03. Linear regression
GS
10/8/2021
library(ISLR)
library(MASS) # for the Boston data set
library(tidymodels)
library(GGally)
library(corrplot)
theme_set(theme_bw())An Introduction to Statistical Learning (2nd ed.)
Labs
Chapter 03
Linear regression
The Boston data set contain various statistics for 506 neighborhoods in Boston. We will build a simple linear regression model that related the median value of owner-occupied homes (medv) as the response with a variable indicating the percentage of the population that belongs to a lower status (lstat) as the predictor.
The Boston data set is quite outdated and contains some really unfortunate variables.
boston <- tibble(Boston)
glimpse(boston)Rows: 506
Columns: 14
$ crim <dbl> 0.00632, 0.02731, 0.02729, 0.03237, 0.06905, 0.02985, 0.08829,~
$ zn <dbl> 18.0, 0.0, 0.0, 0.0, 0.0, 0.0, 12.5, 12.5, 12.5, 12.5, 12.5, 1~
$ indus <dbl> 2.31, 7.07, 7.07, 2.18, 2.18, 2.18, 7.87, 7.87, 7.87, 7.87, 7.~
$ chas <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,~
$ nox <dbl> 0.538, 0.469, 0.469, 0.458, 0.458, 0.458, 0.524, 0.524, 0.524,~
$ rm <dbl> 6.575, 6.421, 7.185, 6.998, 7.147, 6.430, 6.012, 6.172, 5.631,~
$ age <dbl> 65.2, 78.9, 61.1, 45.8, 54.2, 58.7, 66.6, 96.1, 100.0, 85.9, 9~
$ dis <dbl> 4.0900, 4.9671, 4.9671, 6.0622, 6.0622, 6.0622, 5.5605, 5.9505~
$ rad <int> 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4,~
$ tax <dbl> 296, 242, 242, 222, 222, 222, 311, 311, 311, 311, 311, 311, 31~
$ ptratio <dbl> 15.3, 17.8, 17.8, 18.7, 18.7, 18.7, 15.2, 15.2, 15.2, 15.2, 15~
$ black <dbl> 396.90, 396.90, 392.83, 394.63, 396.90, 394.12, 395.60, 396.90~
$ lstat <dbl> 4.98, 9.14, 4.03, 2.94, 5.33, 5.21, 12.43, 19.15, 29.93, 17.10~
$ medv <dbl> 24.0, 21.6, 34.7, 33.4, 36.2, 28.7, 22.9, 27.1, 16.5, 18.9, 15~colSums(is.na(boston)) crim zn indus chas nox rm age dis rad tax
0 0 0 0 0 0 0 0 0 0
ptratio black lstat medv
0 0 0 0 corr <- cor(boston)
corrplot(corr, method = "square", type = "upper")
ggpairs(boston)
ggplot(boston, aes(medv)) +
geom_density(fill = 'lightgrey') +
xlab("Median value") +
ggtitle("Median value of Boston homes in 506 neighborhoods ")
We start by creating a parsnip specification for a linear regression model.
lm_model <-
linear_reg() %>%
set_engine("lm")
lm_modelLinear Regression Model Specification (regression)
Computational engine: lm #While it is unnecessary to set the mode for a linear regression since it can only be regression, we continue to do it in these labs to be explicit.
#The specification doesn’t perform any calculations by itself. It is just a specification of what we want to do.Once we have the specification we can fit it by supplying a formula expression and the data we want to fit the model on. The formula is written on the form y ~ x where y is the name of the response and x is the name of the predictors. The names used in the formula should match the names of the variables in the data set passed to data.
lm_fit <-
fit(lm_model,
medv ~ lstat,
data = boston)
lm_fitparsnip model object
Fit time: 0ms
Call:
stats::lm(formula = medv ~ lstat, data = data)
Coefficients:
(Intercept) lstat
34.55 -0.95 The result of this fit is a parsnip model object. This object contains the underlying fit as well as some parsnip-specific information. If we want to look at the underlying fit object we can access it with lm_fit$fit or with.
lm_fit %>%
pluck("fit") %>%
summary()
Call:
stats::lm(formula = medv ~ lstat, data = data)
Residuals:
Min 1Q Median 3Q Max
-15.168 -3.990 -1.318 2.034 24.500
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 34.55384 0.56263 61.41 <2e-16 ***
lstat -0.95005 0.03873 -24.53 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 6.216 on 504 degrees of freedom
Multiple R-squared: 0.5441, Adjusted R-squared: 0.5432
F-statistic: 601.6 on 1 and 504 DF, p-value: < 2.2e-16We can use packages from the broom package to extract key information out of the model objects in tidy formats.
the tidy() function returns the parameter estimates of a lm object.
tidy(lm_fit)# A tibble: 2 x 5
term estimate std.error statistic p.value
<chr> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) 34.6 0.563 61.4 3.74e-236
2 lstat -0.950 0.0387 -24.5 5.08e- 88and glance() can be used to extract the model statistics.
glance1 <- glance(lm_fit)
glance1 <- glance1 %>%
select(-c(logLik, deviance, nobs)) %>%
mutate(model = "medv ~ lstat")
glance1# A tibble: 1 x 10
r.squared adj.r.squared sigma statistic p.value df AIC BIC df.residual
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
1 0.544 0.543 6.22 602. 5.08e-88 1 3289. 3302. 504
# ... with 1 more variable: model <chr>We need to explicitly supply the data set that the predictions should be performed on via the new_data argument.
predict(lm_fit, boston)# A tibble: 506 x 1
.pred
<dbl>
1 29.8
2 25.9
3 30.7
4 31.8
5 29.5
6 29.6
7 22.7
8 16.4
9 6.12
10 18.3
# ... with 496 more rowsNotice how the predictions are returned as a tibble. This will always be the case for parsnip models, no matter what engine is used. This is very useful since consistency allows us to combine data sets easily.
We can also return other types of predicts by specifying the type argument. Setting type = “conf_int” return a 95% confidence interval.
predict(lm_fit, boston, type = "conf_int")# A tibble: 506 x 2
.pred_lower .pred_upper
<dbl> <dbl>
1 29.0 30.6
2 25.3 26.5
3 29.9 31.6
4 30.8 32.7
5 28.7 30.3
6 28.8 30.4
7 22.2 23.3
8 15.6 17.1
9 4.70 7.54
10 17.7 18.9
# ... with 496 more rowspredict(lm_fit, boston, type = "pred_int")# A tibble: 506 x 2
.pred_lower .pred_upper
<dbl> <dbl>
1 17.6 42.1
2 13.6 38.1
3 18.5 43.0
4 19.5 44.0
5 17.3 41.7
6 17.4 41.8
7 10.5 35.0
8 4.13 28.6
9 -6.18 18.4
10 6.08 30.5
# ... with 496 more rowsYou can get the same results using the augment() function to same you a little bit of typing.
augment(lm_fit, boston) %>%
select(medv, .pred, .resid)# A tibble: 506 x 3
medv .pred .resid
<dbl> <dbl> <dbl>
1 24 29.8 -5.82
2 21.6 25.9 -4.27
3 34.7 30.7 3.97
4 33.4 31.8 1.64
5 36.2 29.5 6.71
6 28.7 29.6 -0.904
7 22.9 22.7 0.155
8 27.1 16.4 10.7
9 16.5 6.12 10.4
10 18.9 18.3 0.592
# ... with 496 more rowsMultiple Linear Regression
lm_fit2 <-
fit(lm_model,
medv ~ lstat + age,
data = boston)
lm_fit2parsnip model object
Fit time: 0ms
Call:
stats::lm(formula = medv ~ lstat + age, data = data)
Coefficients:
(Intercept) lstat age
33.22276 -1.03207 0.03454 lm_fit2 %>%
pluck("fit") %>%
summary()
Call:
stats::lm(formula = medv ~ lstat + age, data = data)
Residuals:
Min 1Q Median 3Q Max
-15.981 -3.978 -1.283 1.968 23.158
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 33.22276 0.73085 45.458 < 2e-16 ***
lstat -1.03207 0.04819 -21.416 < 2e-16 ***
age 0.03454 0.01223 2.826 0.00491 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 6.173 on 503 degrees of freedom
Multiple R-squared: 0.5513, Adjusted R-squared: 0.5495
F-statistic: 309 on 2 and 503 DF, p-value: < 2.2e-16tidy(lm_fit2)# A tibble: 3 x 5
term estimate std.error statistic p.value
<chr> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) 33.2 0.731 45.5 2.94e-180
2 lstat -1.03 0.0482 -21.4 8.42e- 73
3 age 0.0345 0.0122 2.83 4.91e- 3glance2 <- glance(lm_fit2)
glance2 <- glance2 %>%
select(-c(logLik, deviance, nobs)) %>%
mutate(model = "medv ~ lstat + age")
glance2# A tibble: 1 x 10
r.squared adj.r.squared sigma statistic p.value df AIC BIC df.residual
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
1 0.551 0.549 6.17 309. 2.98e-88 2 3283. 3300. 503
# ... with 1 more variable: model <chr>augment(lm_fit2, boston) %>%
select(medv, .pred, .resid)# A tibble: 506 x 3
medv .pred .resid
<dbl> <dbl> <dbl>
1 24 30.3 -6.34
2 21.6 26.5 -4.92
3 34.7 31.2 3.53
4 33.4 31.8 1.63
5 36.2 29.6 6.61
6 28.7 29.9 -1.17
7 22.9 22.7 0.205
8 27.1 16.8 10.3
9 16.5 5.79 10.7
10 18.9 18.5 0.358
# ... with 496 more rowsInteraction terms
Adding interaction terms are quite easy to do using formula expressions. However, the syntax used to describe them isn’t accepted by all engines so we will go over how to include interaction terms using recipes as well.
There are two ways on including an interaction term; x:y and x * y
- x:y will include the interaction between x and y,
- x * y will include the interaction between x and y, x, and y, e.i. it is short for x:y + x + y.
with that out of the way let expand lm_fit2 by adding an interaction term
lm_fit3 <-
fit(lm_model,
medv ~ lstat * age,
data = boston)
lm_fit3parsnip model object
Fit time: 0ms
Call:
stats::lm(formula = medv ~ lstat * age, data = data)
Coefficients:
(Intercept) lstat age lstat:age
36.0885359 -1.3921168 -0.0007209 0.0041560 lm_fit3 %>%
pluck("fit") %>%
summary()
Call:
stats::lm(formula = medv ~ lstat * age, data = data)
Residuals:
Min 1Q Median 3Q Max
-15.806 -4.045 -1.333 2.085 27.552
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 36.0885359 1.4698355 24.553 < 2e-16 ***
lstat -1.3921168 0.1674555 -8.313 8.78e-16 ***
age -0.0007209 0.0198792 -0.036 0.9711
lstat:age 0.0041560 0.0018518 2.244 0.0252 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 6.149 on 502 degrees of freedom
Multiple R-squared: 0.5557, Adjusted R-squared: 0.5531
F-statistic: 209.3 on 3 and 502 DF, p-value: < 2.2e-16tidy(lm_fit3)# A tibble: 4 x 5
term estimate std.error statistic p.value
<chr> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) 36.1 1.47 24.6 4.91e-88
2 lstat -1.39 0.167 -8.31 8.78e-16
3 age -0.000721 0.0199 -0.0363 9.71e- 1
4 lstat:age 0.00416 0.00185 2.24 2.52e- 2glance3 <- glance(lm_fit3)
glance3 <- glance3 %>%
select(-c(logLik, deviance, nobs)) %>%
mutate(model = "medv ~ lstat*age")
glance3# A tibble: 1 x 10
r.squared adj.r.squared sigma statistic p.value df AIC BIC df.residual
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
1 0.556 0.553 6.15 209. 4.86e-88 3 3280. 3301. 502
# ... with 1 more variable: model <chr>augment(lm_fit3, boston) %>%
select(medv, .pred, .resid)# A tibble: 506 x 3
medv .pred .resid
<dbl> <dbl> <dbl>
1 24 30.5 -6.46
2 21.6 26.3 -4.70
3 34.7 31.5 3.24
4 33.4 32.5 0.878
5 36.2 29.8 6.37
6 28.7 30.1 -1.36
7 22.9 22.2 0.723
8 27.1 17.0 10.1
9 16.5 6.79 9.71
10 18.9 18.3 0.574
# ... with 496 more rowsSometimes we want to perform transformations, and we want those transformations to be applied, as part of the model fit as a pre-processing step. We will use the recipes package for this task.
We use the step_interact() to specify the interaction term. Next, we create a workflow object to combine the linear regression model specification lm_spec with the pre-processing specification rec_spec_interact which can then be fitted much like a parsnip model specification.
lm_rec_interact <-
recipe(medv ~ lstat + age, data = boston) %>%
step_interact(~ lstat : age)lm_wf_interact <-
workflow() %>%
add_model(lm_model) %>%
add_recipe(lm_rec_interact)
lm_wf_interact== Workflow ====================================================================
Preprocessor: Recipe
Model: linear_reg()
-- Preprocessor ----------------------------------------------------------------
1 Recipe Step
* step_interact()
-- Model -----------------------------------------------------------------------
Linear Regression Model Specification (regression)
Computational engine: lm fit(lm_wf_interact, boston)== Workflow [trained] ==========================================================
Preprocessor: Recipe
Model: linear_reg()
-- Preprocessor ----------------------------------------------------------------
1 Recipe Step
* step_interact()
-- Model -----------------------------------------------------------------------
Call:
stats::lm(formula = ..y ~ ., data = data)
Coefficients:
(Intercept) lstat age lstat_x_age
36.0885359 -1.3921168 -0.0007209 0.0041560 Non-linear transformations of the predictors
Much like we could use recipes to create interaction terms between values are we able to apply transformations to individual variables as well. If you are familiar with the dplyr package then you know how to mutate() which works in much the same way using step_mutate().
You would want to keep as much of the pre-processing inside recipes such that the transformation will be applied consistently to new data.
lm_rec_pow2 <-
recipe(medv ~ lstat, data = boston) %>%
step_mutate(lstat2 = lstat ^ 2)
lm_wf_pow2 <-
workflow() %>%
add_model(lm_model) %>%
add_recipe(lm_rec_pow2)
lm_fit_pow2 <- fit(lm_wf_pow2, boston)
tidy(lm_fit_pow2)# A tibble: 3 x 5
term estimate std.error statistic p.value
<chr> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) 42.9 0.872 49.1 3.50e-194
2 lstat -2.33 0.124 -18.8 2.55e- 60
3 lstat2 0.0435 0.00375 11.6 7.63e- 28glance4 <- glance(lm_fit_pow2)
glance4 <- glance4 %>%
select(-c(logLik, deviance, nobs)) %>%
mutate(model = "medv ~ lstat ^2")
glance4# A tibble: 1 x 10
r.squared adj.r.squared sigma statistic p.value df AIC BIC
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 0.641 0.639 5.52 449. 1.56e-112 2 3171. 3187.
# ... with 2 more variables: df.residual <int>, model <chr>lm log model
lm_rec_log <-
recipe(
medv ~ lstat, data = boston) %>%
step_log(lstat)
lm_rec_logData Recipe
Inputs:
role #variables
outcome 1
predictor 1
Operations:
Log transformation on lstatlm_wf_log <-
workflow() %>%
add_model(lm_model) %>%
add_recipe(lm_rec_log)
lm_fit_log <- fit(lm_wf_log, boston)
tidy(lm_fit_log)# A tibble: 2 x 5
term estimate std.error statistic p.value
<chr> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) 52.1 0.965 54.0 1.02e-211
2 lstat -12.5 0.395 -31.6 9.28e-122glance5 <- glance(lm_fit_log)
glance5 <- glance5 %>%
select(-c(logLik, deviance, nobs)) %>%
mutate(model = "medv ~ log(lstat)")
glance5# A tibble: 1 x 10
r.squared adj.r.squared sigma statistic p.value df AIC BIC
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 0.665 0.664 5.33 1000. 9.28e-122 1 3133. 3146.
# ... with 2 more variables: df.residual <int>, model <chr>Comparing models
comparison <-
bind_rows(glance1, glance2, glance3, glance4, glance5) %>%
select(model, everything())
comparison# A tibble: 5 x 10
model r.squared adj.r.squared sigma statistic p.value df AIC BIC
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 medv ~ ls~ 0.544 0.543 6.22 602. 5.08e- 88 1 3289. 3302.
2 medv ~ ls~ 0.551 0.549 6.17 309. 2.98e- 88 2 3283. 3300.
3 medv ~ ls~ 0.556 0.553 6.15 209. 4.86e- 88 3 3280. 3301.
4 medv ~ ls~ 0.641 0.639 5.52 449. 1.56e-112 2 3171. 3187.
5 medv ~ lo~ 0.665 0.664 5.33 1000. 9.28e-122 1 3133. 3146.
# ... with 1 more variable: df.residual <int>Qualitative predictors
We will now turn our attention to the Carseats data set. We will attempt to predict Sales of child car seats in 400 locations based on a number of predictors. One of these variables is ShelveLoc which is a qualitative predictor that indicates the quality of the shelving location. ShelveLoc takes on three possible values.
Bad
Medium
Good
If you pass such a variable to lm() it will read it and generate dummy variables automatically using the following convention.
carseats = tibble(Carseats)
glimpse(carseats)Rows: 400
Columns: 11
$ Sales <dbl> 9.50, 11.22, 10.06, 7.40, 4.15, 10.81, 6.63, 11.85, 6.54, ~
$ CompPrice <dbl> 138, 111, 113, 117, 141, 124, 115, 136, 132, 132, 121, 117~
$ Income <dbl> 73, 48, 35, 100, 64, 113, 105, 81, 110, 113, 78, 94, 35, 2~
$ Advertising <dbl> 11, 16, 10, 4, 3, 13, 0, 15, 0, 0, 9, 4, 2, 11, 11, 5, 0, ~
$ Population <dbl> 276, 260, 269, 466, 340, 501, 45, 425, 108, 131, 150, 503,~
$ Price <dbl> 120, 83, 80, 97, 128, 72, 108, 120, 124, 124, 100, 94, 136~
$ ShelveLoc <fct> Bad, Good, Medium, Medium, Bad, Bad, Medium, Good, Medium,~
$ Age <dbl> 42, 65, 59, 55, 38, 78, 71, 67, 76, 76, 26, 50, 62, 53, 52~
$ Education <dbl> 17, 10, 12, 14, 13, 16, 15, 10, 10, 17, 10, 13, 18, 18, 18~
$ Urban <fct> Yes, Yes, Yes, Yes, Yes, No, Yes, Yes, No, No, No, Yes, Ye~
$ US <fct> Yes, Yes, Yes, Yes, No, Yes, No, Yes, No, Yes, Yes, Yes, N~levels(carseats$ShelveLoc)[1] "Bad" "Good" "Medium"carseats %>%
pull(ShelveLoc) %>%
contrasts() Good Medium
Bad 0 0
Good 1 0
Medium 0 1The step_dummy() will perform the same transformation of turning 1 qualitative with C levels into C-1 indicator variables. While this might seem unnecessary right now, some of the engines, later on, do not handle qualitative variables and this step would be necessary. We are also using the all_nominal_predictors() selector to select all character and factor predictor variables. This allows us to select by type rather than having to type out the names.
lm_rec_qual <-
recipe(Sales ~ ., data = carseats) %>%
step_dummy(all_nominal_predictors()) %>%
step_interact(~ Income: Advertising + Price: Age)
lm_wf_qual <-
workflow() %>%
add_model(lm_model) %>%
add_recipe(lm_rec_qual)
lm_wf_qual== Workflow ====================================================================
Preprocessor: Recipe
Model: linear_reg()
-- Preprocessor ----------------------------------------------------------------
2 Recipe Steps
* step_dummy()
* step_interact()
-- Model -----------------------------------------------------------------------
Linear Regression Model Specification (regression)
Computational engine: lm fit(lm_wf_qual, carseats)== Workflow [trained] ==========================================================
Preprocessor: Recipe
Model: linear_reg()
-- Preprocessor ----------------------------------------------------------------
2 Recipe Steps
* step_dummy()
* step_interact()
-- Model -----------------------------------------------------------------------
Call:
stats::lm(formula = ..y ~ ., data = data)
Coefficients:
(Intercept) CompPrice Income
6.5755654 0.0929371 0.0108940
Advertising Population Price
0.0702462 0.0001592 -0.1008064
Age Education ShelveLoc_Good
-0.0579466 -0.0208525 4.8486762
ShelveLoc_Medium Urban_Yes US_Yes
1.9532620 0.1401597 -0.1575571
Income_x_Advertising Price_x_Age
0.0007510 0.0001068 fit(lm_wf_qual, carseats) %>% tidy()# A tibble: 14 x 5
term estimate std.error statistic p.value
<chr> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) 6.58 1.01 6.52 2.22e- 10
2 CompPrice 0.0929 0.00412 22.6 1.64e- 72
3 Income 0.0109 0.00260 4.18 3.57e- 5
4 Advertising 0.0702 0.0226 3.11 2.03e- 3
5 Population 0.000159 0.000368 0.433 6.65e- 1
6 Price -0.101 0.00744 -13.5 1.74e- 34
7 Age -0.0579 0.0160 -3.63 3.18e- 4
8 Education -0.0209 0.0196 -1.06 2.88e- 1
9 ShelveLoc_Good 4.85 0.153 31.7 1.38e-109
10 ShelveLoc_Medium 1.95 0.126 15.5 1.34e- 42
11 Urban_Yes 0.140 0.112 1.25 2.13e- 1
12 US_Yes -0.158 0.149 -1.06 2.91e- 1
13 Income_x_Advertising 0.000751 0.000278 2.70 7.29e- 3
14 Price_x_Age 0.000107 0.000133 0.801 4.24e- 1fit(lm_wf_qual, carseats) %>% glance()# A tibble: 1 x 12
r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 0.876 0.872 1.01 210. 6.14e-166 13 -565. 1159. 1219.
# ... with 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>An Introduction to Statistcial Learning https://www.statlearning.com/
– END
sessionInfo()R version 4.1.0 (2021-05-18)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 19043)
Matrix products: default
locale:
[1] LC_COLLATE=Spanish_Mexico.1252 LC_CTYPE=Spanish_Mexico.1252
[3] LC_MONETARY=Spanish_Mexico.1252 LC_NUMERIC=C
[5] LC_TIME=Spanish_Mexico.1252
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] corrplot_0.90 GGally_2.1.2 yardstick_0.0.8 workflowsets_0.0.2
[5] workflows_0.2.2 tune_0.1.5 tidyr_1.1.3 tibble_3.1.2
[9] rsample_0.1.0 recipes_0.1.16 purrr_0.3.4 parsnip_0.1.6
[13] modeldata_0.1.1 infer_0.5.4 ggplot2_3.3.5 dplyr_1.0.7
[17] dials_0.0.9 scales_1.1.1 broom_0.7.8 tidymodels_0.1.3
[21] MASS_7.3-54 ISLR_1.2
loaded via a namespace (and not attached):
[1] sass_0.4.0 jsonlite_1.7.2 splines_4.1.0 foreach_1.5.1
[5] prodlim_2019.11.13 bslib_0.2.5.1 assertthat_0.2.1 highr_0.9
[9] GPfit_1.0-8 yaml_2.2.1 globals_0.14.0 ipred_0.9-11
[13] pillar_1.6.1 backports_1.2.1 lattice_0.20-44 glue_1.4.2
[17] pROC_1.17.0.1 digest_0.6.27 RColorBrewer_1.1-2 hardhat_0.1.6
[21] colorspace_2.0-2 htmltools_0.5.1.1 Matrix_1.3-3 plyr_1.8.6
[25] timeDate_3043.102 pkgconfig_2.0.3 lhs_1.1.1 DiceDesign_1.9
[29] listenv_0.8.0 gower_0.2.2 lava_1.6.9 farver_2.1.0
[33] generics_0.1.0 ellipsis_0.3.2 withr_2.4.2 furrr_0.2.3
[37] nnet_7.3-16 cli_3.0.0 survival_3.2-11 magrittr_2.0.1
[41] crayon_1.4.1 evaluate_0.14 future_1.21.0 fansi_0.5.0
[45] parallelly_1.26.1 class_7.3-19 prettyunits_1.1.1 tools_4.1.0
[49] lifecycle_1.0.0 stringr_1.4.0 munsell_0.5.0 compiler_4.1.0
[53] jquerylib_0.1.4 rlang_0.4.11 grid_4.1.0 iterators_1.0.13
[57] rstudioapi_0.13 labeling_0.4.2 rmarkdown_2.9 gtable_0.3.0
[61] codetools_0.2-18 reshape_0.8.8 DBI_1.1.1 R6_2.5.0
[65] lubridate_1.7.10 knitr_1.33 utf8_1.2.1 stringi_1.6.2
[69] parallel_4.1.0 Rcpp_1.0.7 vctrs_0.3.8 rpart_4.1-15
[73] tidyselect_1.1.1 xfun_0.24