Statistical inference for generalized additive models with an application to mothers' depression symptoms
HANNULA, ISMO (2011)
2011
Matematiikka/tilastotiede - Mathematics/Statistics
Informaatiotieteiden yksikkö - School of Information Sciences
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Hyväksymispäivämäärä
2011-06-09
Julkaisun pysyvä osoite on
https://urn.fi/urn:nbn:fi:uta-1-21696
https://urn.fi/urn:nbn:fi:uta-1-21696
Tiivistelmä
In statistics, linear modelling techniques are widely used methods to explain one variable by others. Generalized additive model, GAM, has developed from both generalized linear models, GLM, and nonparametric regression methods. This master's thesis aims to provide a solid background view on both of these two, eventually leading to generalized additive model.
Work on generalized linear models provides one part of the statistical theory needed on generalized additive models. With it any exponential family distribution can be used in linear modelling along with different link functions which also play important roles in the GAM. The complicated likelihood equations in GLM are solved by using the derivation chain rule that leads us to a form which allows us to use the Newton's method and iteratively solve the parameters for the model. Nonparametric regression methods, such as cubic splines and thin plate splines, on the other hand allow any form for the explanatory variables to take place. They produce so-called smooth functions that base on data and smoothness selection. Generalized linear models and nonparametric regression are then combined to generalized additive models by imitating nonparametric methods with parametric estimates.Generalized additive models are based largely on generalized linear models. Most of the inference is same and modelling is done in the same way, although with more caution. Smoothness selection in GAM stands out as the biggest problem. It is usually solved by generalized cross-validation criterion.A practical example from the field of psychiatry is presented. Mothers' depression symptoms and adolescents psychosocial problems are modelled by a generalized additive model to illustrate one possible usage of GAM. When modelling adolescent's externalizing problem score mother's depression symptoms are found to be uninfluential on adolescent's symptoms. Although mother's opinion on adolescent's well-being is a predictive factor.
Asiasanat:generalized linear model, nonparametric regression, natural cubic spline, cubic regression spline, thin plate spline, tensor product smoother, Child Behavior Checklist, Young Self Report
Work on generalized linear models provides one part of the statistical theory needed on generalized additive models. With it any exponential family distribution can be used in linear modelling along with different link functions which also play important roles in the GAM. The complicated likelihood equations in GLM are solved by using the derivation chain rule that leads us to a form which allows us to use the Newton's method and iteratively solve the parameters for the model. Nonparametric regression methods, such as cubic splines and thin plate splines, on the other hand allow any form for the explanatory variables to take place. They produce so-called smooth functions that base on data and smoothness selection. Generalized linear models and nonparametric regression are then combined to generalized additive models by imitating nonparametric methods with parametric estimates.Generalized additive models are based largely on generalized linear models. Most of the inference is same and modelling is done in the same way, although with more caution. Smoothness selection in GAM stands out as the biggest problem. It is usually solved by generalized cross-validation criterion.A practical example from the field of psychiatry is presented. Mothers' depression symptoms and adolescents psychosocial problems are modelled by a generalized additive model to illustrate one possible usage of GAM. When modelling adolescent's externalizing problem score mother's depression symptoms are found to be uninfluential on adolescent's symptoms. Although mother's opinion on adolescent's well-being is a predictive factor.
Asiasanat:generalized linear model, nonparametric regression, natural cubic spline, cubic regression spline, thin plate spline, tensor product smoother, Child Behavior Checklist, Young Self Report