To add interactions between covariates in your model, you can add
additional arguments in the pars vector in the
logitr() function separated by the * symbol.
For example, let’s say we want to interact price with
feat in the following model:
library("logitr")
model <- logitr(
data = yogurt,
outcome = 'choice',
obsID = 'obsID',
pars = c('price', 'feat', 'brand')
)To do so, I could add "price*feat" to the
pars vector:
model_price_feat <- logitr(
data = yogurt,
outcome = 'choice',
obsID = 'obsID',
pars = c('price', 'feat', 'brand', 'price*feat')
)The model now has an estimated coefficient for the
price*feat effect:
summary(model_price_feat)
#> =================================================
#>
#> Model estimated on: Fri Nov 01 08:44:07 PM 2024
#>
#> Using logitr version: 1.1.2
#>
#> Call:
#> logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = c("price",
#> "feat", "brand", "price*feat"))
#>
#> Frequencies of alternatives:
#> 1 2 3 4
#> 0.402156 0.029436 0.229270 0.339138
#>
#> Exit Status: 3, Optimization stopped because ftol_rel or ftol_abs was reached.
#>
#> Model Type: Multinomial Logit
#> Model Space: Preference
#> Model Run: 1 of 1
#> Iterations: 19
#> Elapsed Time: 0h:0m:0.02s
#> Algorithm: NLOPT_LD_LBFGS
#> Weights Used?: FALSE
#> Robust? FALSE
#>
#> Model Coefficients:
#> Estimate Std. Error z-value Pr(>|z|)
#> price -0.356909 0.024696 -14.4522 < 2.2e-16 ***
#> feat 1.155206 0.378237 3.0542 0.002257 **
#> brandhiland -3.724702 0.146520 -25.4212 < 2.2e-16 ***
#> brandweight -0.640221 0.054543 -11.7380 < 2.2e-16 ***
#> brandyoplait 0.724315 0.080317 9.0182 < 2.2e-16 ***
#> price:feat -0.086381 0.047275 -1.8272 0.067672 .
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Log-Likelihood: -2655.5403770
#> Null Log-Likelihood: -3343.7419990
#> AIC: 5323.0807540
#> BIC: 5357.8100000
#> McFadden R2: 0.2058178
#> Adj McFadden R2: 0.2040234
#> Number of Observations: 2412.0000000In the above example, both price and feat
were continuous variables, so only a single interaction coefficient was
needed.
In the case of interacting discrete variables, multiple
interactions coefficients will be estimated according to the number of
levels in the discrete attribute. For example, the interaction of
price with brand will require three new
interactions - one for each level of the brand variable
except the first reference level:
model_price_brand <- logitr(
data = yogurt,
outcome = 'choice',
obsID = 'obsID',
pars = c('price', 'feat', 'brand', 'price*brand')
)The model now has three estimated coefficients for the
price*brand effect:
summary(model_price_brand)
#> =================================================
#>
#> Model estimated on: Fri Nov 01 08:44:07 PM 2024
#>
#> Using logitr version: 1.1.2
#>
#> Call:
#> logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = c("price",
#> "feat", "brand", "price*brand"))
#>
#> Frequencies of alternatives:
#> 1 2 3 4
#> 0.402156 0.029436 0.229270 0.339138
#>
#> Exit Status: 3, Optimization stopped because ftol_rel or ftol_abs was reached.
#>
#> Model Type: Multinomial Logit
#> Model Space: Preference
#> Model Run: 1 of 1
#> Iterations: 42
#> Elapsed Time: 0h:0m:0.05s
#> Algorithm: NLOPT_LD_LBFGS
#> Weights Used?: FALSE
#> Robust? FALSE
#>
#> Model Coefficients:
#> Estimate Std. Error z-value Pr(>|z|)
#> price -0.389813 0.045256 -8.6135 < 2.2e-16 ***
#> feat 0.422188 0.122588 3.4440 0.0005732 ***
#> brandhiland -1.692896 0.623122 -2.7168 0.0065917 **
#> brandweight -2.226187 0.561605 -3.9640 7.371e-05 ***
#> brandyoplait 0.441559 0.450562 0.9800 0.3270779
#> price:brandhiland -0.434214 0.115521 -3.7587 0.0001708 ***
#> price:brandweight 0.199624 0.069825 2.8589 0.0042512 **
#> price:brandyoplait 0.033040 0.050385 0.6558 0.5119777
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Log-Likelihood: -2643.2048908
#> Null Log-Likelihood: -3343.7419990
#> AIC: 5302.4097816
#> BIC: 5348.7155000
#> McFadden R2: 0.2095069
#> Adj McFadden R2: 0.2071144
#> Number of Observations: 2412.0000000If you want to include interactions with individual-specific
variables (for example, to assess the difference in an effect between
groups of respondents), you should not include the
individual-specific variable interactions using * in
pars. This is because interactions inside pars
automatically generate the interaction coefficient as well as
coefficients for each covariate.
For example, if you had a group variable that determined
whether individuals belongs to group A or group
B, including price*group in pars
would create coefficients for price, groupA,
and price:groupA, but the groupA coefficient
would be unidentified. In this case, you should only include
price and price:groupA in the model. For now,
the only way to handle this situation is to manually create dummy-coded
interaction variables to include in the model.
To illustrate how one might do this, consider if the
yogurt data frame had two groups of individuals:
A and B. For simple illustration, I’ll define
these groups arbitrarily based on whether or not the obsID
is even or odd:
An interaction between the group variable and
price can be included in the model by first manually
creating a price_groupA interaction variable and then
including it in pars:
# Create dummy coefficients for group interaction with price
yogurt$price_groupA <- yogurt$price*yogurt$groupA
model_price_group <- logitr(
data = yogurt,
outcome = 'choice',
obsID = 'obsID',
pars = c('price', 'feat', 'brand', 'price_groupA')
)The model now has attribute coefficients for price,
feat, and brand as well as an interaction
between the group and price:
summary(model_price_group)
#> =================================================
#>
#> Model estimated on: Fri Nov 01 08:44:08 PM 2024
#>
#> Using logitr version: 1.1.2
#>
#> Call:
#> logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = c("price",
#> "feat", "brand", "price_groupA"))
#>
#> Frequencies of alternatives:
#> 1 2 3 4
#> 0.402156 0.029436 0.229270 0.339138
#>
#> Exit Status: 3, Optimization stopped because ftol_rel or ftol_abs was reached.
#>
#> Model Type: Multinomial Logit
#> Model Space: Preference
#> Model Run: 1 of 1
#> Iterations: 26
#> Elapsed Time: 0h:0m:0.03s
#> Algorithm: NLOPT_LD_LBFGS
#> Weights Used?: FALSE
#> Robust? FALSE
#>
#> Model Coefficients:
#> Estimate Std. Error z-value Pr(>|z|)
#> price -0.3680634 0.0273911 -13.4373 < 2.2e-16 ***
#> feat 0.4915271 0.1200725 4.0936 4.248e-05 ***
#> brandhiland -3.7155231 0.1454216 -25.5500 < 2.2e-16 ***
#> brandweight -0.6411384 0.0544999 -11.7640 < 2.2e-16 ***
#> brandyoplait 0.7345568 0.0806444 9.1086 < 2.2e-16 ***
#> price_groupA 0.0030007 0.0254484 0.1179 0.9061
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Log-Likelihood: -2656.8808982
#> Null Log-Likelihood: -3343.7419990
#> AIC: 5325.7617965
#> BIC: 5360.4911000
#> McFadden R2: 0.2054169
#> Adj McFadden R2: 0.2036225
#> Number of Observations: 2412.0000000Suppose I want to include an interaction between two variables and I
also want one of those variables to be modeled as normally distributed
across the population. The example below illustrates this cases, where a
price*feat interaction is specified and the
feat parameter is modeled as normally distributed by
setting randPars = c(feat = "n"):
model_price_feat_mxl <- logitr(
data = yogurt,
outcome = 'choice',
obsID = 'obsID',
pars = c('price', 'feat', 'brand', 'price*feat'),
randPars = c(feat = "n")
)In this case, the price*feat interaction parameter is
interpreted as a difference in the feat_mu parameter and
price; that is, it an interaction in the mean feat
parameter and price:
summary(model_price_feat_mxl)
#> =================================================
#>
#> Model estimated on: Fri Nov 01 08:44:08 PM 2024
#>
#> Using logitr version: 1.1.2
#>
#> Call:
#> logitr(data = yogurt, outcome = "choice", obsID = "obsID", pars = c("price",
#> "feat", "brand", "price*feat"), randPars = c(feat = "n"))
#>
#> Frequencies of alternatives:
#> 1 2 3 4
#> 0.402156 0.029436 0.229270 0.339138
#>
#> Exit Status: 3, Optimization stopped because ftol_rel or ftol_abs was reached.
#>
#> Model Type: Mixed Logit
#> Model Space: Preference
#> Model Run: 1 of 1
#> Iterations: 34
#> Elapsed Time: 0h:0m:1s
#> Algorithm: NLOPT_LD_LBFGS
#> Weights Used?: FALSE
#> Robust? FALSE
#>
#> Model Coefficients:
#> Estimate Std. Error z-value Pr(>|z|)
#> price -0.388123 0.027026 -14.3612 < 2.2e-16 ***
#> feat 0.829004 0.552278 1.5011 0.1333
#> brandhiland -3.991559 0.165894 -24.0609 < 2.2e-16 ***
#> brandweight -0.662086 0.055779 -11.8698 < 2.2e-16 ***
#> brandyoplait 0.787718 0.086232 9.1348 < 2.2e-16 ***
#> price:feat -0.076733 0.071069 -1.0797 0.2803
#> sd_feat -2.341507 0.493413 -4.7455 2.08e-06 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Log-Likelihood: -2645.2196929
#> Null Log-Likelihood: -3343.7419990
#> AIC: 5304.4393858
#> BIC: 5344.9569000
#> McFadden R2: 0.2089044
#> Adj McFadden R2: 0.2068109
#> Number of Observations: 2412.0000000
#>
#> Summary of 10k Draws for Random Coefficients:
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> feat -Inf -0.747793 0.8307929 0.8322819 2.409918 Inf