class: center, middle, inverse, title-slide # Prediction Intervals ### Dr. D’Agostino McGowan --- layout: true <div class="my-footer"> <span> Dr. Lucy D'Agostino McGowan </span> </div> --- ## Predictions Once we have built a model, `\(\hat{\mathbf{y}} = \mathbf{X}\hat\beta\)`, we can calculate _predicted_ y, `\(\hat{\mathbf{y}}_0\)` values for a new set of _predictors_, `\(\mathbf{x}_0\)`. -- `$$\hat{\mathbf{y}}_0=\mathbf{x}_0^T\hat\beta$$` -- For example, if we fit a model `\(\hat{y} = 1.2+2.5x_1+3x_2\)` And would like to know the predicted value for someone with `\(x_1 = 3\)` and `\(x_2 = 2\)`, we would calculate -- `$$\hat{\mathbf{y}}_0 = \begin{bmatrix}1&3&2\end{bmatrix} \begin{bmatrix}1.2\\2.5\\3\end{bmatrix}$$` -- `$$\hat{\mathbf{y}}_0 = 14.7$$` --- class: inverse ## <svg style="height:0.8em;top:.04em;position:relative;" viewBox="0 0 640 512"><path d="M512 64v256H128V64h384m16-64H112C85.5 0 64 21.5 64 48v288c0 26.5 21.5 48 48 48h416c26.5 0 48-21.5 48-48V48c0-26.5-21.5-48-48-48zm100 416H389.5c-3 0-5.5 2.1-5.9 5.1C381.2 436.3 368 448 352 448h-64c-16 0-29.2-11.7-31.6-26.9-.5-2.9-3-5.1-5.9-5.1H12c-6.6 0-12 5.4-12 12v36c0 26.5 21.5 48 48 48h544c26.5 0 48-21.5 48-48v-36c0-6.6-5.4-12-12-12z"/></svg> `Application Exercise` We are interested in predicting a chicken's weight based on their diet using the `chickwts` dataset * Fit the model of interest and extract the estimated `\(\beta\)` coefficients * Construct `\(\mathbf{x}_0\)` for a chicken that is eating "sunflower". * Find the predicted weight for a chicken eating sunflowers. --- ## Predictions There are ✌️ kinds of predictions that can be made from regression models -- * A predicted _mean response_ -- * A prediction of a _future observation_ -- .definition[ This matters for estimating the **uncertainty** ] --- ## Example * What would a chicken who eats sunflowers weigh **on average**? -- * Suppose you want to feed your chicken sunflowers, what will your chicken's predicted weight be? -- .question[ What is the difference? ] -- * one is a prediction for an **average** one is for an **individual** --- ## Prediction of the mean response .definition[ Example: What would a chicken who eats sunflowers weigh **on average**? ] The prediction is `\(\mathbf{x}_0^T\beta\)`, estimated by `\(\mathbf{x}_0^T\hat\beta\)`. -- .question[ What is the variance of this prediction? ] --- class: inverse ## <svg style="height:0.8em;top:.04em;position:relative;" viewBox="0 0 576 512"><path d="M402.6 83.2l90.2 90.2c3.8 3.8 3.8 10 0 13.8L274.4 405.6l-92.8 10.3c-12.4 1.4-22.9-9.1-21.5-21.5l10.3-92.8L388.8 83.2c3.8-3.8 10-3.8 13.8 0zm162-22.9l-48.8-48.8c-15.2-15.2-39.9-15.2-55.2 0l-35.4 35.4c-3.8 3.8-3.8 10 0 13.8l90.2 90.2c3.8 3.8 10 3.8 13.8 0l35.4-35.4c15.2-15.3 15.2-40 0-55.2zM384 346.2V448H64V128h229.8c3.2 0 6.2-1.3 8.5-3.5l40-40c7.6-7.6 2.2-20.5-8.5-20.5H48C21.5 64 0 85.5 0 112v352c0 26.5 21.5 48 48 48h352c26.5 0 48-21.5 48-48V306.2c0-10.7-12.9-16-20.5-8.5l-40 40c-2.2 2.3-3.5 5.3-3.5 8.5z"/></svg> `Application Exercise` Show that the variance of `\(\mathbf{x}_0^T\hat\beta\)` is `$$\textrm{var}(\mathbf{x}_0^T\hat\beta) = \mathbf{x}_0^T(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{x}_0\sigma^2$$` --- ## Confidence interval for a mean response `$$\hat{\mathbf{y}_0}\pm t*\hat\sigma\sqrt{\mathbf{x}_0^T(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{x}_0}$$` --- ## Prediction of a future value .definition[ Example: Suppose you want to feed your chicken sunflowers, what will your chicken's predicted weight be? ] The prediction is `\(\mathbf{x}_0^T\beta + \epsilon\)`. .question[ What is the expected value? What is the variance? ] --- class: inverse ## <svg style="height:0.8em;top:.04em;position:relative;" viewBox="0 0 576 512"><path d="M402.6 83.2l90.2 90.2c3.8 3.8 3.8 10 0 13.8L274.4 405.6l-92.8 10.3c-12.4 1.4-22.9-9.1-21.5-21.5l10.3-92.8L388.8 83.2c3.8-3.8 10-3.8 13.8 0zm162-22.9l-48.8-48.8c-15.2-15.2-39.9-15.2-55.2 0l-35.4 35.4c-3.8 3.8-3.8 10 0 13.8l90.2 90.2c3.8 3.8 10 3.8 13.8 0l35.4-35.4c15.2-15.3 15.2-40 0-55.2zM384 346.2V448H64V128h229.8c3.2 0 6.2-1.3 8.5-3.5l40-40c7.6-7.6 2.2-20.5-8.5-20.5H48C21.5 64 0 85.5 0 112v352c0 26.5 21.5 48 48 48h352c26.5 0 48-21.5 48-48V306.2c0-10.7-12.9-16-20.5-8.5l-40 40c-2.2 2.3-3.5 5.3-3.5 8.5z"/></svg> `Application Exercise` * What is the expected value of `\(\mathbf{x}_0^T\beta + \epsilon\)`? * Show that the variance is `$$(1 + \mathbf{x}_0^T(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{x}_0)\sigma^2$$` --- ## Prediction Intervals `$$\hat{\mathbf{y}}_0\pm t^*\hat\sigma\sqrt{1+\mathbf{x}_0^T(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{x}_0}$$` --- ## Prediction Intervals * There is an **important conceptual difference here** -- * _parameters_ (like `\(\beta_1\)`, `\(\beta_2\)`, etc) are considered **fixed** but **unknown** (they are **not random**) which is why we interpret confidence intervals like we do -- * A _future observation_ **is** a **random variable**. Therefore, we are saying there is a 95% chance that the future value falls within this interval -- * **THIS IS NOT the correct interpretation of a parameter's confidence interval**. It **is** the correct interpretation of a prediction interval --- ## Prediction Intervals Which is larger? `$$(\mathbf{x}_0^T(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{x}_0)\sigma^2$$` or `$$(1 + \mathbf{x}_0^T(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{x}_0)\sigma^2$$` -- * _prediction intervals_ tend to be wider than confidence intervals for a _mean response_