models. This can benefit model interpretation, shorten fitting time, and improve generalization (by reducing overfitting).

There are many different methods. The appropriate method for a given problem will depend on the model type, the data, the objective, and the theoretical rationale.

The parameters package implements a helper that will automatically pick a method deemed appropriate for the provided model, run the variables selection and return the optimal formula, which you can then re-use to update the model.

## Simple linear regression

### Fit a powerful model

If you are familiar with R and the formula interface, you know of the possibility of including a dot (.) in the formula, signifying “all the remaining variables”. Curiously, few are aware of the possibility of additionally easily adding “all the interaction terms”. This can be achieved using the .*. notation.

Let’s try that with the linear regression predicting Sepal.Length with the iris dataset, included by default in R.

model <- lm(Sepal.Length ~ . * ., data = iris)
summary(model)
#>
#> Call:
#> lm(formula = Sepal.Length ~ . * ., data = iris)
#>
#> Residuals:
#>    Min     1Q Median     3Q    Max
#> -0.726 -0.210  0.014  0.213  0.713
#>
#> Coefficients:
#>                                Estimate Std. Error t value Pr(>|t|)
#> (Intercept)                      1.6998     1.0576    1.61   0.1103
#> Sepal.Width                      0.8301     0.3047    2.72   0.0073 **
#> Petal.Length                     0.3178     0.7852    0.40   0.6863
#> Petal.Width                      2.5827     1.5874    1.63   0.1061
#> Speciesversicolor               -2.7193     1.6201   -1.68   0.0956 .
#> Speciesvirginica                -6.1704     3.2045   -1.93   0.0563 .
#> Sepal.Width:Petal.Length        -0.0149     0.2185   -0.07   0.9458
#> Sepal.Width:Petal.Width         -0.6112     0.4536   -1.35   0.1801
#> Sepal.Width:Speciesversicolor    0.4300     0.6666    0.65   0.5200
#> Sepal.Width:Speciesvirginica     0.8288     1.0031    0.83   0.4101
#> Petal.Length:Petal.Width        -0.1195     0.3300   -0.36   0.7178
#> Petal.Length:Speciesversicolor   0.7398     0.5166    1.43   0.1544
#> Petal.Length:Speciesvirginica    0.9034     0.6938    1.30   0.1951
#> Petal.Width:Speciesversicolor   -1.0070     1.2454   -0.81   0.4202
#> Petal.Width:Speciesvirginica    -0.2864     1.5065   -0.19   0.8495
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.3 on 135 degrees of freedom
#> Multiple R-squared:  0.882,  Adjusted R-squared:  0.87
#> F-statistic: 72.1 on 14 and 135 DF,  p-value: <2e-16

Wow, that’s a lot of parameters! And almost none of them are significant!

Which is weird, considering that gorgeous $$R^2$$ of 0.882!

I wish I had that in my research!

### Too many parameters?

As you might know, having a model that is too performant is not always a good thing. For instance, it can be a marker of overfitting: the model corresponds too closely to a particular set of data, and may therefore fail to predict future observations reliably. In multiple regressions, in can also fall under the Freedman’s paradox: some predictors that have actually no relation to the dependent variable being predicted will be spuriously found to be statistically significant.

Let’s run a few checks using the performance package:

library(performance)

check_normality(model)
#> OK: residuals appear as normally distributed (p = 0.612).
check_heteroscedasticity(model)
#> OK: Error variance appears to be homoscedastic (p = 0.118).
check_autocorrelation(model)
#> OK: Residuals appear to be independent and not autocorrelated (p = 0.574).
check_collinearity(model)
#> # Check for Multicollinearity
#>
#> High Correlation
#>
#>                      Term      VIF           VIF 95% CI Increased SE Tolerance
#>               Sepal.Width    29.42 [   22.31,    38.91]         5.42      0.03
#>              Petal.Length  3204.78 [ 2414.74,  4253.41]        56.61  3.12e-04
#>               Petal.Width  2442.10 [ 1840.11,  3241.14]        49.42  4.09e-04
#>                   Species 3.98e+05 [3.00e+05, 5.28e+05]       630.97  2.51e-06
#>  Sepal.Width:Petal.Length  2183.98 [ 1645.63,  2898.55]        46.73  4.58e-04
#>   Sepal.Width:Petal.Width  1866.48 [ 1406.42,  2477.15]        43.20  5.36e-04
#>       Sepal.Width:Species 3.49e+05 [2.63e+05, 4.64e+05]       591.13  2.86e-06
#>  Petal.Length:Petal.Width  4032.80 [ 3038.60,  5352.41]        63.50  2.48e-04
#>      Petal.Length:Species 1.23e+06 [9.27e+05, 1.63e+06]      1109.09  8.13e-07
#>       Petal.Width:Species 3.77e+05 [2.84e+05, 5.01e+05]       614.38  2.65e-06
#>  Tolerance 95% CI
#>      [0.03, 0.04]
#>      [0.00, 0.00]
#>      [0.00, 0.00]
#>      [0.00, 0.00]
#>      [0.00, 0.00]
#>      [0.00, 0.00]
#>      [0.00, 0.00]
#>      [0.00, 0.00]
#>      [0.00, 0.00]
#>      [0.00, 0.00]

The main issue of the model seems to be the high multicollinearity. This suggests that our model might not be able to give valid results about any individual predictor, nor tell which predictors are redundant with respect to others.

### Parameters selection

Time to do some variables selection! This can be easily done using the select_parameters() function in parameters. It will automatically select the best variables and update the model accordingly. One way of using that is in a tidy pipeline (using %>%), using this output to update a new model.

lm(Sepal.Length ~ . * ., data = iris) %>%
select_parameters() %>%
summary()
#>
#> Call:
#> lm(formula = Sepal.Length ~ Sepal.Width + Petal.Length + Petal.Width +
#>     Species + Sepal.Width:Petal.Width + Petal.Length:Species +
#>     Petal.Width:Species, data = iris)
#>
#> Residuals:
#>     Min      1Q  Median      3Q     Max
#> -0.7261 -0.2165  0.0021  0.2191  0.7439
#>
#> Coefficients:
#>                                Estimate Std. Error t value Pr(>|t|)
#> (Intercept)                       2.090      0.528    3.95  0.00012 ***
#> Sepal.Width                       0.734      0.130    5.66  8.3e-08 ***
#> Petal.Length                      0.232      0.260    0.89  0.37310
#> Petal.Width                       1.051      0.532    1.98  0.04993 *
#> Speciesversicolor                -1.047      0.547   -1.92  0.05754 .
#> Speciesvirginica                 -2.682      0.638   -4.21  4.6e-05 ***
#> Sepal.Width:Petal.Width          -0.232      0.103   -2.24  0.02667 *
#> Petal.Length:Speciesversicolor    0.660      0.298    2.22  0.02837 *
#> Petal.Length:Speciesvirginica     0.720      0.273    2.63  0.00941 **
#> Petal.Width:Speciesversicolor    -1.112      0.550   -2.02  0.04528 *
#> Petal.Width:Speciesvirginica     -0.499      0.460   -1.09  0.27934
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.3 on 139 degrees of freedom
#> Multiple R-squared:  0.88,   Adjusted R-squared:  0.872
#> F-statistic:  102 on 10 and 139 DF,  p-value: <2e-16

That’s still a lot of parameters, but as you can see, almost all of them are now significant, and the $$R^2$$ did not change much.

Although appealing, please note that these automated selection methods are quite criticized, and should not be used in place of theoretical or hypothetical reasons (i.e., you should have a priori hypotheses about which parameters of your model you want to focus on).

## Mixed models

For simple linear regressions as above, the selection is made using the step() function (available in base R). This performs a stepwise selection. However, this procedures is not available for other types of models, such as mixed models.

### Mixed models

For mixed models (of class merMod), stepwise selection is based on cAIC4::stepcAIC(). This step function only searches the “best” model based on the random effects structure, i.e. select_parameters() adds or excludes random effects until the cAIC can’t be improved further.

This is what our initial model looks like.

library(lme4)
data("qol_cancer")

# initial model
lmer(
QoL ~ time + phq4 + age + (1 + time | hospital / ID),
data = qol_cancer
) %>%
summary()
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: QoL ~ time + phq4 + age + (1 + time | hospital/ID)
#>    Data: qol_cancer
#>
#> REML criterion at convergence: 4647
#>
#> Scaled residuals:
#>    Min     1Q Median     3Q    Max
#> -3.581 -0.393  0.082  0.507  2.505
#>
#> Random effects:
#>  Groups      Name        Variance Std.Dev. Corr
#>  ID:hospital (Intercept) 202.49   14.23
#>              time         12.54    3.54    -0.72
#>  hospital    (Intercept)   8.49    2.91
#>              time          1.27    1.13    -1.00
#>  Residual                143.11   11.96
#> Number of obs: 564, groups:  ID:hospital, 188; hospital, 2
#>
#> Fixed effects:
#>             Estimate Std. Error t value
#> (Intercept)  70.6498     3.1302   22.57
#> time          1.4902     1.2203    1.22
#> phq4         -4.7687     0.3235  -14.74
#> age          -0.0173     0.1889   -0.09
#>
#> Correlation of Fixed Effects:
#>      (Intr) time   phq4
#> time -0.954
#> phq4  0.014 -0.010
#> age  -0.017  0.003  0.107
#> optimizer (nloptwrap) convergence code: 0 (OK)
#> boundary (singular) fit: see help('isSingular')

This is the model selected by select_parameters(). Please notice the differences in the random effects structure between the initial and the selected models:

## TODO: this is currently broken due to an issue in package cAIC4

# multiple models are checked, however, initial models
# already seems to be the best one...
lmer(
QoL ~ time + phq4 + age + (1 + time | hospital / ID),
data = qol_cancer
) %>%
select_parameters() %>%
summary()