Calculating risk of disease Suppose we are doing a cross sectional study on heart disease in males and females. If $16/10000$ males have heart disease and $10/10000$ females have heart disease, can we conclude that men have a much higher risk of heart disease than women? Why or why not?
I think you cannot since we are comparing proportions. We don't have an odds ratio.
 A: The odds ratio is not the only epidemiological effect measure that exists. Given you have a cross sectional study, and thus a prevalence study, the most appropriate measure is arguably the Prevalence Ratio - though for very low prevalence like the example study, Henry's approach, which is a Prevalence Odds Ratio, will work as an adequate approximation.
Prevalence in Females:  10/10000 = 0.001 
Prevalence in Males:  16/10000 = 0.0016 
Prevalence Ratio (Females as Referent Group):  0.0016/0.001 = 1.6 
This is however a relatively imprecise ratio - the 95% CI is 0.73 to 3.52 - due to the low number of cases.
"can we conclude that men have a much higher risk of heart disease than women? Why or why not?"
Two thoughts on this:


*

*What is "much higher"? On a relative scale, yes, I'd argue that having 60% more prevalent heart disease in one population is considerably higher - enough that if this was the finding of a properly conducted study with better power, I'd likely be pleased with the magnitude of the effect. But in an absolute sense? Not really - 6 more cases in 10,000 isn't much. And it's possible to have relative measures that are massively higher, but on an absolute scale are nearly identical for very rare diseases. Consider if your number of cases was 1 and 2. Here, the ratio is 2 (higher) but the difference in cases is now 1 case per 10,000. Herein lies the argument between risk ratios and risk differences.

*You can't say much about the risk - which usually implies something about incidence. Prevalence is a composite measure of both Incidence and Duration of Disease. All you can say is that males have a higher prevalence of heart disease - they may actually get more heart disease, or they may simply have it for longer (for example, if heart disease tends to kill women swiftly).

A: You can easily turn this into an odds ratio by calculating 
$$\frac{\; \frac{16}{10000-16} \;}{ \;\frac{10}{10000-10} \; }$$ 
and although this is not exactly $1.6$, it is close at about $1.60096$.
As to whether "men have a much higher risk of heart disease than women", I would not say so: 


*

*because I think it sometimes misleading to say that two very small numbers are very different (similarly I would not say that somebody with net assets worth $\$10$ was "much richer" than somebody with net assets of $\$5$); and 

*because the fact that out of $26$ people found to have heart disease $16$ were men is not significantly different from a hypothesis that half might be male, using for example the binomial test 
