So far in this course, we've looked at ANOVAs for a parametric test, analyses of variants, and all of these have fallen into the category of linear models, which we'll abbreviate LM. And the ANOVAs that we've used are examples of linear models. But there's a generalization of linear models called, appropriately, generalized linear models. How do they generalize the linear model? Well, we'll leave the math aside, but conceptually what you can think of is that the linear model relates our factors, or what are called predictor variables, the things we're manipulating in an experiment, it relates them to a response that's generally considered to be normal in nature. You'll remember our normal distribution from before. The generalized linear model goes beyond the relationship of predictors to our response being normal to allow for those other kinds of distributions that we looked at previously. I won't draw them all again, but we'll remember the log normal distribution, and we'll remember things like the exponential distribution, the Poisson distribution, gamma distribution, binomial or multinomial distributions, and we talked about all those previously. When we generalize the model to these other distributions, we're able to analyze data that would otherwise perhaps violate the assumptions of ANOVA or not be suitable to just the regular linear model. So we're going to use GLMs to revisit some of the data that we analyzed previously in other ways. And we'll show that these generalized linear models are quite powerful and versatile. The generalized linear model, it's important to recognize, can only handle between-subjects factors. So we'll be looking at just between-subjects situations for the generalized linear model. And later in the course, we'll consider the generalized linear mixed model and the linear mixed model, which add the opportunity to do within-subjects analyses of experiments. For now we'll be between-subjects with the generalized linear model. And we'll look at three examples. We'll look, again, at our prefs ABC sex data, where we saw website preference responses from men and women in three categories, websites A, B, or C. Originally we analyzed that with a chi-squared test. We analyzed it with a two-sample chi-squared test. Now we'll use something called multinomial logistic regression. And you'll recall the word multinomial is related to the multinomial distribution that we discussed previously. It's when we have multiple categories of responses. We also will look again at our search scroll and voice data, and in particular the effort ratings on the Likert scales. So we had a 1 to 7 effort score that people gave. You might recall that this was about using search, scrolling, and voice on a smartphone to find contacts in a contacts manager. Now this was originally a within-subjects rating. We'll recode it to pretend it's a between-subjects variable and comply with the between-subjects requirement for generalized linear models. Previously we analyzed this then with a Friedman test. With a between-subjects approach to this, we'll analyze it with something called ordinal logistic regression. Ordinal because we have an ordinal response variable. It's related to multinomial logistic regression, but for ordinal responses. And last we're going to also look in search scroll and voice, but at the error counts. And the error counts were the errors made while finding contacts in the contacts manager. We originally analyzed that also with a Friedman test. Here we'll assume again they were between subjects, and we'll use Poisson regression to analyze those counts. You might recall we did test early on, knowing this was coming, to see whether those counts were Poisson distributed. All of these, Poisson, ordinal, and multinomial, are forms of generalized linear models. And we'll see the power of these models to analyze non-numeric responses in these cases, and non-normally distributed data, discrete data distributed in a Poisson model in this case. So let's go now to our R code and take a look at how we'd carry out these generalized linear models.
