Arnold Steven F, Moschopoulos Panagis G
Pennsylvania State University, State College, PA, United States.
J Stat Plan Inference. 2012 Nov;142(11):2965-2975. doi: 10.1016/j.jspi.2012.04.017.
We consider inference for row effects in the presence of possible interactions in a two-way fixed effects model when the numbers of observations are themselves random variables. Let be the number of observations in the (, ) cell, be the probability that a particular observation is in that cell and be the expected value of an observation in that cell. We assume that the { } have a joint multinomial distribution with parameters n and { }. Then . = Σ /Σ is the expected value of a randomly chosen observation in the th row. Hence, we consider testing that the . are equal. With the { } unknown, there is no obvious sum of squares and F-ratio computed by the widely available statistical packages for testing this hypothesis. Let .. be the sample mean of the observations in the th row. We show that .. is an MLE of ., is consistent and is conditionally unbiased. We then find the asymptotic joint distribution of the .. and use it to construct a sensible asymptotic size test of the equality of the . and asymptotic simultaneous (1 - ) confidence intervals for contrasts in the ..
当观测值的数量本身为随机变量时，我们考虑在双向固定效应模型中存在可能的交互作用的情况下对行效应进行推断。设(n_{ij})为((i,j))单元格中的观测值数量，(p_{ij})为特定观测值在该单元格中的概率，(\mu_{ij})为该单元格中观测值的期望值。我们假设({n_{ij}})具有参数为(n)和({p_{ij}})的联合多项分布。那么(\mu_{i.}=\sum_{j}\mu_{ij}/\sum_{j}p_{ij})是第(i)行中随机选择的观测值的期望值。因此，我们考虑检验(\mu_{i.})是否相等。由于({p_{ij}})未知，对于检验该假设，广泛使用的统计软件包没有明显的平方和与(F)比率计算方法。设(\bar{y}{i.})为第(i)行观测值的样本均值。我们证明(\bar{y}{i.})是(\mu_{i.})的极大似然估计，是一致的且条件无偏。然后我们找到(\bar{y}{i.})的渐近联合分布，并使用它来构建关于(\mu{i.})相等性的合理渐近水平为(\alpha)的检验以及关于(\bar{y}_{i.})对比的渐近同时((1 - \alpha))置信区间。
