Young Robin L, Weinberg Janice, Vieira Verónica, Ozonoff Al, Webster Thomas F
Department of Biostatistics, Boston University School of Public Health, Boston, MA, USA.
Comput Stat Data Anal. 2011 Jan 1;55(1):366-374. doi: 10.1016/j.csda.2010.05.004.
Generalized additive models (GAMs) have distinct advantages over generalized linear models as they allow investigators to make inferences about associations between outcomes and predictors without placing parametric restrictions on the associations. The variable of interest is often smoothed using a locally weighted regression (LOESS) and the optimal span (degree of smoothing) can be determined by minimizing the Akaike Information Criterion (AIC). A natural hypothesis when using GAMs is to test whether the smoothing term is necessary or if a simpler model would suffice. The statistic of interest is the difference in deviances between models including and excluding the smoothed term. As approximate chi-square tests of this hypothesis are known to be biased, permutation tests are a reasonable alternative. We compare the type I error rates of the chi-square test and of three permutation test methods using synthetic data generated under the null hypothesis. In each permutation method a distribution of differences in deviances is obtained from 999 permuted datasets and the null hypothesis is rejected if the observed statistic falls in the upper 5% of the distribution. One test is a conditional permutation test using the optimal span size for the observed data; this span size is held constant for all permutations. This test is shown to have an inflated type I error rate. Alternatively, the span size can be fixed a priori such that the span selection technique is not reliant on the observed data. This test is shown to be unbiased; however, the choice of span size is not clear. A third method is an unconditional permutation test where the optimal span size is selected for observed and permuted datasets. This test is unbiased though computationally intensive.
广义相加模型（GAMs）相较于广义线性模型具有明显优势，因为它们使研究者能够推断结果与预测变量之间的关联，而无需对这些关联施加参数限制。通常使用局部加权回归（LOESS）对感兴趣的变量进行平滑处理，并且可以通过最小化赤池信息准则（AIC）来确定最优跨度（平滑程度）。使用GAMs时的一个自然假设是检验平滑项是否必要，或者一个更简单的模型是否就足够了。感兴趣的统计量是包含和平滑项的模型与不包含平滑项的模型之间偏差的差异。由于已知该假设的近似卡方检验存在偏差，排列检验是一种合理的替代方法。我们使用在原假设下生成的合成数据，比较卡方检验和三种排列检验方法的I型错误率。在每种排列方法中，从999个排列数据集获得偏差差异的分布，如果观察到的统计量落在分布的上5%，则拒绝原假设。一种检验是使用观察数据的最优跨度大小的条件排列检验；该跨度大小在所有排列中保持不变。结果表明该检验的I型错误率过高。或者，可以事先固定跨度大小，使得跨度选择技术不依赖于观察数据。结果表明该检验是无偏的；然而，跨度大小的选择并不明确。第三种方法是无条件排列检验，其中为观察数据集和排列数据集选择最优跨度大小。该检验是无偏的，尽管计算量很大。
