Explain how to perform a two-sample z-test for the difference between two population proportions

• What is two-proportions z-test?
• Research questions and statistical hypotheses
• Formula of the test statistic
  • Case of large sample sizes
  • Case of small sample sizes
• Compute two-proportions z-test in R
  • R functions: prop.test[]
  • Compute two-proportions z-test
  • Interpretation of the result
  • Access to the values returned by prop.test[] function
• See also
• Infos

What is two-proportions z-test?

The two-proportions z-test is used to compare two observed proportions. This article describes the basics of two-proportions *z-test and provides pratical examples using R sfoftware**.

For example, we have two groups of individuals:

• Group A with lung cancer: n = 500
• Group B, healthy individuals: n = 500

The number of smokers in each group is as follow:

• Group A with lung cancer: n = 500, 490 smokers, \[p_A = 490/500 = 98%\]
• Group B, healthy individuals: n = 500, 400 smokers, \[p_B = 400/500 = 80%\]

In this setting:

• The overall proportion of smokers is \[p = frac{[490 + 400]}{500 + 500} = 89%\]
• The overall proportion of non-smokers is \[q = 1-p = 11%\]

We want to know, whether the proportions of smokers are the same in the two groups of individuals?

Research questions and statistical hypotheses

Typical research questions are:

1. whether the observed proportion of smokers in group A [\[p_A\]] is equal to the observed proportion of smokers in group [\[p_B\]]?
2. whether the observed proportion of smokers in group A [\[p_A\]] is less than the observed proportion of smokers in group [\[p_B\]]?
3. whether the observed proportion of smokers in group A [\[p_A\]] is greater than the observed proportion of smokers in group [\[p_B\]]?

In statistics, we can define the corresponding null hypothesis [\[H_0\]] as follow:

1. \[H_0: p_A = p_B\]
2. \[H_0: p_A \leq p_B\]
3. \[H_0: p_A \geq p_B\]

The corresponding alternative hypotheses [\[H_a\]] are as follow:

1. \[H_a: p_A \ne p_B\] [different]
2. \[H_a: p_A > p_B\] [greater]
3. \[H_a: p_A < p_B\] [less]

Note that:

• Hypotheses 1] are called two-tailed tests
• Hypotheses 2] and 3] are called one-tailed tests

Formula of the test statistic

Case of large sample sizes

The test statistic [also known as z-test] can be calculated as follow:

\[ z = \frac{p_A-p_B}{\sqrt{pq/n_A+pq/n_B}} \]

where,

• \[p_A\] is the proportion observed in group A with size \[n_A\]
• \[p_B\] is the proportion observed in group B with size \[n_B\]
• \[p\] and \[q\] are the overall proportions

• if \[|z| < 1.96\], then the difference is not significant at 5%
• if \[|z| \geq 1.96\], then the difference is significant at 5%
• The significance level [p-value] corresponding to the z-statistic can be read in the z-table. We’ll see how to compute it in R.

Note that, the formula of z-statistic is valid only when sample size [\[n\]] is large enough. \[n_Ap\], \[n_Aq\], \[n_Bp\] and \[n_Bq\] should be \[\geq\] 5.

Case of small sample sizes

The Fisher Exact probability test is an excellent non-parametric technique for comparing proportions, when the two independent samples are small in size.

Compute two-proportions z-test in R

R functions: prop.test[]

The R functions prop.test[] can be used as follow:

• x: a vector of counts of successes
• n: a vector of count trials
• alternative: a character string specifying the alternative hypothesis
• correct: a logical indicating whether Yates’ continuity correction should be applied where possible

Note that, by default, the function prop.test[] used the Yates continuity correction, which is really important if either the expected successes or failures is < 5. If you don’t want the correction, use the additional argument correct = FALSE in prop.test[] function. The default value is TRUE. [This option must be set to FALSE to make the test mathematically equivalent to the uncorrected z-test of a proportion.]

Compute two-proportions z-test

We want to know, whether the proportions of smokers are the same in the two groups of individuals?