class: center, middle, inverse, title-slide # IDS 702: Module 3.3 ## Multinomial logistic regression ### Dr. Olanrewaju Michael Akande --- ## Recall logistic regression - Recall that for logistic regression, we had .block[ .small[ $$ y_i | x_i \sim \textrm{Bernoulli}(\pi_i); \ \ \ \textrm{log}\left(\dfrac{\pi_i}{1-\pi_i}\right) = \beta_0 + \beta_1 x_i $$ ] ] for each observation `\(i = 1, \ldots, n\)`. -- - To get `\(\pi_i\)`, we solved the logit equation above to get .block[ .small[ `$$\pi_i = \dfrac{e^{\beta_0 + \beta_1 x_i}}{1 + e^{\beta_0 + \beta_1 x_i}}$$` ] ] -- - Consider `\(Y=0\)` a baseline category. Suppose `\(\Pr[y_i = 1 | x_i] = \pi_{i1}\)` and `\(\Pr[y_i = 0 | x_i] = \pi_{i0}\)`. Then, the logit expression is essentially .block[ .small[ `$$\textrm{log}\left(\dfrac{\pi_{i1}}{\pi_{i0}}\right) = \beta_0 + \beta_1 x_i$$` ] ] -- - `\(e^{\beta_1}\)` is thus the (multiplicative) change in odds of `\(y = 1\)` over the baseline `\(y = 0\)` when increasing `\(x\)` by one unit. --- ## Multinomial logistic regression - Suppose we have a nominal-scale response variable `\(Y\)` with `\(J\)` categories. First, for the .hlight[random component], we need a distribution to describe `\(Y\)`. -- - A standard option for this is the .hlight[multinomial distribution], which is essentially a generalization of the binomial distribution. Read about the multinomial distribution [here](https://www.olanrewajuakande.com/STA111-Summer2018-Course-Website/Lectures/Lecture6.pdf) and [here](https://en.wikipedia.org/wiki/Multinomial_distribution). -- - .hlight[Multinomial distribution] gives us a way to characterize .block[ .small[ `$$\Pr[y_i = 1] = \pi_1, \ Pr[y_i = 2] = \pi_2, \ \ldots, \ \Pr[y_i = J] = \pi_J, \ \ \ \textrm{where} \ \ \ \sum^J_{j=1} \pi_j = 1.$$` ] ] -- - When there are no predictors, the best guess for each `\(\pi_j\)` is the sample proportion of cases with `\(y_i = j\)`, that is, .block[ .small[ `$$\hat{\pi}_j = \dfrac{\mathbf{1}[y_i = j]}{n}$$` ] ] -- - When we have predictors, then we want .block[ .small[ `$$\Pr[y_i = 1 | \boldsymbol{x}_i] = \pi_{i1}, \ \Pr[y_i = 2 | \boldsymbol{x}_i] = \pi_{i2}, \ \ldots, \ \Pr[y_i = J | \boldsymbol{x}_i] = \pi_{iJ}.$$` ] ] --- ## Multinomial logistic regression - That is, we want the `\(\pi_j\)`'s to be functions of the predictors, like in logistic regression. -- - Turns out we can use the same .hlight[link function], that is the logit function, if we set one of the levels as the baseline. -- - Pick a baseline outcome level, say `\(Y=1\)`. -- - Then, the multinomial logistic regression is defined as a set of logistic regression models for each probability `\(\pi_j\)`, compared to the baseline, where `\(j\geq 2\)`. That is, .block[ .small[ `$$\textrm{log}\left(\dfrac{\pi_{ij}}{\pi_{i1}}\right) = \beta_{0j} + \beta_{1j} x_{i1} + \beta_{2j} x_{i2} + \ldots + \beta_{pj} x_{ip},$$` ] ] where `\(j\geq 2\)`. -- - We therefore have `\(J-1\)` .hlight[separate logistic regressions] in this setup. --- ## Multinomial logistic regression - The equation for each `\(\pi_{ij}\)` is given by .block[ .small[ `$$\pi_{ij} = \dfrac{e^{\beta_{0j} + \beta_{1j} x_{i1} + \beta_{2j} x_{i2} + \ldots + \beta_{pj} x_{ip}}}{1 + \sum^J_{j=2} e^{\beta_{0j} + \beta_{1j} x_{i1} + \beta_{2j} x_{i2} + \ldots + \beta_{pj} x_{ip}}} \ \ \ \textrm{for} \ \ \ j > 1$$` ] ] and .block[ .small[ `$$\pi_{i1} = 1-\sum^J_{j=2} \pi_{ij}$$` ] ] -- - Also, we can extract the log odds for comparing other pairs of the response categories `\(j\)` and `\(j^\star\)`, since .block[ .small[ $$ `\begin{split} \textrm{log}\left(\dfrac{\pi_{ij}}{\pi_{ij^\star}}\right) & = \textrm{log}\left(\pi_{ij}\right) - \textrm{log}\left(\pi_{ij^\star}\right) \\ & = \textrm{log}\left(\pi_{ij}\right) - \textrm{log}\left(\pi_{i1}\right) - \textrm{log}\left(\pi_{ij^\star}\right) + \textrm{log}\left(\pi_{i1}\right) \\ & = \left[ \textrm{log}\left(\pi_{ij}\right) - \textrm{log}\left(\pi_{i1}\right) \right] - \left[ \textrm{log}\left(\pi_{ij^\star}\right) - \textrm{log}\left(\pi_{i1}\right) \right] \\ & = \textrm{log}\left(\dfrac{\pi_{ij}}{\pi_{i1}}\right) - \textrm{log}\left(\dfrac{\pi_{ij^\star}}{\pi_{i1}}\right). \end{split}` $$ ] ] --- ## Multinomial logistic regression - Each coefficient has to be interpreted relative to the baseline. -- - That is, for a continuous predictor, + `\(\beta_{1j}\)` is the .hlight[increase (or decrease) in the log-odds] of `\(Y=j\)` versus `\(Y=1\)` when increasing `\(x_1\)` by one unit. + `\(e^{\beta_{1j}}\)` is the .hlight[multiplicative increase (or decrease) in the odds] of `\(Y=j\)` versus `\(Y=1\)` when increasing `\(x_1\)` by one unit. -- - Whereas, for a binary predictor, + `\(\beta_{1j}\)` is the .hlight[log-odds] of `\(Y=j\)` versus `\(Y=1\)` for the group with `\(x_1 = 1\)`, compared to the group with `\(x_1 = 0\)`. + `\(e^{\beta_{1j}}\)` is the .hlight[odds] of `\(Y=j\)` versus `\(Y=1\)` for the group with `\(x_1 = 1\)`, compared to the group with `\(x_1 = 0\)`. -- - Exponentiate confidence intervals from log-odds scale to get on the odds scale. --- ## Significance tests - For multinomial logistic regression, use the change in deviance test to compare models and test significance, just like we had for logistic regression. -- - Fit model with and without some predictor `\(x_k\)`. -- - Perform a change in deviance test to compare the two models. -- - Interpret p-value as evidence about whether the coefficients excluded from the smaller model are equal to zero. --- ## Model diagnostics - Use binned residuals like in logistic regression. -- - Each outcome level has its own raw residual. For each outcome level `\(j\)`, -- + make an indicator variable equal to one whenever `\(Y = j\)` and equal to zero otherwise -- + compute the predicted probability that `\(Y=j\)` for each record (using the `fitted` command) -- + compute the raw residual = indicator value - predicted probability -- - For each outcome level, make bins of predictor values and plot average value of predictor versus the average raw residual. Look for patterns. -- - We can still compute .hlight[accuracy] just like we did for the logistic regression. -- - ROC on the other hand is not so straightforward; we can draw a different ROC curve for each level of the response variable. We can also draw pairwise ROC curves. --- ## Implementation in R - Install the package .hlight[nnet] from CRAN. - Load the library: `library(nnet)`. - The command for running the multinomial logistic regression in R looks like: ```r Modelfit <- multinom (response ~ x_1 + x_2 + ... + x_p, data = Data) ``` - Use `fitted(Modelfit)` to get predicted probabilities for observed cases. -- - We will see an example in the next module. --- class: center, middle # What's next? ### Move on to the readings for the next module!