## φ intervals by inverted Liebetrau Gaussian s(φ)

### 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 ≡ (adbc) / √(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
N
{4Σ
i j
pi,j³
pi+² p+j²
– 3Σ
i
1
pi+
(Σ
j
pi,j²
pi+ p+j
)² – 3Σ
j
1
p+j
(Σ
i
pi,j²
pi+ p+j
)²

+2Σ
i j
[ pi,j
pi+ p+j
(Σ
k
pk,j²
pk+ p+j
)(Σ
l
pi,l²
pi+ p+l
)]}, for φ ≠ 0, (3)

where pi,j = fi,j / N and pi+, p+j, etc. represent row and column (prior) probabilities (Bishop, Fienberg and Holland 1975: 386). If φ = 0 we adjust the table by a small delta.

## UCL Summer School in English Corpus Linguistics 2019

I am pleased to announce the seventh annual Summer School in English Corpus Linguistics to be held at University College London from 1-3 July.

The Summer School is a short three-day intensive course aimed at PhD-level students and researchers who wish to get to grips with Corpus Linguistics.

Please note that this course is very popular, and numbers are deliberately limited on a first-come, first-served basis! You will be taught in a small group by a teaching team.

Each day begins with a theory lecture, followed by a guided hands-on workshop with corpora, and a more self-directed and supported practical session in the afternoon.

## UCL Summer School in English Corpus Linguistics 2017

I am pleased to announce the fifth annual Summer School in English Corpus Linguistics to be held at University College London from 5-7 July.

The Summer School is a short three-day intensive course aimed at PhD-level students and researchers who wish to get to grips with Corpus Linguistics. Numbers are deliberately limited on a first-come, first-served basis. You will be taught in a small group by a teaching team.

Each day begins with a theory lecture, followed by a guided hands-on workshop with corpora, and a more self-directed and supported practical session in the afternoon.