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Generalized additive models with flexible response functions

机译：具有灵活响应功能的通用加性模型

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摘要

Common generalized linear models depend on several assumptions: (i) the specified linear predictor, (ii) the chosen response distribution that determines the likelihood and (iii) the response function that maps the linear predictor to the conditional expectation of the response. Generalized additive models (GAM) provide a convenient way to overcome the restriction to purely linear predictors. Therefore, the covariates may be included as flexible nonlinear or spatial functions to avoid potential bias arising from misspecification. Single index models, on the other hand, utilize flexible specifications of the response function and therefore avoid the deteriorating impact of a misspecified response function. However, such single index models are usually restricted to a linear predictor and aim to compensate for potential nonlinear structures only via the estimated response function. We will show that this is insufficient in many cases and present a solution by combining a flexible approach for response function estimation using monotonic P-splines with additive predictors as in GAMs. Our approach is based on maximum likelihood estimation and also allows us to provide confidence intervals of the estimated effects. To compare our approach with existing ones, we conduct extensive simulation studies and apply our approach on two empirical examples, namely the mortality rate in SAo Paulo due to respiratory diseases based on the Poisson distribution and credit scoring of a German bank with binary responses.
机译：通用的广义线性模型取决于几个假设：（i）指定的线性预测变量，（ii）确定似然性的选定响应分布，以及（iii）将线性预测变量映射到响应的条件期望的响应函数。通用加性模型（GAM）提供了一种克服纯线性预测变量限制的便捷方法。因此，可以将协变量包括为灵活的非线性或空间函数，以避免因错误指定而引起的潜在偏差。另一方面，单索引模型利用了响应函数的灵活规范，因此避免了错误指定响应函数的恶化影响。但是，这种单指标模型通常仅限于线性预测器，旨在仅通过估计的响应函数来补偿潜在的非线性结构。我们将证明，这在许多情况下是不够的，并且通过将灵活的方法（如GAM中使用单调P样条和加性预测变量组合起来）来提供响应函数估计的解决方案。我们的方法基于最大似然估计，并且还允许我们提供估计效果的置信区间。为了将我们的方法与现有方法进行比较，我们进行了广泛的模拟研究，并将我们的方法应用于两个经验示例，即基于Poisson分布的圣保罗因呼吸道疾病导致的死亡率和具有二元响应的德国银行的信用评分。

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