10.3 Tests for ORs

Singh and Prasad (2016) examined the mortality rates of lower-limb amputation, and factors that may be associated with mortality. As part of the study, the researchers recorded the five-year mortality rate (the number of amputees who died within five years of amputation) and whether or not the person used an artificial limb.

The RQ was:

For this type of amputees, is the odds of being alive after five years the same for those who used an artificial limb, and those who did not?

A total of $105$ subjects were used: $35$ died within five years, and $70$ were still alive after five years. In addition, $65$ subjects used an artificial limb, and $40$ did not.

After five years, $49$ people using an artificial limb were alive. Construct the $2\times 2$ table (Table 10.3) displaying the number of people alive or dead after five years (in columns, say) and whether or not they used an artificial limb or not (in rows).

TABLE 10.3: Five-year mortality for artifical limb users
	Alive	Dead	Total
Used artificial limb
Did not use artificial limb
Total			105

For a person using an artificial limb, compute the odds of being alive after five years. For a person not using an artificial limb, compute the odds of being alive after five years. Then compute the odds ratio, complete Table 10.4, and carefully explain what the OR means.
Write down the $95$ % CI for the population OR using the software output (Fig. 10.3).
Perform a hypothesis test to compare the odds of being alive after five years for both groups (Fig. 10.3):
1. Write the hypotheses.
2. Write down the value of $\chi^2$ .
3. What $z$ -score is this equivalent to?
4. What is the approximate $P$ -value using the $68$ -- $95$ -- $99.7$ rule?
5. What is the exact $P$ -value as reported by software?
6. Write a conclusion.

TABLE 10.4: Five-year mortality and use of an artificial limb: Numerical summary
	Percentage alive	Odds of being alive	Sample
	after 5 years	after 5 years	size
Use artificial limb
Did not use artifical limb
	Diff in percentages:	OR:

FIGURE 10.3: The jamovi output for the question on lower-limb amputees

References

Singh RK, Prasad G. Long-term mortality after lower-limb amputation. Prosthetics and Orthotics International. 2016;40(5).