## Archive for November 8th, 2022

### What about the notion of Probability, Statistics, and odd? What is your odd evaluation of another nuclear bomb be tested on people?

Posted on: November 8, 2022

Note: Since 1945, hundreds of nuclear bombs have been tested above ground and under ground by all the colonial powers. Thousands of intercontinental missiles with nuclear bombs have been produced and what outdated missiles that have been dismantled…we have no details of how they have been “decontaminated” and disposed off.

The colonial powers claims that environments in older sites of testing are safe from contamination, but they never tell us of what about the ground, sand, dirt and pebbles…whether they are “contamination-free”.

Mind you that many people in islands and regions of nuclear sites were Not told of an imminent testing. Many of them were contaminated many hundred of kilometers away. Even citizens of the testing nations who prepared for the sites were left in shorts and T-shirts behind makeshift bunkers.

Like all these “tourists” who have no “pieces of intelligence” of how close they are from nuclear sites testing… and return Home with high-risk of all sort of cancer…

What are the odds and probability of contamination of “citizens” in every country to be at health risk?

On Probability, Odds, and Odds Ratios

Rob Taylor

Probability is a measure of how likely an outcome is to occur and takes on a value between 0 and 1.

For example, given some outcome A, we can assign the probability P(A) that we actually observe this result. What this implies is that, at any time prior to observing an outcome, we can’t know exactly whether A will occur or not; all we know is that A is possible.

We just consider processes that produce two possible outcomes, so the complementary outcome is that A does not occur, to which we can assign probability P(¬A) = 1 — P(A)

In statistics, odds are interpreted as the relative likelihood of observing a particular outcome. Formally, it’s the number of times an outcome occurs to the number of times that it doesn’t (this differs somewhat from how odds are interpreted in gambling, just so you know).

For example, let’s suppose we make 20 observations with outcome A occurring on 12 occasions. The odds in favour of outcome A is then 12:8 (read as “twelve-to-eight”). Another way to state this is that A is 1.5 times (12 / 8) more likely to occur relative to it not occurring.

But if we’re just counting the number of times an event occurs, isn’t that just probability? Not quite, but they’re related.

The odds of outcome A (image by author)

As it turns out, there’s a very simple relationship between probability and odds

The odds, then, is simply the probability that outcome A occurs over the probability that it does not.

If we use our toy data again, we can calculate both P(A) = 12 / 20 = 0.6 and P(¬A) = 8 / 20 = 0.4. Let’s plug these into the equation above and see what happens:

The odds of outcome A expanded (image by author).

This just gets us back to where we started! It’s also useful to know that the odds of outcome A is the reciprocal of the odds for ¬A (and vice versa):

Finally, if you want to convert your odds back to probabilities, it’s pretty straightforward:

Odds to probability (image by author). This probably goes without saying, but if the outcomes are equiprobable, then the odds are 1, or even.

Log Odds

If you’re not familiar with log odds, they’re literally the log of the odds: The log odds, or logit (image by author). For those of you well versed in logistic regression, you’ll recognise the above equation as the logit. I’m not going to discuss the logit in great detail here — I’ll save that for another blog — but the logit is a very useful function because it maps the domain [0,1] to the real line [-∞, ∞].

In addition, the logit transform is symmetrical around zero, so outcomes with even odds are represented by 0, rather than 1.

To get a sense of how these quantities relate to one another, check out the figure below. Probability, odds, and log odds.

Probability, odds, and log odds. The red lines denote even odds (image by author).

Odds Ratio

The odds of an outcome is the ratio of two mutually exclusive probabilities. The odds ratio is a ratio of odds. Suppose we are now interested in another outcome, B. Let’s go ahead and denote the probability of this outcome P(B) and calculate the odds in favour of outcome B:

The odds of outcome B (image by author).

So now we have two sets of odds: the odds in favour of outcome A and the odds in favour of outcome B. Suppose we now wanted to know if the odds of observing outcome B are influenced by the presence of A. What the odds ratio tells us is how much more, or less, likely an outcome is to occur in the presence of another.

For example, the odds of outcome B relative to outcome A are measured as follows: The odds of outcome B relative to outcome A (image by author).Or, if you wanted to flip things around and measure outcome A relative to outcome B, then

Like above, but flipped around (image by author).Conveniently, and much like odds, we can use reciprocals to calculate odds ratios, too: Reciprocal of odds ratios (image by author).

In essence, the odds ratio is a measure of the association between two events. If there is absolutely no relationship between outcomes A and B, then the odds ratio is equal to 1 (or zero, if you log transform it).

However, if there is an association, the odds of each outcome shift by some factor.

What’s important to remember is that outcomes A and B can either occur or not, and the odds measure the relative probability that each outcome does indeed occur.

The odds ratio, on the other hand, is a measure of how much the presence of one outcome affects the odds of another.

Reciprocal of odds ratios (image by author).

Example of an epidemiological problem

.Let’s imagine a small proportion of the population has been exposed to an environmental contaminant that is suspected to increase the risk of a rare disease.

There are only 100 people in this population, so assume we surveyed everyone to assess disease prevalence among those who were exposed and those who were not.

The results of the survey are tabulated below:#

```# Example Survey Results            Positive Negative
Exposed            4       36
Not Exposed        1       59```

What we want to know is whether being exposed to the contaminant increased the odds of developing the disease.

.First, we’ll calculate the odds of having the disease among those who were exposed. Now, exposed individuals either have the disease or don’t. So the odds are simply the number of exposed persons who tested positive to the number that did not:

Odds of having the disease for those exposed (image by author).

We now do the same thing for those who were not exposed: Odds of having the disease for those not exposed (image by author).

it’s looking like the odds of having the disease are quite a bit smaller for those who were not exposed.

Let’s calculate the odds ratio:

The odds ratio (image by author)..

it looks like those exposed to the contaminant are approximately 6.6 times more likely to develop the disease relative to those who were not exposed. However, all the usual caveats around correlation and causation apply when interpreting odds ratios, and in the absence of adequate controls, the best we can usually say is that the two outcomes are associated.

For example, our survey may not have collected information about other exogenous factors that affect health outcomes. Perhaps those who developed the disease after exposure had poorer health in general

let’s just compute the odds ratio again, but expand things out a little:

That’s pretty crazy! You can just multiply the cells along the major and minor diagonals!

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