Bike rentals

Description

From Example 6.3 (Bike Rentals)

Daily bike-rental records with columns for season, working-day indicator, weather variables (humidity etc.), and rental counts split into casual, registered, and total (cnt). Used as the running example through L6 Introduction to SAS and revisited in L9 Linear Regression to model the relationship between registered and casual users.

File: data/bike2.csv

Numerical summaries

From Example 6.4 (5-number summaries)

In SAS Studio: Tasks > Statistics > Summary Statistics. Analysis variable cnt, classification variable season. Under options, check lower/upper quartiles and comparative boxplots.

Observations: median count is highest in fall, then summer, winter, spring. IQRs are similar (~2000) across seasons; spring’s middle 50% is the most right-skewed.

Scatterplot

From Example 6.5 (Casual vs Registered Scatterplot)

In SAS: Tasks > Graphs > Scatter Plot. x = casual, y = registered, group = workingday.

Observation: two distinct relationships appear between casual and registered counts — one for working days and one for non-working days. This motivates the interaction model fit in L9 Linear Regression.

Histograms

From Example 6.6 (Casual Users Distribution)

In SAS: Tasks > Graphs > Histogram. Analysis variable casual, group workingday, with Normal density overlaid.

Observation: right-skewed in both cases; non-working-day counts extend further (to ~3500).

Boxplots

From Example 6.7 (Boxplots for Casual Users, by Season)

In SAS: Tasks > Box Plot. Analysis variable casual, category season, subcategory workingday. To force calendar ordering:

proc sgplot data=ST2137.BIKE2;
    vbox casual / category=season group=workingday grouporder=ascending;
    xaxis values=('spring' 'summer' 'fall' 'winter');
    yaxis grid;
run;

Re-running on a log scale (APPEARANCE tab) tames the skew and reveals the seasonal pattern more clearly.

QQ plots

From Example 6.8 (Normality Check for Humidity)

In SAS: Tasks > Statistics > Distribution Analysis. Analysis variable hum; under options add the Normal curve, kernel density estimate, and the Normal QQ-plot.

Observation: humidity values are close to Normal except for a single low observation.

For the theory behind QQ-plots, see QQ-plots.

Chi-squared test

From Example 6.9 (Chi-square Test for Independence)

Shown in the bike-rentals chapter but applied to the Student performance dataset (variables address vs paid). The workflow is: Tasks > Table Analysis, select one variable as column and another as row, check “Chi-square statistics” under OPTIONS.

See Chi-Square & Fisher for the formula and the general test, and Chi-squared Test for Independence for theory.

Regression F-test

From Example 9.4 (Bike Data F-test)

Fitting a simple linear regression of registered on casual, restricted to non-working days:

R code

bike2 <- read.csv("data/bike2.csv")
bike2_sub <- bike2[bike2$workingday == "no", ]
lm_reg_casual <- lm(registered ~ casual, data=bike2_sub)
anova(lm_reg_casual)

Python code

import pandas as pd
import statsmodels.api as sm
from statsmodels.formula.api import ols
 
bike2 = pd.read_csv("../data/bike2.csv")
bike2_sub = bike2[bike2.workingday == "no"]
 
lm_reg_casual = ols('registered ~ casual', bike2_sub).fit()
anova_tab = sm.stats.anova_lm(lm_reg_casual)
print(anova_tab)

From the ANOVA table: $S{S}_{\text{Reg}}=237,654,556$, $S{S}_{\text{Res}}=147,386,970$, giving $F_0=369.25$ with p-value $\approx 2\times 10^{-16}$. Strong evidence against $H_0:{\beta }_1=0$ — the casual count is a significant predictor of registered count on non-working days.

Registered vs. casual, by workday

From Example 9.7 (Bike Data Working Day)

Adding the categorical workingday indicator (dummy coded) allows separate intercepts for the two day types while sharing a slope:

R code

lm_reg_casual2 <- lm(registered ~ casual + workingday, data=bike2)
summary(lm_reg_casual2)

Python code

lm_reg_casual2 = ols('registered ~ casual + workingday', bike2).fit()
print(lm_reg_casual2.summary())

Fitted model: $Y=605+1.72X_1+2330X_2$, where $X_2=1$ on working days. This splits into:

$$Y=\{\begin{array}{ll}605+1.72X_1,& \text{non-working days}\\ 2935+1.72X_1,& \text{working days}\end{array}$$

The benefit vs. fitting two separate regressions is that the entire dataset is used to estimate the common variability.

Interaction term

From Example 9.8 (Bike Data Working Day)

To fit separate slopes and intercepts, include an interaction between casual and workingday:

R code

lm_reg_casual3 <- lm(registered ~ casual * workingday, data=bike2)
summary(lm_reg_casual3)

Python code

lm_reg_casual3 = ols('registered ~ casual * workingday', bike2).fit()
print(lm_reg_casual3.summary())

Adjusted $R^2$ rises from 50.8% to 60.7%. The fitted models are:

$$Y=\{\begin{array}{ll}1362+1.16X_1,& \text{non-working days}\\ 2168+2.97X_1,& \text{working days}\end{array}$$

On working days each additional casual user is associated with a much larger bump in registered users (2.97 vs. 1.16).

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See also: L6 Introduction to SAS · L9 Linear Regression · L3 Exploring Quantitative Data