Sparrow Wing Length Regression Analysis

Step 1: Loading Data and Visualizing Groups

First, we load the required packages (Stat2Data, dplyr, ggplot2) and load the Sparrows dataset. We then construct two scatterplots to explore the overall relationship between weight and wing length across treatment groups.

# Load packages
library(Stat2Data)
library(dplyr)
library(ggplot2)

# Load dataset
data("Sparrows")

# Plain scatterplot
ggplot(Sparrows, aes(x = Weight, y = WingLength)) +
  geom_point() +
  labs(
    title = "Sparrow Wing Length vs. Weight",
    x = "Weight (grams)",
    y = "Wing Length (mm)"
  )

# Scatterplot colored and shaped by group (Treatment)
ggplot(Sparrows, aes(x = Weight, y = WingLength, color = Treatment, shape = Treatment)) +
  geom_point() +
  labs(
    title = "Sparrow Wing Length vs. Weight by Treatment",
    x = "Weight (grams)",
    y = "Wing Length (mm)"
  )

Observation: There are some outliers, but the groups seem to clump up at about 15g and 30 mm.

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Step 2: Control Group Exploration & Correlation

Next, we filter the dataset to isolate the control group, construct a scatterplot with transparent points, and compute the correlation coefficient (\(r\)).

# Filter data for the control group
my_group <- Sparrows %>%
  filter(Treatment == "control")

# Scatterplot of my_group
ggplot(my_group, aes(x = Weight, y = WingLength)) +
  geom_point(alpha = 0.6) +
  labs(
    title = "Control Sparrows: Wing Length vs. Weight",
    x = "Weight (grams)",
    y = "Wing Length (mm)"
  )

# Calculate correlation coefficient r
cor(my_group$Weight, my_group$WingLength, use = "complete.obs")

## [1] 0.8059326

Analysis of Association: The relationship between weight and wing length for control sparrows is positive, linear, and strong, with a correlation coefficient of \(r = 0.81\).

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Step 3: Fitting the Linear Model

We fit a simple linear regression model using WingLength as the response variable and Weight as the explanatory variable.

# Fit the linear model
model <- lm(WingLength ~ Weight, data = my_group)

# View the coefficients
coef(model)

## (Intercept)      Weight 
##    8.669793    1.309223

Model Results: - Intercept: 8.66 - Slope: 1.30

Slope Interpretation: For every 1 gram increase in a sparrow’s weight, the wing length is predicted to increase by 1.30 mm.

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Step 4: Visualizing the Regression Line

Finally, we overlay our fitted regression line onto the scatterplot using our calculated slope and intercept.

# Scatterplot with regression line overlay
ggplot(my_group, aes(x = Weight, y = WingLength)) +
  geom_point(alpha = 0.6) +
  geom_abline(intercept = 8.66, slope = 1.30, color = "steelblue", linewidth = 1) +
  labs(
    title = "Control Sparrows: Linear Regression Model",
    subtitle = "WingLength = 8.66 + 1.30 * Weight (r = 0.81)",
    x = "Weight (grams)",
    y = "Wing Length (mm)"
  )

Visual Verification: The regression line runs directly through the middle of the scatterplot data points as expected.