class: center, middle, inverse, title-slide # IDS 702: Module 1.9 ## Special predictors, F-tests, and multicollinearity ### Dr. Olanrewaju Michael Akande --- class: center, middle # Special predictors --- ## Special predictors: higher order terms - We have already seen that the relationships between a response variable and some of the predictors can be potentially nonlinear. -- - Sometimes our outcome of interest can appear to have quadratic or even higher order polynomial trends with some predictors. -- - Whenever this is the case, we should look to include squared terms or higher order powers for predictors to capture trends. -- - In the baseline salary example, we included squared terms for both age and experience. -- - .block[General practice: include all lower order terms when including higher order ones (even if the lower order terms are not significant). This aids interpretation. ] -- - As we have seen before, the best way to present results when including quadratic/polynomial trends is to plot the predicted average of `\(Y\)` for different values of `\(X\)`. --- ## Special predictors: indicator/dummy variables - From the Harris Trust and Savings Bank example, we have also seen how to include binary variables in a MLR model with the variable `sex`. -- - In the example, we could actually have used the variable `fsex` (where 1=female and 0=male) instead of `sex` to give us the same exact results. -- - That means that we also could have made a variable equal to `\(1\)` for all males and `\(0\)` for all females, instead. -- - The value of that coefficient would be `\(767\)` instead of `\(-767\)` like we had. All other statistics stay the same (SE, t-stat, p-value). Other coefficients also remain the same. -- - Turns out that we cannot include indicator variables for the two values of the same binary variable when we also include the intercept. --- ## Special predictors: indicator/dummy variables - It is not possible to estimate all three of these parameters in the same model uniquely. -- - The exact same problem arises for any set of predictors such that one is an exact linear combination of the others. -- - Example: Consider a regression model with dummy variables for both males and females, plus an intercept. .block[ .small[ `$$y_i = \beta_0 + \beta_1 \textrm{M}_i + \beta_2 \textrm{F}_i + \epsilon_i = \beta_0*1 + \beta_1 \textrm{M}_i + \beta_2 \textrm{F}_i + \epsilon_i$$` ] ] -- - Note that `\(\textrm{M}_i + \textrm{F}_i = 1\)` for all cases. Thus, .block[ .small[ `$$y_i = \beta_0*(\textrm{M}_i + \textrm{F}_i) + \beta_1 \textrm{M}_i + \beta_2 \textrm{F}_i + \epsilon_i = (\beta_0+\beta_1) \textrm{M}_i + (\beta_0+\beta_2) \textrm{F}_i + \epsilon_i.$$` ] ] We can estimate `\((\beta_0+\beta_1)\)` and `\((\beta_0+\beta_2)\)` but not all three uniquely. -- - .block[Side note: there is no need to mean center dummy variables, since they have a natural interpretation at zero.] --- ## Special predictors: indicator/dummy variables - What if a categorical variable has `\(k > 2\)` levels? -- - Make `\(k\)` dummy variables, one for each level. -- - Use only `\(k-1\)` of the levels in the regression model, since we cannot uniquely estimate all `\(k\)` at once if we also include an intercept (see previous slide). -- - Excluded level is called the baseline. -- - R will actually do this for you automatically; that is, make the `\(k-1\)` dummy variables and set the first level as the baseline. -- - Values of coefficients of dummy variables are interpreted as changes in average `\(Y\)` over the baseline. -- - We will go through an example soon. --- ## Special predictors: interaction terms - Sometimes the relationship of some predictor with `\(Y\)` depends on values of other predictors. This is called an .hlight[interaction effect]. -- - Sometimes, the question we wish to answer would require including interactions in the model, even though they might not be significant. -- - An example of interaction effect for the Harris Bank dataset would be if the effect of age on baseline income was different for male versus female. -- - That is, what if older males are paid more starting salaries than younger males but the reverse is actually the case for females? -- - How do we account for such interaction effects? Make an interaction predictor: .hlight[multiply one predictor times the other predictor]. Ideally, one of them should be a factor variable. -- - .block[General practice is to include all main effects (each variable without interaction) when including interactions.] --- ## Testing if groups of coefficients are equal to zero - With so many variables (polynomial terms, dummy variables and interactions) in a linear model, we may want to test if multiple coefficients are equal to zero or not. -- - We can do so using an F test (a nested F test in this case). -- - First, we fit a MLR model with all `\(p\)` predictors. That is, .block[ .small[ `$$\textrm{M}_1: \ y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \ldots + \beta_p x_{ip} + \epsilon_i; \ \ \epsilon_i \overset{iid}{\sim} N(0, \sigma^2).$$` ] ] -- - We can compute the sum of squares of the errors `\((\textrm{SSE}_1)\)` or residual sum of squares `\((\textrm{RSS}_1)\)` for the FULL model, that is, .block[ .small[ `$$\textrm{RSS}_1 = \sum^n_{i=1} \left(y_i - \hat{y}_i \right)^2.$$` ] ] --- ## Testing if groups of coefficients are equal to zero - Now suppose we want to test that a particular subset of `\(q\)` of the coefficients are zero. .block[ .small[ `$$H_0: \beta_{p-q+1} = \beta_{p-q+2} = \ldots = \beta_p = 0.$$` ] ] -- - We fit a reduced model that uses all the variables except the last `\(q\)`, that is, .block[ .small[ `$$\textrm{M}_{0}: \ y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \ldots + \beta_p x_{i(p-q)} + \epsilon_i; \ \ \epsilon_i \overset{iid}{\sim} N(0, \sigma^2).$$` ] ] -- - Let's call the residual sum of squares for that model `\(\textrm{RSS}_0\)`. -- <div class="question"> Which of the two RSS values would be larger? Why? </div> -- - Then the appropriate F-statistic is .block[ .small[ `$$F = \dfrac{(\textrm{RSS}_0 - \textrm{RSS}_1)/q}{\textrm{RSS}_1/(n-p-1)}.$$` ] ] --- ## Testing if groups of coefficients are equal to zero - To calculate the p-value, look for the area under the `\(F\)` curve with `\(q\)` degrees of freedom in the numerator, and `\((n-p-1)\)` degrees of freedom in the denominator. -- - Guess what? As is the case with pretty much everything else we do in this class, this is so easy to do in R! -- <img src="img/surprised-baby.jpeg" height="300px" style="display: block; margin: auto;" /> --- class: center, middle # Multicollinearity --- ## The problem of multicollinearity - Just like we had with the dummy variables, you cannot include two variables with a perfect linear association as predictors in regression. -- - Example: suppose the true population line is .block[ .small[ `$$\textrm{Avg. y} = 3 + 4x.$$` ] ] -- - Suppose we try to include `\(x\)` and `\(z = x/10\)` as predictors in our own model, - Example: suppose the true population line is .block[ .small[ `$$\textrm{Avg. y} = \beta_0 + \beta_1 x + \beta_2 z,$$` ] ] and estimate all coefficients. Since `\(z = x/10\)`, we have .block[ .small[ `$$\textrm{Avg. y} = \beta_0 + \beta_1 x + \beta_2 \dfrac{x}{10} = \beta_0 + \left( \beta_1 + \dfrac{\beta_2}{10} \right) x$$` ] ] -- - We could set `\(\beta_1\)` and `\(\beta_2\)` to ANY two numbers such that `\(\beta_1 + \beta_2/10 = 4\)`. The data cannot pick from the possible combinations. --- ## The problem of multicollinearity - In real data, when we get “close” to perfect colinearities we see standard errors inflate, sometimes massively. -- - When might we get close: + Very high correlations `\((|\rho| > 0.9)\)` among two (or more) predictors in modest sample sizes. + When one or more variables are nearly a linear combination of the others. + Including quadratic terms as predictors without first mean centering the values before squaring. + Including interactions involving continuous variables. --- ## The problem of multicollinearity - How to diagnose: + Look at a correlation matrix of all the predictors (including dummy variables). Look for values near -1 or 1. + If you are suspicious that some predictor is a near linear combination of others, run a regression of that predictor on all other predictors (not including Y) to see if R squared is near 1. + If the R squared is near 1, you should think about centering your variables or maybe even excluding that variable from your regression in some cases. + Take a look at the .hlight[variance inflation factor]. + Variance inflation factor measures how much the multicollinearity between a variable and other variables in the model inflates the variance of the regression coefficient for that variable. --- ## Variance inflation factor - .block[ .small[ `$$\textrm{VIF}_j = \dfrac{1}{1-R^2_{X_j | X_{-j}}}$$` ] ] where `\(R^2_{X_j | X_{-j}}\)` is the R-squared from the regression of predictor `\(X_j\)` on all other predictors `\((X_1, \ldots,X_{j-1},X_{j+1},\ldots, X_{p})\)`. -- - Since R-squared always lies between 0 and 1, + the denominator `\(1-R^2_{X_j | X_{-j}} \leq 1\)` + which implies that `\(\textrm{VIF} \geq 1\)` -- - Generally, VIF of + 1 = not correlated. .hlight[Why?] + between 1 and 5 = moderately correlated. + greater than 5 = highly correlated. - Typically, we start to get worried when VIF > 10. --- ## We see multicollinearity... so what? - Multicollinearity is really only a problem if standard errors for the involved coefficients are too large to be useful for interpretation, and you actually care about interpreting those coefficients. -- - In the Harris Bank example, + The main coefficient of interest is the one for `sex`. + The remaining variables are really just "control variables". That is, those variables may be correlated with both `bsal` and `sex`, and so we want to account for their effects in our model. + Recall that the correlation between `age` and `exper` was actually 0.8. + Even with this correlation, it is still okay to keep both in the model since we want to simply account for them but do not care about interpreting either. - Another scenario is prediction: including highly correlated predictors can increase prediction uncertainty. --- ## What to do about multicollinearity? - What if you do want to interpret the coefficients involved in the multicollinearity, and the SEs are inflated substantially because of it? - Easiest remedy: remove one of the "offending" predictors. - Keep the one that is easiest to explain or that has the largest T-statistic. -- - Better remedy: + Mean center (or scale) your variables. It helps but may not always solve the problem. + Use a Bayesian regression model with an informative prior distribution on the parameters (take STA 602). + Get more data! Multicollinearity tends to be unimportant in large sample sizes. --- class: center, middle # What's next? ### Move on to the readings for the next module!