Conceptual difference between probability vs percentages

Question

Suppose there is a medical study which finds that having some Z gene is relate to a disease Y by a by 50%. Now, would it be correct to interpret this is as a probabilistic result?

That is, there is a fifty percent chance, if you take a new random person with Z gene to have a disease Y. I feel not, because in a genuine theoritical probability calculation, then experiments are supposed to be the exact same, while here, the subjects of the experiment could be of totally different nature (height, weight and psychiatric state etc.

Answer 1

If it is established that a Z gene is present in 50% of all people with disease Y, then that is an average over all people, whatever their height, weight, etc. However...

1. In practice, we cannot examine all people, so a percentage like that will be derived from samples. We do our best to acquire unbiased samples, but it is notoriously difficult to ensure a lack of bias in sampling. This is especially true when sampling human beings.

2. Averages over all people, even if we knew them exactly, are not all that useful, since usually we want to take more specific information into account. To change the example, consider the probability that a person will die tomorrow. The number of people who die each day is fairly steady, and it is approximately 1 in 15,000 of the total human population. But it does not follow that the probability that you will die tomorrow is 1 in 15,000. How likely it is that you will die depends on all kinds of things particular to you: your age, state of health, location, occupation, etc. That is the kind of information an insurance company will take into account when selling you life insurance.

3. Using an average to make a prediction is fraught with difficulties. No matter how good your frequency data is, it is based on a finite quantity of observations and these cannot guarantee what frequencies will be observed in future. There may also be unknown factors or confounding variables. There is also the frame problem: it is difficult to determine which variables are relevant when making a prediction.

Answer 2

There are some points about your question that need to be clarified, as follows...

A probability can be expressed as a percentage, but a percentage can also be used to express ratios other than probabilities. For example, if I say that the price of my new mansion was reduced by 87%, I am not referring to a probability.

If I tell you that 87% of the answers I have posted on Philosophy SE are about the philosophical works of Cary Grant, I am not expressing a probability. However, if you read one of my answers at random, there is an 87% probability that it will concern Cary Grant. You should be able to see from that example that in the first sentence I am using a percentage to indicate the proportion of one type of thing in a wider set that includes other types of thing. In the second sentence, I am using a percentage to express something else, namely the likelihood of a particular type of outcome relative to all the possible outcomes, which is what we call a probability.

Answer 3

"Now, would it be correct to interpret this is as a probabilistic result?"

Yes. There is a mathematical equivalence between 0.5 probability and 50% chance.

However, strictly speaking, one cannot assign a probability on the basis of statistical information, for the reason that you point out in your last sentence. So what statisticians provide is an estimated probability to allow for the uncertainty that attaches to empirical premises. They allow for variability in the results by assigning a "confidence interval". So one would assign 50% chance to people with Z gene and a confidence interval which indicates how variable that figure is likely to be.

This is all very fine if one is talking about groups of people. It is less obvious what it means if one tries to apply it to a single case. Indeed some people think it is meaningless to apply a probability to a single case.

However Bayes' theory of probability does exactly that. You can look up the details Wikipedia -- Bayesian probability. Intuitively, I feel that the 50% version of probability is less appropriate here, in a single case. But few people agree with me and would insist that probability=50% and probability=0.5 mean exactly the same.

So your question becomes "what does it mean to say that there is a fifty percent chance, if you take a new random person with Z gene to have a disease Y?"

This could be the basis of a bet, but not of a prediction. But predictions, yes or no, are not everything. Probability is not useless. If the probability of rain tomorrow is 50% and you're going out, it makes sense to take an umbrella. It makes less sense if the probability of rain is 5%. Probability combined with expected utility justifies taking out insurance on your life, your house, or your car. Probability justifies locking the door when you leave home. and so on.

I haven't specified the meaning in the usual sense, but I have given some examples of the uses of probability. Whether those are the same thing or not would have to be argued elsewhere.

Answer 4

There is a saying : Lies, Damned Lies and Statistics.

What you describe as "relate to" is correlation. Correlation is a relationship, an association between two variables. However, seeing two variables related or associated, does NOT imply that one variable causes the other to occur. This is described as "correlation does not imply causation".

A strong correlation might indicate causality, but there could be other explanations too :

• random chance : the variables appear to be related, but there is no true underlying relationship or it is not clear which on affects the other.
• third variable : there exists another variable that makes the relationship appear stronger or weaker than it actually is, or affects both variables separately.

It is supposed that causation can be demonstrated with controlled experiments. These experiments test hypotheses, to establish causality in one direction at a time.

It is obvious from the above that since there exists much subjectivity in this process and also a hypotheses is needed in order to establish causality, it is easy to manipulate (either on purpose or by wrong assessments) the outcome.

The bottom line for me is that the probability result of these kind of studies means nothing at all. It is just a number used to justify or promote a specific kind of thinking.

Answer 5

All probabilities are frame-dependent. If your study says "People with gene Z have a 50% of developing disease Y", then the frame is merely 'people with gene Z' with no other qualifications. To use this result, you are not allowed to change the frame: to consider height, weight, age, race, gender, etc, etc. Considering such would require a new study. It's just like saying that 9% of people have blue eyes. That means that if we pick a random person from the entire world population, there's a 9% chance they will have blue eyes. But if we change the frame — select a random person from a specified subgroup — the probabilities will naturally change. If we select a random person from China or the Republic of Congo, the probability of blue eyes will drop well below 1%; if we choose a random person from the USA or Sweden, the probability will jump to 20% or 30%.

It's a common rule in statistics that assumptions produce statistical power. If we assume nothing about our population we will get a gross mean with wide variability. If we start assuming characteristics — changing our frame — our mean will become more exact and our variability will shrink, but it will only apply to groups that have those characteristics.