Define the multiple linear regression model as
$$\hat{y} = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_k x_k$$
where there are $k$ predictors (explanatory variables).
Interpret the estimate for the intercept ($b_0$) as the expected value of $y$ when all predictors are equal to 0, on average.
Interpret the estimate for a slope (say $b_1$) as “All else held constant, for each unit increase in $x_1$, we would expect $y$ to increase/decrease on average by $b_1$.”
Define collinearity as a high correlation between two independent variables such that the two variables contribute redundant information to the model – which is something we want to avoid in multiple linear regression.
Note that $R^2$ will increase with each explanatory variable added to the model, regardless of whether or not the added variables is a meaningful predictor of the response variable. Therefore we use adjusted $R^2$, which applies a penalty for the number of predictors included in the model, to better assess the strength of a multiple linear regression model:
$$R^2 = 1 - \frac{Var(e_i) / (n - k - 1)}{Var(y_i) / (n - 1)}$$
where $Var(e_i)$ measures the variability of residuals ($SS_{Err}$), $Var(y_i)$ measures the total variability in observed $y$ ($SS_{Tot}$), $n$ is the number of cases and $k$ is the number of predictors.
Note that adjusted $R^2$ will only increase if the added variable has a meaningful contribution to the amount of explained variability in $y$, i.e. if the gains from adding the variable exceeds the penalty.
Define model selection as identifying the best model for predicting a given response variable.
Note that we usually prefer simpler (parsimonious) models over more complicated ones.
Define the full model as the model with all explanatory variables included as predictors.
Note that the p-values associated with each predictor are conditional on other variables being included in the model, so they can be used to assess if a given predictor is significant, given that all others are in the model.
These p-values are calculated based on a $t$ distribution with $n - k - 1$ degrees of freedom.
The same degrees of freedom can be used to construct a confidence interval for the slope parameter of each predictor:
$$b_i \pm t^\star_{n - k - 1} SE_{b_i}$$
Stepwise model selection (backward or forward) can be done based based on adjusted $R^2$ (choose the model with higher adjusted $R^2$).
The general idea behind backward-selection is to start with the full model and eliminate one variable at a time until the ideal model is reached.
i. Start with the full model.
ii. Refit all possible models omitting one variable at a time, and choose the model with the highest adjusted $R^2$.
iii. Repeat until maximum possible adjusted $R^2$ is reached.
The general idea behind forward-selection is to start with only one variable and adding one variable at a time until the ideal model is reached.
i. Try all possible simple linear regression models predicting $y$ using one explanatory variable at a time. Choose the model with the highest adjusted $R^2$.
ii. Try all possible models adding one more explanatory variable at a time, and choose the model with the highest adjusted $R^2$.
iii. Repeat until maximum possible adjusted $R^2$ is reached.
Adjusted $R^2$ method is more computationally intensive, but it is more reliable, since it doesn’t depend on an arbitrary significant level.
List the conditions for multiple linear regression as
linear relationship between each (numerical) explanatory variable and the response - checked using scatterplots of $y$ vs. each $x$, and residuals plots of $residuals$ vs. each $x$
nearly normal residuals with mean 0 - checked using a normal probability plot and histogram of residuals
constant variability of residuals - checked using residuals plots of $residuals$ vs. $\hat{y}$, and $residuals$ vs. each $x$
independence of residuals (and hence observations) - checked using a scatterplot of $residuals$ vs. order of data collection (will reveal non-independence if data have time series structure)
Note that no model is perfect, but even imperfect models can be useful.