# Logistic regression using z-standardized values

library(tidyverse)  # data wrangling
library(easystats)

# 2 Data

data(mtcars)

# 3 Motivation

In this post, we’ll investigate the consequence of z-standardizing the predictor variables, and in addition the outcome variable in a simple logistic regression setting.

Do some coefficients change as a result of standardizing the values?

# 4 EDA

mtcars |>
group_by(am) |>
summarise(mpg_Avg = mean(mpg))
#> # A tibble: 2 × 2
#>      am mpg_Avg
#>   <dbl>   <dbl>
#> 1     0    17.1
#> 2     1    24.4

As we can see, am=1, i.e., manual (gear shifting) cars have a better mpg value.

# 5 Model with raw values

mod_raw <- glm(am ~ mpg, data = mtcars, family = "binomial")
parameters(mod_raw, exponentiate = TRUE)
#> Parameter   | Odds Ratio |       SE |       95% CI |     z |     p
#> ------------------------------------------------------------------
#> (Intercept) |   1.36e-03 | 3.19e-03 | [0.00, 0.06] | -2.81 | 0.005
#> mpg         |       1.36 |     0.16 | [1.13, 1.80] |  2.67 | 0.008

The odds ratio of 1.36 means that for every one-unit increase in mpg, the odds of a car having an manual transmission increase by 36%.

Note that the logistic regression (in R) models the second level of the outcome variable (see here for more information).

# 6 Model with am as factor-Variable

mtcars <-
mtcars |>
mutate(am_f = factor(am))

levels(mtcars$am_f) #> [1] "0" "1" mod_raw_f <- glm(am ~ mpg, data = mtcars, family = "binomial") parameters(mod_raw, exponentiate = TRUE) #> Parameter | Odds Ratio | SE | 95% CI | z | p #> ------------------------------------------------------------------ #> (Intercept) | 1.36e-03 | 3.19e-03 | [0.00, 0.06] | -2.81 | 0.005 #> mpg | 1.36 | 0.16 | [1.13, 1.80] | 2.67 | 0.008 Identical! # 7 Visualizing pred_df <- tibble( mpg = seq(min(mtcars$mpg), max(mtcars\$mpg), by = .1),
am_pred = predict(mod_raw, type = "response", newdata = tibble(mpg))
)

ggplot(mtcars) +
aes(x = mpg, y = am) +
geom_point() +
geom_line(data = pred_df, aes(x = mpg, y = am_pred), color = "blue") +
labs(title = "Predicting manual gear shifting",
subtitle = "Logistic model")

# 8 Standardizing predictors

mtcars_z <-
mtcars |>
mutate(across(c(everything(),-am), ~standardize(.x)))

# 9 Model with z-scaled predictors

mod_z <- glm(am ~ mpg, data = mtcars_z, family = "binomial")
parameters(mod_z, exponentiate = TRUE)
#> Parameter   | Odds Ratio |   SE |        95% CI |     z |     p
#> ---------------------------------------------------------------
#> (Intercept) |       0.65 | 0.29 | [0.25,  1.58] | -0.96 | 0.338
#> mpg         |       6.36 | 4.40 | [2.09, 34.49] |  2.67 | 0.008

# 10 Model with all variables z-scaled

Note that it makes no sense to z-scale the outcome variable of a logistic regression.

# 11 Conclusion

As can be seen the Odds ratio gets really big after standardization.

# 12 Reproducibility

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