Page 47 - Clinical Small Animal Internal Medicine

2 Statistical Interpretation for Practitioners 15

that the rate of the outcome occurring under the numer- hyperthyroidism is 1.5 (but there is no percentage
VetBooks.ir ator condition (such as treatment A) is faster than the equivalent). This leads to two related but important
findings. First, as the probability approaches 1, the odds
rate under the denominator condition (such as treatment
B); conversely, an incidence rate ratio between 0 and 1
of 0.95 corresponds to an odds of 19, while a probability
indicates that the rate of the outcome occurring under becomes unintelligibly large (for example, a probability
the numerator condition is slower than the rate under of 0.999 corresponds to an odds of 999). Second, when
the denominator condition. the probability is small (between 0 and 0.05), such as
Returning to the above example, if the rate of resorp- with the incidence of a rare disease, the odds is similarly
tion in Yorkshire terriers is 30 cases/100 dog‐years, and small (between 0 and 0.053, respectively). This implies
the rate of resorption in border collies is 10 cases/100 that for rare events, probabilities and odds are nearly
dog‐years, then the incidence rate ratio is 3.0. Not only equal, and for interpretive purposes are essentially
does this indicate that the rate of developing tooth interchangeable.
resorption is three times faster in Yorkshire terriers com- While odds, just like probabilities, can be expressed as
pared to border collies, but it also means that the time to unconditional (not depending on the values of other fac-
developing tooth resorption is less in Yorkshire terriers, tors) estimates, it is more common to see them expressed
and that the proportion of dogs remaining free of tooth as conditional odds; for example, the odds of developing
resorption (i.e., “surviving”) is greater in border collies at tooth resorption among Yorkshire terriers is a condi-
all times. tional statement because it only applies to Yorkshire ter-
riers and not other dog breeds. Just as ratios of two
different cumulative incidences and two different inci-
dence rates can be calculated to measure proportionate
Incidence Odds and Incidence changes in incidence between two levels of a variable, so
Odds Ratios can ratios of odds. Returning to the earlier example of
the cumulative incidence of tooth resorption, if the five‐
Perhaps the least understood yet ubiquitously found year cumulative incidence in Yorkshire terriers is 0.3 and
measure of association in the veterinary medical litera- the five‐year cumulative incidence in border collies is
ture is the odds ratio. Although odds ratios arise under 0.1, then the five‐year incidence odds in the two breeds
different study designs (both experimental and nonex- are 0.43 and 0.11, respectively. Therefore, the odds ratio
perimental), and their interpretations can vary, these dif- relating five‐year incidence of tooth resorption in
ferences are less critical than the properties they share as Yorkshire terriers compared to border collies is 0.43/0.11
measures of association and potential impact of factors = 3.86. This can be contrasted with the cumulative inci-
on the incidence of health outcomes. In order to under- dence ratio of 3.00, and the disparity between the two
stand odds ratios, it is necessary to first understand the measures primarily arises because the odds in (0.43)
statistical meaning of an odds. does not closely approximate the cumulative incidence
Probabilities are measures of the likelihood of an event (0.3) in Yorkshire terriers because the latter is not rare.
occurring, and are only strictly between (and including) Just as cumulative incidence, as a probability, is much
0 and 1; a probability of 0 implies impossibility and a more comprehensible than an incidence odds, so the
probability of 1 implies inevitability. The odds is another cumulative incidence ratio is more comprehensible than
measure of likelihood, and is calculated as the probabil- an incidence odds ratio. Knowing in the above example
ity of an event occurring divided by the probability of an that the cumulative incidence ratio = 3.0 makes the inter-
event not occurring (note that the sum of these two pretation straightforward: the five‐year cumulative inci-
probabilities must equal 1). When the probability of an dence in Yorkshire Terriers is three times the incidence
event occurring equals 0, the odds equals 0. However, in border collies. In the absence of any biases, we could
when the probability of an event occurring equals 1, then attach a causal interpretation to this and claim that the
the probability of an event not occurring equals 0, and average five‐year risk of dental resorption in Yorkshire
the odds (which would equal 1 divided by 0) is undefined terriers is three times the corresponding risk in border
but infinitely large. collies.
The restricted range for probabilities versus the infi- In contrast, no such interpretation can be attributed to
nitely large range for odds necessarily makes the former the odds ratio of 3.86 above, because it is not in general
more understandable and desirable when communicat- interpretable either as a relative measure of either risks or
ing biostatistical findings. For example, if the probabil- incidence rates. While it is legitimate to interpret it liter-
ity of a 14‐year‐old cat developing hyperthyroidism in ally as a relative odds, other interpretations that include
the ensuing years of its life is 0.60 (which can equiva- “risk,” “likelihood,” “probability,” and “cumulative inci-
lently be expressed as 60%), then the odds of developing dence” are incorrect and should not be used. This begs