Introductory Biostatistics

the two eventsðX¼þÞandðY¼þÞare said to beindependent(because the
conditionY¼þdoes not change the probability ofX¼þ) and we have the
multiplication rulefor probabilities of independent events:

PrðX¼þ;Y¼þÞ¼PrðX¼þÞPrðY¼þÞ

If the two events are not independent, they have a statistical relationship or we
say that they arestatistically associated. For the screening example above,

PrðX¼þÞ¼ 0 : 021

PrðX¼þjY¼þÞ¼ 0 : 406

clearly indicating a strong statistical relationship [because PrðX¼þj
Y¼þÞ 0 PrðX¼þÞ]. Of course, it makes sense to have a strong statistical
relationship here; otherwise, the screening is useless. However, it should be
emphasized that a statistical association does not necessarily mean that there is
a cause and an e¤ect. Unless a relationship is so strong and repeated so con-
stantly that the case is overwhelming, a statistical relationship, especially one
observed from a sample (because the totality of population information is
rarely available), is only a clue, meaning that more study or confirmation is
needed.
It should be noted that there are several di¤erent ways to check for the
presence of a statistical relationship.

1.Calculation of the odds ratio. WhenXandY are independent, or not

associated statistically, the odds ratio equals 1. Here we refer to the odds

ratio value for the population; this value is defined as

odds ratio¼

PrðX¼þjY¼þÞ=ðPrðX¼
jY¼þÞÞ

PrðX¼þjY¼
Þ=ðPrðX¼
jY¼
ÞÞ

and can be expressed, equivalently, in terms of the joint probabilities as

odds ratio¼

PrðX¼þ;Y¼þÞPrðX¼
;Y¼
Þ

PrðX¼þ;Y¼
ÞPrðX¼
;Y¼þÞ

and the example above yields

OR¼

ð 0 : 006 Þð 0 : 970 Þ

ð 0 : 015 Þð 0 : 009 Þ

¼ 43 : 11

clearly indicating a statistical relationship.

114 PROBABILITY AND PROBABILITY MODELS