And, while such models are very explicit in the black box approach, tools for interpretability have been much more accessible these days. Such models are usually found among machine learning techniques, any number of which might be utilized in a number of disciplines. As Venables and Ripley note, generalized additive models might be thought of as falling somewhere in between the fully parametric and highly interpretable models of linear regression and more black box techniques. Indeed, there are even algorithmic approaches which utilize GAMs as part of their approach.
Note that just as generalized additive models are an extension of the generalized linear model, there are generalizations of the basic GAM beyond the settings described. In particular, random effects can be dealt with in this context as they can with linear and generalized linear models, and there is an interesting connection between smooths and random effects in general.
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This amounts to an interaction of the sort we demonstrated with two continuous variables. See the appendix for details. Additive models also provide a framework for dealing with spatially correlated data as well. As an example, a Markov Random Field smooth can be implemented for discrete spatial structure, such as countries or states Incidentally, this same approach would potentially be applicable to network data as well, e.
For the continuous spatial domain, one can use the 2d smooth as was demonstrated previously, e. Again one can consult the appendix for demonstration, and see also the Gaussian process paragraph. The combination of random effects, spatial effects, etc. It is the penalized regression approach that makes this possible, where we have a design matrix that might include basis functions or an indicator matrix for groups, and an appropriate penalty matrix.
With those two components, we can specify the models in almost identical fashion, and combine such effects within a single model. This results in a very powerful regression modeling strategy. Generalized additive models for location, scale, and shape GAMLSS allow for distributions beyond the exponential family 31 , and modeling different parameters besides the mean.
In addition, there are boosted, ensemble and other machine learning approaches that apply GAMs. See the GAMens package for example. Also, boosted models are GAMs. Generalized smoothing splines are built on the theory of reproducing kernel Hilbert spaces. This connection lends itself to a tie between GAMs and e. We can also approach modeling by using generalizations of the Gaussian distribution. Where the Gaussian distribution is over vectors and defined by a mean vector and covariance matrix, a Gaussian Process is a distribution over functions. They have a close tie to RKHS methods, and generalize commonly used models in spatial modeling.
The reader is encouraged to consult Rasmussen and Williams for the necessary detail. The text is free for download, and Rasmussen also provides a nice and brief intro. I also have some R code for demonstration based on his Matlab code, as well as Bayesian examples in Stan.
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Suffice it to say in this context, it turns out that generalized additive models with a tensor product or cubic spline smooth are maximum a posteriori MAP estimates of Gaussian processes with specific covariance functions and a zero mean function. In that sense, one might segue nicely to Gaussian processes if familiar with additive models.
The mgcv package also allows one to use a spline form of Gaussian process. Venables, William N.
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Modern Applied Statistics with S. Rasmussen, Carl Edward, and Christopher K. I Williams. Gaussian Processes for Machine Learning. Cambridge, Mass. Third, what contribution do individual predictors make to the model? In order to assess models, different model fit statistics would be examined.
The likelihood-ratio test can be employed for model building in general, for examining what happens when effects in a model are allowed to vary, and when testing a dummy-coded categorical variable as a single effect. When testing non-nested models, comparisons between models can be made using the Akaike information criterion AIC or the Bayesian information criterion BIC , among others. Multilevel models have the same assumptions as other major general linear models e. The assumption of linearity states that there is a rectilinear straight-line, as opposed to non-linear or U-shaped relationship between variables.
The assumption of normality states that the error terms at every level of the model are normally distributed. However, most statistical software allows one to specify different distributions for the variance terms, such as a Poisson, binomial, logistic. The multilevel modelling approach can be used for all forms of Generalized Linear models. The assumption of homoscedasticity , also known as homogeneity of variance, assumes equality of population variances.
Independence is an assumption of general linear models, which states that cases are random samples from the population and that scores on the dependent variable are independent of each other. The type of statistical tests that are employed in multilevel models depend on whether one is examining fixed effects or variance components. When examining fixed effects, the tests are compared with the standard error of the fixed effect, which results in a Z-test. When computing a t-test, it is important to keep in mind the degrees of freedom, which will depend on the level of the predictor e.
For a level 2 predictor, the degrees of freedom are based on the number of level 2 predictors and the number of groups. Statistical power for multilevel models differs depending on whether it is level 1 or level 2 effects that are being examined.
Power for level 1 effects is dependent upon the number of individual observations, whereas the power for level 2 effects is dependent upon the number of groups. However, the number of individual observations in groups is not as important as the number of groups in a study. In order to detect cross-level interactions, given that the group sizes are not too small, recommendations have been made that at least 20 groups are needed. The concept of level is the keystone of this approach. In an educational research example, the levels for a 2-level model might be:. However, if one were studying multiple schools and multiple school districts, a 4-level model could be:.
The researcher must establish for each variable the level at which it was measured. In this example "test score" might be measured at pupil level, "teacher experience" at class level, "school funding" at school level, and "urban" at district level. As a simple example, consider a basic linear regression model that predicts income as a function of age, class, gender and race.
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It might then be observed that income levels also vary depending on the city and state of residence. A simple way to incorporate this into the regression model would be to add an additional independent categorical variable to account for the location i. This would have the effect of shifting the mean income up or down—but it would still assume, for example, that the effect of race and gender on income is the same everywhere.
In reality, this is unlikely to be the case—different local laws, different retirement policies, differences in level of racial prejudice, etc. A multilevel model, however, would allow for different regression coefficients for each predictor in each location. Essentially, it would assume that people in a given location have correlated incomes generated by a single set of regression coefficients, whereas people in another location have incomes generated by a different set of coefficients.
Meanwhile, the coefficients themselves are assumed to be correlated and generated from a single set of hyperparameters. Additional levels are possible: For example, people might be grouped by cities, and the city-level regression coefficients grouped by state, and the state-level coefficients generated from a single hyper-hyperparameter.
Modeling risk using generalized linear models.
Multilevel models are a subclass of hierarchical Bayesian models , which are general models with multiple levels of random variables and arbitrary relationships among the different variables. Multilevel analysis has been extended to include multilevel structural equation modeling , multilevel latent class modeling , and other more general models. Multilevel models have been used in education research or geographical research, to estimate separately the variance between pupils within the same school, and the variance between schools.
In psychological applications, the multiple levels are items in an instrument, individuals, and families. In sociological applications, multilevel models are used to examine individuals embedded within regions or countries. In organizational psychology research, data from individuals must often be nested within teams or other functional units. Different covariables may be relevant on different levels. They can be used for longitudinal studies, as with growth studies, to separate changes within one individual and differences between individuals. Cross-level interactions may also be of substantive interest; for example, when a slope is allowed to vary randomly, a level-2 predictor may be included in the slope formula for the level-1 covariate.
For example, one may estimate the interaction of race and neighborhood so that an estimate of the interaction between an individual's characteristics and the context.
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There are several alternative ways of analyzing hierarchical data, although most of them have some problems. First, traditional statistical techniques can be used. One could disaggregate higher-order variables to the individual level, and thus conduct an analysis on this individual level for example, assign class variables to the individual level.
The problem with this approach is that it would violate the assumption of independence, and thus could bias our results. This is known as atomistic fallacy. The problem with this approach is that it discards all within-group information because it takes the average of the individual level variables.
Another way to analyze hierarchical data would be through a random-coefficients model. This model assumes that each group has a different regression model—with its own intercept and slope. This allows for an analysis in which one can assume that slopes are fixed but intercepts are allowed to vary.
This also allows for an analysis in which the slopes are random; however, the correlations of the error terms disturbances are dependent on the values of the individual-level variables. Multilevel models have two error terms, which are also known as disturbances.