Summary: Regression Analysis

Read the summary and the most important questions on Regression Analysis

• 5 Chapter 5: Transformation and Weighing to Correct Model Inadequacy

• 5.1 Introduction

• What are some basic assumptions regarding regression modeling?

  Regression model fitting assumptions:
  

  1. Model errors have mean zero, constant variance, and are uncorrelated.
  2. Model errors have a normal distribution, used for hypothesis testing and construct CI, errors are independent.
  3. Form of the model is correct, includes specification of regressors.

• What methods and procedures can be used when assumptions are violated?

  • Data transformation can be used when the response or regressor variables are expressed in the correct scale of measurement or metric, violation of assumptions.
  • Subject matter knowledge will choose choice of metric.
  • Weighted Least Squares when some assumptions are violated.

• 5.2 Variance Stabilizing Transformations

• What is the basic requirement for regression analysis?

  Constant variance assumption
  

  • Often violated when y follows a probability distribution where variance is functionally related to mean

• How is variance stabilizing transformation used?

  • Variance stabilizing transformation is used when y is proportional to x
  • Strength of transformation depends on curvature that it induces
  • Transformation applied on narrow range = little effect
  • Larger range has a dramatic effect on the analysis
  • Based on scatter diagram or residual analysis is used to determine transformation
  • Appropriate selection made empirically
  • If non-constant variance is not corrected, Least Square (LS) estimators will be unbiased but no longer will have minimum variance property
  • Cause regression coefficients to have larger standard error
  • Transformation help improve more precise estimate of model parameters and ↑ sensitivity for statistical tests

• 5.3 Transformation to Linearize the Model

• How can nonlinearity be detected?

  Lack of Fit (LOF) test, scatter diagrams, matrices of scatterplots, or residual plots from partial regression plots.

• What are nonlinear functions that can be linearized using transformations called?

  Nonlinear functions can be linearized by using transformations; nonlinear models are called intrinsically or transformably linear.

• 5.4.1 Transformation on y: Box-Cox Method

• What method is used for selecting a transformation in Chapter 5.4?

  Analytical Methods for selecting a transformation

• 6 Chapter 6: Diagnostics for Leverage and Influence

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• Why do leverage points affect model summary stats?

  Due to the unusual x values, the leverage point, it can disproportion the influencing parameters. The points can strengthen correlation or alter the slope coefficient (Affects Slope/Intercept, R², p-values), mainly affect R².

• How are influential points and leverage points related?

  Leverage point has the potential to be influential. Can be measured using hat matrix. Influential point significantly impact the model fit (i.e. how well the model represents the observed data; accuracy of predictions).

• Why are influential observations important in Regression Analysis?

  Influential observations can significantly affect the model fit by altering the intercept/slope, reducing accuracy, and can also show an artificially high R² value and altered p-value.