Welcome

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Congratulations on finishing Exam 1!

Topics

Mean-centering quantitative predictors
Using indicator variables for categorical predictors
Using interaction terms

Computational setup

# load packages
library(tidyverse)
library(tidymodels)
library(openintro)
library(patchwork)
library(knitr)
library(kableExtra)
library(colorblindr)

# set default theme and larger font size for ggplot2
ggplot2::theme_set(ggplot2::theme_minimal(base_size = 20))

Introduction

Data: Peer-to-peer lender

Today’s data is a sample of 50 loans made through a peer-to-peer lending club. The data is in the loan50 data frame in the openintro R package.

# A tibble: 50 × 4
   annual_income debt_to_income verified_income interest_rate
           <dbl>          <dbl> <fct>                   <dbl>
 1         59000         0.558  Not Verified            10.9 
 2         60000         1.31   Not Verified             9.92
 3         75000         1.06   Verified                26.3 
 4         75000         0.574  Not Verified             9.92
 5        254000         0.238  Not Verified             9.43
 6         67000         1.08   Source Verified          9.92
 7         28800         0.0997 Source Verified         17.1 
 8         80000         0.351  Not Verified             6.08
 9         34000         0.698  Not Verified             7.97
10         80000         0.167  Source Verified         12.6 
# … with 40 more rows

Variables

Predictors:

annual_income: Annual income
debt_to_income: Debt-to-income ratio, i.e. the percentage of a borrower’s total debt divided by their total income
verified_income: Whether borrower’s income source and amount have been verified (Not Verified, Source Verified, Verified)

Outcome: interest_rate: Interest rate for the loan

Outcome: `interest_rate`

min	median	max
5.31	9.93	26.3

Predictors

Data manipulation 1: Rescale income

loan50 <- loan50 %>%
  mutate(annual_income_th = annual_income / 1000)

ggplot(loan50, aes(x = annual_income_th)) +
  geom_histogram(binwidth = 20) +
  labs(title = "Annual income (in $1000s)")

Outcome vs. predictors

Fit regression model

int_fit <- linear_reg() %>%
  set_engine("lm") %>%
  fit(interest_rate ~ debt_to_income + verified_income  + annual_income_th,
      data = loan50)

Summarize model results

term	estimate	std.error	statistic	p.value	conf.low	conf.high
(Intercept)	10.726	1.507	7.116	0.000	7.690	13.762
debt_to_income	0.671	0.676	0.993	0.326	-0.690	2.033
verified_incomeSource Verified	2.211	1.399	1.581	0.121	-0.606	5.028
verified_incomeVerified	6.880	1.801	3.820	0.000	3.253	10.508
annual_income_th	-0.021	0.011	-1.804	0.078	-0.043	0.002

Describe the subset of borrowers who are expected to get an interest rate of 10.726% based on our model. Is this interpretation meaningful? Why or why not?

Mean-centered variables

Mean-centering

If we are interested in interpreting the intercept, we can mean-center the quantitative predictors in the model.

We can mean-center a quantitative predictor \(X_j\) using the following:

\[X_{j_{Cent}} = X_{j}- \bar{X}_{j}\]

If we mean-center all quantitative variables, then the intercept is interpreted as the expected value of the response variable when all quantitative variables are at their mean value.

Data manipulation 2: Mean-center numeric predictors

loan50 <- loan50 %>%
  mutate(
    debt_inc_cent = debt_to_income - mean(debt_to_income), 
    annual_income_th_cent = annual_income_th - mean(annual_income_th)
    )

Visualize mean-centered predictors

Using mean-centered variables in the model

How do you expect the model to change if we use the debt_inc_cent and annual_income_cent in the model?

# A tibble: 5 × 7
  term                  estimate std.error statistic  p.value conf.low conf.high
  <chr>                    <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
1 (Intercept)             9.44      0.977      9.66  1.50e-12   7.48    11.4    
2 debt_inc_cent           0.671     0.676      0.993 3.26e- 1  -0.690    2.03   
3 verified_incomeSourc…   2.21      1.40       1.58  1.21e- 1  -0.606    5.03   
4 verified_incomeVerif…   6.88      1.80       3.82  4.06e- 4   3.25    10.5    
5 annual_income_th_cent  -0.0205    0.0114    -1.80  7.79e- 2  -0.0434   0.00238

Original vs. mean-centered model

term	estimate
(Intercept)	10.726
debt_to_income	0.671
verified_incomeSource Verified	2.211
verified_incomeVerified	6.880
annual_income_th	-0.021

term	estimate
(Intercept)	9.444
debt_inc_cent	0.671
verified_incomeSource Verified	2.211
verified_incomeVerified	6.880
annual_income_th_cent	-0.021

Indicator variables

Dummy variables

Suppose there is a categorical variable with \(K\) categories (levels)
We can treat them as continuous variables, Or
We can make \(K-1\) indicator variables (by default)
An indicator variable takes values 1 or 0
- 1 if the observation belongs to that category
- 0 if the observation does not belong to that category
- all 0s if the observation belong to the benchmark category

Indicator variables

We can also make \(K\) indicator variables - one indicator for each category
An indicator variable takes values 1 or 0
- 1 if the observation belongs to that category
- 0 if the observation does not belong to that category
No intercept in this case

Data manipulation 3: Create indicator variables for `verified_income`

Since

loan50 <- loan50 %>%
  mutate(
    not_verified = if_else(verified_income == "Not Verified", 1, 0),
    source_verified = if_else(verified_income == "Source Verified", 1, 0),
    verified = if_else(verified_income == "Verified", 1, 0)
  )

# A tibble: 3 × 4
  verified_income not_verified source_verified verified
  <fct>                  <dbl>           <dbl>    <dbl>
1 Not Verified               1               0        0
2 Verified                   0               0        1
3 Source Verified            0               1        0

Indicators in the model

We will use \(K-1\) of the indicator variables in the model.
The baseline is the category that doesn’t have a term in the model.
The coefficients of the indicator variables in the model are interpreted as the expected change in the response compared to the baseline, holding all other variables constant.
This approach is also called dummy coding.

# A tibble: 3 × 3
  verified_income source_verified verified
  <fct>                     <dbl>    <dbl>
1 Not Verified                  0        0
2 Verified                      0        1
3 Source Verified               1        0

Interpreting `verified_income`

term	estimate	std.error	statistic	p.value	conf.low	conf.high
(Intercept)	9.444	0.977	9.663	0.000	7.476	11.413
debt_inc_cent	0.671	0.676	0.993	0.326	-0.690	2.033
verified_incomeSource Verified	2.211	1.399	1.581	0.121	-0.606	5.028
verified_incomeVerified	6.880	1.801	3.820	0.000	3.253	10.508
annual_income_th_cent	-0.021	0.011	-1.804	0.078	-0.043	0.002

The baseline category is Not verified.
People with source verified income are expected to take a loan with an interest rate that is 2.211% higher, on average, than the rate on loans to those whose income is not verified, holding all else constant.
People with verified income are expected to take a loan with an interest rate that is 6.880% higher, on average, than the rate on loans to those whose income is not verified, holding all else constant.

Interaction terms

Sometimes the relationship between a predictor variable and the response depends on the value of another predictor variable.
This is an interaction effect.
To account for this, we can include interaction terms in the model.

Interest rate vs. annual income

The lines are not parallel indicating there is an interaction effect. The slope of annual income differs based on the income verification.

Interaction term in model

int_cent_int_fit <- linear_reg() %>%
  set_engine("lm") %>%
  fit(interest_rate ~ debt_inc_cent  +  debt_inc_cent + 
        annual_income_th_cent + verified_income * annual_income_th_cent,
      data = loan50)

term	estimate	std.error	statistic	p.value
(Intercept)	9.484	0.989	9.586	0.000
debt_inc_cent	0.691	0.685	1.009	0.319
annual_income_th_cent	-0.007	0.020	-0.341	0.735
verified_incomeSource Verified	2.157	1.418	1.522	0.135
verified_incomeVerified	7.181	1.870	3.840	0.000
annual_income_th_cent:verified_incomeSource Verified	-0.016	0.026	-0.643	0.523
annual_income_th_cent:verified_incomeVerified	-0.032	0.033	-0.979	0.333

Interpreting interaction terms

What the interaction means: The effect of annual income on the interest rate differs by -0.016 when the income is source verified compared to when it is not verified, holding all else constant.
Interpreting annual_income for source verified: If the income is source verified, we expect the interest rate to decrease by 0.023% (-0.007 + -0.016) for each additional thousand dollars in annual income, holding all else constant.

Data manipulation 4: Create interaction variables

Defining the interaction variable in the model formula as verified_income * annual_income_th_cent is an implicit data manipulation step as well

Rows: 50
Columns: 9
$ `(Intercept)`                                          <dbl> 1, 1, 1, 1, 1, …
$ debt_inc_cent                                          <dbl> -0.16511719, 0.…
$ annual_income_th_cent                                  <dbl> -27.17, -26.17,…
$ `verified_incomeNot Verified`                          <dbl> 1, 1, 0, 1, 1, …
$ `verified_incomeSource Verified`                       <dbl> 0, 0, 0, 0, 0, …
$ verified_incomeVerified                                <dbl> 0, 0, 1, 0, 0, …
$ `annual_income_th_cent:verified_incomeNot Verified`    <dbl> -27.17, -26.17,…
$ `annual_income_th_cent:verified_incomeSource Verified` <dbl> 0.00, 0.00, 0.0…
$ `annual_income_th_cent:verified_incomeVerified`        <dbl> 0.00, 0.00, -11…

Transformation

Data manipulation 5: Transformation on variables

Linearity is with respect of \(\beta\): \(y = \beta_0+\beta_1 x^2\) is also a linear regression
For right-skewed long tail distributed variable (financial data), log transformation : decrease the variability of data and make data conform more closely to the normal distribution

Data manipulation 5: log transformation

Interpretation on log-scale

As annual income increase by 10% (\(\log(x+0.1x)=\log(1.1\times x)=log(1.1)+\log x\)), the interest rate is expected to increase by \(\beta \times \log(1.1)\) on average, hold … constant.

term	estimate	std.error	statistic	p.value
(Intercept)	35.144	14.200	2.475	0.017
debt_to_income	0.725	0.671	1.081	0.286
verified_incomeSource Verified	2.140	1.397	1.532	0.133
verified_incomeVerified	7.032	1.809	3.888	0.000
annual_income_log	-2.338	1.260	-1.855	0.070

Wrap up

Recap

Mean-centering quantitative predictors
Using indicator variables for categorical predictors
Using interaction terms

Looking backward

Data manipulation, with dplyr (from tidyverse):

loan50 %>%
  select(interest_rate, annual_income, debt_to_income, verified_income) %>%
  mutate(
    # 1. rescale income
    annual_income_th = annual_income / 1000,
    # 2. mean-center quantitative predictors
    debt_inc_cent = debt_to_income - mean(debt_to_income),
    annual_income_th_cent = annual_income_th - mean(annual_income_th),
    # 3. create dummy variables for verified_income
    source_verified = if_else(verified_income == "Source Verified", 1, 0),
    verified = if_else(verified_income == "Verified", 1, 0),
    # 4. create interaction variables
    `annual_income_th_cent:verified_incomeSource Verified` = annual_income_th_cent * source_verified,
    `annual_income_th_cent:verified_incomeVerified` = annual_income_th_cent * verified
  )

Looking forward

Feature engineering, with recipes (from tidymodels):

loan_rec <- recipe( ~ ., data = loan50) %>%
  # 1. rescale income
  step_mutate(annual_income_th = annual_income / 1000) %>%
  # 2. mean-center quantitative predictors
  step_center(all_numeric_predictors()) %>%
  # 3. create dummy variables for verified_income
  step_dummy(verified_income) %>%
  # 4. create interaction variables
  step_interact(terms = ~ annual_income_th:verified_income)

Recipe

loan_rec

Recipe

Inputs:

      role #variables
 predictor         25

Operations:

Variable mutation for annual_income / 1000
Centering for all_numeric_predictors()
Dummy variables from verified_income
Interactions with annual_income_th:verified_income

Welcome

Announcements

Topics

Computational setup

Introduction

Data: Peer-to-peer lender

Variables

Outcome: interest_rate

Predictors

Data manipulation 1: Rescale income

Outcome vs. predictors

Fit regression model

Summarize model results

Mean-centered variables

Mean-centering

Data manipulation 2: Mean-center numeric predictors

Visualize mean-centered predictors

Using mean-centered variables in the model

Original vs. mean-centered model

Indicator variables

Dummy variables

Indicator variables

Data manipulation 3: Create indicator variables for verified_income

Indicators in the model

Interpreting verified_income

Interaction terms

Interaction terms

Interest rate vs. annual income

Interaction term in model

Interpreting interaction terms

Data manipulation 4: Create interaction variables

Transformation

Data manipulation 5: Transformation on variables

Data manipulation 5: log transformation

Interpretation on log-scale

Wrap up

Recap

Looking backward

Looking forward

Recipe

Outcome: `interest_rate`

Data manipulation 3: Create indicator variables for `verified_income`

Interpreting `verified_income`