Some Tests for Categorical Data

摘要: We shall be concerned with experimental data given in the form of frequencies cells determined by a multiway cross-classification, predefined categories along each way classification. Roy and Bhapkar [10] have posed hypotheses, which might considered generalizations appropriate to this set up usual hypotheses classical "normal" univariate "fixed effects" analysis variance, multivariate variance various kinds independence. Large sample tests for such are offered here. The large suggested based on $\chi^2$-test Karl Pearson [8]. general probability model is that product several multinomial distributions. According as marginal any dimension held fixed or left free, said associated "factor" "response" (or variable). \begin{equation*}\tag{1}\prod_j \frac{n_{oj}!}{n_{ij}!} \prod p^{n_{ij}}_{ij},\end{equation*} where $\sum_i p_{ij} \equiv p_{oj} = 1$ n_{ij} n_{oj}$ fixed. Thus $i$ refers response while $j$ factor. $n_{oj}$ denotes preassigned sample-size $j$th factor-category, out $n_{ij}$ happen lie $i$th response-category. It should noticed may multiple subscript, say $i_1, i_2, \cdots, i_k; j$ also $j_1, j_2 j_l$. then speak $k$-response $k$-variate) $l$-factor problem real numbers not classification (factor response), will structured unstructured. well-known (for example, Neyman [6]) if hypothesis $H_o$ certain constraints $p_{ij}$'s, test statistic under (1) $\chi^2$ $\sum_{ij} (n_{ij} - n_{oj}\hat p_{ij})^2/(n_{oj}\hat p_{ij}),$ $\chi^2_1$ p_{ij})^2/n_{ij}$, $\hat p_{ij}$'s BAN estimates [6]. In particular case when linear $p$'s, method minimum permits reduction solution system equations hence more convenient. Reiersol [9] considers binomial experiments makes use results [6] determine factorial experiments. Mitra [5] only generalizes Reiersol's theorems experiments, but avoids his restriction parameter-sets different forms occurring nonoverlapping. prove cover cases cannot treated these theorems. Section 2, obtained hypotheses. further shown that, specifies functions $p$'s known some unknown parameters, statistic, estimates, exactly same sum squares residuals least technique estimate parameters. This applied derive criteria proposed [3] [10].