Day 35

Math 216: Statistical Thinking

Bastola

Describing associations between two quantitative variables

Data: each case \(i\) has two measurements

A scatterplot is the plot of \((x_i, y_i)\)

positive association: as \(x\) increases, \(y\) increases

negative association: as \(x\) increases, \(y\) decreases

Correlation coefficients, denoted \(r\) (sample) or \(\rho\) (population), measure the linear relationship between two variables.
Strength varies as \(r \approx \pm 1\) (strong), \(r \approx 0\) (weak).
Direction: Positive (\(r > 0\)) or negative (\(r < 0\)) linear association.
Formula: \[ r = \frac{\sum_{i=1}^n \left(\frac{x_i - \bar{x}}{s_x}\right) \left(\frac{y_i - \bar{y}}{s_y}\right)}{n-1} \]
- Interpretation: \(r = 1\) (perfect positive), \(r = -1\) (perfect negative), \(r = 0\) (no relationship).
Visualization: Scatterplots reveal the clustering around the regression line; outliers can heavily influence \(r\).

# Example R code to calculate correlation
cor(data$x, data$y)  # Note: Order of x and y doesn't affect the outcome.

Deterministic Model: Assumes a perfect, predictable relationship without error, e.g., \(y = 1.5x\).
Probabilistic Model: Incorporates randomness, modeling \(y\) as: \[ y = 1.5x + \text{random error} \]
- General Form: \[ y = \text{Deterministic component} + \text{Random error} \]
- Assumes mean of random error is 0, aligning \(E(y)\) with the deterministic component.

Goal: To find a straight line that best fits the data in a scatterplot.

Regression Equation: \(\hat{y} = b_0 + b_1x\)
- \(x\): explanatory variable \(\qquad\) \(\hat{y}\): predicted response variable
Parameters:
- Slope (\(b_1\)): Increase in predicted \(y\) for every unit increase in \(x\). \[ b_1 = \frac{\text{change }\hat{y}}{\text{change } x} \]
- Intercept (\(b_0\)): Predicted \(y\) value when \(x = 0\). \[ \hat{y} = b_0 + b_1(0) = b_0 \]

Steps:
1. Hypothesize the deterministic component (e.g., \(E(y) = \beta_0 + \beta_1x\)).
2. Estimate model parameters using least squares.
3. Assess the fit and use the model for prediction.
Residuals:
- Geometrically, a residual is the vertical distance from each point to the regression line, helping measure model fit.

Example: Understanding the relationship between temperature and cricket chirp rate.
- Model cricket chirp rate as a function of temperature.
Methodology:
- Utilize observed data to fit a linear model that predicts cricket chirp rate based on temperature.
- Calculate deviations to assess model fit and refine parameters as needed.