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Analysing a binary variable

Test: Binomial test

In the example we noticed that 26% of the respondents indicated to be Female, and 74% Male. This might appear as a big difference, but is it a ‘significant’ difference, i.e. will there also be a difference in the population.

As discussed in the general section on significance (see here), the significance is the probability of a result as in the sample or even more extreme, if the assumption about the population is true.

With only two options to choose from, most often the assumption about the population is here that both groups are equal. This would mean that if we pick a random person from the population, the chance of him/her belonging to either category is 0.5 (50%).

The result in the sample was that we had 12 Female respondents. ‘More extreme’ would be less than 12. What we can ‘easily’ determine now is the probability of getting 12 or less Female out of 46, if in the population the chance of picking a female is 0.5. This can be done using a so-called binomial distribution. The chance for this is 0.0008. However, ‘more extreme’ can also mean that we have a similar over-representation of females. If the expected proportion is 0.5 (50%) then we can simply double the result, so the significance is 2 x .0008 = 0.0016.

We could report this as:

An exact binomial test indicated that the percentage of female (N_f = 12, 26%), was significantly different from the male percentage (N_m = 34, 76%), p = .002.

		# libraries needed
		import pandas as pd
		from scipy.stats import binom_test
		
		# some data
		myDf = pd.read_csv('../../Data/csv/StudentStatistics.csv', sep=';')
		myCd = myDf['Gen_Gender'].value_counts()
		
		# the test
		binom_test(myCd.values[0], sum(myCd.values), 1/2, alternative='two-sided')
		

		#one sample binomial test
		
		#Preparation
		#Getting some data
		#install.packages("foreign")
		library(foreign)
		myData <- read.spss("../Data Files/StudentStatistics.sav", to.data.frame = TRUE)
		
		#Remove na's
		myVar <- na.omit(myData$Gen_Gender)
			
		#Determine number of successes
		k <- sum(myVar==myVar[1])
			
		#Determine total sample size
		n <- length(myVar)

		#Test if expected both groups to be equal  
		#Perform binomial test
		binom.test(k,n)
		
		#Or use binomial distribution directly
		2*pbinom(k,n,.5)
		

Note that if you have a different expectation about the population than the 0.50, we cannot simply double the result anymore. Well we could for a quick approximation, provided the sample size is large, but usually a more complex technique is then used, known as the ‘method of small p’, and in some cases another method known as ‘equal distance’. How these methods work is discussed in the end notes at the bottom of this page.

The binomial test can be computational heavy, so sometimes an approximation is used. The approximation is then either done using the Normal distribution, or a goodness-of-fit test. In both cases so-called continuity corrections can be applied and there are different variations on these corrections.

In short the binomial test has the following steps:

1. The assumption about the population (the null hypothesis (H₀)) is that the proportion of one of the two categories will be some amount (e.g. 0.5).
   
   
2. The alternative is that it isn't (H_a) (e.g. the proportion in the population is not 0.5). This would be the so-called two-tailed test.
   
   
3. Perform the binomial test and find the p-value (sig.).
   
   
4. If the p-value is less than .05, the chance of a result as in the sample or even rarer if the assumption is true, is considered so low, that the assumption is probably NOT true. The proportion in the population is then probably NOT the one assumed at step 1. This is then called a significant result.
   
   
5. If the p-value is .05 or more, the chance of a result as in the sample or even rarer if the assumption is true, is considered not low enough, that the assumption could be true. We don't have enough evidence to reject the assumption. This is then called a non-significant result.
   
   

The test informs us that there most likely will also be a difference in the population, it however does not say anything about if it is a big or small difference. For that we need a so-called effect size, which is the topic for the next section.

Appendix

Single binary variable 

Introduction

Impression

Visualisation (na)

Center & Dispersion (na)

Test

Effect Size

Report

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