Comparisons of linear regression and survival analysis

Understanding the differences between linear regression and survival analysis is crucial as they address different types of data and research questions and applying the correct method ensures accurate modeling, better predictions, and more informed decision-making in fields like healthcare, engineering, and finance.

In this article, we are going to explore the differences between linear regression and survival analysis.

Linear Regression

Linear Regression is a fundamental statistical technique used to model the relationship between a dependent variable and one or more independent variables. It aims to identify the linear relationship between these variables, which allows for predictions and insights into the data.

Linear regression fits a line (in simple linear regression) or a hyperplane (in multiple linear regression) to the data. The equation of this line is given by:

Y = \beta_0 + \beta_1 X + \epsilon

• \beta_0​: Intercept of the line.
• \beta_1​: Slope of the line, representing the effect of the independent variable on the dependent variable.
• \epsilon: Error term, accounting for the variance in Y not explained by X.

The goal is to find the values of \beta_0​ and \beta_1​ that minimize the difference between the observed values and the values predicted by the model. This is achieved by minimizing the Sum of Squared Errors (SSE):

\text{SSE} = \sum_{i=1}^n (Y_i - \hat{Y}_i)^2

where \hat{Y}_i​ is the predicted value of Y_i.

Survival Analysis

Survival Analysis is a statistical technique used to analyze the time until an event of interest occurs. It is commonly used in fields such as medicine, engineering, and social sciences to study time-to-event data, like the duration until a patient experiences a relapse or a machine fails.

Similarities Between Linear Regression and Survival Analysis

1. Statistical Modeling: Both build models to explain relationships within data.
2. Predictive Analysis: Both are used for predicting outcomes—continuous for Linear Regression and time-to-event for Survival Analysis.
3. Variable Relationships: Both analyze how independent variables affect a dependent variable.
4. Model Evaluation: Both use statistical measures to assess model fit and performance.
5. Handling of Covariates: Both include multiple covariates to understand their impact.
6. Inferential Statistics: Both provide estimates of parameters and assess their significance.
7. Assumption Testing: Both involve validating model assumptions.
8. Visualization: Both use visual tools to aid in the interpretation of results

Difference Between Linear Regression and Survival Analysis

1. Data Types

Linear Regression requires continuous outcome data where the goal is to model the relationship between a dependent variable and one or more independent variables. The data is typically assumed to be normally distributed, and the relationship between variables is linear.

Survival Analysis deals with time-to-event data, focusing on the duration until a specific event occurs. This type of data often includes censored observations, where the event of interest has not occurred by the end of the study period. Unlike linear regression, survival analysis can handle data where the time to an event is of primary interest, rather than a continuous outcome.

2. Model Assumptions

Linear Regression assumes:

• Linearity: The relationship between dependent and independent variables is linear.
• Independence: Observations are independent of each other.
• Homoscedasticity: Constant variance of errors across all levels of the independent variable.
• Normality: Errors are normally distributed.

Survival Analysis assumes:

• Non-negativity: Time-to-event data cannot be negative.
• Proportional Hazards (for Cox model): The effect of covariates on the hazard is constant over time.
• Independence: Survival times are independent, though the data can be censored.

The key difference is that survival analysis does not assume a linear relationship or normally distributed errors, focusing instead on the time until an event and handling censoring.

3. Handling of Censoring

Linear Regression does not account for censoring, as it deals with continuous outcomes without considering incomplete data. Censoring is not a feature of the data structure in linear regression.

Survival Analysis explicitly addresses censoring. Censoring occurs when the event of interest has not happened by the end of the observation period, leading to incomplete data for those subjects. Survival analysis methods, such as the Kaplan-Meier estimator and Cox proportional-hazards model, are designed to handle such incomplete information effectively.

4. Interpretation of Results

Linear Regression:

• Results are interpreted in terms of the relationship between independent and dependent variables.
• Coefficients represent the change in the dependent variable for a one-unit change in the independent variable, assuming other variables are held constant.
• The focus is on how well the model explains the variation in the outcome.

Survival Analysis:

• Results are interpreted in terms of survival probabilities and hazard rates.
• The survival function provides the probability of surviving beyond a certain time, while the hazard function indicates the instantaneous risk of the event occurring.
• The focus is on the timing of events and the impact of covariates on survival.

5. Use Cases

Linear Regression is suitable when:

• The outcome variable is continuous and normally distributed.
• The primary interest is in the relationship between predictors and the outcome.
• There are no time-to-event considerations or censored data.

Survival Analysis is appropriate when:

• The outcome variable is the time until an event occurs.
• The data includes censored observations where some subjects have not yet experienced the event by the end of the study.
• The focus is on understanding the duration of time until the event and the effect of covariates on this duration.

6. Example Scenarios:

• Linear Regression: Predicting a student’s final exam score based on hours studied and attendance.
• Survival Analysis: Evaluating the time until patients experience a relapse after starting a new treatment, considering that some patients may not have relapsed by the study’s end.

Summary Table: Linear Regression vs. Survival Analysis

- Linearity: Linear relationship

- Independence: Observations independent

- Homoscedasticity: Constant variance of errors

- Normality: Errors are normally distributed

Practical Implementation of Linear Regression

Steps for implementing Linear Regression with the California Housing dataset:

• Load Dataset: Use fetch_california_housing() to load the dataset.
• Prepare Data:
  • Split the dataset into features (X) and target (y).
  • Split data into training and testing sets using train_test_split().
• Build Model: Create a LinearRegression model then, fit the model on the training data.
• Make Predictions: Predict the target values for the test set.
• Evaluate Model: Calculate the Mean Squared Error (MSE) between predicted and actual values.
• Output Results: Print the MSE to assess model performance.

from sklearn.datasets import fetch_california_housing
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error

# Load dataset
california_housing = fetch_california_housing()
X = california_housing.data
y = california_housing.target

# Split data
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)

# Build model
model = LinearRegression()
model.fit(X_train, y_train)

# Make predictions
y_pred = model.predict(X_test)

# Evaluate model
mse = mean_squared_error(y_test, y_pred)
print(f'Mean Squared Error: {mse}')

import matplotlib.pyplot as plt

# Plot the true values vs. predicted values
plt.figure(figsize=(10, 6))
plt.scatter(y_test, y_pred, alpha=0.5)
plt.plot([y_test.min(), y_test.max()], [y_test.min(), y_test.max()], 'k--', lw=2)
plt.xlabel('True Values')
plt.ylabel('Predicted Values')
plt.title('True vs. Predicted Values for Linear Regression')
plt.show()

Output:

Mean Squared Error: 0.5558915986952422

Practical Implementation of Survival Analysis

In this implementation, we are going to analyze survival times using the lung dataset, ensuring that all missing values are addressed.

The steps are discussed below:

1. Data Collection: use the 'lung' dataset from the lifelines library.
2. Data Cleaning: Handle missing values in the dataset.
3. Model Building:
   • Use Kaplan-Meier estimator to visualize survival functions.
   • Fit the Cox Proportional-Hazards model after cleaning the data.
4. Model Evaluation: Interpret the results of the Cox model.

from lifelines import KaplanMeierFitter, CoxPHFitter
from lifelines.datasets import load_lung
import pandas as pd

# Load dataset
data = load_lung()

# Data Cleaning: Drop rows with NaN values in relevant columns
data = data.dropna(subset=['time', 'status'])

# Kaplan-Meier Estimator
kmf = KaplanMeierFitter()
kmf.fit(durations=data['time'], event_observed=data['status'])
kmf.plot_survival_function()
print('Kaplan-Meier Survival Function plotted.')

# Check for any remaining NaN values in the dataset
print("Checking for NaN values in the dataset:")
print(data.isnull().sum())

# Cox Proportional-Hazards Model
# Drop any remaining NaNs
data_clean = data.dropna()
cph = CoxPHFitter()
cph.fit(data_clean, duration_col='time', event_col='status')
cph.print_summary()

Output:

Kaplan-Meier Survival Function plotted.
Checking for NaN values in the dataset:
inst          1
time          0
status        0
age           0
sex           0
ph.ecog       1
ph.karno      1
pat.karno     3
meal.cal     47
wt.loss      14
dtype: int64

This summary provides an overview of how each covariate affects survival, as well as the overall fit and performance of the Cox model.

Conclusion

Linear regression and survival analysis serve distinct purposes and are suited to different types of data. Linear regression models the relationship between continuous variables, assuming linearity and normality. Survival analysis, on the other hand, focuses on time-to-event data, handling censoring and providing insights into survival probabilities and hazard rates. Selecting the appropriate method based on your data and research goals is essential for accurate modeling and decision-making.