Logistic Regression Model Selection

Author

Kabinga Chanda

1 Introduction

1.1 Background

The sinking of the RMS Titanic on April 15, 1912, remains one of the most infamous maritime disasters in history. The tragedy claimed over 1,500 lives and has since become a subject of extensive historical and statistical analysis. Understanding the factors that influenced passenger survival can provide insights into the social dynamics and emergency response protocols of the era.

1.2 Objective

This analysis employs logistic regression modeling to identify and quantify the factors associated with passenger survival on the Titanic. Specifically, we aim to:

  1. Examine the individual (univariable) associations between demographic and socioeconomic variables and survival outcomes
  2. Develop a multivariable model to assess the adjusted effects of these predictors while controlling for potential confounding
  3. Demonstrate a systematic approach to model selection in logistic regression

1.3 Dataset

The analysis uses the Titanic dataset available in R, which contains information on passengers’ survival status, sex, age (child or adult), and passenger class (1st, 2nd, 3rd, or crew). This dataset has been widely used in statistical education to illustrate classification and regression techniques.

1.4 Analytical Approach

We follow a two-stage modeling strategy commonly used in epidemiological and clinical research:

  1. Univariable screening: Each predictor is examined individually to identify variables with sufficient evidence of association (p < 0.2) for inclusion in the multivariable model
  2. Multivariable modeling: Variables that pass the screening threshold are included in a comprehensive model to estimate adjusted associations

This approach balances statistical efficiency with the need to control for confounding, providing a robust framework for understanding the complex relationships between passenger characteristics and survival.

2 Univariable Model

The univariable analysis examines the association between each predictor and survival independently. This screening step identifies variables with p-values < 0.2, which are then candidates for inclusion in the multivariable logistic regression model.

3 Univariable Model

The univariable analysis examines the association between each predictor and survival independently. This screening step identifies variables with p-values < 0.2, which are then candidates for inclusion in the multivariable logistic regression model.

Show the code 
df %>%
  select(Survived, Sex, Age, Class) %>%
  tbl_uvregression(
    method = glm,
    y = Survived,
    method.args = list(family = binomial),
    exponentiate = TRUE,
    label = list(
        Age ~ "Age (Years)",
        Sex ~ "Sex of the Passengers"
    )
  ) |> 
    bold_labels() |> 
    bold_p() |> 
    as_gt() |>  # Use as_gt() instead of gt()
    tab_header(
        title = md("**Univariable Logistic Regression Model**"),
        subtitle = md("Fate of passengers on the fatal maiden
                      voyage of the ocean liner 'Titanic'")
    ) |> 
    tab_source_note(
        source_note = md("Dawson, Robert J. MacG. (1995), The ‘Unusual Episode’ Data Revisited. Journal of Statistics Education, 3. *doi:10.1080/10691898.1995.11910499.*") 
    ) |> 
    opt_align_table_header(
        align = "left") |> 
     tab_options(
        table_body.hlines.color = "transparent",
        table_body.vlines.color = "transparent"
    )
Univariable Logistic Regression Model
Fate of passengers on the fatal maiden voyage of the ocean liner ‘Titanic’
Characteristic N OR 95% CI p-value
Sex of the Passengers 2,201


    Male
— —
    Female
10.1 8.05, 12.9 <0.001
Age (Years) 2,201


    Child
— —
    Adult
0.41 0.28, 0.61 <0.001
Class 2,201


    1st
— —
    2nd
0.42 0.31, 0.59 <0.001
    3rd
0.20 0.15, 0.27 <0.001
    Crew
0.19 0.14, 0.25 <0.001
Abbreviations: CI = Confidence Interval, OR = Odds Ratio
Dawson, Robert J. MacG. (1995), The ‘Unusual Episode’ Data Revisited. Journal of Statistics Education, 3. doi:10.1080/10691898.1995.11910499.

4 Fit Initial Multivaribale Model

After identifying significant predictors in the univariable analysis, we fit a multivariable logistic regression model to examine the adjusted associations while controlling for confounding variables.

Show the code 
full_model <- glm(
  Survived ~ Sex + Age + Class,
  data = df,
  family = binomial
)


tbl_regression(
    full_model,
     exponentiate = T,
    label = list(
        Age ~ "Age (Years)",
        Sex ~ "Sex of the Passengers"
    )
) |> 
    bold_labels() |> 
    modify_header(estimate ~ "**AOR**") |> 
    bold_p() |> 
    as_gt() |>  # Use as_gt() instead of gt()
    tab_header(
        title = md("**Mulativariable Logistic Regression Model**"),
        subtitle = md("Fate of passengers on the fatal maiden
                      voyage of the ocean liner 'Titanic'")
    ) |> 
    tab_source_note(
        source_note = md("Dawson, Robert J. MacG. (1995), The ‘Unusual Episode’ Data Revisited. Journal of Statistics Education, 3. *doi:10.1080/10691898.1995.11910499.*") 
    ) |> 
    opt_align_table_header(
        align = "left") |> 
     tab_options(
        table_body.hlines.color = "transparent",
        table_body.vlines.color = "transparent"
    )
Mulativariable Logistic Regression Model
Fate of passengers on the fatal maiden voyage of the ocean liner ‘Titanic’
Characteristic AOR 95% CI p-value
Sex of the Passengers


    Male — —
    Female 11.2 8.57, 14.9 <0.001
Age (Years)


    Child — —
    Adult 0.35 0.21, 0.56 <0.001
Class


    1st — —
    2nd 0.36 0.25, 0.53 <0.001
    3rd 0.17 0.12, 0.24 <0.001
    Crew 0.42 0.31, 0.58 <0.001
Abbreviations: CI = Confidence Interval, OR = Odds Ratio
Dawson, Robert J. MacG. (1995), The ‘Unusual Episode’ Data Revisited. Journal of Statistics Education, 3. doi:10.1080/10691898.1995.11910499.

4.1 Interpretation

When interpreting the multivariable model, compare the adjusted odds ratios (AOR) to the crude odds ratios from the univariable analysis. Key considerations include:

  • Confounding: Variables whose effect estimates change substantially (typically >10-20%) when adjusted for other variables may indicate confounding.
  • Effect modification: Assess whether associations differ across subgroups.
  • Statistical significance: Variables that were significant in univariable analysis but become non-significant in the multivariable model may be explained by confounding.

4.2 Model Selection Decision

If all variables in the multivariable model remain statistically significant and are theoretically justified, no further variable selection is necessary. The model adequately represents the relationships between predictors and the outcome. However, if some variables become non-significant, consider:

  1. Retaining variables that are clinically or theoretically important, regardless of statistical significance
  2. Using backward elimination or other selection methods to arrive at a parsimonious model
  3. Assessing model fit using criteria such as AIC, BIC, or likelihood ratio tests