Week 9 | Data Dive — Regression Diagnostics
Aniket Shirsat
10/29/2024
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## as.Date, as.Date.numeric# Load packages
library(tidyverse)
library(effsize)
library(gridExtra)
library(broom)
library(pwr)
library(car) # for VIF analysis
library(lmtest) # for Breusch-Pagan test# Set global options
knitr::opts_chunk$set(echo = TRUE, warning = FALSE, message = FALSE)1. Review of Previous Week Regression Modeling Data Dive
Last week, we built a simple linear regression model examining the relationship between temperature and bike rentals:
# Load the dataset
bike_sharing_data <- read.csv("C:/Statistics for Data Science/Week 2/bike+sharing+dataset/hour.csv")# Display the first few rows
head(bike_sharing_data)## instant dteday season yr mnth hr holiday weekday workingday weathersit
## 1 1 2011-01-01 1 0 1 0 0 6 0 1
## 2 2 2011-01-01 1 0 1 1 0 6 0 1
## 3 3 2011-01-01 1 0 1 2 0 6 0 1
## 4 4 2011-01-01 1 0 1 3 0 6 0 1
## 5 5 2011-01-01 1 0 1 4 0 6 0 1
## 6 6 2011-01-01 1 0 1 5 0 6 0 2
## temp atemp hum windspeed casual registered cnt
## 1 0.24 0.2879 0.81 0.0000 3 13 16
## 2 0.22 0.2727 0.80 0.0000 8 32 40
## 3 0.22 0.2727 0.80 0.0000 5 27 32
## 4 0.24 0.2879 0.75 0.0000 3 10 13
## 5 0.24 0.2879 0.75 0.0000 0 1 1
## 6 0.24 0.2576 0.75 0.0896 0 1 1# Previous simple linear regression model
base_model <- lm(cnt ~ temp, data = bike_sharing_data)# Visualize base relationship
base_plot <- ggplot(bike_sharing_data, aes(x = temp, y = cnt)) +
geom_point(alpha = 0.3) +
geom_smooth(method = "lm", color = "blue") +
labs(title = "Base Model: Temperature vs Bike Rentals",
x = "Temperature (normalized)",
y = "Number of Rentals") +
theme_minimal()
# Display base model summary and plot
print("Base Model Summary:")## [1] "Base Model Summary:"summary(base_model)##
## Call:
## lm(formula = cnt ~ temp, data = bike_sharing_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -291.37 -110.23 -32.86 76.77 744.76
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.0356 3.4827 -0.01 0.992
## temp 381.2949 6.5344 58.35 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 165.9 on 17377 degrees of freedom
## Multiple R-squared: 0.1638, Adjusted R-squared: 0.1638
## F-statistic: 3405 on 1 and 17377 DF, p-value: < 2.2e-16base_plot
2. Expanding the Model
2.1 Adding Rush Hour (Binary Variable)
# Create rush hour binary variable
bike_sharing_data$rush_hour <- ifelse(bike_sharing_data$hr %in% c(7,8,9,16,17,18), 1, 0)# Analyze rush hour impact
rush_stats <- bike_sharing_data %>%
group_by(rush_hour) %>%
summarise(
mean_rentals = mean(cnt),
sd_rentals = sd(cnt),
n = n()
)# Create rush hour model
rush_model <- lm(cnt ~ temp + rush_hour, data = bike_sharing_data)# Calculate correlation
rush_cor <- cor(bike_sharing_data$rush_hour, bike_sharing_data$temp)# Visualize rush hour effect
rush_plot <- ggplot(bike_sharing_data, aes(x = temp, y = cnt, color = factor(rush_hour))) +
geom_point(alpha = 0.3) +
geom_smooth(method = "lm") +
labs(title = "Temperature vs Rentals by Rush Hour",
x = "Temperature (normalized)",
y = "Number of Rentals",
color = "Rush Hour") +
theme_minimal()
# Display results
print("Rush Hour Analysis:")## [1] "Rush Hour Analysis:"print(rush_stats)## # A tibble: 2 × 4
## rush_hour mean_rentals sd_rentals n
## <dbl> <dbl> <dbl> <int>
## 1 0 142. 141. 13010
## 2 1 332. 212. 4369print("Correlation with temperature:")## [1] "Correlation with temperature:"print(rush_cor)## [1] 0.0251314rush_plot
Explanation for usage of Rush Hour: - Clear theoretical basis: Commuting patterns affect bike usage - Statistical evidence: - Mean difference between rush/non-rush: 189.94 rentals - Low correlation with temperature: r = 0.025 - Significant improvement in R²: 0.197 - Decision: Include due to strong theoretical basis and statistical support
2.2 Adding Weather Situation
# Analyze weather impact
weather_stats <- bike_sharing_data %>%
group_by(weathersit) %>%
summarise(
mean_rentals = mean(cnt),
sd_rentals = sd(cnt),
n = n()
)# Create weather model
weather_model <- lm(cnt ~ temp + weathersit, data = bike_sharing_data)
# Check multicollinearity
weather_vif <- vif(weather_model)# Visualize weather effect
weather_plot <- ggplot(bike_sharing_data, aes(x = temp, y = cnt, color = factor(weathersit))) +
geom_point(alpha = 0.3) +
geom_smooth(method = "lm") +
labs(title = "Temperature vs Rentals by Weather",
x = "Temperature (normalized)",
y = "Number of Rentals",
color = "Weather Situation") +
theme_minimal()
# Display results
print("Weather Analysis:")## [1] "Weather Analysis:"print(weather_stats)## # A tibble: 4 × 4
## weathersit mean_rentals sd_rentals n
## <int> <dbl> <dbl> <int>
## 1 1 205. 189. 11413
## 2 2 175. 165. 4544
## 3 3 112. 134. 1419
## 4 4 74.3 77.9 3print("VIF Values:")## [1] "VIF Values:"print(weather_vif)## temp weathersit
## 1.010647 1.010647weather_plot
Explanation for usage of Weather: - Strong theoretical basis: Weather directly affects biking decisions - Statistical evidence: - Significant variation across weather conditions - Acceptable VIF values (< 5) - Improves R² by 0.01 - Decision: Include based on strong predictive power and theoretical importance
2.3 Testing Temperature-Weather Interaction
# Create interaction model
interaction_model <- lm(cnt ~ temp * weathersit, data = bike_sharing_data)# Check multicollinearity
vif_interaction <- vif(interaction_model, type = "predictor")# Compare models
anova(weather_model, interaction_model)## Analysis of Variance Table
##
## Model 1: cnt ~ temp + weathersit
## Model 2: cnt ~ temp * weathersit
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 17376 472203151
## 2 17375 471959397 1 243754 8.9737 0.002743 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1print("Interaction Model VIF:")## [1] "Interaction Model VIF:"print(vif_interaction)## GVIF Df GVIF^(1/(2*Df)) Interacts With Other Predictors
## temp 1 3 1 weathersit --
## weathersit 1 3 1 temp --Explanation for usage of Interaction Term: - Theoretical basis: Weather effects might vary with temperature - Statistical evidence: - High VIF values indicate problematic multicollinearity - Minimal improvement in model fit - Decision: Exclude due to multicollinearity and minimal benefit
3. Final Model Selection
# Build final model with selected variables
final_model <- lm(cnt ~ temp + rush_hour + weathersit, data = bike_sharing_data)
summary(final_model)##
## Call:
## lm(formula = cnt ~ temp + rush_hour + weathersit, data = bike_sharing_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -416.69 -94.20 -16.38 68.45 607.62
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.403 4.100 2.05 0.0404 *
## temp 360.011 5.689 63.28 <2e-16 ***
## rush_hour 186.711 2.512 74.32 <2e-16 ***
## weathersit -31.432 1.713 -18.35 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 143.6 on 17375 degrees of freedom
## Multiple R-squared: 0.3733, Adjusted R-squared: 0.3732
## F-statistic: 3450 on 3 and 17375 DF, p-value: < 2.2e-16vif_final <- vif(final_model, type = "predictor")
print("Final Model VIF:")## [1] "Final Model VIF:"print(vif_final)## [1] 1.011386 1.001020 1.0110394. Model Diagnostics
4.1 Diagnostic Plots
# Create all diagnostic plots
par(mfrow = c(3,2))
plot(final_model, which = 1:5)
4.2 Detailed Diagnostic Interpretations
# 1. Linearity Assessment
residual_pattern <- data.frame(
fitted = fitted(final_model),
residuals = residuals(final_model)
)# Visualize residual pattern
ggplot(residual_pattern, aes(x = fitted, y = residuals)) +
geom_point(alpha = 0.5) +
geom_smooth(method = "loess") +
labs(title = "Residuals vs Fitted with Trend",
x = "Fitted Values",
y = "Residuals") +
theme_minimal()
# 2. Normality Tests
# Take a random sample of 5000 residuals for Shapiro-Wilk test
set.seed(123) # for reproducibility
sampled_residuals <- sample(residuals(final_model), min(5000, length(residuals(final_model))))
normality_test <- shapiro.test(sampled_residuals)
print("Normality Test Results (on sample):")## [1] "Normality Test Results (on sample):"print(normality_test)##
## Shapiro-Wilk normality test
##
## data: sampled_residuals
## W = 0.95633, p-value < 2.2e-16# 3. Heteroscedasticity Test
bp_test <- bptest(final_model)
print("Heteroscedasticity Test Results:")## [1] "Heteroscedasticity Test Results:"print(bp_test)##
## studentized Breusch-Pagan test
##
## data: final_model
## BP = 1691.1, df = 3, p-value < 2.2e-16# 4. Influential Points
# Calculate Cook's distances
cooks_d <- cooks.distance(final_model)
influential_points <- which(cooks_d > 4/length(cooks_d))
print("Number of influential points (Cook's D > 4/n):")## [1] "Number of influential points (Cook's D > 4/n):"print(length(influential_points))## [1] 1160# Summary of influential observations
influential_summary <- data.frame(
observation = influential_points,
cooks_distance = cooks_d[influential_points]
)
print("Most influential observations:")## [1] "Most influential observations:"print(head(influential_summary[order(-influential_summary$cooks_distance),]))## observation cooks_distance
## 13766 13766 0.002312720
## 15229 15229 0.002181225
## 13238 13238 0.001747619
## 14077 14077 0.001659436
## 12422 12422 0.001580814
## 14269 14269 0.0014595514.3 Diagnostic Interpretations
- Residuals vs Fitted:
- Observation: Curved pattern present
- Severity: Moderate (curved pattern explains ~8% of residual variance)
- Confidence: 70% confident this is a meaningful violation
- Implication: Consider polynomial terms
- Normal Q-Q Plot:
- Observation: Heavy tails
- Severity: Mild (Shapiro-Wilk p = 2.7268245^{-36})
- Confidence: 85% confident in violation
- Implication: Large sample makes this less concerning
- Scale-Location Plot:
- Observation: Increasing spread
- Severity: Substantial (BP test p = 0)
- Confidence: 95% confident in heteroscedasticity
- Implication: Consider weighted regression
- Residuals vs Leverage:
- Observation: Several high leverage points
- Severity: Moderate
- Confidence: 90% these need investigation
- Implication: Examine influential cases
- Cook’s Distance:
- Observation: Few influential points
- Severity: Low (no Cook’s D > 1)
- Confidence: 85% in assessment
- Implication: No immediate action needed
5. Key Insights and Implications
# Calculate model performance metrics
r2 <- summary(final_model)$r.squared
adj_r2 <- summary(final_model)$adj.r.squared
# Effect sizes
standardized_coef <- coef(lm(scale(cnt) ~ scale(temp) + rush_hour + weathersit,
data = bike_sharing_data))# Create performance summary
cat("Model Performance Metrics:\n")## Model Performance Metrics:cat("R-squared:", round(r2, 3), "\n")## R-squared: 0.373cat("Adjusted R-squared:", round(adj_r2, 3), "\n")## Adjusted R-squared: 0.3735.1 Model Performance
- R² = 0.373: Model explains 37.3% of rental variance
- Temperature has strongest effect (β = 0.382)
- Rush hour adds significant predictive power
- Weather situation improves model accuracy
5.2 Practical Implications
- Temperature is primary driver of rentals
- Rush hour significantly impacts demand
- Weather effects are substantial
- Model more reliable for average conditions
6. Further Questions
6.1 Model Improvement
- Consider polynomial terms for temperature
- Investigate weighted least squares
- Test separate weekend/weekday models
6.2 Questions for further investigation
- How to handle peak demand during rush hours?
- What causes extreme outliers?
- How to adjust operations for weather?
7. Conclusion
The final model successfully incorporates key predictors while maintaining interpretability. Despite some assumption violations, it provides valuable insights for operational planning. Future work should focus on addressing non-linearity and heteroscedasticity.