Binomial test

Enter the name for this tabbed section: Description
The binomial test is used to test if the proportion of individuals that falls into 2 categories is driven by chance only. It is used when we want to know if the observed frequencies of a binary variable (gender, yes/no, sick/not sick) differ from the frequencies that correspond to a binomial distribution.
When the number of trials is large, we could also use a Chi-square test or a G-test, but with small sample size, the approximation between these test break down.
Enter the name for this tabbed section: R Syntax
In R, the test can be performed using the syntax :

binom.test {stats}

binom.test(x,p = 0.5, alternative = c(“two.sided, “less”, “greater”), conf.level = 0.95)
  • x is the number of observed successes (defined by the context of the test
  • p is the expected probability of a success. This value is determined by the type of problem in hand, it is the results of the theoretical model underlying the null Hypothesis.
  • alternative indicates whether it is the bilateral or unilateral tests that are to be used.
  • Finally, the conf.level defines the confidence level of the test (0.95 is typical, but other values can be used).
Enter the name for this tabbed section: Code Example
Binomial test

The binomial test examines whether a number of “successful” events correspond to a binomial distribution. For instance, after throwing a regular dice 232 times, we obtain 51 sixes, is this “normal” or is this too much ?. Should we suspect that the owner of the dice cheated ? On a normal dice, the probability of obtaining a 6 (success here) is 1/6.

binom.test(51,232,1/6, alternative = "greater")
## 
##  Exact binomial test
## 
## data:  51 and 232
## number of successes = 51, number of trials = 232, p-value =
## 0.02128
## alternative hypothesis: true probability of success is greater than 0.1666667
## 95 percent confidence interval:
##  0.1758254 1.0000000
## sample estimates:
## probability of success 
##              0.2198276

We used a unilateral test, because here we are interested to know the the dice was “tricked” to produce more 6 than expected by the model. If we were to be only interested in knowing if the observed results were compaticle with a “normal” dice, we would have used a bilateral test:

binom.test(51,232,1/6, alternative = "two.sided")
## 
##  Exact binomial test
## 
## data:  51 and 232
## number of successes = 51, number of trials = 232, p-value =
## 0.03423
## alternative hypothesis: true probability of success is not equal to 0.1666667
## 95 percent confidence interval:
##  0.1682745 0.2786909
## sample estimates:
## probability of success 
##              0.2198276

In both cases, we have to reject the null hypothesis: the dice is likely loaded. Note also that nearly the whole output is similar (normal), but that the p-value is different.