Power Comparisons in Contingency Tables

Abstract

It is an important inferential problem to test no association between two binary variables based on data. Tests based on the sample odds ratio are commonly used. We bring in a competing test based on the Pearson correlation coefficient. In particular, the odds ratio does not extend to higher order contingency tables, whereas Pearson correlation does. It is important to understand how Pearson correlation stacks against the odds ratio in 2 x 2 contingency tables. Another measure of association is the canonical correlation. In this paper, we examine how competitive Pearson correlation in relation to odds ratio in terms of power in the binary context, contrasting further with both the Wald Z and Rao score tests. We generated an extensive collection of joint distributions of the binary variables and estimated the power of the tests under each joint alternative distribution based on random samples. The consensus is that none of the tests dominates the other.

Appendix

Asymptotic variance of the maximum likelihood estimator of Pearson correlation $\phi$ Steps:

• 1. Joint distribution of X and Y

  $$\begin{aligned} Q= \begin{pmatrix} {a} &{} {b} \\ {c} &{} {d} \\ \end{pmatrix} \end{aligned}$$

• 2. Pearson correlation

  $$\begin{aligned} \begin{aligned} \rho&= \frac{ad - bc}{\sqrt{(}a+b)(a+c)(c+d)(b+d) } \\&= \phi \\&= UV^{-0.5}, {\text{ where}}, U = ad - bc \,\, \text{and}\,\, V = (a+b)(a+c)(c+d)(b+d) \end{aligned} \end{aligned}$$

• 3. Generate data

  $$\begin{aligned} D= \begin{pmatrix} n_{11} &{} n_{12} \\ n_{21} &{} n_{22} \\ \end{pmatrix} \end{aligned}$$

• 4. Estimator of Q ,

  $$\begin{aligned} {{\widehat{Q}}}= \begin{pmatrix} \frac{n_{11}}{n} &{} \frac{n_{12}}{n} \\ \frac{n_{21}}{n} &{} \frac{n_{22}}{n} \\ \end{pmatrix} \end{aligned}$$

  For ease, in the description of the asymptotic formula, use a simple notation for the entries of ${\widehat{Q}}$

  $$\begin{aligned} {{\widehat{Q}}}= \begin{pmatrix} j &{} k \\ l &{} m \\ \end{pmatrix} \end{aligned}$$

• 5. Estimate of $\rho$,

  $$\begin{aligned} \begin{aligned} {\widehat{\rho }}&= \frac{jm - lk}{\sqrt{(}j+k)(j+l)(l+m)(k+m) } \\&= f(j,k,l,m) \\&= x.y^{-0.5}, {\text{ where}},\, {x} = jm - lk \,\, {\text{and}} \,\, V = (j+k)(j+l)(l+m)(k+m) \end{aligned} \end{aligned}$$

• 6. Asymptotic variance of ${{\widehat{\rho }}}$ using the delta method evaluated at their expectations, $j= E(j), k= E(k), l= E(l), m= E(m)$

  $$\begin{aligned} \begin{aligned}& {\text{AsymptoticVariance}} = \left( \frac{df}{dj} \right) ^{2} * \text{var}(j) + \left( \frac{df}{dk} \right) ^{2} * \text{var}(k) + \left( \frac{df}{dl} \right) ^{2} * {\text{var}}(l) \\& + \left( \frac{df}{dm} \right) ^{2} * {\text{var}}(m) + 2 \left( \frac{df}{dj} \right) * \left( \frac{df}{dk} \right) * {\text{cov}}(j,k) \\& + 2 \left( \frac{df}{dj} \right) * \left( \frac{df}{dl} \right) * {\text{cov}}(j,l) + 2 \left( \frac{df}{dj} \right) * \left( \frac{df}{dm} \right) * {\text{cov}}(j,m) \\& + 2 \left( \frac{df}{dk} \right) * \left( \frac{df}{dl} \right) * {\text{cov}}(k,l) + 2 \left( \frac{df}{dk} \right) * \left( \frac{df}{dm} \right) * {\text{cov}}(k,m) \\ & + 2 \left( \frac{df}{dl} \right) * \left( \frac{df}{dm} \right) * {\text{cov}}(l,m) \end{aligned} \end{aligned}$$

• 7. Calculate the variances and covariances,

  $$\begin{aligned}\begin{array}{l} {\mathop {\mathrm {var}}} \left( j \right) = \frac{{a\left( {1 - a} \right) }}{n};{\mathop {\mathrm {var}}} \left( k \right) = \frac{{b\left( {1 - b} \right) }}{n}\\ {\mathop {\mathrm {var}}} \left( l \right) = \frac{{c\left( {1 - c} \right) }}{n};{\mathop {\mathrm {var}}} \left( m \right) = \frac{{d\left( {1 - d} \right) }}{n} {\mathop {\mathrm {cov}}} \left( {j,k} \right) = - \frac{{ab}}{n};{\mathop {\mathrm {cov}}} \left( {j,l} \right) = - \frac{{ac}}{n}\\ {\mathop {\mathrm {cov}}} \left( {j,m} \right) = - \frac{{ad}}{n};{\mathop {\mathrm {cov}}} \left( {k,l} \right) = - \frac{{bc}}{n}\\ {\mathop {\mathrm {cov}}} \left( {k,m} \right) = - \frac{{bd}}{n};{\mathop {\mathrm {cov}}} \left( {l,m} \right) = - \frac{{cd}}{n} \end{array}\end{aligned}$$

• 8. $\begin{aligned} \begin{aligned} \frac{df}{dj}&= x \left( \frac{dy^{-0.5}}{dj} \right) + y^{- 0.5} \left( \frac{dx}{dj} \right) \\&= x(-0.5)y^{- \frac{3}{2}} \frac{dy}{dj} + y ^{- 0.5} \left( \frac{dx}{dj} \right) \\&= -(0.5)xy ^{- 0.5}y ^{- 1}(2j+k+l)(l+m)(k+m) + y ^{- 1}m \end{aligned} \end{aligned}$

• 9. $\begin{aligned} {\left( {\frac{{\partial f}}{{\partial j}}} \right) _{j = {\mathop {\mathrm {E}}\nolimits } \left( j \right) ,k = {\mathop {\mathrm {E}}\nolimits } \left( k \right) ,l = {\mathop {\mathrm {E}}\nolimits } \left( l \right) ,m = {\mathop {\mathrm {E}}\nolimits } \left( m \right) }} &= - {\textstyle {1 \over 2}}u{v^{ - {\scriptstyle 1} /{\scriptstyle 2}}}{v^{ - 1}}\left( {2a + b + c} \right) \left( {c + d} \right) \left( {b + d} \right) + {v^{ - {\scriptstyle 1} /{\scriptstyle 2}}}d\\ & = - {\textstyle {1 \over 2}}\rho {v^{ - 1}}\left( {2a + b + c} \right) \left( {c + d} \right) \left( {b + d} \right) + {v^{ - {{\scriptstyle 1} / {\scriptstyle 2}}}}d \end{aligned}$

• 10. $\begin{aligned}&{\left( {\frac{{\partial f}}{{\partial k}}} \right) _{j = {\mathop {\mathrm { E}}\nolimits } \left( j \right) ,k = {\mathop {\mathrm {E}}\nolimits } \left( k \right) ,l = {\mathop {\mathrm {E}}\nolimits } \left( l \right) ,m = {\mathop {\mathrm {E}}\nolimits } \left( m \right) }} = - {\textstyle {1 \over 2}}\rho {v^{ - 1}}\left( {2b + a + d} \right) \left( {a + c} \right) \left( {c + d} \right) - {v^{ - {{\scriptstyle 1} /{\scriptstyle 2}}}}c\end{aligned}$

• 11. $\begin{aligned}&{\left( {\frac{{\partial f}}{{\partial l}}} \right) _{j = {\mathop {\mathrm {E}}\nolimits } \left( j \right) ,k = {\mathop {\mathrm {E}}\nolimits } \left( k \right) ,l = {\mathop {\mathrm {E}}\nolimits } \left( l \right) ,m = {\mathop {\mathrm {E}}\nolimits } \left( m \right) }} = - {\textstyle {1 \over 2}}\rho {v^{ - 1}}\left( {2c + a + d} \right) \left( {a + b} \right) \left( {b + d} \right) - {v^{ - {{\scriptstyle 1} /{\scriptstyle 2}}}}b\end{aligned}$

• 12. $\begin{aligned}&{\left( {\frac{{\partial f}}{{\partial m}}} \right) _{j = {\mathop {\mathrm {E}}\nolimits } \left( j \right) ,k = {\mathop {\mathrm {E}}} \left( k \right) ,l = {\mathrm{E}} \left( l \right) ,m = {\mathrm{E}} \left( m \right) }} = - {\textstyle {1 \over 2}}\rho {v^{ - 1}}\left( {2b + b + c} \right) \left( {a + b} \right) \left( {a + c} \right) + {v^{ - {{\scriptstyle 1} /{\scriptstyle 2}}}}a\end{aligned}$

• 13. The expression derived in steps 1 through 12 is plugged into the asymptotic variance formula in Step 6.

• 14. if $\rho = 0$ then asymptotic variance $\left( {\widehat{\rho }} \right) = \frac{1}{n}$