Introduction
The following is a presentation of the rvif package that
aims to facilitate its use. To this end, the document is divided into
four sections, each of them dedicated to each of the functions that make
up the package.
This package is focused on determining whether or not the degree of
approximate multicollinearity in a multiple linear regression model is
of concern, meaning that it affects the statistical analysis
(i.e. individual significance tests) of the model.
Please note that the results presented here are based on the articles
by Salmerón, García and García (2025) and Salmerón, García and García
(working paper), see bibliography at the end of the document.
We start by loading the rvif library, which in turn loads
the multiColl package.
library(rvif)
#> Cargando paquete requerido: multiColl
#> Cargando paquete requerido: car
#> Warning: package 'car' was built under R version 4.4.3
#> Cargando paquete requerido: carData
#> Warning: package 'carData' was built under R version 4.4.3
cv_vif function
Given the multiple linear regression model \(\mathbf{y} = \mathbf{X} \boldsymbol{\beta}
+ \mathbf{u},\) where the design matrix \(\mathbf{X}\) contains the observations of
each independent variable in columns, the function cv_vif
calculates the Coefficients of Variation (CV) and Variance Inflation
Factors (VIF) of each variable (column) contained in \(\mathbf{X}\).
The command and arguments of this function are as follows:
Both CVs and VIFs are calculated using existing functions in the
multiColl package. Since these functions require the intercept
of the model to be in the first column of \(\mathbf{X}\), the design function matrix
\(\mathbf{X}\) has to be entered as the
first argument of the cv_vif function so that the intercept is
in the first column.
This issue is crucial for ensuring the function works correctly.
A first example: how to use the function
To illustrate this issue and how this function works, we will first
use the Wissel data available in the rvif package. For more
details use help(Wissel).
head(Wissel, n=5)
#> t D cte C I CP
#> 1 1996 3.8051 1 4.7703 4.8786 808.23
#> 2 1997 3.9458 1 4.7784 5.0510 798.03
#> 3 1998 4.0579 1 4.9348 5.3620 806.12
#> 4 1999 4.1913 1 5.0998 5.5585 865.65
#> 5 2000 4.3585 1 5.2907 5.8425 997.30
As can be seen, the design matrix \(\mathbf{X}\) in this case is formed by the
last four columns. To calculate the CVs and VIFs of each independent
variable, the following code must be executed:
x = Wissel[,-c(1,2)]
cv_vif(x)
#> CV VIF
#> Variable 2 0.1718940 589.7540
#> Variable 3 0.2482804 281.8862
#> Variable 4 0.3607848 189.4874
An alternative approach is to obtain the design matrix of the
independent variables after specifying the corresponding model using the
lm command and the model.matrix function:
attach(Wissel)
reg_W = lm(D~C+I+CP)
detach(Wissel)
x = model.matrix(reg_W)
cv_vif(x)
#> CV VIF
#> Variable 2 0.1718940 589.7540
#> Variable 3 0.2482804 281.8862
#> Variable 4 0.3607848 189.4874
As can be seen, the same results are obtained in both cases.
According to Salmerón, Rodríguez and García (2020), the Coefficient
of Variation (CV) is useful for detecting non-essential
multicollinearity (the relationship between the intercept and at least
one of the remaining independent variables in a linear model). They
state specifically that if the CV of an independent variable is less
than 0.1002506, then it is related to the intercept and needs to be
centered to eliminate this relationship.
As can be seen from the Wissel data, the degree of non-essential
multicollinearity is not strong.
On the other hand, according to Salmerón, García and García (2018),
the VIF is only able to detect multicollinearity of the essential type
(linear relationship between at least two independent variables of the
linear model excluding the intercept). As VIF values suggest the
presence of this type of multicollinearity, the results clearly indicate
that it exists in the Wissel data model.
A second example: construction of the design matrix
Since the concept of multicollinearity only affects the independent
variables of the model, it is possible to detect multicollinearity
without considering the dependent variable.
When constructing the design matrix directly, it is important to note
that the first column must correspond to the intercept, as previously
mentioned.
The following simulation illustrates the situation:
set.seed(2025)
obs = 100
cte = rep(1, obs)
x2 = rnorm(obs, 5, 0.01) # variable with very little variability
x3 = rnorm(obs, 5, 10)
x4 = x3 + rnorm(obs, 5, 1) # fourth variable related to the third
x5 = rnorm(obs, -1, 30)
x = cbind(cte, x2, x3, x4, x5) # the first column has to be the intercept
cv_vif(x)
#> CV VIF
#> Variable 2 0.002030169 1.025399
#> Variable 3 1.886093419 100.123352
#> Variable 4 0.961634537 100.320601
#> Variable 5 10.725968638 1.025810
In this case, it is observed that the second variable causes
non-essential multicollinearity, whereas the relation between the second
and third variables cause essential multicollinearity (this is how the
simulated data has been designed).
A third example: checking some errors
In what follows, we will use the data on soil characteristics as
predictors of forest diversity from Bondell and Reich’s paper. This data
set is available in the rvif package. For more details, use
help(soil).
This data set contains a variable called Sum Cations which is the sum
of the concentrations of calcium (Ca), magnesium (Mg), potassium (K) and
sodium (Na). For this reason, the VIF calculation produces errors in
this case.
head(soil, n=5)
#> BaseSat SumCation CECbuffer Ca Mg K Na P Cu Zn
#> 1 2.34 0.1576 0.614 0.0892 0.0328 0.0256 0.010 0.000 0.080 0.184
#> 2 1.64 0.0970 0.516 0.0454 0.0218 0.0198 0.010 0.000 0.064 0.112
#> 3 5.20 0.4520 0.828 0.3306 0.0758 0.0336 0.012 0.240 0.136 0.350
#> 4 4.10 0.3054 0.698 0.2118 0.0536 0.0260 0.014 0.030 0.126 0.364
#> 5 2.70 0.2476 0.858 0.1568 0.0444 0.0304 0.016 0.384 0.078 0.376
#> Mn HumicMatter Density pH ExchAc Diversity
#> 1 3.200 0.1220 0.0822 0.516 0.466 0.2765957
#> 2 2.734 0.0952 0.0850 0.512 0.430 0.2613982
#> 3 4.148 0.1822 0.0746 0.554 0.388 0.2553191
#> 4 3.728 0.1646 0.0756 0.546 0.408 0.2401216
#> 5 4.756 0.2472 0.0692 0.450 0.624 0.1884498
x = soil[,-16]
cv_vif(x)
#> System is computationally singular. Modify the design matrix before running the code.
The second argument of this function sets the threshold used to
determine whether the system is computationally singular. Eliminating
the second variable allows the results to be obtained without any
problem.
x = soil[,-c(2,16)]
cv_vif(x)
#> CV VIF
#> Variable 2 0.26254488 65655.631992
#> Variable 3 0.93022708 49093.141906
#> Variable 4 0.56511827 2096.203209
#> Variable 5 0.24604041 78.315177
#> Variable 6 0.23242890 7.017204
#> Variable 7 1.41496855 5.451706
#> Variable 8 0.37215759 10.065882
#> Variable 9 0.42380972 8.879019
#> Variable 10 0.25216470 3.132565
#> Variable 11 0.28426257 31.738328
#> Variable 12 0.07203395 23.370488
#> Variable 13 0.06026917 22.221129
#> Variable 14 0.14073646 7609.600200
It is now observed that the design matrix has 14 variables, but the
output only provides information on 13 of these (from 2 to 14). This is
because the code is designed to eliminate the first column of the design
matrix (which corresponds to the intercept). In this case, the design
matrix entered does not have an intercept in its first column. This has
resulted in the elimination of the first variable.
This can be resolved either by entering the constant manually into
the design matrix, or by obtaining it from the specification of the
linear regression model.
y = soil[,16]
reg_S = lm(y~as.matrix(x))
cv_vif(model.matrix(reg_S))
#> CV VIF
#> Variable 2 0.52570466 196.097486
#> Variable 3 0.26254488 87577.945861
#> Variable 4 0.93022708 63996.254093
#> Variable 5 0.56511827 2478.282664
#> Variable 6 0.24604041 92.107236
#> Variable 7 0.23242890 8.320798
#> Variable 8 1.41496855 6.807994
#> Variable 9 0.37215759 11.966375
#> Variable 10 0.42380972 9.795579
#> Variable 11 0.25216470 3.476432
#> Variable 12 0.28426257 34.800524
#> Variable 13 0.07203395 26.743959
#> Variable 14 0.06026917 27.459519
#> Variable 15 0.14073646 10412.458323
It can be seen that variables 13 (pH) and 14 (ExchAc) are causing
non-essential multicollinearity, while all the other variables, except
for 7 (Na), 8 (P), 10 (Zn) and 11 (Mn), are causing essential
multicollinearity.
If the design matrix is constructed directly but the intercept is not
positioned in the first column of the design matrix \(\mathbf{X}\), the following error message
occurs:
cte = rep(1, length(y))
x = cbind(x, cte)
cv_vif(x)
#> There is a constant variable. Delete it before running the code or, if it is the intercept, it must be the first column of the design matrix.
A fourth example: the special case of the simple linear regression
model
In the special case of the simple linear regression model, the design
matrix \(\mathbf{X}\) consists of an
intercept and a second variable.
In this case, it makes no sense to calculate the VIF since it is
always equal to 1 regardless of the observed values of the second
variable (see Salmerón, Rodríguez and García (2020) for more
details).
For this reason, to execute the cv_vif function, the design
matrix (the first argument of the function) must have more than two
columns:
cte = rep(1, obs)
set.seed(2025)
x2 = rnorm(obs, 3, 4)
x = cbind(cte, x2)
cv_vif(x)
#> At least 3 independent variables are needed (including the interceptin the first column) to carry out the calculations.
Final warning regarding the presence of binary variables in the
model
The code is not designed to distinguish between quantitative and
binary independent variables, so users should be especially careful when
interpreting results involving binary variables. Note that the VIF for a
binary variable is calculated using the coefficient of determination
from a regression in which the binary variable is treated as the
dependent variable. However, the coefficient of determination is
generally considered inappropriate in such cases, which is why models
such as logit or probit are typically recommended for binary
outcomes.
The following example illustrates this:
set.seed(2025)
x3 = rbinom(obs, 1, 0.5)
x = cbind(cte, x2, x3)
head(x, n=5)
#> cte x2 x3
#> [1,] 1 5.483027 1
#> [2,] 1 3.142566 0
#> [3,] 1 6.092618 1
#> [4,] 1 8.089956 0
#> [5,] 1 4.483902 1
cv_vif(x)
#> CV VIF
#> Variable 2 1.3630182 1.066598
#> Variable 3 0.9416966 1.066598
All calculations involving a binary variable in the design matrix, X,
are performed without any problem. However, we would like to point out
to the reader that the associated VIF should not be interpreted as a
quantitative variable. We recommend ignoring it.
cv_vif_plot function
The cv_vif_plot function is intended to facilitate the
interpretation of the results provided by the cv_vif function.
For this reason, the input of this function (first argument) is the
output from the cv_vif function. Consequently, it represents
the scatter plot of the Coefficient of Variation (CV) and the Variance
Inflation Factor (VIF) for the independent variables (excluding the
intercept) of a multiple linear regression model.
The command and arguments of this function are as follows:
# cv_vif_plot(x, limit = 40)
It should be noted that the distinction between essential and
non-essential multicollinearity and the limitations of each measure (CV
and VIF) for detecting the different kinds of multicollinearity, can be
very useful to detect if there is a troubling degree of
multicollinearity, what kind of multicollinearity it is and what
variables are causing it. For this purpose, it is important to include
in the representation of the scatter plot of the CV and VIF the lines
corresponding to the established thresholds for each measure: dashed
vertical line for 0.1002506 (CV) and dotted horizontal line for 10
(VIF).
These lines determine four regions (see the following graphical
representation) which can be interpreted as follows: A: existence of
troubling non-essential and non-troubling essential multicollinearity;
B: existence of troubling essential and non-essential multicollinearity;
C: existence of non-troubling non-essential and troubling essential
multicollinearity; D: non-troubling degree of existing multicollinearity
(essential and non-essential).
plot(-2:20, -2:20, type = "n", xlab="Coefficient of Variation", ylab="Variance Inflation Factor")
abline(h=10, col="red", lwd=3, lty=2)
abline(h=0, col="black", lwd=1)
abline(v=0.1002506, col="red", lwd=3, lty=3)
text(-1.25, 2, "A", pos=3, col="blue")
text(-1.25, 12, "B", pos=3, col="blue")
text(10, 12, "C", pos=3, col="blue")
text(10, 2, "D", pos=3, col="blue")
Some examples
When the cv_vif_plot command is applied to the Wissel data,
it is clear that the three independent variables have a CV greater than
0.1002506 (dashed vertical line) and a VIF greater than 10 (dotted
horizontal line). Therefore, it can be concluded immediately that the
degree of multicollinearity of the non-essential type is not strong, but
that of the essential type is.
x = Wissel[,-c(1,2)]
cv_vif_plot(cv_vif(x))
When the cv_vif_plot function is applied to the simulation
of the second example of the cv_vif function, it can be
observed that the second variable causes non-essential multicollinearity
while the third and fourth variables cause essential
multicollinearity:
set.seed(2025)
obs = 100
cte = rep(1, obs)
x2 = rnorm(obs, 5, 0.01) # variable with very little variability
x3 = rnorm(obs, 5, 10)
x4 = x3 + rnorm(obs, 5, 1) # fourth variable related to the third
x5 = rnorm(obs, -1, 30)
x = cbind(cte, x2, x3, x4, x5) # the first column has to be the intercept
cv_vif_plot(cv_vif(x))
cv_vif_plot(cv_vif(x), limit=0) # note how the 'limit' argument works
rvifs function
In their paper, Salmerón, García and García (2025) propose a
redefinition of the variance inflation factor (VIF) by modifying the
reference orthogonal model. The redefined VIF (RVIF) is defined as
follows: \[RVIF(i) = \frac{\mathbf{X}_{i}^{t}
\mathbf{X}_{i}}{\mathbf{X}_{i}^{t} \mathbf{X}_{i} - \mathbf{X}_{i}^{t}
\mathbf{X}_{-i} \cdot \left( \mathbf{X}_{-i}^{t} \mathbf{X}_{-i} \right)
\cdot \mathbf{X}_{-i}^{t} \mathbf{X}_{i}},\] which shows, among
other questions, that the RVIF is defined for \(i=1,2,\dots,k\) (suppose the design matrix
\(\mathbf{X}\) has \(k\) independent variables (columns)). In
contrast to the VIF, the RVIF can therefore be calculated for the
intercept of the linear regression model.
The following other considerations should be taken into account:
If the data are expressed in unit length, using the same
transformation used to calculate the condition number, then: \[RVIF(i) = \frac{1}{1 - \mathbf{X}_{i}^{t}
\mathbf{X}_{-i} \cdot \left( \mathbf{X}_{-i}^{t} \mathbf{X}_{-i} \right)
\cdot \mathbf{X}_{-i}^{t} \mathbf{X}_{i}}, \quad
i=1,\dots,k.\]
In this case (data expressed in unit length), when \(\mathbf{X}_{i}\) is orthogonal to \(\mathbf{X}_{-i}\), it is verified that
\(\mathbf{X}_{i}^{t} \mathbf{X}_{-i} =
\mathbf{0}\) and, consequently \(RVIF(i) = 1\) for \(i=1,\dots,k\). That is, the RVIF is always
greater than or equal to 1 and its minimum value is indicative of the
absence of multicollinearity.
Denoted by \(a_{i}= \mathbf{X}_{i}^{t}
\mathbf{X}_{i} \cdot \left( \mathbf{X}_{-i}^{t} \mathbf{X}_{-i} \right)
\cdot \mathbf{X}_{-i}^{t} \mathbf{X}_{i}\), \(RVIF(i) = \frac{1}{1-a_{i}}\), where \(a_{i}\) can be interpreted as the
percentage of approximate multicollinearity due to the variable \(\mathbf{X}_{i}\). Note the similarity of
this expression to the VIF expression: \(VIF(i) = \frac{1}{1-R_{i}^{2}}\).
Finally, Salmerón, García and García (2025) show, from a
simulation for \(k=3\), that if \(a_{i} > 0.826\), then the degree of
multicollinearity is worrying. In any case, this value should be refined
by considering a higher number of independent variables.
In short, while it is not necessary to transform the data, it is
advisable to do so for a better interpretation of the results (as with
the condition number).
The rvifs function calculates the \(RVIF(i)\) and \(a_{i}\) values associated with the
independent variable \(i=1,2,\dots,k\).The command and arguments
of this function are as follows:
# rvifs(x, ul = TRUE, intercept = TRUE, tol = 1e-30)
As can be seen:
The first argument to the function refers to the design matrix
\(\mathbf{X}\).
The novelty here is that the design matrix, \(\mathbf{X}\), may or may not have an
intercept. This is specified by the third argument. If an intercept is
present, as in previous cases, it must correspond to the first column of
\(\mathbf{X}\).
The second argument indicates whether the data has been
transformed to a unit length (the default value) or whether no
transformation has been performed.
The last argument is the threshold that determines whether the
system is computationally singular, similar to the cv_vif
function.
Some examples
When the rvifs function is applied to the simulation of the
second example of the cv_vif function, it can be observed that
all the variables (except the last one) have high RVIF values and a high
percentage of approximate multicollinearity:
set.seed(2025)
obs = 100
cte = rep(1, obs)
x2 = rnorm(obs, 5, 0.01) # variable with very little variability
x3 = rnorm(obs, 5, 10)
x4 = x3 + rnorm(obs, 5, 1) # fourth variable related to the third
x5 = rnorm(obs, -1, 30)
x = cbind(cte, x2, x3, x4, x5) # the first column has to be the intercept
rvifs(x)
#> RVIF %
#> Intercept 2.495289e+05 99.9996
#> Variable 2 2.487886e+05 99.9996
#> Variable 3 1.282689e+02 99.2204
#> Variable 4 2.088057e+02 99.5211
#> Variable 5 1.034727e+00 3.3561
However, when applied to data on soil characteristics, where there is
no intercept in the design matrix, high values are observed for all
variables:
x = soil[,-16]
rvifs(x, intercept=FALSE)
#> System is computationally singular. Modify the design matrix before running the code.
rvifs(x[,-2], intercept=FALSE)
#> RVIF %
#> Variable 1 9.055351e+02 99.8896
#> Variable 2 1.346429e+06 99.9999
#> Variable 3 1.363495e+05 99.9993
#> Variable 4 1.019537e+04 99.9902
#> Variable 5 1.613634e+03 99.9380
#> Variable 6 1.483139e+02 99.3258
#> Variable 7 1.019762e+01 90.1938
#> Variable 8 9.835658e+01 98.9833
#> Variable 9 6.422169e+01 98.4429
#> Variable 10 5.591258e+01 98.2115
#> Variable 11 4.432249e+02 99.7744
#> Variable 12 4.653218e+03 99.9785
#> Variable 13 5.570325e+03 99.9820
#> Variable 14 5.334982e+05 99.9998
Finally, by applying the rvif to the simple linear
regression model, we can determine the degree of non-essential
multicollinearity:
cte = rep(1, obs)
set.seed(2025)
x2 = rnorm(obs, 3, 4)
x = cbind(cte, x2)
rvifs(x)
#> RVIF %
#> Intercept 1.538266 34.9917
#> Variable 2 1.538266 34.9917
A high degree of non-essential multicollinearity is detected if a
variable with little variability is generated.
cte = rep(1, obs)
set.seed(2025)
x2 = rnorm(obs, 3, 0.04)
x = cbind(cte, x2)
rvifs(x)
#> RVIF %
#> Intercept 5459.413 99.9817
#> Variable 2 5459.413 99.9817
If this variable is centered to solve the detected problem, the RVIFs
coincide with their minimum value, indicating orthogonality between the
elements of the design matrix:
x2 = x2 - mean(x2)
x = cbind(cte, x2)
rvifs(x)
#> RVIF %
#> Intercept 1 0
#> Variable 2 1 0
If the data are not transformed into unit length, the percentages of
multicollinearity due to each variable remain equal to zero. However,
since there is no universal minimum reference value, interpreting the
RVIF becomes more complicated:
rvifs(x, ul=FALSE)
#> RVIF %
#> Intercept 0.010000 0
#> Variable 2 6.065757 0
This last example illustrates why it is advisable to transform the
data into unit length when calculating the RVIF.
multicollinearity function
Until now, measures of the degree of multicollinearity in multiple
linear regression models have been used. However, it has never been
proven that this degree affects model analysis.
In their working paper, Salmerón, García and García propose a
statistical test that determines whether the degree of multicollinearity
affects the statistical analysis of the model, based on a region of
non-rejection (which depends on a level of significance).
More specifically, the test determines whether the non-rejection of
the null hypothesis in the individual significance tests of the
coefficient associated with each independent variable is due to the
degree of multicollinearity.
Ultimately, the decision rule for the test is determined by the
following result:
Given the multiple linear regression model \(\mathbf{y} = \mathbf{X} \boldsymbol{\beta}
+ \mathbf{u}\), the degree of multicollinearity affects its
statistical analysis (with a level of significance of \(\alpha\%\)) if there is a variable \(i\), with \(i=1,\dots,k\), that verifies \(RVIF(i) > \max \{ c_{0}(i), c_{3}(i)
\}\) where: \[c_{0}(i) = \left(
\frac{\widehat{\beta}_{i}}{\widehat{\sigma} \cdot t_{n-k}(1-\alpha/2)}
\right)^{2}\] and \[c_{3}(i) = \left(
\frac{t_{n-k}(1-\alpha/2)}{\widehat{\beta}_{i,o}} \right)^{2} \cdot
\widehat{var} \left( \widehat{\beta}_{i} \right),\] where \(\widehat{\beta}_{i}\) is the OLS estimate
of \(\beta_{i}\), \(\widehat{var} \left( \widehat{\beta}_{i}
\right)\) its estimated variance, \(\widehat{\beta}_{i,o}\) is the OLS estimate
of \(\beta_{i}\) in the orthogonal
model, \(\sigma^{2}\) is the variance
of the random disturbance, and \(t_{n-k}(1-\alpha/2)\) is the theoretical
value associated with the individual significance test.
The multicollinearity function allows you to check the
condition of the theorem using the following code:
# multicollinearity(y, x, alpha = 0.05)
The first argument of the function refers to the dependent variable
of the linear regression model. The second relates to the design matrix
containing the independent variables and the third to the significance
level considered in the statistical test.
Note that, as with the cv_vif function, the rvif
command will only work correctly if the design matrix has the intercept
in its first column.
It should also be noted that, in this case, the RVIF is calculated
without the data being transformed into unit length, as the results
obtained in the working paper by Salmerón, García and García do not
assume any transformation..
A first example: Wissel data
In Wissel’s data on outstanding mortgage debt, the model is
considered jointly valid at the same time that is not rejected the null
hypothesis that each coefficient of the independent variables is
zero:
summary(reg_W)
#>
#> Call:
#> lm(formula = D ~ C + I + CP)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -1.2729 -0.7138 0.2853 0.7033 1.0711
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 5.469264 13.016791 0.420 0.681
#> C -4.252429 5.135058 -0.828 0.423
#> I 3.120395 2.035671 1.533 0.149
#> CP 0.002879 0.005764 0.499 0.626
#>
#> Residual standard error: 0.9325 on 13 degrees of freedom
#> Multiple R-squared: 0.9235, Adjusted R-squared: 0.9058
#> F-statistic: 52.3 on 3 and 13 DF, p-value: 1.629e-07
Let us remember that a high degree of essential multicollinearity was
determined due to the three independent variables. However, when the
multicollinearity command is used, it can be observed that only
the second of these is affected by the detected problem:
y = Wissel[,2]
x = Wissel[,3:6]
multicollinearity(y, x)
#> RVIFs c0 c3 Scenario Affects
#> 1 1.948661e+02 7.371069e+00 1.017198e+00 b.1 Yes
#> 2 3.032628e+01 4.456018e+00 9.157898e-01 b.1 Yes
#> 3 4.765888e+00 2.399341e+00 1.053598e+01 b.2 No
#> 4 3.821626e-05 2.042640e-06 7.149977e-04 b.2 No
In other words, it can be concluded that the failure to reject the
null hypothesis in the test associated with the personal consumption (C)
variable is due to the degree of multicollinearity in the model.
A second example: Klein and Goldberger data
The multiColl package contains data from Klein and
Goldberger on consumption and wage income.
data(KG)
attach(KG)
reg_KG = lm(consumption~wage.income+non.farm.income+farm.income)
detach(KG)
summary(reg_KG)
#>
#> Call:
#> lm(formula = consumption ~ wage.income + non.farm.income + farm.income)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -13.494 -1.847 1.116 2.541 6.460
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 18.7021 6.8454 2.732 0.0211 *
#> wage.income 0.3803 0.3121 1.218 0.2511
#> non.farm.income 1.4186 0.7204 1.969 0.0772 .
#> farm.income 0.5331 1.3998 0.381 0.7113
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 6.06 on 10 degrees of freedom
#> Multiple R-squared: 0.9187, Adjusted R-squared: 0.8943
#> F-statistic: 37.68 on 3 and 10 DF, p-value: 9.271e-06
Once again, at a 5% significance level, the null hypothesis is not
rejected in the individual significance tests, while the null hypothesis
is rejected in the joint significance test. This situation is considered
a sign of strong multicollinearity.
Using the multicollinearity command, it is possible to
determine that the statistical analysis of the second variable is
affected by the degree of multicollinearity in the model:
head(KG, n=5)
#> consumption wage.income non.farm.income farm.income
#> 1 62.8 43.41 17.10 3.96
#> 2 65.0 46.44 18.65 5.48
#> 3 63.9 44.35 17.09 4.37
#> 4 67.5 47.82 19.28 4.51
#> 5 71.3 51.02 23.24 4.88
y = KG[,1]
x = model.matrix(reg_KG)
multicollinearity(y, x)
#> RVIFs c0 c3 Scenario Affects
#> 1 1.275947615 1.9183829079 0.0021892653 a.1 No
#> 2 0.002652862 0.0007931658 0.0001206694 b.1 Yes
#> 3 0.014130621 0.0110372472 0.0187393601 b.2 No
#> 4 0.053354814 0.0015584988 1.8265885762 b.2 No
A third example: the case of the simple linear regresion
The work of Salmerón, García and García (2025) illustrates the
limitations of the vif command in the car package for
detecting the degree of multicollinearity in a simple linear regression
model.
The data used in this work for this purpose are available in the
rvif package. For more details, use help(SLM1) and
help(SLM2).
If the first one is used, the vif command effectively
indicates that the model has fewer than two terms and does not perform
any calculations:
head(SLM1, n=5)
#> y1 cte V
#> 1 82.392059 1 19.001420
#> 2 -1.942157 1 -1.733458
#> 3 7.474090 1 1.025146
#> 4 -12.303381 1 -4.445014
#> 5 30.378203 1 6.689864
attach(SLM1)
#> The following object is masked _by_ .GlobalEnv:
#>
#> cte
reg_SLM1 = lm(y1~V)
detach(SLM1)
library(car)
vif(reg_SLM1)
#> Error in vif.default(reg_SLM1): model contains fewer than 2 terms
In other words, it denies the possibility that the intercept could
form part of the model as another independent variable and,
consequently, there could be a problem of multicollinearity.
However, if the multicollinearity command is used, it is
possible to determine the degree of multicollinearity in the model and
whether it affects the statistical analysis of the model:
y = SLM1[,1]
x = SLM1[,-1]
multicollinearity(y, x)
#> RVIFs c0 c3 Scenario Affects
#> 1 0.0403049717 0.6454323 1.045802e-05 a.1 No
#> 2 0.0002675731 0.8383436 8.540101e-08 a.1 No
While the degree of multicollinearity does not affect the statistical
analysis of the model in the first data set, it does affect the
second:
head(SLM2, n=5)
#> y2 cte Z
#> 1 43.01204 1 9.978211
#> 2 40.04163 1 9.878235
#> 3 40.17086 1 9.924592
#> 4 40.79076 1 10.019123
#> 5 44.72774 1 10.104728
y = SLM2[,1]
x = SLM2[,-1]
multicollinearity(y, x)
#> RVIFs c0 c3 Scenario Affects
#> 1 187.800878 21.4798003 0.03277691 b.1 Yes
#> 2 1.879296 0.3687652 9.57724567 b.2 No
It can be observed that when the significance level is changed, the
values of the non-rejection region (determined by the values \(c_{0}(i)\) and $c_{3}(i)) change:
multicollinearity(y, x, alpha=0.01)
#> RVIFs c0 c3 Scenario Affects
#> 1 187.800878 12.0701590 0.0583291 b.1 Yes
#> 2 1.879296 0.2072205 17.0434643 b.2 No
A fourth example: soil characteristics data
Using the rvif command on the soil characteristics data
revealed that all the variables had high RVIF values and a high
percentage of multicollinearity.
Meanwhile, using the cv_vif command revealed that the pH and
ExchAc variables caused a non-essential multicollinearity problem, while
all other variables except Na, P, Zn nad Mn caused essential
multicollinearity.
Using the *multicollinearity** command, it is observed that only the
statistical analysis of the intercept, CECbuffer and HumicMatter is
affected by the degree of multicollinearity present:
y = soil[,16]
x = soil[,-16]
intercept = rep(1, length(y))
x = cbind(intercept, x) # the design matrix has to have the intercept in the first column
multicollinearity(y, x)
#> System is computationally singular. Modify the design matrix before running the code.
multicollinearity(y, x[,-3]) # eliminating the problematic variable (SumCation)
#> RVIFs c0 c3 Scenario Affects
#> 1 4.407184e+02 6.150190e-03 1.480048e+00 b.1 Yes
#> 2 3.828858e+00 1.142356e-02 7.653413e+00 b.2 No
#> 3 1.093791e+05 1.254955e+02 7.236491e+04 b.1 Yes
#> 4 9.883235e+04 3.938383e+01 2.237445e+05 b.2 No
#> 5 1.767758e+05 1.101028e+03 3.609837e+05 b.2 No
#> 6 1.150029e+05 1.627349e+03 1.976176e+05 b.2 No
#> 7 4.627807e+04 5.960870e+02 2.033176e+06 b.2 No
#> 8 1.338591e+01 6.062571e-01 4.060382e+02 b.2 No
#> 9 3.113066e+02 4.089095e+01 5.246698e+05 b.2 No
#> 10 5.177176e+01 6.371216e+00 8.094828e+02 b.2 No
#> 11 1.905089e-01 3.907589e-02 9.787963e-01 b.2 No
#> 12 3.379360e+02 4.534540e+01 2.861964e+02 b.1 Yes
#> 13 4.761238e+04 8.453066e+01 3.828016e+08 b.2 No
#> 14 1.502903e+03 7.901580e+01 9.961215e+03 b.2 No
#> 15 1.066711e+05 2.369347e+02 4.802466e+07 b.2 No
names(x[,-3])
#> [1] "intercept" "BaseSat" "CECbuffer" "Ca" "Mg"
#> [6] "K" "Na" "P" "Cu" "Zn"
#> [11] "Mn" "HumicMatter" "Density" "pH" "ExchAc"