## Distribution of the difference between two binomial variables

For a project I’m working on, I needed to work with the probabilities associated with the difference between two binomial variables. I thought I’d share the code for four functions for calculating the probability mass and the cumulative probabilites.

The functions d.diff.binom (probability mass) and p.diff.binom (cumulative distribution) are functions for calculating the distribution of the differences for equal sample sizes N and equal “succes” probability.  The arguments of the functions are the quantiles k of the distribution of the differences, the number of trials N and the probability of succes p.

The functions d.diff.binom.un (probability mass) and p.diff.binom.un (cumulative distribution) are functions for calculating the distribution of the differences for sample sizes and succes probabilities that may (or may not) differ between the two populations. The arguments of the functions are the quantiles k of the distribution of the differences, the number of trials N.1 and N.2 for the two groups and the probability of succes p.1 and p.2 for each group.

#equal N and p:
#probability mass:
d.diff.bin = function(k, N, p) {
diff = outer(0:N, 0:N, "-")
prob = outer(dbinom(0:N, N, p, log=TRUE),
dbinom(0:N, N, p, log=TRUE), "+")
p = sum(exp(prob))
return(p)
}
#cumulative probability:
p.diff.bin = function(k, N, p) {
diff = outer(0:N, 0:N, "-")
prob = outer(dbinom(0:N, N, p, log=TRUE),
dbinom(0:N, N, p, log=TRUE), "+")
p = sum(exp(prob))
return(p)
}
#examples
d.diff.bin(0, 30, .50)

## [1] 0.1025782

p.diff.bin(0, 30, .50)

## [1] 0.5512891

#mean of distribution:
N = 30
m = sum(sapply(-N:N, d.diff.bin, N = 30, p= .50)*(-N:N))
m

## [1] -3.084642e-20

all.equal(0, m)

## [1] TRUE

#(un)equal N and p:
#probability mass:
d.diff.bin.un = function(k, N.1, N.2, p.1, p.2) {
diff = outer(0:N.1, 0:N.2, "-")
prob = outer(dbinom(0:N.1, N.1, p.1, log=TRUE),
dbinom(0:N.2, N.2, p.2, log=TRUE), "+")
p = sum(exp(prob))
return(p)
}

#cumulative distribion:
p.diff.bin.un = function(k, N.1, N.2, p.1, p.2) {
diff = outer(0:N.1, 0:N.2, "-")
prob = outer(dbinom(0:N.1, N.1, p.1, log=TRUE),
dbinom(0:N.2, N.2, p.2, log=TRUE), "+")
p = sum(exp(prob))
return(p)
}

#examples
d.diff.bin.un(0, 30, 20, .60, .70)

## [1] 0.05896135

p.diff.bin.un(0, 30, 20, .60, .70)

## [1] 0.1497043


## Decisions are not evidence

The thinking that lead to this post began with trying to write something about what Kline (2013) calls the filter myth. The filter myth is the arguably – in the sense that it depends on who you ask – mistaken belief in NHST practice that the p-value discriminates between effects that are due to chance (null-hypothesis not rejected) and those that are real (null-hypothesis rejected). The question is whether decisions to reject or not reject can serve as evidence for the existence of an effect.

Reading about the filter myth made me wonder whether NHST can be viewed as a screening test (diagnostic test), much like those used in medical practice. The basic idea is that if the screening test for a particular condition gives a positive result, follow-up medical research will be undertaken to figure out whether that condition is actually present. (We can immediately see, by the way, that this metaphor does not really apply to NHST, because the presumed detection of the effect is almost never followed up by trying to figure out whether the effect actually exists, but the detection itself is, unlike the screening test, taken as evidence that the effect really exists; this is simply the filter myth in action).

Let’s focus on two properties of screening tests. The first property is the Positive Likelihood Ratio (PLR). The PLR is the ratio of the probability of a correct detection to the probability of a false alarm. In NHST-as-screening-test, the PLR  equals the ratio of the power of the test to the probability of a type I error: PLR = (1 – β) / α. A high value of the PLR means, basically, that a rejection is more likely to be a rejection of a false null than a rejection of a true null, thus the PLR means that a rejection is more likely to be correct than incorrect.

As an example, if β = .20, and α = . 05, the PLR equals 16. This means that a rejection is 16 times more likely to be correct (the null is false) than incorrect (the null is true).

The second property I focused on is the Negative Likelihood Ratio (NLR). The NLR is the ratio of the frequency of incorrect non-detections to the frequency of correct non-detections. In NHST-as-screening-test, the NLR equals the ratio of the probability of a type II error to the probability of a correct non-rejection: NLR = β / (1 – α). A small value of the NLR means, in essence, that a non-rejection is less likely to occur when the null-hypothesis is false than when it is true.

As an example, if β = .20, and α = . 05, the NLR equals .21. This means that a non-rejection is .21 times more likely (or 4.76 (= 1/.21) times less likely) to occur when the null-hypothesis is false, than when it is true.

The PLR and the NLR can be used to calculate the likelihood ratio of the alternative hypothesis to the null-hypothesis given that you have rejected or given that you have not-rejected, the posterior odds of the alternative to the null. All you need is the likelihood ratio of the alternative to the null before you have made a decision and you multiply this by the PLR after you have rejected, and by the NLR after you have not rejected.

Suppose that we repeatedly (a huge number of times) take a random sample from a population of null-hypotheses in which 60% of them are false and 40% true. If we furthermore assume that a false null means that the alternative must be true, so that the null and the alternative cannot both be false, the prior likelihood of the alternative to the null equals p(H1)/p(H0) = .60/.40 = 1.5. Thus, of all the randomly selected null-hypotheses, the proportion of them that are false is 1.5 times larger than the proportion of  null-hypotheses that are true. Let’s also repeatedly sample (a huge number of times) from the population of decisions. Setting β = .20, and α = . 05, the proportion of rejections equals p(H1)*(1 – β) + p(H0)*α = .60*.80 + .40*.05 = .50 and the proportion of non-rejections equals p(H1)*β + p(H0)*(1 – α) = .60*.20 + .40*.95 = .50. Thus, if we sample repeatedly from the population of decisions 50% of them are rejections and 50% of them are non-rejections.

First, we focus only on the rejections. So, the probability of a rejection is now taken to be 1.0.  The posterior odds of the alternative to the null, given that the probability of a rejection is 1.0, is the prior likelihood ratio multiplied by the PLR: 1.5 * 16 = 24. Thus, we have a huge number of rejections (50% of our huge number of randomly sampled decisions) and within this huge number of rejections the proportion of rejections of false nulls is 24 times larger than the proportion of rejections of true nulls. The proportion of rejections of false nulls equals the posterior odds / (1 + posterior odds) = 24 / 25 = .96. (Interpretation: If we repeatedly sample a null-hypothesis from our huge number of rejected null-hypotheses, 96% of those samples are false null-hypotheses).

Second, we focus only on the non-rejections. Thus, the probability of a non-rejection is now taken to be 1.0. The posterior odds of the alternative to the null, given that the probability of a non-rejection is 1.0, is the prior odds multiplied by the NLR: 1.5 * 0.21 = 0.32. In other words, we have a huge number of non-rejections (50% of our huge sample of randomly selected decisions) and the proportion of non-rejections of false nulls is 0.32 times as large as the proportion of non-rejections of true nulls. The proportion of non-rejections of false nulls equals 0.32 / ( 1 + 0.32) =  .24. (Interpretation: If we repeatedly sample a null-hypothesis from our huge number of non-rejected hypotheses, 24% of them are false nulls).

So, based on the assumptions we made, NHST seems like a pretty good screening test, although in this example NHST is much better at detecting false null-hypothesis than ruling out false alternative hypotheses. But how about the question of decisions as evidence for the reality of an effect? I will first write a little bit about the interpretation of probabilities, then I will show you that decisions are not evidence.

Sometimes, these results are formulated as follows. The probability that the alternative is true given a decision to reject is .96 and the probability that the alternative hypothesis is true given a decision to not-reject  is .24.  If you want to correctly interpret such a statement, you have to keep in mind what “probability” means in the context of this statement, otherwise it is very easy to misinterpret the statement’s meaning. That is why I included interpretations of these results that are consistent with the meaning of the term probability as it used in our example. (In conceptual terms, the limit of the relative frequency of an event (such as reject or not-reject) as the number of random samples (the number of decisions) goes to infinity).

A common (I believe) misinterpretation (given the sampling context described above) is that rejecting a null-hypothesis makes the alternative hypothesis likely to be true. This misinterpretation is easily translated to the incorrect conclusion that a significant test result (that leads to a rejection) makes the alternative hypothesis likely to be true. Or, in other words, that a significant result is some sort of evidence for  the alternative hypothesis (or against the null-hypothesis).

The mistake can be described as confusing the probability of a single result with the long term (frequentist) probabilities associated with the decision or estimation procedure. For example, the incorrect interpretation of the p-value as the probability of a type I error or the incorrect belief that an obtained 95% confidence interval contains the true value of a parameter with  probability .95.

A quasi-simple example may serve to make the mistake clear. Suppose I flip a fair coin, keep the result hidden from you, and let you guess whether the result is heads or tails (we assume that the coin will not land on it’s side). What is the probability that your guess is correct?

My guess is that you believe that the probability that your guess is correct is .50. And my guess is also that you will find it hard to believe that you are mistaken. Well, you are mistaken if we define probabilities as long term relative frequencies. Why? The result is either heads or tails. If the result is heads, the long term relative frequency of that result is heads. That is to say, the result is a constant and constants do not vary in the long run. Your guess is also a constant, if you guess heads, it will stay heads in the long run. So, if the result is heads, the probability that your guess (heads) is correct is 100%, however, if your guess is tails, the probability of you being correct is 0%. Likewise, if the result is tails, it will stay tails forever, and the probability that your guess is correct is 0% if your guess is heads and 100% if your guess is tails. So, the probability that your guess is correct is either 0 or 1.0, depending on the result of the coin flip, and not .50.

The probability of guessing correct is .50, however, if we repeatedly (a huge number of times) play our game and both the result of the coin flip and your guess are the result of random sampling. Let’s assume that of all the guesses you do 50% are heads and 50% are tails.  In the long run, then, there is a probability of .25 of the result being heads and your guess being heads and a probability of .25 of the result being tails and your guess being tails. The probability of a correct decision is therefore, .25 + .25 = .50

Thus, if  both the results and your guesses are the result of random sampling and we repeated the game a huge number of times, the probability that you are correct is .50. But if we play our game only once, the probability of you being correct is 0 or 1.0, depending on the result of the coin flip.

Let’s return to the world of hypotheses and decisions. If we play the decision game once, the probability that your decision is correct is 0 or 1.0, depending on whether the null-hypothesis in question is true (with probability 0 or 1.0) or false (with probability 0 or 1.0). Likewise, the probability that the null-hypothesis is true given that you have rejected is also 0 or 1.0, depending on whether the null-hypothesis in question is true or false. But if we play the decision game a huge number of times, the probability that a null-hypothesis is false, given that you have decided to reject is .96 (in the context of the situation described above).

In sum, from the frequentist perspective we can only assign probabilities 0 or 1.0 to a single hypothesis given we have a made single decision about it, and this probability depends on whether that single hypothesis is true or false.  For this reason, a significant result cannot be magically translated to the probability that the alternative hypothesis is true given that the test result is significant. That probability is 0 or 1.0 and there’s nothing that can change that.

The consequence of all this is as follows. If we define the evidence for or against our alternative hypothesis in terms of the likelihood ratio of the alternative to the null-hypothesis after obtaining the evidence, no decision can serve as evidence if our decision procedure is based on frequentist probabilities. Decisions are not evidence.

References
Kline, R.B. (2013). Beyond significance testing. Statistics reform in the behavioral sciences. Second Edition. Washington: APA.