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I major in Bioinformatics. Now,I am in a problem: we all know that temperature changes during a year , I find that a disease incidence is really high when temperature is relatively high , while it becomes really low when the temperature is relatively low , that is to say ,they are related. So ,I want to find out a way to prove that they are related ,not just intuitively feel that they are related. So, any suggestions?

wuchang
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    From statistical standpoint it's called Correlation. You may calculate correlation coefficient and show that it's close to $1$, which establishes that two random variables are actually related. – Kaster Jan 16 '14 at 02:15
  • You should also investigate other factors, influencing disease incidence, which evolve closely with the temperature change, either caused by the temperature change or independent to it. If there are few or none, then your point is stronger. If you only have data about the influence of temperature change, then you have to be careful with your conclusions. – Olivier Jan 16 '14 at 02:37
  • @Mathlovin : This isn't just about computing correlation. If $X$ and $Y$ are uncorrelated with each other in a population, and one takes a small sample, then for most variables, with probability $1$ they will be correlated in the sample. If that correlation is so extreme that it would be improbable that it would be that large, given no correlation in the population, then it's statistically significant. How big that has to be depends on the sample size. And if $Y=X^2$ then the correlation may be exactly $0$ even though they're very must related. – Michael Hardy Jan 16 '14 at 02:40
  • . . . and what if, for example, the probability $p$ that a patient dies varies with the concentration $x$ of a toxin in the patient's blood according to $\displaystyle\log\frac{p}{1-p} = ax+b$? Then it's not just correlation in the most usual sense. That equation is actually a fairly frequently used model. – Michael Hardy Jan 16 '14 at 02:41

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If you are trying to prove that higher temperatures cause the disease incidence to increase (vs merely being associated with it) and vice versa, then even a correlation of 1 will not show that, merely that they co-occurr.

To show causation, you have to rule out the possibility that other variables are responsible for increase in disease incidence. For example, perhaps air conditioner use tends to increase the probability of lung infection. Air conditioners are used almost exclusively in the summer, hence you would expect infection rates to be positively correlated with temperature, perhaps to a very high degree, even though heat, per se, does not increase infection rates.

The gold standard for causation is the double-blind randomzied trial. Absent that, you need to show that other plausible causes admit of counterexamples where a cause is present but the effect is absent. Most epidemiology texts will discuss options for observational studies, but hard evidence of causation requires controlled trials.