11. Using Generalized Linear (Mixed) Models in HCI

The practice of fitting GLMs is also briefly discussed in Chap. 6 as a method for dealing with non-normal dependent data.

See also Chap. 3 of this volume for more on data visualization.

This is quite obvious since in this particular case we generated the data using a second-order polynomial, \(y \,=\, 320 \, +\, 25x \,-\, 0.3x^2 \,+\, \varepsilon \). However, in a real study one would not know the exact data generating model and visual inspection of the data will help to understand the data.

More formally, finding the “best” of “closest” line can be (and is often) defined by minimizing the squared error \(\sum _{i\,=\,1}^n (y\, -\, X\beta )^2\) where, using matrix notation, X is the \(n \times k\) design matrix, and \(\beta \) the vector of coefficients of length k. For example, for model M1 (\(\hat{y} \,=\, \beta _0 + \beta _1 x\), see body text), X is a \(n \times 2\) matrix of which the first column contains only 1’s (for the intercept) and the second column contains the values \(x_{1, \,\ldots , n}\) respectively. We are looking for the vector \(\beta \) that minimizes the square error which is relatively easy to do by taking the gradient (vector of first order partial derivatives) of the error function and setting it to 0. Minimizing the squared error gives the same solution for \(\beta \) as maximizing the likelihood using a probabilistic framework. Likelihood maximization provides an estimation method that scales more easily to more complex models then the minimixation of the squared error. To use maximum likelihood estimation we would assume \(y | X \sim \mathscr {N}(X\beta , \sigma ^2)\) where \(\sigma ^2\) denotes the residual variance. Hence, in this model we assume the dependent variable y to be distributed normal conditional on X. The likelihood of the dataset, given that we assume our observations to be independent and identically distributed (i.i.d), is just the product over the likelihood for each datapoint. This we can maximize by taking its derivative and setting to zero. Often, for practical purposes, we take the derivative of the log of the likelihood which results in a summation over datapoints instead of a product and is thus easier to differentiate (For more info see, e.g., Gelman and Hill 2007; Millar 2011). In the case of simple linear models (LMs) an exact solution for \(\beta \) exists and is given, in matrix notation, by: \(\hat{\beta } \,=\, (X^T X)^{-1} X^T y\).

Please be cautious using these types of comparisons: a “good” fit, does not mean the model is true. This chapter is too short to properly cover model selection methods. More on the topic of model selection can be found in (e.g., Bozdogan 1987).

Model M0 is a 0th order polynomial, M1 is a first order polynomial of age (the linear term), and M2 a second order polynomial of age (the quadratic term). The poly function easily generates higher order polynomial. The model we fit here thus looks as follows: \(y \,=\, \beta _0 \,+\, \beta _1 \texttt {age} \,+\, \beta _2 \texttt {age}^2 +\, \cdots \,+\, \beta _{30} \texttt {age}^{30}\).

This procedure outlines the standard methods that are used to asses overfitting: by splitting up a dataset into a training-set and a test-set one can fit models on the training-set, and subsequently evaluate them on the test-set. If a model performs well on the training-set, but badly on the test-set (in terms of error), then the model likely overfits the data.

Regularization changes the definition of “best” line to one in which we minimize a term that looks like \(\sum _{i\,=\,1}^n (y\, -\,X\beta )^2 \,+\, \lambda f(\beta )\). Here, \(f({\beta })\) is some function whose output grows as the size of the elements of \(\beta \) grow. Using the sum of the absolute elements of \(\beta \) for f(x) is called the “Lasso”, while using the L2 norm is called “ridge” regression. Here, \(\lambda \) is a tuning parameter which determines the magnitude of the penalty.

Note the use of jitter to display the values of adopt. Since these observations are \(\in \{ 0 ,1 \}\), plotting the values directly would clutter the figure. The jitter command adds a slight random variation to the observed values.

For example the fact that for the linear model, one of the assumptions—in the maximum likelihood framework—is the fact that the conditional expected value and the variance of the observed variable are unrelated: while this is true for normally distributed outcomes, it is not generally true for many other outcome types.

For more info on the actual methods of finding a “best” line in this context, using Maximum Likelihood estimation, see (Hastie et al. 2013).

Often, in a complete unpooled approach, the analyst actually fits multiple, completely independent models. Then, the slopes would also differ per country. Here we focus only on the intercept for illustration purposes.

From the above it can be seen why also more formally mixed models can be regarded “in-between” pooled and unpooled models: A pooled model is the special case of a mixed model where \(\beta _{[k]}\,\sim \,\mathscr {N}(\beta _0, 0)\), and an unpooled model is the special case where \(\beta _{[k]}\,\sim \,\mathscr {N}(\beta _0, \infty )\).

The analyst could also include different distributional assumptions regarding the “batches” of coefficient. This however is outside the scope of this article.