Reading multiple regression output: model form, β, t, p, \(R^2\), F–test, residuals, standard error. Concept only, no calculator.
1) In multiple linear regression, the variable we are trying to predict is usually called the response and is denoted by \(Y\).
2) The intercept \(\\beta_0\) is the predicted value of \(Y\) when all predictors \(X_1, X_2, \dots\) are equal to 1.
3) A slope \(\\beta_1\) describes how much the predicted \(Y\) changes when \(X_1\) increases by 1 unit, holding the other predictors fixed.
4) If a slope has a p-value of 0.80, this is strong evidence that the predictor is useful in the model.
5) A very small p-value for a slope always guarantees that the model will make highly accurate predictions for new students.
6) The overall F–test in the ANOVA table tests \(H_0\!:\\beta_1=\\beta_2=\\cdots=0\) versus \(H_1\!:\) at least one slope is nonzero.
7) The coefficient of determination \(R^2\) always decreases when we add more predictors to the model.
8) Adjusted \(R^2\) can go down when we add a predictor that does not really help explain \(Y\).
9) The standard error of the regression (Root MSE) is measured in squared units of \(Y\).
10) A residual is defined as the predicted value minus the actual value, \(\\hat y - y\).
11) If the 95% confidence interval for a slope includes 0, then that slope will not be significant at the 5% level (its p-value will be > 0.05).
12) A p-value is the probability that the null hypothesis is true given the data.
13) In a good regression model, the residual plot should show a clear curved pattern rather than a random cloud around zero.
14) In our homework-style examples, we usually treat \(n\) as the number of students and \(p\) as the number of parameters (intercept plus slopes) in the model.
15) In a multiple regression model with three predictors and \(n=23\) students, the residual degrees of freedom are 23.
16) To predict a new student’s FinalGrade from an estimated regression equation, we plug that student’s StudyHW, TutorHours, and ExamPrep values into \(\\hat y = b_0 + b_1 X_1 + b_2 X_2 + b_3 X_3\).
17) If a slope has a very small p-value (for example 0.0001), that guarantees the effect is large in size and very important in practice.
18) Multiple linear regression can only be used when every predictor enters the model as a straight line. If we include a term like \(X^2\), it is no longer a linear regression model.
19) Deleting a single unusual outlier can sometimes change the estimated slopes and p-values a lot in a multiple regression.
20) An \(R^2\) value close to 0 means that the regression model does not reduce the sum of squared errors very much compared with just using the sample mean of \(Y\).
21) When we interpret a slope as “holding other predictors fixed,” we allow the other X’s to change in whatever way they normally do in the data.
22) If one predictor has a slope of 50 and another has a slope of 0.5, then the first predictor must be 100 times more important in the model, no matter what units are used.
23) If we change the units of a predictor from minutes to hours, the p-value for its slope will automatically change, even though the underlying data set is the same.
24) For a fixed data set and model, if we lower the significance level from \(\alpha=0.05\) to \(\alpha=0.01\), some slopes that used to be “significant” at 5% may no longer be called significant.
25) If a regression model has a very high \(R^2\), then it is automatically good for real-world decision-making, even if the predictors do not really make sense in context.