Headbanger,

Yes, statistics help us determine whether an observed phenomenon occurs by chance, or because of some other factor, for example, treatment with a vaccine. There are several statistical methods that are used to answer the very question you pose. I will mention one that is quite common in biomedical experiments and clinical trials. I centers around the odds-ratio.

Lets say we estimate the probability that an individual in the placebo group develops HIV as

p1 = n11/n12 = 74/8201

and the corresponding probability for the vaccine group as

p2 = n21/n22 = 51/8201

Then the odds for the placebo group is

o1 = p1 / (1-p1)

and the odds for the vaccine group is

o2 = p2 / (1-p2)

In this case o2 = 0.00626, and o1 = 0.00911. We say that the "odds" of developing HIV, given that you are in the placebo group, is 0.00911. The odds-ratio is just that, the ratio of the two odds, or

o1 / o2 = 1.455. We say that the "odds" of developing HIV is 1.455 times larger in the placebo group than in the vaccine group.

Of course, this is just a summary measure for what you already know. To test if this value is "statistically significant", or not completely due to change, we can apply a statistical test to the odds-ratio, or the log(odds-ratio).

Under the hypothesis that there is no difference in the probability of developing HIV between the two groups, we would expect that the odds-ratio be close to the value 1, and the log(odds-ratio) to be close to the value 0. In more advanced statistics, we show that the probability distribution of the log(odds-ratio) under this hypothesis is approximately normal (Gaussian) with mean m=0 and standard deviation sd=sqrt(1/n11 + 1/n12 + 1/n21 + 1/n22) = sqrt(1/51 + 1/8201 + 1/8201 + 1/74).

With this information, we can construct a confidence interval for log(odds-ratio) or the odds-ratio, or conduct a statistical test. The 95% confidence interval for odds-ratio in this example is

( exp( log(odds-ratio) - 1.96*sd ), exp( log(odds-ratio) + 1.96*sd ) )

=( exp( 1.455 - 1.96*0.1827 ), exp( 1.455 + 1.96*0.1827 ) )

=( 1.017, 2.081 )

Hence, we are 95% confident that the "true" odds-ratio lies within these bounds. Since the lower bound is greater than 1, we can conclude, with 95% confidence, that observing this odds-ratio was NOT entirely due to chance.

Hope this gets you on the right path.

BioStatMatt