The package permits the covariate effects of trinomial regression models to be represented graphically by means of a ternary plot. The aim of the plots is helping the interpretation of regression coefficients in terms of the effects that a change in regressors’ values has on the probability distribution of the dependent variable. Such changes may involve either a single regressor, or a group of them (composite changes), and the package permits both cases to be handled in a user-friendly way. Methodological details are illustrated and discussed in Santi, Dickson, and Espa (2019).

The package can read the results of **both categorical and ordinal trinomial logit** regression fitted by various functions (see the next section) and creates a `field3logit`

object which may be represented by means of functions `gg3logit`

and `stat_field3logit`

.

The `plot3logit`

package inherits graphical classes and methods from the package `ggtern`

(Hamilton and Ferry 2018) which, in turn, is based on the package `ggplot2`

(Wickham 2016).

Graphical representation based on **standard graphics** is made available through the package `Ternary`

(Smith 2017) by functions `plot3logit`

and `TernaryField`

, and by the `plot`

method of `field3logit`

objects.

See the help of `field3logit`

for representing composite effects and `multifield3logit`

for drawing multiple fields.

Function `field3logit`

of package `plot3logit`

can read trinomial regression estimates from the output of the following functions:

`polr`

of package`MASS`

(ordinal logit regression);`mlogit`

of package`mlogit`

(logit regression);`multinom`

of package`nnet`

(logit regression);`clm`

and`clm2`

of package`ordinal`

(ordinal logit regression);`vgam`

and`vglm`

of package`VGAM`

(logit regression).

Explicit estimates can be passed to `field3logit()`

by means of a named list too (see the help of `field3logit`

); moreover, users can easily extend the range of objects that can be processed by `field3logit()`

by implementing the S3 methods of the generic `extract3logit`

(see the help).

In the following, an example is provided for each object `field3logit`

can work with. First, however, package `plot3logit`

must be attached and dataset `cross_1year`

must be loaded:

```
library(plot3logit)
data(cross_1year)
str(cross_1year)
#> 'data.frame': 3282 obs. of 7 variables:
#> $ employment_sit: Factor w/ 3 levels "Employed","Unemployed",..: 1 1 1 3 1 1 1 1 1 2 ...
#> $ gender : Factor w/ 2 levels "Male","Female": 2 1 2 2 2 1 2 1 1 2 ...
#> $ finalgrade : Factor w/ 3 levels "Average","Low",..: 3 3 1 1 2 3 1 1 2 3 ...
#> $ duration : Factor w/ 3 levels "Average","Short",..: 3 3 3 3 3 3 1 3 3 3 ...
#> $ social_class : Factor w/ 5 levels "Working class",..: 1 2 4 4 3 3 1 4 1 2 ...
#> $ irregularity : Factor w/ 3 levels "Average","Low",..: 1 1 3 3 3 3 1 3 3 3 ...
#> $ hsscore : num 100 95 82 64 69 ...
```

`MASS`

Function `polr`

of package `MASS`

(Venables and Ripley 2002) fits a *proportional odds logistic regression* by means of a fairly simple syntax, however the order of labels of the dependent variable must be explicitly stated:

```
library(MASS)
<- cross_1year
mydata $finalgrade <- factor(
mydatax = mydata$finalgrade,
levels = c('Low', 'Average', 'High'),
ordered = TRUE
)
<- polr(finalgrade ~ gender + irregularity, data = mydata)
mod0 field3logit(mod0, 'genderFemale')
```

`mlogit`

Package `mlogit`

of package `mlogit`

(Croissant 2020) can fit a wide range of models. The current implementation of `plot3logit`

only works with pure trinomial models, where only specific coefficients are considered:

```
library(mlogit)
<- mlogit.data(cross_1year, choice = 'employment_sit', shape = 'wide')
mydata <- mlogit(employment_sit ~ 0 | gender + finalgrade, data = mydata)
mod0 field3logit(mod0, 'genderFemale')
```

`nnet`

Function `multinom`

of package `nnet`

(Venables and Ripley 2002) fits a multinomial model by means of a very simple syntax:

```
library(nnet)
<- multinom(employment_sit ~ gender + finalgrade, data = cross_1year)
mod0 field3logit(mod0, 'genderFemale')
```

`ordinal`

Package `ordinal`

(Christensen 2019) permits ordinal multinomial logistic models to be fitted by means of two functions: `clm`

and `clm2`

. The latter is “a new improved implementation” of the former (see the help of `clm2`

), in both cases the syntax is simple and standard. Unlike `polr`

of package `MASS`

(Venables and Ripley 2002), the dependent variable may be an unordered factor, and in that case both `clm`

and `clm2`

consider the vector of the labels as ordered: if this is not the case, the dependent variable should be properly redefined.

```
library(ordinal)
$finalgrade <- factor(mydata$finalgrade, c('Low', 'Average', 'High'))
mydata
<- clm(finalgrade ~ gender + irregularity, data = mydata)
mod0 field3logit(mod0, 'genderFemale')
<- clm2(finalgrade ~ gender + irregularity, data = mydata)
mod0 field3logit(mod0, 'genderFemale')
```

`VGAM`

Package `VGAM`

(Yee 2010) permits multinomial logistic models to be fitted by means of two functions: `vgam`

and `vglm`

. In case of multinomial logistic models they share the same syntax:

```
library(VGAM)
<- vgam(
mod0 formula = employment_sit ~ gender + finalgrade,
family = multinomial(),
data = cross_1year
)field3logit(mod0, 'genderFemale')
<- vglm(
mod0 formula = employment_sit ~ gender + finalgrade,
family = multinomial(),
data = cross_1year
)field3logit(mod0, 'genderFemale')
```

`list`

Point estimates and other information on a fitted model may be passed as a named list to `field3logit`

. In case of a non-ordinal trinomial model, the matrix of coefficients (component `B`

) should be a matrix with named rows and two columns, whereas possible levels of the dependent variable should be included as a separate component (`levels`

) which should be a character vector of length three, whose first component is the reference level of the model (see the help of `extract3logit`

for details):

```
<- list(
mod0 B = matrix(
data = c(-2.05, 0.46, -2.46, 0.37, -1.2, 0.8, 0.7, 0.4),
ncol = 2,
dimnames = list(c('Intercept', 'X1', 'X2', 'X3'))
),levels = c('Class A', 'Class B', 'Class C')
)field3logit(mod0, 'X1')
```

Fit a trilogit model by means of package `nnet`

where the student’s employment situation is analysed with respect to all variables in the dataset `cross_1year`

:

```
library(plot3logit)
data(cross_1year)
library(nnet)
<- multinom(employment_sit ~ ., data = cross_1year)
mod0 #> # weights: 42 (26 variable)
#> initial value 3605.645531
#> iter 10 value 2167.042903
#> iter 20 value 2136.782685
#> iter 30 value 2134.363158
#> final value 2134.352162
#> converged
```

The gender effect is analysed by means of a ternary plot which is generated in two steps.

Firstly, the vector field is computed:

`<- field3logit(mod0, 'genderFemale') field0 `

Secondly, the field is represented on a ternary plot, using either `gg`

-graphics:

`gg3logit(field0) + stat_field3logit()`

or standard graphics:

`plot(field0)`

Ternary plots represent the effect of a change in covariate values on the probability distribution of the dependent variable. The function `field3logit`

permits such change to be specified in three different ways: explicitly, as a numeric vector, or implicitly, either by means of a named numeric vector, or by means of an R expressions. All three methods are briefly illustrated below.

As an example, the following subsections refer to this trinomial logistic regression model:

```
library(plot3logit)
library(nnet)
<- multinom(employment_sit ~ finalgrade + irregularity + hsscore, cross_1year)
mod0 #> # weights: 21 (12 variable)
#> initial value 3605.645531
#> iter 10 value 2187.709284
#> iter 20 value 2157.087955
#> final value 2157.087854
#> converged
mod0#> Call:
#> multinom(formula = employment_sit ~ finalgrade + irregularity +
#> hsscore, data = cross_1year)
#>
#> Coefficients:
#> (Intercept) finalgradeLow finalgradeHigh irregularityLow
#> Unemployed -0.4481761 0.05551765 -0.07810893 -0.01874354
#> Trainee -1.3751140 0.14456683 -0.26849829 0.05764144
#> irregularityHigh hsscore
#> Unemployed 0.15691595 -0.016619227
#> Trainee -0.03477569 -0.009964381
#>
#> Residual Deviance: 4314.176
#> AIC: 4338.176
```

This method for specifying the change in the covariate values requires the vector \(\Delta x\) to be explicitly defined, thus it may be suitable when \(\Delta x\) results from some calculations. On the other hand, it is less user-friendly than implicit syntax, as it depends on the order of regressors in the design matrix.

If the effect of a *high final grade* has to be assessed, the vector of changes \(\Delta x\) can be set according to the position of the dummy variable `finalgradeHigh`

in the matrix of coefficients of the model `mod0`

:

```
coef(mod0)
#> (Intercept) finalgradeLow finalgradeHigh irregularityLow
#> Unemployed -0.4481761 0.05551765 -0.07810893 -0.01874354
#> Trainee -1.3751140 0.14456683 -0.26849829 0.05764144
#> irregularityHigh hsscore
#> Unemployed 0.15691595 -0.016619227
#> Trainee -0.03477569 -0.009964381
```

in this case, we have that \[
\Delta x=[0, 0, 1, 0, 0, 0]'
\] since `finalgradeHigh`

is the fourth coefficient (including the intercept) of the matrix of coefficients.

It follows that the function `field3logit`

can be invoked as it follows:

```
field3logit(mod0, c(0, 0, 1, 0, 0, 0))
#> Object of class "field3logit"
#> -------------------------------
#> Label : <empty>
#> Possible outcomes : Employed; Unemployed; Trainee
#> Reference level : Employed
#> Type of model : categorical
#> Effect : 0 0 1 0 0 0
#> Model has been read from : nnet::multinom
#> Number of stream lines : 8
#> Number of arrows : 182
#> Covariance matrix : available
#> Confidence regions : not available
```

It is also possible to set \(\Delta x\) so as to consider changes involving more than one regressor, as well as fractional changes. In such cases, \(\Delta x\) will consist in a vector where there are several non-zero elements which may take any positive or negative value.

Assume, for example, that we want to study the effect of a decrease by 10 in the high school final score, associated to an high final grade. In such a case, we have that: \[ \Delta x =[0, 0, 1, 0, 0, -10]'\,, \] thus:

```
field3logit(mod0, c(0, 0, 1, 0, 0, -10))
#> Object of class "field3logit"
#> -------------------------------
#> Label : <empty>
#> Possible outcomes : Employed; Unemployed; Trainee
#> Reference level : Employed
#> Type of model : categorical
#> Effect : 0 0 1 0 0 -10
#> Model has been read from : nnet::multinom
#> Number of stream lines : 8
#> Number of arrows : 166
#> Covariance matrix : available
#> Confidence regions : not available
```

Unlike the explicit method, the implicit syntaxes allows the user to initialise the vector \(\Delta x\) by specifying only the covariates which should vary. There are two possible syntaxes: one is based on named numeric vectors, the other is based on `R`

code. In the following, both of them are illustrated.

If the effect of a *high final grade* has to be assessed, the implicit syntaxes which allow to assess the effect of a unitary change of `finalgradeHigh`

are the following:

```
# Named numeric vectors:
field3logit(mod0, c(finalgradeHigh = 1))
#> Object of class "field3logit"
#> -------------------------------
#> Label : <empty>
#> Possible outcomes : Employed; Unemployed; Trainee
#> Reference level : Employed
#> Type of model : categorical
#> Effect : finalgradeHigh
#> Explicit effect : 0 0 1 0 0 0
#> Model has been read from : nnet::multinom
#> Number of stream lines : 8
#> Number of arrows : 182
#> Covariance matrix : available
#> Confidence regions : not available
# R code:
field3logit(mod0, 'finalgradeHigh')
#> Object of class "field3logit"
#> -------------------------------
#> Label : <empty>
#> Possible outcomes : Employed; Unemployed; Trainee
#> Reference level : Employed
#> Type of model : categorical
#> Effect : finalgradeHigh
#> Explicit effect : 0 0 1 0 0 0
#> Model has been read from : nnet::multinom
#> Number of stream lines : 8
#> Number of arrows : 182
#> Covariance matrix : available
#> Confidence regions : not available
```

Note that the console output produced by printing the output of `field3logit`

shows both the implicit effect (line `Effect`

) and the associated vector \(\Delta x\) (line `Explicit effect`

).

If we want to study the effect of a decrease by 10 in the high school final score, associated to an high final grade, the implicit syntaxes are:

```
# Named numeric vectors:
field3logit(mod0, c(finalgradeHigh = 1, hsscore = -10))
#> Object of class "field3logit"
#> -------------------------------
#> Label : <empty>
#> Possible outcomes : Employed; Unemployed; Trainee
#> Reference level : Employed
#> Type of model : categorical
#> Effect : finalgradeHigh - 10 * hsscore
#> Explicit effect : 0 0 1 0 0 -10
#> Model has been read from : nnet::multinom
#> Number of stream lines : 8
#> Number of arrows : 166
#> Covariance matrix : available
#> Confidence regions : not available
# R code:
field3logit(mod0, 'finalgradeHigh - 10 * hsscore')
#> Object of class "field3logit"
#> -------------------------------
#> Label : <empty>
#> Possible outcomes : Employed; Unemployed; Trainee
#> Reference level : Employed
#> Type of model : categorical
#> Effect : finalgradeHigh - 10 * hsscore
#> Explicit effect : 0 0 1 0 0 -10
#> Model has been read from : nnet::multinom
#> Number of stream lines : 8
#> Number of arrows : 166
#> Covariance matrix : available
#> Confidence regions : not available
```

Compare the line `Explicit effect`

of this output to the line `Effect`

of the same example in the previous section: as expected, they are the same.

When effects of multiple changes have to be compared at a time, multiple fields should be computed and represented on the same plot. This task can be easily done by creating a `multifield3logit`

object and directly representing it.

Since objects `multifield3logit`

result by putting together two or more `field3logit`

objects, the package `plot3logit`

allows the user to create a `multifield3logit`

object by adding up two or more `filed3logit`

or `multifield3logit`

objects using standard sum operator `+`

.

Here it is an example. The following command fit a trilogit model where all available variables are used as regressors. Then four `fields3logit`

objects are computed for assessing the effects of a some combined changes in the duration of studies and in students’ final degree score.

Note that each field is computed just with respect to a single probability distribution (`refpoint`

) of the dependent variable, and only one arrow is computed. The reason of this is that we have to represent four fields on the same plot, thus olny a small number of arrows can be drawn in order to preserve the clarity of the graph.

```
data(cross_1year)
<- nnet::multinom(employment_sit ~ ., data = cross_1year)
mod0
<- list(c(0.7, 0.15, 0.15))
refpoint
<- field3logit(mod0, 'durationShort', label = 'Short duration', p0 = refpoint, narrows = 1)
field_Sdur <- field3logit(mod0, 'durationLong', label = 'Long duration', p0 = refpoint, narrows = 1)
field_Ldur <- field3logit(mod0, 'finalgradeHigh', label = 'High final grade', p0 = refpoint, narrows = 1)
field_Hfgr <- field3logit(mod0, 'finalgradeLow', label = 'Low final grade', p0 = refpoint, narrows = 1) field_Lfgr
```

Now the `multifield3logit`

object can be created by adding all the `field3logit`

objects up together:

```
<- field_Sdur + field_Ldur + field_Lfgr + field_Hfgr
mfields
mfields#> Object of class "multifield3logit"
#> ------------------------------------
#> Number of fields : 4
#> Labels
#> 1. Short duration (dX: durationShort)
#> 2. Long duration (dX: durationLong)
#> 3. Low final grade (dX: finalgradeLow)
#> 4. High final grade (dX: finalgradeHigh)
```

and the `multifield3logit`

object `mfield`

can be represented in a graph:

```
gg3logit(mfields) +
stat_field3logit(aes(colour = label)) +
theme_zoom_L(0.45)
```

The code needed for generating the object `mfields`

may be conveniently made shorter in this way (see the help of `field3logit`

for details on syntax):

```
<- list(
depo list(delta = 'durationShort', label = 'Short duration'),
list(delta = 'durationLong', label = 'Long duration'),
list(delta = 'finalgradeHigh', label = 'High final grade'),
list(delta = 'finalgradeLow', label = 'Low final grade')
)
<- field3logit(mod0, delta = depo, p0 = refpoint, narrows = 1)
mfields
mfields#> Object of class "multifield3logit"
#> ------------------------------------
#> Number of fields : 4
#> Labels
#> 1. Short duration (dX: durationShort)
#> 2. Long duration (dX: durationLong)
#> 3. High final grade (dX: finalgradeHigh)
#> 4. Low final grade (dX: finalgradeLow)
```

It is also possible to rely on syntax based on delimiters `<<...>>`

(see the help of `field3logit`

, section **Details**) in order to create a `multifield3logit`

object consisting of fields based on the effect of each dummy variable generated by the same `factor`

regressor:

```
field3logit(mod0, delta = '<<finalgrade>>', p0 = refpoint, narrows = 1)
#> Object of class "multifield3logit"
#> ------------------------------------
#> Number of fields : 2
#> Labels
#> 1. Low (dX: `finalgradeLow`)
#> 2. High (dX: `finalgradeHigh`)
```

The package `plot3logit`

allows also to draw the confidence regions associated to each effect, both in case of `field3logit`

and `multifield3logit`

objects.

The confidence regions can be computed when the function `field3logit`

is called by setting the argument `conf`

. Otherwise, they can be added later through the function `add_confregions`

as it follows:

```
<- add_confregions(field0, conf = 0.95)
field0
field0#> Object of class "field3logit"
#> -------------------------------
#> Label : <empty>
#> Possible outcomes : Employed; Unemployed; Trainee
#> Reference level : Employed
#> Type of model : categorical
#> Effect : genderFemale
#> Explicit effect : 0 1 0 0 0 0 0 0 0 0 0 0 0
#> Model has been read from : nnet::multinom
#> Number of stream lines : 8
#> Number of arrows : 111
#> Covariance matrix : available
#> Confidence regions : 95%
```

and through the same syntax in case of `multifield3logit`

objects:

`<- add_confregions(mfields, conf = 0.95) mfields `

The statistic `stat_conf3logit`

permits confidence regions to be drawn, if available:

`gg3logit(field0) + stat_field3logit() + stat_conf3logit()`

and

```
gg3logit(mfields) +
stat_field3logit(aes(colour = label)) +
stat_conf3logit(aes(fill = label)) +
theme_zoom_L(0.45)
```

Christensen, R. H. B. 2019. *ordinal — Regression Models for Ordinal Data*. https://CRAN.R-project.org/package=ordinal.

Croissant, Y. 2020. *mlogit: Multinomial Logit Models*. https://CRAN.R-project.org/package=mlogit.

Hamilton, N. E., and M. Ferry. 2018. “ggtern: Ternary Diagrams Using ggplot2.” *Journal of Statistical Software, Code Snippets* 87 (3): 1–17. https://doi.org/10.18637/jss.v087.c03.

Santi, F., M. M. Dickson, and G. Espa. 2019. “A Graphical Tool for Interpreting Regression Coefficients of Trinomial Logit Models.” *The American Statistician* 73 (2): 200–207. https://doi.org/10.1080/00031305.2018.1442368.

Smith, M. R. 2017. “Ternary: An r Package for Creating Ternary Plots.” *Zenodo*.

Venables, W. N., and B. D. Ripley. 2002. *Modern Applied Statistics with s*. Fourth. New York: Springer-Verlag. https://www.stats.ox.ac.uk/pub/MASS4/.

Wickham, H. 2016. *ggplot2: Elegant Graphics for Data Analysis*. New York: Springer-Verlag.

Yee, T. W. 2010. “The VGAM Package for Categorical Data Analysis.” *Journal of Statistical Software, Articles* 32 (10): 1–34. https://doi.org/10.18637/jss.v032.i10.