### Introduction

Experimenting with deriving accurate 2 × 2 φ intervals, I also considered using Liebetrau’s population standard deviation estimate.

To recap: Cramér’s φ (Cramér 1946) is a probabilistic intercorrelation for contingency tables based on the χ² statistic. An unsigned φ score is defined by

Cramér’s φ = √χ²/*N*(*k* – 1)(1)

where χ² is the *r* × *c* test for homogeneity (independence), *N* is the total frequency in the table, and *k* the minimum number of values of variables *X* and *Y*, i.e. *k* = min(*r*, *c*). For 2 × 2 tables, *k* – 1 = 1, so φ = √χ²/*N* is often quoted.

An alternative formula for 2 × 2 tables obtains a signed result, where a negative sign implies that the table tends towards the opposite diagonal.

signed 2 × 2 φ ≡ (*ad* – *bc*) / √(*a* + *b*)(*c* + *d*)(*a* + *c*)(*b* + *d*),(2)

where *a*, *b*, *c* and *d* are cell frequencies. However, Equation (2) cannot be applied to larger tables.

The method I discuss here is potentially extensible to other effect sizes and other published estimates of standard deviations.

We employ Liebetrau’s best estimate of the population standard deviation of φ for *r* × *c* tables:

*s*(φ) ≈ 1

2φ*N* {4Σ

*i j**p _{i,j}*³

*p*²

_{i+}*p*² – 3Σ

_{+j}*i*1

*p*(Σ

_{i+}*j*

*p*²

_{i,j}*p*

_{i+}*p*)² – 3Σ

_{+j}*j*1

*p*(Σ

_{+j}*i*

*p*²

_{i,j}*p*

_{i+}*p*)²

_{+j}+2Σ

*i j*[ *p _{i,j}*

*p*(Σ

_{i+}p_{+j}*k*

*p*²

_{k,j}*p*)(Σ

_{k+}p_{+j}*l*

*p*²

_{i,l}*p*)]}, for φ ≠ 0, (3)

_{i+}p_{+l}where *p _{i,j}* =

*f*/

_{i,j}*N*and

*p*,

_{i+}*p*, etc. represent row and column (prior) probabilities (Bishop, Fienberg and Holland 1975: 386). If φ = 0 we adjust the table by a small delta.

_{+j}Continue reading “φ intervals by inverted Liebetrau Gaussian s(φ)”