Section 13 Model selection

The aim of linear regression is to find a model with the following properties:

  1. The model gives good predictions \(\hat y\).
  2. The model is easy to interpret.
  3. The modelling assumptions are satisfied.

These three aims can sometimes contradict: models with fewer parameters are often easier to interpret, whereas good predictions sometimes require a large number of parameters. Some trade-off between these constraints must be made. In this section we discuss how to choose the inputs for a model in a systematic way.

13.1 Candidates Models

Here we assume that we are given data with input variables \(x_1, \ldots, x_p\). The standard model which we have discussed so far adds an intercept to these inputs, so that there are \(p+1\) model parameters in total. We can modify this model in various ways:

  • When we discussed hypothesis tests, we allowed for the possibility that some inputs do not influence the output. A reasonable approach would be to fit a model with these inputs omitted. Since each input may either be included in or excluded from the model, independent of the others, and similarly the intercept may be included or not, there are \(2^{p+1}\) possible models to consider.

  • In the section about Improving the Model Fit we have seen that sometimes it is useful to transform input variables before using them in the model. This can either happen individually, e.g. \(x_2' = \log x_2\), or collectively, e.g. \(x^\ast = x_1 x_2\). The number of models we may obtain in this way is unlimited.

If we want to compare these candidate models in a systematic way, we need to use efficient methods for comparison, since a often a large number of models needs to be considered.

13.2 Misspecified Models

The model is said to be misspecified, if the data we are using comes from a different model than the one we use to estimate the coefficients.

13.2.1 Missing Variables

Assume that the data comes from the model \[\begin{equation*} Y = \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_p x_p + \varepsilon, \end{equation*}\] (we can include the intercept as \(x_1 = 1\) if needed), but we are only using the first \(q\) variables \(x_1, \ldots, x_q\), where \(q < p\), to estimate the coefficients. Then the least squares estimate of \(\beta = (\beta_1, \ldots, \beta_q)\) is given by \[\begin{equation*} \hat\beta = (X^\top X)^{-1} X^\top y \end{equation*}\] where \(X \in \mathbb{R}^{n\times q}\), and our estimate for the error variance \(\sigma^2\) is \[\begin{equation*} \hat\sigma^2 = \frac{1}{n-q} y^\top (I - H) y, \end{equation*}\] where \(H = X (X^\top X)^{-1} X^\top\) is the hat matrix computed without using \(x_{q+1}, \ldots, x_p\).

If we write \(\tilde X \in \mathbb{R}^{n\times (p-q)}\) for the matrix with columns \(x_{q+1}, \ldots, x_p\) and \(\tilde\beta = (\beta_{q+1}, \ldots, \beta_p) \in \mathbb{R}^{p-q}\), then the full model can be written as \[\begin{equation*} Y = X \beta + \tilde X \tilde\beta + \varepsilon \end{equation*}\] and we get \[\begin{align*} \hat\beta &= (X^\top X)^{-1} X^\top Y \\ &= (X^\top X)^{-1} X^\top (X \beta + \tilde X \tilde\beta + \varepsilon) \\ &= (X^\top X)^{-1} X^\top X \beta + (X^\top X)^{-1} X^\top \tilde X \tilde\beta + (X^\top X)^{-1} X^\top \varepsilon\\ &= \beta + (X^\top X)^{-1} X^\top \tilde X \tilde\beta + (X^\top X)^{-1} X^\top \varepsilon. \end{align*}\] Thus, we have \[\begin{equation*} \mathbb{E}(\hat\beta) = \beta + (X^\top X)^{-1} X^\top \tilde X \tilde\beta \end{equation*}\] and \[\begin{equation*} \mathop{\mathrm{bias}}(\hat\beta) = (X^\top X)^{-1} X^\top \tilde X \tilde\beta. \end{equation*}\] This shows that now the estimate is biased in general. There are two special cases when omitting variables does not introduce a bias: The first is when all of the omitted coefficients equal zero, i.e. when we have \(\tilde\beta = 0\). The second case is when the omitted inputs are orthogonal to the included inputs, since then we have \(X^\top \tilde X = 0\).

Using the formula for \(\hat\beta\) we find \[\begin{align*} \mathop{\mathrm{Cov}}( \hat\beta ) &= \mathop{\mathrm{Cov}}\Bigl( \beta + (X^\top X)^{-1} X^\top \tilde X \tilde\beta + (X^\top X)^{-1} X^\top \varepsilon\Bigr) \\ &= \mathop{\mathrm{Cov}}\Bigl( (X^\top X)^{-1} X^\top \varepsilon\Bigr) \\ &= \sigma^2 (X^\top X)^{-1} X^\top I X (X^\top X)^{-1} \\ &= \sigma^2 (X^\top X)^{-1}. \end{align*}\] Similar to our derivation in the section Estimating the Error Variance one can show that \[\begin{equation*} \mathbb{E}\bigl( \hat\sigma^2 \bigr) = \sigma^2 + \frac{\tilde\beta^\top \tilde X^\top (I - H) \tilde X \tilde \beta}{n-q}. \end{equation*}\] This shows that the estimate of the error variance is in general also biased.

This shows that our estimates can become biased if some of the inputs are missing from our model. As in the previous section, it may happen that the MSE of the parameter estimates in the reduced model is smaller than the MSE in the correct model; this is the case if the variance of the estimates decreases enough to compensate for the introduced bias.

13.2.2 Unnecessary Variables

If the data comes from a model with fewer inputs than the ones we are using, the model is not “misspecified” in the strict sense. It is just the case that some of the true \(\beta_j\) exactly equal zero. In this case, our previous results show that the least squares estimate is unbiased.

Including additional variables into a model still can cause problems: One can show that each unnecessary variable added increases the variance of the estimates \(\hat\beta_j\). Instead of giving a proof of this fact, we illustrate the effect using a numerical example.

Example 13.1 Here we consider the stackloss dataset with an added column of noise. We have seen that \[\begin{equation*} \mathop{\mathrm{Var}}( \hat\beta_j ) = \sigma^2 \bigl( X^\top X \bigr)^{-1}_{jj}. \end{equation*}\] While we don’t know the true value of \(\sigma^2\), we can determine the relative change in variance when adding a column, since this process does not change \(\sigma^2\) and thus \(\sigma^2\) will cancel when we determine the relative change in variance.

We first compute the diagonal elements of \(\bigl( X^\top X \bigr)^{-1}\) for the original dataset:

X1 <- model.matrix(stack.loss ~ ., data = stackloss)
d1 <- diag(solve(t(X1) %*% X1))
d1
 (Intercept)     Air.Flow   Water.Temp   Acid.Conc. 
13.452726695  0.001728874  0.012875424  0.002322167 

Now we add a column of noise and re-compute the values:

set.seed(20211115)
n <- nrow(X1)
X2 <- cbind(X1, rnorm(n))
d2 <- diag(solve(t(X2) %*% X2))
d2
 (Intercept)     Air.Flow   Water.Temp   Acid.Conc.              
14.397515774  0.001730570  0.015195467  0.002796487  0.064828744 

Finally, we determine the change in variance in percent:

round(100 * (d2[-5] - d1) / d1, 3)
(Intercept)    Air.Flow  Water.Temp  Acid.Conc. 
      7.023       0.098      18.019      20.426 

We see that the variances of the \(\hat\beta_j\) increased by up to \(20\%\) when we added the unnecessary input.

The example illustrates that it is important to keep the model as small as possible.

13.3 Assessing Models

A variety of criteria is used in practice to select variables to include in a regression model.

  1. The coefficient of multiple determination, \(R^2\), can be used to compare models. “Good” models have large \(R^2\). If comparisons are made between models with different numbers of inputs then the adjusted \(R^2\)-value, \(R^2_\mathrm{adj}\), must be used.

  2. The estimate for the error variance, \[\begin{equation*} \hat\sigma^2 = \frac{1}{n - q} \sum_{i=1}^n (y_i - \hat y_i)^2 \end{equation*}\] can be used, where \(q\) is the number of columns of the design matrix. (For the full model with an intercept we have \(q = p+1\).) “Good” models have small \(\hat\sigma^2\). Since \[\begin{align*} R^2_\mathrm{adj} &= 1 - \frac{\frac{1}{n-q}\mathrm{s}_{\hat\varepsilon}^2}{\frac{1}{n-1}\mathrm{s}_y^2} \\ &= 1 - \frac{\frac{n-1}{n-q}\mathrm{s}_{\hat\varepsilon}^2}{\mathrm{s}_y^2} \\ &= 1 - \frac{\frac{1}{n-q} \sum_{i=1}^n (y_i - \hat y_i)^2}{\mathrm{s}_y^2} \\ &= 1 - \frac{\hat\sigma^2}{\mathrm{s}_y^2}, \end{align*}\] where \(\mathrm{s}_{\hat\varepsilon}^2\) and \(\mathrm{s}_y^2\) are the sample variances as before, minimising \(\hat\sigma^2\) is equivalent to maximising \(R^2_\mathrm{adj}\). Thus, this criterion is equivalent to the first one.

  3. The Prediction Error Sum of Squares (PRESS) compares each output \(y_i\) to the fitted value \(\hat y^{(i)}_i\) for observation i, computed without using sample \(i\): \[\begin{equation*} \mathrm{PRESS} := \sum_{i=1}^n \bigl( y_i - \hat y^{(i)}_i \bigr)^2. \end{equation*}\] In the comparison, \(\hat y^{(i)}_i\) is used instead of \(\hat y_i\) so that in models with too many parameters, overfitting does not result in overoptimistic estimates. “Good” models have small PRESS. The following lemma helps to compute PRESS without having to fit \(n\) separate models.

    Lemma 13.1 We have \[\begin{equation*} \mathrm{PRESS} = \sum_{i=1}^n \Bigl( \frac{\hat\varepsilon_i}{1 - h_{ii}} \Bigr)^2, \end{equation*}\] where \(h_{ii}\) denotes the diagonal elements of the hat matrix \(H\).

    Proof. From equation (9.3) we know how the estimated regression coefficients change, when observation \(i\) is deleted from the model: \[\begin{equation*} \hat\beta^{(i)} - \hat\beta = - (X^\top X)^{-1} x_i \frac{\hat\varepsilon_i}{1 - h_{ii}}. \end{equation*}\] Thus we have \[\begin{align*} \hat y^{(i)}_i - \hat y_i &= \bigl( X \hat\beta^{(i)} - X \hat\beta \bigr)_i \\ &= - \bigl( X (X^\top X)^{-1} x_i \frac{\hat\varepsilon_i}{1 - h_{ii}} \bigr)_i \\ &= - x_i^\top (X^\top X)^{-1} x_i \frac{\hat\varepsilon_i}{1 - h_{ii}} \\ &= - \frac{h_{ii}}{1 - h_{ii}} \hat\varepsilon_i. \end{align*}\] From this we get \[\begin{align*} y_i - \hat y^{(i)}_i &= y_i - \hat y_i + \hat y_i - \hat y^{(i)}_i \\ &= \hat\varepsilon_i + \frac{h_{ii}}{1 - h_{ii}} \hat\varepsilon_i \\ &= \frac{1}{1 - h_{ii}} \hat\varepsilon_i. \end{align*}\] Substituting this expression into the definition of PRESS completes the proof.

  4. We can consider \[\begin{equation*} C_q := \frac{1}{\hat{\sigma}^2} \sum_{i=1}^n (y_i - \hat y_i)^2 - n + 2q. \end{equation*}\] The subscript \(q\) in \(C_q\) denotes the number of parameters in the model. This quantity is called Mallows’ Cp or sometimes Mallows’ Ck. “Good” models have \(C_q \approx q\).

    The form of \(C_q\) is explained by the fact that it approximates the quantity \[\begin{equation*} \Gamma_q := \frac{1}{\sigma^2} \sum_{i=1}^n \mathop{\mathrm{MSE}}\nolimits(\hat y_i). \end{equation*}\] To see that \(C_q \approx \Gamma_q\) we first rewrite \(\Gamma_q\) as \[\begin{align*} \Gamma_q &= \sum_{i=1}^n \Bigl( \mathop{\mathrm{Var}}(\hat y_i) + \mathop{\mathrm{bias}}(\hat y_i)^2 \Bigr) \\ &= \frac{1}{\sigma^2} \sum_{i=1}^n \mathop{\mathrm{bias}}(\hat y_i)^2 + \frac{1}{\sigma^2} \sum_{i=1}^n \mathop{\mathrm{Var}}(\hat y_i). \end{align*}\] Since \(\hat y = HY \sim \mathcal{N}(HX\beta, \sigma^2 HIH^\top) = \mathcal{N}(X\beta, \sigma^2 H)\) we have \(\mathop{\mathrm{Var}}(\hat y_i) = \sigma^2 h_{ii}\) were \(h_{ii}\) are the diagonal elements of the hat matrix. Furthermore, using the trace operator (see appendix A.2.5) we find \[\begin{align*} \sum_{i=1}^n h_{ii} &= \mathop{\mathrm{tr}}(H) \\ &= \mathop{\mathrm{tr}}\bigl(X (X^\top X)^{-1} X^\top \bigr) \\ &= \mathop{\mathrm{tr}}\bigl((X^\top X)^{-1} X^\top X \bigr) \\ &= \mathop{\mathrm{tr}}(I) \\ &= q, \end{align*}\] and thus \[\begin{equation} \Gamma_q = \frac{1}{\sigma^2} \sum_{i=1}^n \mathop{\mathrm{bias}}(\hat y_i)^2 + q. \tag{13.1} \end{equation}\] This shows that for a model where the \(\hat y_i\) are unbiased, we have \(\Gamma_q = q\), and if there is a bias we have \(\Gamma_q > q\). We now show that \(C_q \approx \Gamma_q\).

    We also have \(\hat y = \mathop{\mathrm{bias}}(\hat y) + HY\) and thus \[\begin{equation*} Y - \hat y = (I - H)Y - \mathop{\mathrm{bias}}(\hat y) = (I - H) \varepsilon- \mathop{\mathrm{bias}}(\hat y). \end{equation*}\] Using this relation, one can show that \[\begin{align*} \mathbb{E}\Bigl( \sum_{i=1}^n (y_i - \hat y_i)^2 \Bigr) &= (Y - \hat y)^\top (Y - \hat y) \\ &= \dots \\ &= \sum_{i=1}^n \mathop{\mathrm{bias}}(\hat y_i)^2 + (n-q) \sigma^2. \end{align*}\] Substituting this result into the formula for \(\Gamma_q\) we get \[\begin{align*} \Gamma_q &= \frac{1}{\sigma^2} \mathbb{E}\Bigl( \sum_{i=1}^n (y_i - \hat y_i)^2 \Bigr) - (n-q) + q \\ &\approx \frac{1}{\sigma^2} \sum_{i=1}^n (y_i - \hat y_i)^2 - n + 2q \\ &\approx \frac{1}{\hat\sigma^2} \sum_{i=1}^n (y_i - \hat y_i)^2 - n + 2q \\ &= C_q. \end{align*}\]

    A plot of \(C_q\) vs. \(q\) will show adequate models as points close to the line \(C_q=q\). Points far from the line are not recommended, and, of those which lie close to the line, we generally prefer smaller models. The method relies on having a good estimate of \(\sigma^2\).

  5. Akaike’s Information Criterion (AIC) is defined as \[\begin{equation*} \mathrm{AIC} = 2q - 2 \log(\hat L), \end{equation*}\] where \(q\) is the number of parameters in the model, and \(\hat L\) is the maximum of the likelihood function when these \(q\) parameters are chosen optimally. “Good” models have small AIC.

    Since we assume \(\varepsilon_i \sim \mathcal{N}(0, \sigma^2)\), i.i.d. and we have \(\varepsilon_i = y_i - x_i^\top\beta\), the likelihood function for our model is \[\begin{align*} L(\beta, \sigma^2; X, y) &= \prod_{i=1}^n \frac{1}{\sqrt{2\pi\sigma^2}} \exp\Bigl( - \frac{(y_i - x_i^\top\beta)^2}{2\sigma^2} \Bigr) \\ &= \frac{1}{(2\pi\sigma^2)^{n/2}} \exp\Bigl( - \sum_{i=1}^n \frac{(y_i - x_i^\top\beta)^2}{2\sigma^2} \Bigr). \end{align*}\] Taking logarithms we get \[\begin{align*} \log L(\beta, \sigma^2; X, y) &= -\frac{n}{2} \log(2\pi\sigma^2) - \sum_{i=1}^n \frac{(y_i - x_i^\top\beta)^2}{2\sigma^2} \\ &= -\frac{n}{2} \log(2\pi\sigma^2) - \frac{1}{2\sigma^2} r(\beta), \end{align*}\] where \(r(\beta)\) is the residual sum of squares, as introduced in the section about Least Squares Estimates. To compute the AIC we need to maximise this expression. The values of \(\beta\) and \(\sigma^2\) where the maximum is taken are called the maximum likelihood estimate (MLE) for \(\beta\) and \(\sigma^2\). From lemma 2.1 we know that for fixed \(\sigma^2\), \(L\) takes its maximum when \(\beta\) equals the least squares estimate \(\hat\beta\). Taking derivatives we can then find the optimal value of \(\sigma^2\). The result is \[\begin{equation*} \hat\sigma^2_\mathrm{MLE} = \frac1n r(\hat\beta) = \frac{1}{n} \sum_{i=1}^n (y_i - x_i^\top \hat\beta)^2 = \frac{n-q}{n} \hat\sigma^2. \end{equation*}\] Thus we get \[\begin{align*} \mathrm{AIC} &= 2q + n \log(2\pi\hat\sigma^2_\mathrm{MLE}) + \frac{1}{\hat\sigma^2_\mathrm{MLE}} r(\hat\beta) \\ &= 2q + n \log\bigl( 2\pi r(\hat\beta) / n \bigr) + n \\ &= 2q + n \log\bigl( \|\hat\varepsilon\|^2 \bigr) + n + n \log( 2\pi / n ) \end{align*}\]

Summary

  • We have discussed how different models can be considered by omitting existing variables or by adding non-linear functions of the inputs as new variables.
  • We have seen that removing too many variables can introduce a bias.
  • We have seen that adding too many variables can increase the variance of the estimate.
  • We have considered different criteria for choosing the “best” model.