Regression Analysis and Linear Regression

Modeling Basics

• Response variable (Y) and predictive variables (Xs) are combined into a model.
• Modeling involves extra steps like variable selection.
• Complex models like machine learning can handle many variables, but regression models require careful selection.
• Too many variables in regression can reduce the model's interpretability.
• Calibration and validation are crucial steps in real-life settings.
• Train the model on a subset of the data and test its generalizability on the remaining data.
• A good model should be generalizable to new data, not just fit the training data well.

Course Overview

• The lecture provides a refresher for ENVX one double o two.
• It aims to provide necessary knowledge even for those without prior statistics coursework.
• Today's lecture serves as a cheat sheet for regression.

Regression Fundamentals

• Regression involves one response variable (Y) and one or more predictive variables (Xs).
• Relationships between variables can be linear or nonlinear.
• The course focuses on simple linear regression (one predictor) for theoretical understanding.
• The majority of the course covers multiple linear regression (multiple predictors).

Reasons for Regression

1. Determine if a relationship exists between predictors and response.
2. Estimate new values of the response based on existing data.
   • Example: Estimate cow weights based on existing data from 100 cows.
3. Test hypotheses about relationships between variables.
   • Test if the model is significant.

The Essence of Regression

• Regression fits a line of best fit to the data points.
• Technically, the goal is to minimize the sum of squared residuals.

Lecture Outline: Model Development

1. Developing the Model
   • Visualizing and summarizing data.
   • Fitting a simple model.
   • Checking assumptions.
   • Interpreting the output.
   • Applying transformations if necessary.

Historical Context

• Galton and others developed the theory of least squares.
• Residual: The distance between the observed data point and the regression line.
• Sum of Squared Residuals: Squaring the residuals to eliminate negative values and summing them.
• The goal is to minimize the sum of squared residuals to get the best fit.

Simple Linear Regression

• Involves one predictive variable and one response variable.
• Example: Galton's data on parent and child heights.

Galton's Data

• Measures the height of parents (average of mother and father) and their children.
• Investigates the influence of parent height on child height.
• Each pair of parents had approximately four children.
• Height is measured in inches.
• Data is binned into 0.5-inch size classes.

Checking for Linearity

• Visually inspect the data to see if the relationship appears linear.

• Use Pearson's correlation coefficient $r$ to quantify the linear relationship.

• Formula for Pearson's correlation coefficient (will be provided in exams):

  • r = \frac{\sum{(xi - \bar{x})(yi - \bar{y})}}{\sqrt{\sum{(xi - \bar{x})^2 \sum{(yi - \bar{y})^2}}}

• $r$ ranges from -1 to +1.

  • Positive values indicate a positive relationship (as one increases, the other increases).
  • Negative values indicate a negative relationship (as one increases, the other decreases).
  • The absolute value indicates the strength of the relationship (1 is a perfect linear relationship).

• Caveat: Pearson's correlation coefficient doesn't distinguish patterns well.

Examples of Correlation Coefficients

• Coefficient of 0.8 can occur in various data patterns.
  • Nonlinear data.
  • Data with outliers.
  • Quadratic relationships.
  • Linear data.
• Always check the plot in addition to the correlation coefficient.

Equation

• $y = \beta<em>0 + \beta</em>1x + \epsilon$
• y is the response variable.
• $\beta_0$ is the intercept.
• $\beta_1$ is the slope.
• x is the predictor variable.
• $\epsilon$ is the error term (residuals).

Understanding the Equation

• y is the result of the equation.
• $\beta<em>0$ and $\beta</em>1$ are constants (fixed once known).
• x is a known value.
• The error term is associated with the residual.

Model Fitting

• $\hat{y}$ represents the predicted value.
• $\hat{y} = \beta<em>0 + \beta</em>1x$
• Residual is the difference between the actual y and the predicted $\hat{y}$.
• $Residual = y - \hat{y}$
• $Residual = y - (\beta<em>0 + \beta</em>1x)$
• The method of least squares minimizes the squared residual.
• $\text{Minimize } \sum{(y - \hat{y})^2}$

Analytical vs. Numerical Fitting

• Analytical: Solving for $\beta<em>1$ and $\beta</em>0$ using equations.
  • $\beta<em>1 = \frac{\sum{(x</em>i - \bar{x})(y<em>i - \bar{y})}}{\sum{(x</em>i - \bar{x})^2}}$
  • $\beta<em>0 = \bar{y} - \beta</em>1\bar{x}$
• Numerical: Estimating $\beta<em>1$ and $\beta</em>0$ iteratively and tweaking them to reduce the residual.
• Computers use numerical fitting, especially for large datasets.

Model Fitting in R

• Use the lm function for simple and multiple linear regression.
• lm(y ~ x, data = galton_data)
• y is the response variable (child height).
• x is the predictor variable (average parent height).
• galton_data is the dataset.

Hypothesis Testing

• Revisiting the hypothesis testing process (hat PC).
• H: Hypothesis
• A: Assumptions
• T: Test
• P: p-values
• C: Conclusions

Hypothesis for Simple Linear Regression

• Null hypothesis ($H<em>0$): The slope is equal to zero ($\beta</em>1 = 0$).
• Alternative hypothesis ($H<em>1$): The slope is not equal to zero ($\beta</em>1 \neq 0$).
• If slope is zero, it tests the model that has a fit of a horizontal line, which is at the average of the y response variable.
• Simple linear regression expresses whether there is a slope (model with slope is better than the mean of the y response).

Assumptions of Linear Regression (LINE)

• L: Linearity
• I: Independence
• N: Normality
• E: Equal Variance (Homoscedasticity)
• The assumptions apply to the residuals, not the data itself.
• If the data is non-linear, the residuals will also be non-linear.

Checking Assumptions

• In base R, use par(mfrow = c(2, 2)) to arrange plots.
• Use the plot(model) function to generate diagnostic plots.

Diagnostic Plots

1. Residuals vs. Fitted: Points should be randomly scattered around the line.
2. QQ Plot: Residuals should follow the one-to-one line.
3. Scale-Location: Red line should be flat, and points should be evenly scattered.
4. Residuals vs. Leverage: Check for points outside Cook's distance lines.

Alternative Packages for Checking Assumptions

• ggfortify package: Provides prettier plots but lacks Cook's distance lines.
• performance package: Provides instructions for each assumption plot.

Linearity Assessment

• Assess whether residuals are linear.
• Check if the green reference line is flat and horizontal near zero.
• If violated, consider transforming the data or using a non-linear model.

Independence

• Addressed during experimental design.
• Violations can occur with paired data, time series data (autocorrelation), or similar predictive variables (multicollinearity).

Multicollinearity

Definition

• High correlation between predictive variables (e.g., > 0.9).

Effects

• Unstable model results.
• Difficulty in determining the importance of individual predictors.

Solutions

• Remove one of the highly correlated predictors (usually the less important one).

Normality

• Residuals should be normally distributed at every stage of the line.
• Problems occur with skewed distributions or fanning data.
• Use histogram and QQ plot from the performance package.
• Histogram: Points should follow the green line.
• QQ Plot: Points should follow the flat line.

Skewness

Heavy vs. Light Tails

• Heavy tails: Very skewed.
• Light tails: Slightly skewed.

Left vs. Right Skew

• Left skewed: Tail on the left.
• Right skewed: Tail on the right.
• Interpret the skew based on the shape of the distribution.

Equal Variances (Homoscedasticity)

• Check for fanning in the data.
• Use the homogeneity of variance plot.
• Green line should be relatively flat, with evenly scattered points.
• Standardized residuals should be less than 2.
• Leverage can distort the model fit.
• A single point shifted far-off may shift the line.

Standardized Residuals

• Residual divided by the standard error of the residual.
• Mean of residuals should be zero.
• Values above 2 (positive or negative) indicate potential outliers.
• Be cautious if variance is not constant.
• Particular shape such as a U or a W indicates non-linearity and questionable residuals.

Model Fit and ANOVA

• ANOVA (Analysis of Variance) is a variation of linear regression.
• Both partition variance into sums of squares for the residual and error.
• Components form the F-statistic.

ANOVA Table and Regression Output

• Can generate ANOVA table from a linear regression model.
• anova(model)
• Outputs an F-value and a p-value.
• F-value: $\frac{\text{Sums of squares for the parent variable}}{\text{Mean square of the residuals}}$

Interpreting the Regression Output

• Use summary(model) to get detailed regression output.
• The F-statistic and p-value in the regression output are the same as in the ANOVA table.
• The F statistic will tell how much does the parent's height explain the child's height.
• When you look at the significance of the F statistic is that it's the same value.

Writing out the Equation

• With function summary, we can write out the equation
• y = intercept + slope * parent
• Y = estimate + Beta 1 (slope) * parent.

Interpreting the Relationship

• For every unit change in the predictor (parent), we expect a corresponding change in the response (child).

R-squared Values

• Two types: multiple R-squared and adjusted R-squared.
• Multiple R-squared: Use when only one predictor variable is available.
• Adjusted R-squared: Use when multiple predictor are available, with a penalty based on keeping extra variables around.
• Adjusted R-squared is always going to be lower than Multiple R-squared

Making Predictions

• Plug in any value for the predictor variable and calculate an estimation from training data.
• Use the equation or the "predict" function
• predict(model, newdata = data.frame(parent = 70))

Generalizability

• Assess whether the model can apply to a new data set.
• Galton's study may not relate to more modern data due to improved eating etc.

Transformations

• Apply transformations when assumptions are not met, particularly linearity, equal variance, and normality.
• Transform either the response (y) or predictor (x) variable.
• Transforming the y is easier when all the variables show that it's not linearity.
• Easier to transform just individual x variables
• Beware: the equation becomes harder to read when transformed.

Example: Air Quality Data

• Measure the amount of ozone in New York in 1973.
• Fit a simple linear regression with temperature as the predictor.

Checking Assumptions

• With the performance package, assumptions still may not be fulfilled
• Even if a bit subjective, it's good enough to pass
• Still may exhibit waviness, values over standard deviation,