MLR: model building and selection

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# MLR: model building and selection
### Dr. Olanrewaju Michael Akande
### Sept 12, 2019

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## Announcements

- More office hours
  + Mon 3:00 - 4:00pm
  + Tue 2:00 - 4:00pm
  + Thur 2:00 - 3:00pm

## Outline

- Questions from last lecture

- Prediction vs. interpretation

- Model selection criterion: AIC, BIC

- Common selection strategies

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# Prediction vs. interpretation

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## Which predictors should be in your model?

- This is a very hard question and one of intense statistical research.

- Different people have different opinions on how to answer the question.

- It also depends on the goal of your analysis: prediction vs. interpretation or association.

- We will not focus on answering the question on which is the best "overall".

- Instead, we will focus on how to approach the problem and the most common methods used.

- See Section 6.1 of [An Introduction to Statistical Learning with Applications in R](http://faculty.marshall.usc.edu/gareth-james/ISL/) for more details on the methods we will cover.

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## What variables should you include?

- .hlight[Goal]: prediction
  + Include only variables that are strong predictors of the outcome.
  + Excluding irrelevant variables can reduce the widths of the prediction intervals.
  
- .hlight[Goal]: interpretation and association
  + Include all variables that you thought a apriori were related to the outcome of interest, even if they are not statistically significant.
  + This improves interpretation of coefficients of interest.

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# Model selection criterion

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## Model selection criterion

The most common are:
- Adjusted R-squared:
.block[
.small[
$$\textrm{Adj.}R^2 = 1 - (1-R^2) \left[\dfrac{n-1}{n-p-1} \right] $$
]
]

- Akaike's Information Criterion (AIC):
.block[
.small[
$$\textrm{AIC} = n \textrm{ln}(\textrm{RSS}) - n \textrm{ln}(n)  + 2(p+1) $$
]
]

- Bayesian Information Criterion (BIC) or Schwarz Criterion:
.block[
.small[
`$$\textrm{BIC} = n \textrm{ln}(\textrm{RSS}) - n \textrm{ln}(n)  + (p+1) \textrm{ln}(n)$$`
]
]
  
  where `$n$` is the number of observations, `$p$` is the number of variables (or parameters) excluding the intercept, and RSS is the residual sum of squares, that is,
  .block[
.small[
`$$\textrm{RSS} = \sum^n_{i=1} \left(y_i - \hat{y}_i \right)^2.$$`
]
]

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## Model selection criterion

- Note:
  + Large `$\textrm{Adj.}R^2$` = .hlight[good!]
  + Small AIC = .hlight[good!]
  + Small BIC = .hlight[good!]

- Notice that BIC generally places a heavier penalty on models with many variables for `$n > 8$` since
.block[
.small[
$$\textrm{ln}(n) (p+1) > 2(p+1) $$
]
]
for fixed `$p$` and  `$n > 8$`.

- Thus, BIC can result in the selection of smaller models than AIC.

- *Note: the formulas for `$\textrm{Adj.}R^2$`, AIC and BIC in Section 6.1 of [An Introduction to Statistical Learning with Applications in R](http://faculty.marshall.usc.edu/gareth-james/ISL/) take slightly different forms but are equivalent to those given here when comparing models.*

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# Common selection strategies

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## Backward selection

- Start with the full model that includes all `$p$` available predictors.

- Drop variables one at a time that are deemed irrelevant based on some criterion.
  + Drop the variable with the largest p-value
  + Drop variables (possibly all at once) with p-value over some threshold (for example, 0.10).
  + Drop the variable that leads to the smallest value of AIC or BIC, or the largest value of `$\textrm{Adj.}R^2$`.  
    *You might even consider using average MSE from k-fold cross-validation if the goal is prediction.*

- Stop when removing variables no longer improve the model, based on the chosen criterion.

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## Forward selection

- Start with the model that only includes the intercept.

- Add variables one at a time based on some criterion.
  + Add the variable with the smallest p-value using some threshold (for example, 0.10).
  + Add the variable that leads to the smallest value of AIC or BIC, or the largest value of `$\textrm{Adj.}R^2$`.  
    *Again, you might consider using average MSE from k-fold cross-validation if the goal is prediction.*

- Stop when adding variables no longer improves the model, based on the chosen criterion.

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## Stepwise selection

- Start with the model that only includes the intercept.

- Potentially do one forward step to enter a variable in the model, using some criterion to decide if it is worth including the variable.

- From the current model, potentially do one backwards step, using some criterion to decide if it is worth dropping one of the variables in the model.

- Repeat these steps until the model does not change.

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## Model Selection in R

- .hlight[step] function: forward, backward, and stepwise selection using AIC/BIC.

- .hlight[regsubsets] function (.hlight[leaps] package): forward, backward, and stepwise selection using `$\textrm{Adj.}R^2$` or BIC.
  
  
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## Other options: shrinkage methods

- Fit a model containing all `$p$` available predictors, then use a technique that shrinks the coefficient estimates towards zero.

- The two most common methods are:
  + Ridge regression
  + Lasso regression (performs variable selection)

- We will not cover these methods in this course.

- Consider taking STA521 if you are interested in learning about how they work!