1. Background
2. Data Overview
   
3. Exploratory Plot 1 - Distribution of Tumor Thickness
4. Exploratory Plot 2 - Relationship Between Tumor Thickness & Survival Time
5. Model Specification
6. Estimation & Inference
   
7. Code & Model Output (Continued.)
8. Diagnostics - Residuals
   
9. Diagnostics - Interactive (3D View)
10. Results & Interpretation
    
11. Limitations & Next Steps

• Malignant melanoma is a severe form of skin cancer with high variability in patient outcomes.
• Prognosis is influenced by several clinical measures, with tumor thickness recognized as one of the most important indicators.
• Understanding how thickness relates to survival time can help illustrate how statistical models connect medical measures with patient outcomes.
• Linear regression provides a straightforward way to quantify this relationship, test statistical significance, and build intuition for more advanced survival models.
• The goal is to demonstrate regression in a medical context by modeling survival time as a function of tumor thickness.

• Dataset: boot::melanoma
  
• 205 patients diagnosed with malignant melanoma
• Key Measures:
  • time: Survival time in days.
  • thickness: Tumor thickness in millimeters.
    
  • age: Age at operation.
    
  • ulcer: Indicator of ulceration.
  • status: Survival outcome. (Censored, Alive, Cause of Death)
• Each row represents one patient.

• Tumor thickness is a key clinical measure used in prognosis.
• Distribution helps us understand how values are spread across patients.
• Provides context before modeling survival time against thickness.

• Survival time generally decreases as tumor thickness increases.
• Scatter Plot with regression line shows the overall trend.
  
• Provides evidence of a negative association.

• Survival time is modeled as a linear function of tumor thickness.

\[ Y_i = f(X_i,\beta) + \varepsilon_i \;=\; \beta_0 + \beta_1 X_i + \varepsilon_i \]

• Where: \(Y_i\) = Survival Time; \(X_i\) = Tumor Thickness; \(\beta_0\) = Intercept; \(\beta_1\) = Slope; \(\varepsilon_i\) = Error Term

• Point Estimate - Slope (\(\hat{\beta}_1\)) = Change in survival per mm of thickness.

• Hypothesis Test
  \[ H_0: \beta_1 = 0 \quad \text{(No relationship.)}, \quad H_A: \beta_1 \neq 0 \quad \text{(Thickness affects survival.)} \]

• t-test - Evaluates if slope differs significantly from 0.
  

• p-value - Probability of observing results this extreme if \(H_0\) were true.
  

• Confidence Interval - Range of plausible values for \(\beta_1\).

• Fit linear regression of survival time vs. tumor thickness.

model <- lm(time ~ thickness, data = melanoma)
summary(model)

## 
## Call:
## lm(formula = time ~ thickness, data = melanoma)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2325.4  -707.6  -210.6   744.9  3410.4 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2413.41     107.39  22.473  < 2e-16 ***
## thickness     -89.25      25.86  -3.451 0.000679 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1093 on 203 degrees of freedom
## Multiple R-squared:  0.05542,    Adjusted R-squared:  0.05076 
## F-statistic: 11.91 on 1 and 203 DF,  p-value: 0.0006793

coefs <- summary(model)$coefficients
cis <- confint(model)
out <- cbind(Term = rownames(coefs), coefs, cis)

knitr::kable(
  out,
  digits = 3,
  row.names = FALSE,
  col.names = c("Term","Estimate","Std. Error","t value","p-value","2.5 %","97.5 %")
)

• Residuals help check whether regression assumptions hold.
• A good model should show:
  • Residuals scattered randomly around zero.
  • No clear trend or curve.
  • Roughly equal spread across fitted values.

• Interactive view of survival time vs tumor thickness, with age as a third dimension.
• Points colored by ulceration status to spot clusters/outliers.
  
• Use your mouse to rotate, zoom, and inspect values.

• Direction & Size: Estimated slope = -89 days per mm thickness.
  • 95% CI: [-125, -54] days; p-value < 0.001
    
• Interpretation: Each 1 mm increase in tumor thickness is associated with a decrease in expected survival time.
• Discussion:
  • Confirms tumor thickness as a strong prognostic factor in melanoma.
  • Even modest increases in thickness translate to meaningful differences in survival.
  • Simple regression highlights the core relationship, while further models could include age and ulceration.
• Note: Data details and diagnostics are presented on other slides.

• Simplified Model - Only tumor thickness was used as a predictor.
  
• Assumption Limits - Linearity, normal errors, and constant variance may not fully hold.
  
• Censoring Ignored - Survival data often requires Cox regression or Kaplan–Meier analysis.
  
• Outliers - Extreme values in survival time or thickness may affect results.

Next Steps
- Expand model to include additional predictors such as age, ulceration status, or sex.
- Explore log transformation of survival time to stabilize variance.
- Apply survival specific methods such as Cox proportional hazards for more realistic analysis.
- Validate results with larger datasets or external studies.