Section 6.6: Inferences about Regression Parameters

1. Construct a \(1 - \alpha\) confidence interval for a regression coefficient \(\beta_{j}\) under the MLRGN model.
2. Perform a significance level \(\alpha\) test for a regression coefficient \(\beta_{j}\) under the MLRGN model.

Standardizing Regression Coefficients (Lecture Notes for Lecture 15)

1. State the regression function for a Multiple Linear Regression (MLR) with standardized predictor variables.
2. State how the coefficients of a MLR with standardized predictor variables are related to the coefficients of a MLR with unstandardized predictor variables.
3. Explain why coefficients of a MLR with standardized predictor variables are more comparable than coefficients from a MLR with unstandardized predictor variables.
4. Interpet the coefficients of a MLR with standardized predictor variables in the context of a given problem.
5. Standardize the predictor variables in a data frame using the mutate_* functions from the dplyr package.

Coefficient Plots (Lecture Notes for Lecture 15)

1. Interpret coefficient plots for a fitted multiple linear regression.
2. Create coefficient plots using plot.lm.coef from the confcurve package.

What Makes a Coefficient Estimate “Significantly Different from 0”? (Lecture Notes for Lecture 15)

1. Recognize that estimates, not parameters, are statistically significant.
2. State the null hypothesis used by R for the \(P\)-values reported by summary.
3. Never, ever (ever) make the mistake of saying / writing (/ thinking?):
   • “a significant coefficient estimate means the associated predictor is important for prediction”
   • “a non-significant coefficient estimate means the associated predictor is unimportant for prediction”
4. Reason about what makes a coefficient estimate significantly different from zero in terms of:
   • the model(s) and underlying population parameter
   • the estimated noise variance
   • the standard deviation of the predictor
   • the correlation between the predictor and the other predictors in the model
5. Explain why a coefficient estimate can be statistically significant with one set of predictors and non-significant with another set of predictors.
6. Explain in what way a \(P\)-value mixes together the size of a population coefficient and the precision with which that population coefficient has been estimated.

Confidence Curves for Population Coefficients Under the MLRGN Model (Lecture Notes for Lecture 15)

1. State how to construct a confidence curve for a population coefficient under the MLRGN Model.
2. Interpret and use a confidence curve for a population coefficient under the MLRGN Model to do all the usual things:
   • Identify the point estimate
   • Construct confidence intervals
   • Perform two-sided hypothesis tests