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

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.

R functions: prop.test()

The R functions prop.test() can be used as follow:

prop.test(x, n, p = NULL, alternative = "two.sided",
          correct = TRUE)

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?

res <- prop.test(x = c(490, 400), n = c(500, 500))

# Printing the results
res 

    2-sample test for equality of proportions with continuity correction

data:  c(490, 400) out of c(500, 500)
X-squared = 80.909, df = 1, p-value < 2.2e-16
alternative hypothesis: two.sided
95 percent confidence interval:
 0.1408536 0.2191464
sample estimates:
prop 1 prop 2 
  0.98   0.80 

prop.test(x = c(490, 400), n = c(500, 500),
           alternative = "less")

prop.test(x = c(490, 400), n = c(500, 500),
              alternative = "greater")

Interpretation of the result

The p-value of the test is 2.36310^{-19}, which is less than the significance level alpha = 0.05. We can conclude that the proportion of smokers is significantly different in the two groups with a p-value = 2.36310^{-19}.

Note that, for 2 x 2 table, the standard chi-square test in chisq.test() is exactly equivalent to prop.test() but it works with data in matrix form.

Access to the values returned by prop.test() function

The result of prop.test() function is a list containing the following components:

The format of the R code to use for getting these values is as follow:

# printing the p-value
res$p.value

[1] 2.363439e-19

# printing the mean
res$estimate

prop 1 prop 2 
  0.98   0.80 

# printing the confidence interval
res$conf.int

[1] 0.1408536 0.2191464
attr(,"conf.level")
[1] 0.95