## 31.2 Hypotheses and notation: Comparing odds

For two-way tables of counts, the parameter is the population odds ratio. As usual, the null hypothesis is the ‘no difference, no change, no relationship’ position. So in this context:

• $$H_0$$: The population OR is one; or (equivalently):
The population odds are the same in each group.

This hypothesis proposes that the sample odds are not the same due to sampling variation. This is the initial assumption.

The alternative hypothesis is

• $$H_1$$: The population OR is not one; or (equivalently):
The population odds are not the same in each group.

The alternative hypothesis is always two-tailed for analysing two-way tables of counts.

For analysing two-way tables of counts, the alternative hypotheses are always two-tailed.

The hypotheses can also be written in terms of differences in percentages (or proportions), though the software output is usually expressed in terms of odds. The hypotheses can also be written in terms of associations:

• $$H_0$$: In the population, there is no association between the two variables
• $$H_1$$: In the population, there is an association between the two variables
The RQ and hypothesis only needs to be given in one of these ways. The RQ and hypotheses should be consistent (for example, if the RQ is written in terms of odds, the hypotheses should be written in terms of odds).

As usual, following the decision-making process, start by assuming that the null hypothesis is true: that the population odds ratio is one.