Multiway Tables

• Multi-way tables
  • Sometimes a researcher is interested in fitting a table with more than two dimensions. Generally, there are two reasons to fit three-way or higher order tables:
    1. The researcher is interested in whether the association between two variables varies in the context of a third variable (interaction).
    2. The researcher is interested in whether the association between two variables is explained by a third (controlling).
  • Let's say for example that we had mobility tables for eight industrialized countries. We would then have a three-way table of father's occuation by son's occupation by county with dimensions 5x5x8. One interesting question for such tables is whether patterns of mobility are uniform across industrialized nations once we have controlled different marginal distributions.

    In other words, our comparison would be between the following models:

    $$\log(F_{ijk})=\lambda+\lambda_{i}+\lambda_{j}+\lambda_{k}+\lambda_{ij}+\lambda_{ik}+\lambda_{jk}$$
    So $\lambda_{ij}$ gives the association between father's and son's occupation in general, and the $\lambda_{ik}$ and $\lambda_{jk}$ give the different distributions of son's and father's occupations across countries, respectively. It is important to control for these different distributions to get the measure of association right.
    $$\log(F_{ijk})=\lambda+\lambda_{i}+\lambda_{j}+\lambda_{k}+\lambda_{ij}+\lambda_{ik}+\lambda_{jk}+\lambda_{ijk}$$
    This final model is the saturated model. $\lambda_{ijk}$ gives the difference in mobility across countries. So if our first model is not preferred to the saturated model by standard tests, then we need to explore the ways in which mobility differs between nation-states. Optimally we would find some constrained version of $\lambda_{ijk}$.
  • Models for three way tables can be summarized into several categories:

    Mutual independence

    There is no relationship between any of the variables:

    $$\log(F_{ijk})=\lambda+\lambda_{i}+\lambda_{j}+\lambda_{k}$$

    Sometimes this is written as (R,C,L)

    Joint Independence

    This model assumes that there is only one two-way interaction in the table:

    $$\log(F_{ijk})=\lambda+\lambda_{i}+\lambda_{j}+\lambda_{k}+\lambda_{ij}$$

    In this format we have (RC,L). This model is collapsible over the L dimension. There is no relationship between RL, CL, or RCL.

    Conditional Independence

    This model assumes that there are two two-way interactions in the table:

    $$\log(F_{ijk})=\lambda+\lambda_{i}+\lambda_{j}+\lambda_{k}+\lambda_{ik}+\lambda_{jk}$$

    This model assumes that there is no association between RC once we have accounted for the distribution of R and C at each level of L (RL, CL).

    no three-way interaction

    $$\log(F_{ijk})=\lambda+\lambda_{i}+\lambda_{j}+\lambda_{k}+\lambda_{ij}+\lambda_{ik}+\lambda_{jk}$$

    This model assumes that there is an association between all three variables, but that the association between each of them is not affected by the level of the third. This is the model we start with in trying to explain country-specific levels of mobility.

  • Powers and Xie give us an example of graduate admissions at UC Berkeley by gender and major. This can be modeled as a three-way table of sex by admissions outcome by major (2x2x8). The overall table of sex by admissions indicates a serious sex imbalance.

    Sex	No Admit	Admit
    Men	1480	1211
    Women	1285	551

    But, does this finding hold conditional on differences in major choice between men and women? Is there an association between sex and admittance once major is accounted for? To answer this question, Powers and Xie fit the conditional independence model (RL, CL). This model had a $G^{2}=21.13$ and $df=6$. It did not fit by $LRT$ standards, although $BIC$ liked it well enough ( $BIC=-29.37$).

    By examining the actual admission rates by major, Xie and Powers noted that the admission rate only seemed to differ by gender for Major A. Therefore they fit the following model:

    $$\log(F_{ijk})=\lambda+\lambda_{i}+\lambda_{j}+\lambda_{k}+\lambda_{ik}+\lambda_{jk}+\beta x_{ijk}$$
    Where $x_{ijk}$ was equal to one if i=female, j=admitted, and k=major A, and o otherwise. This model is the same as the conditional independence model, except that it does allow for gender differences in admission rates for major A.

    This model fits the data well ( $G^{2}=2.81$, $df=5$) indicating that except for Major A, there is no difference in admission rates by gender, once the association between major choice and gender (RL) and admission rates and major (CL) are accounted for. Furthermore, the difference by Major A actually favored women ( $\beta=1.05$).