Project Overview
This project analyzes the survival times of ice cream machines and investigates how warranty period and manufacturer are associated with machine failure risk.
The dataset contains censored survival data: some machines failed during observation, while others were replaced before an observed failure. The analysis combines non-parametric survival estimation, semi-parametric Cox regression, time-dependent covariates, parametric relative risk modeling, hypothesis testing, and simulation.
Research Motivation
For manufacturers, machine reliability is directly connected to warranty costs, customer satisfaction, and product design. A key question is whether machines show different failure behavior depending on warranty length and manufacturer.
Main research goal: evaluate the effect of warranty period and manufacturer on the survival time and failure risk of ice cream machines.
Dataset
The dataset contains survival times for more than 100 ice cream machines. Machines that were still functional at the end of a year were replaced, so censoring times occur at multiples of 365 days.
Each machine belongs to one of three manufacturer groups and has either a standard 2-year warranty or a premium 5-year warranty.
| Variable |
Description |
Role in Analysis |
| id |
Unique identifier for each ice cream machine |
Machine-level index |
| days |
Days until failure or replacement |
Survival time |
| status |
Failed or replaced at the end of the year |
Event / censoring indicator |
| warranty |
Warranty period in years: 2-year standard or 5-year premium |
Main explanatory variable |
| manufacturer |
Manufacturer group coded as 1, 2, or 3 |
Group comparison factor |
Problem Definition
The project treats machine lifetime as a survival analysis problem. The event of interest is machine failure, while machines replaced before failure are treated as censored observations.
The central statistical question is whether warranty and manufacturer affect the hazard rate, meaning the instantaneous risk that a machine fails at a given time, conditional on having survived until that time.
Statistical Methods
Several survival-analysis methods were used to understand the failure process from different perspectives. The workflow starts with exploratory diagnostics and then moves to Cox regression, time-dependent covariates, parametric relative risk regression, and simulation-based validation.
| Method |
Purpose |
Why It Matters |
| Nelson-Aalen Estimator |
Estimate cumulative hazard non-parametrically |
Used for diagnostics and visual comparison |
| Cox Proportional Hazards Model |
Model hazard as a function of warranty and manufacturer |
Core semi-parametric survival model |
| Breslow Estimator |
Estimate cumulative baseline hazard |
Supports Cox model interpretation and baseline risk estimation |
| Time-Dependent Covariates |
Model changes in risk after the warranty period |
Checks whether failure behavior changes over time |
| Parametric Relative Risk Regression |
Fit a parametric relative risk model |
Provides an alternative model-based view of failure risk |
| Simulation |
Validate method behavior using simulated data |
Connects theory with empirical model behavior |
Key Results & Interpretation
The analysis showed how survival analysis can be used to model machine reliability under censoring. The Cox model provided a structured way to quantify the relationship between covariates and failure risk, while non-parametric estimators helped visualize group differences.
The project emphasized interpretation rather than only prediction: hazard ratios, cumulative hazards, censoring mechanisms, and time-dependent effects were all considered when evaluating the machine failure process.