class: center, middle, inverse, title-slide # <b> Generalised Linear Model </b> ## Data Analysis for Psychology in R 2<br><br> ### dapR2 Team ### Department of Psychology<br>The University of Edinburgh --- # Weeks Learning Objectives + Develop awareness of the glm for modelling non-continuous outcomes + Understand the types of missing data and common causes. + Develop awareness of approaches to dealing with missing data. --- # Topics for today + Generalised Linear Model + Basic structure + Types of outcome and associated GLM --- # Why do we need the GLM? - We saw last week that when we have a binary outcome, the linear model is not appropriate. - There are many other different types of outcome variable for which the same is true. - The GLM provides a unified framework to understand how we can analyse outcome data of different types. - Binary - multiple categories (ordered or unordered) - count data - continuous data with non-normal distributions (response times) --- # Structure of GLM 1. ***Random component*** that specifies the conditional distribution of `\(Y_i\)` given the values of the predictors (or 2). - Sounds scary, but remember this: `\(\epsilon \sim N(0, \sigma)\)` from linear model - This is just another way of saying that the conditional distribution of `\(Y_i\)` is normal or, - `\(Y_i \sim N(\beta_0 + \beta1x_1, \sigma)\)` -- 2. ***Linear function of predictors***. `\(\eta_i = \beta_0 + \beta1x_1 ... +\beta_kx_k\)` - This is what we have seen for a majority of this course. - And previously called the deterministic element of the linear model -- 3. A ***linearizing link function*** `\(g(.)\)` which links `\(\eta_i\)` to mean of response (or `\(E(Y_i))\)` ) - This is a transformation of (1) to (2) - This can take lots of forms (we wont look at all in detail) --- # Structure of GLM - This leads to two ways to think about the GLM. 1. A linear function predicting a transformed response variable. `$$g(E(Y_i)) = \eta_i$$` 2. A non-linear model for the response. `$$E(Y_i) = g^{-1}(\eta_i)$$` - Note this mirrors the two ways we looked at the logistic model last week. --- # Returning to our types of data - The conditional distribution of `\(Y_i\)` for different data types is not normal. - Hence the linear model is not appropriate. <table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:left;"> Data Type </th> <th style="text-align:left;"> Example </th> <th style="text-align:left;"> Distribution </th> <th style="text-align:left;"> Link Name </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;"> ~Continuous Normal </td> <td style="text-align:left;"> Cognitive scores </td> <td style="text-align:left;"> Normal </td> <td style="text-align:left;"> Identity </td> </tr> <tr> <td style="text-align:left;"> ~Continuous non-normal </td> <td style="text-align:left;"> Response time </td> <td style="text-align:left;"> Exponential or Gamma </td> <td style="text-align:left;"> Negative Inverse </td> </tr> <tr> <td style="text-align:left;"> Count (unbounded) </td> <td style="text-align:left;"> Road traffic accidents </td> <td style="text-align:left;"> Poisson </td> <td style="text-align:left;"> Log </td> </tr> <tr> <td style="text-align:left;"> Binary </td> <td style="text-align:left;"> Hiring </td> <td style="text-align:left;"> Bernoulli or Binomial </td> <td style="text-align:left;"> Logit </td> </tr> <tr> <td style="text-align:left;"> 2+ categories </td> <td style="text-align:left;"> Occupational Choice </td> <td style="text-align:left;"> Multinomial </td> <td style="text-align:left;"> Logit </td> </tr> </tbody> </table> --- # Linear model within GLM - We can apply this idea to the standard linear model. - In the previous slide we saw reference to the **identity** link. - This simply means that the function returns the same value as the input. - Why? - Remember, the purpose of `\(g(.)\)` is to be a linearizing function. - With a standard continuous predictor, the linear model ***is*** linear. - There is no transformation needed. `$$\eta_i = g(E(Y_i)) = E(Y_i)$$` --- # Looking back to logistic - We can also link our logistic model from last week. - One step which hopefully connects is to emphasize: `$$E(Y_i) = P(Y=1)$$` - `\(g(.)\)` is thr logistic function we discussed last week. `$$g(.) = ln \left (\frac{P(Y=1)}{1-P(Y=1)} \right)$$` - So: `$$g(E(Y_i)) = \eta_i$$` - Is the logistic model from last week: `$$ln \left (\frac{P(Y=1)}{1-P(Y=1)} \right) = \beta_0 + \beta_1x_1 + \beta_2x_2 ... + \beta_kx_k$$` --- # Other forms of GLM <table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;"> <thead> <tr> <th style="text-align:left;"> Data Type </th> <th style="text-align:left;"> Example </th> <th style="text-align:left;"> Distribution </th> <th style="text-align:left;"> Link Name </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;"> Continuous Normal </td> <td style="text-align:left;"> A </td> <td style="text-align:left;"> Normal </td> <td style="text-align:left;"> Identity </td> </tr> <tr> <td style="text-align:left;"> Continuous non-normal </td> <td style="text-align:left;"> B </td> <td style="text-align:left;"> Exponential or Gamma </td> <td style="text-align:left;"> Negative Inverse </td> </tr> <tr> <td style="text-align:left;"> Count </td> <td style="text-align:left;"> C </td> <td style="text-align:left;"> Poisson </td> <td style="text-align:left;"> Log </td> </tr> <tr> <td style="text-align:left;"> Binary </td> <td style="text-align:left;"> D </td> <td style="text-align:left;"> Bernoulli or Binomial </td> <td style="text-align:left;"> Logit </td> </tr> <tr> <td style="text-align:left;"> 2+ categories </td> <td style="text-align:left;"> E </td> <td style="text-align:left;"> Multinomial </td> <td style="text-align:left;"> Logit </td> </tr> </tbody> </table> - This table represents a small subset of the most commonly used models. - But as a framework the GLM has been extended into a huge array of different models. --- # Estimation and evaluation - **Estimation**: For all cases you will likely encounter, it is maximum likelihood. - As noted last week, there is an excellent introduction in the Enders Missing Data book (see reading) - And there is a short introduction in week 8 lab. - **Evaluation**: If we have used ML, we have the deviance. - And so we can use the likelihood ratio (or `\(\chi^2\)` difference or drop in deviance) tests - AIC - BIC - *z* (sometimes called Wald) tests --- # How much of this do I need to remember? - Remember is exists! - Remember that you know about the tools needed to run and evaluate these models: - maximum likelihood - likelihood ratio test - AIC and BIC - *z* or Wald test - linear function of `\(x\)` and `\(\beta\)` - `glm()` - And after this, remember that if you think this is the way you need to go for a project, you know what to ask about!