Renáta Németh, Dávid Simon

ELTE

Some other indices to show the connection between two nominal variables:

odds ratio

Rogoff ratio

**Notation:**

Let’s take two nominal variables with two values each.

Gender: male/female

Height: *taller than* 180 cm / *shorter
than* 180 cm

tall |
short |
sum of row | |

female |
f |
f |
f |

male |
f |
f |
f |

sum of column |
f |
f |
f |

thus:

f_{11} no. of tall women

f_{+1} no. of tall respondents

f_{++} total no. of cases

Rogoff ratio:

where the second part of the formula is the number of cases in cell f11 with the given marginal distributions if the two variables are independent. That is, how great is the difference compared to the independence.

**Characteristics:**

symmetric

its minimum and maximum values depend on the marginal disribution (variationally not independent)

it’s always 1 if (and only if) the variables are independent

the table can be easily reconstructed knowing only the marginals

Gender: male/female

Height: *taller than* 180 cm / *shorter
than* 180 cm

talll |
short |
sum of row | |

female |
f |
f |
f |

male |
f |
f |
f |

sum of column |
f |
f |
f |

thus:

f_{11} no. of tall women

f_{+1} no. of tall respondents

f_{++} total no. of cases

the odds ratio (α):

**Interpretation**: The ratio of two frequencies (or
probabilities) are called odds. Think of bookies: what are the odds that the horse
called Nick Carter is going to win? If it’s 3:1, it means it’s going to win once in
every four cases. The odds ratio shows how much greater the odds of one event is
than that of another.

Characteristics:

symmetrical

minimum value: 0

maximum value: +

its value if and only if independent: 1

if we take its logarithm, the same absolute values mean ’the same strength’ connection

if we know the marginals, the table can be reconstructed but it’s complicated

(variationally independent: it’s value doesn’t depend on the marginal distribution)

Which associational index shows the ’direction’ of the connection as well?

Why can’t lambda be negative?

What are the advantages and disadvantages of the individual indices?

Which associational indices require us to specify a dependent and an independent variable?

Why is it ’bad’ if the original table can’t be reconstructed knowing the associational index and the marginals?

Why doesn’t the value of the odds ratio depend on the individual distribution of the variables (i.e. the marginals?)

Why does the Rogoff ratio depend on the marginals?

When is lambda = 0 ?