Variable transformations

Log-transformed predictor

Prof. Maria Tackett

Oct 17, 2022

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Other notes:

  • We will review office hours schedule to make sure they are scheduled during times that don’t have major conflicts

  • Grading

    • Wording in statistics matters! For example - these are two different statements:

      • Correct: For every one month in age, we expect the respiratory rate to decrease by 0.659 breaths per minute, on average.
      • Incorrect: For every one month in age, the respiratory rate will decrease by 0.659 breaths per minute.

    • Full credit is awarded for (1) using the most appropriate methods (e.g., appropriate summary statistics given a distribution), (2) comprehensively and accurately justifying response, (3) consistency in response and explanation.

    • There is an example in the lecture notes, application exercises, and/or readings.

Regrade requests

  • Dates they are available in email from Gradescope
  • Review solutions and ask during office hours first
  • Do not submit regrade requests to dispute point values
  • Question is completely regraded by Prof. Tackett or Head TA
  • Policy in syllabus

Topics

  • Log transformation on the response variable

  • Log transformation on the predictor variable

Computational set up

library(tidyverse)
library(tidymodels)
library(knitr)
library(Sleuth3) 
library(patchwork)

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

Recap

Respiratory Rate vs. Age

  • A high respiratory rate can potentially indicate a respiratory infection in children. In order to determine what indicates a “high” rate, we first want to understand the relationship between a child’s age and their respiratory rate.

  • The data contain the respiratory rate for 618 children ages 15 days to 3 years. It was obtained from the Sleuth3 R package and is originally form a 1994 publication “Reference Values for Respiratory Rate in the First 3 Years of Life”.

  • Variables:

    • Age: age in months
    • Rate: respiratory rate (breaths per minute)

Rate vs. Age

Training + test sets

set.seed(101222)
# iniital split 
resp_split <- initial_split(respiratory)

# training set
resp_train <- training(resp_split)

# test set
resp_test <- testing(resp_split)

Model 1: Rate vs. Age

resp_fit <- linear_reg() |>
  set_engine("lm") |>
  fit(Rate ~ Age, data = resp_train)

tidy(resp_fit) |>
  kable(digits = 3)
term estimate std.error statistic p.value
(Intercept) 46.458 0.589 78.924 0
Age -0.659 0.034 -19.498 0

Model 1: Residuals

Consider different transformations…

Model 2: log(Rate) vs. Age

term estimate std.error statistic p.value
(Intercept) 3.831 0.015 259.086 0
Age -0.018 0.001 -21.243 0


  • Slope: For each additional month in a child’s age, the median respiratory rate is expected to multiply by a factor of 0.982 [exp(-0.018)].

  • Intercept: The median respiratory rate for children who are 0 months old is expected to be 29.4 [exp(3.381)].

Model 2: Residuals

Compare residual plots

Log transformation on a predictor variable

Log Transformation on \(X\)

Try a transformation on \(X\) if the scatterplot shows some curvature but the variance is constant for all values of \(X\)

Rate vs. log(Age)

Model with Transformation on \(X\)

Suppose we have the following regression equation:

\[\hat{Y} = \hat{\beta}_0 + \hat{\beta}_1 \log(X)\]

  • Intercept: When \(X = 1\) \((\log(X) = 0)\), \(Y\) is expected to be \(\hat{\beta}_0\) (i.e. the mean of \(Y\) is \(\hat{\beta}_0\))

  • Slope: When \(X\) is multiplied by a factor of \(\mathbf{C}\), the mean of \(Y\) is expected to increase by \(\boldsymbol{\hat{\beta}_1}\mathbf{\log(C)}\) units

    • Example: when \(X\) is multiplied by a factor of 2, \(Y\) is expected to increase by \(\boldsymbol{\hat{\beta}_1}\mathbf{\log(2)}\) units

Model 3: Rate vs. log(Age)

term estimate std.error statistic p.value
(Intercept) 49.397 0.755 65.436 0
log_age -5.668 0.311 -18.248 0


Interpret the slope and intercept in the context of the data.

04:00

Model 3: Residuals

Choose a model

Recall the goal of the analysis:

In order to determine what indicates a “high” rate, we first want to understand the relationship between a child’s age and their respiratory rate.


Which is the preferred metric to compare the models - \(R^2\) or RMSE?

Compare models on testing data

Rate vs. Age log(Rate) vs. Age Rate vs. log(Age)
0.549 0.596 0.559


Which model would you choose?

Learn more

See Log Transformations in Linear Regression for more details about interpreting regression models with log-transformed variables.

🔗 Week 08