Tests of Single Proportions

Calculating odds and risk ratios only gives an indication of whether a potential cause is related to the outcome. To be more specific, we can do tests on groups with different exposures with regard to their outcomes. First, let us introduce the idea of testing for proportions, from the simplest scenario.

Tests of single proportions are generally based on the binomial distribution with size parameter N and probability parameter p. For large sample sizes, this can be well approximated by a normal distribution with mean N*p and variance N*p(1 − p). As a rule of thumb, the approximation is satisfactory when the expected numbers of "successes" and "failures" are both larger than 5. The normal approximation can be somewhat improved by the Yates correction (aka continuity correction), which shrinks the observed value by half a unit towards the expected value when calculating the test statistic (by default, this correction is used; it can also be turned off by using "correct = F").

In the outbreak data set, 447 of the 998 individuals who ate beef curry were observed to have food poisoning symptoms, and one may want to test the hypothesis that the probability of a "random individual who ate beef curry" having food poisoning is 0.1.

  • H0: The proportion of individuals who eat beefcurry and get sick is 0.1: true p = 0.1
  • Ha: The proportion of individuals who eat beefcurry and get sick is not 0.1: true p ≠ 0.1

These hypotheses can be tested using prop.test. The three arguments to prop.test are the number of positive outcomes, the total number, and the (theoretical) probability parameter that you want to test for. The latter is 0.5 by default (OK for symmetric problems).

> prop.test(447, 998, .1)

        1-sample proportions test with continuity correction

data:  447 out of 998, null probability 0.1

X-squared = 1338.242, df = 1, p-value < 2.2e-16

alternative hypothesis: true p is not equal to 0.1

95 percent confidence interval:

 0.4168064 0.4793912

sample estimates:

        p

0.4478958

Conclusion:

We reject the null hypothesis (χ1 2 = 1338.242, df = 1, p-value < 2.2e-16). The estimated proportion of people who ate beef curry is 0.448 (95% CI: 0.42, 0.49).

Tests for Two Independent Proportions

The function prop.test can also be used to compare two or more proportions, which can help answer more interesting questions for the outbreak data. For comparing two proportions, the arguments are given as two vectors, where the first vector contains the number of positive outcomes in each group, and the second vector the total number for each group.

Suppose we want to test the hypothesis that gender is associated with developing food poisoning based on the outbreak data. Specifically, we are interested in determining whether men are at a higher risk for developing food poisoning than women (this should be our "test" hypothesis). The relevant hypotheses are as follows:

  • H0: The proportion of males who have gastrointestinal illness is less than or equal to the proportion of females who have gastrointestinal illness.
  • Ha: The proportion of males who have gastrointestinal illness is greater than the proportion of females who have gastrointestinal illness.

We need to construct two vectors first:

> male.cases = length(which(case == 1 & sex == 1))

> female.cases = length(which(case == 1 & sex == 0))

> people.cases = c(male.cases, female.cases)

> male.total = length(which(sex==1))

> female.total = length(which(sex==0))

> people.total= c(male.total, female.total)

Now we will do a two-sample test for proportions (note the one-sided alternative here!)

 > prop.test(people.cases, people.total, alternative =  "greater")

        2-sample test for equality of proportions with continuity correction

data:  people.cases out of people.total

X-squared = 8.3383, df = 1, p-value = 0.001941

alternative hypothesis: greater

95 percent confidence interval:

 0.03998013 1.00000000

sample estimates:

   prop 1    prop 2

0.4604716 0.3672922

Conclusion: We reject the null hypothesis, and conclude that the proportion of males who have gastrointestinal illness is greater than the proportion of females with gastrointestinal illness (χ1 2 = 8.34, p-value = 0.0019). The estimated proportion of males with gastrointestinal illness is 0.46, while the estimated proportion of females with gastrointestinal illness is 0.37. The 95% CI for the difference between the proportions is (0.04, 1.00). Note that this CI excludes 0, and so is concordant with our decision to reject the null based on the p-value.

The above test uses approximations, which may not be accurate if the sample sizes are small. If you want to be sure that at least the p-value is correct, you can use Fisher's exact test. The relevant function is fisher.test , which requires that data be given in matrix form. The second column of the table needs to be the number of negative outcomes, not the total number of observations. This is obtained as follows:

> cases.matrix = matrix(c(male.cases, female.cases, male.total - male.cases, female.total - female.cases),2,2)

> fisher.test(cases.matrix, alternative="greater)

           Fisher's Exact Test for Count Data

data:  cases.matrix

p-value = 0.001881

alternative hypothesis: true odds ratio is greater than 1

95 percent confidence interval:

 1.176021 Inf

sample estimates:

odds ratio

  1.469679

Notice that in this case the p-values from Fisher's exact test and the normal approximation are very close, as expected by the large sample sizes.

The standard chi-square (χ2) test in chisq.test performs chi-squared contingency table tests and goodness-of-fit tests. It works with data in matrix form, just as fisher.test does. For a 2×2 table the test is exactly equivalent to prop.test (except that this is always for a two-sided alternative!).

 > chisq.test(cases.matrix)

        Pearson's Chi-squared test with Yates' continuity correction

data:  cases.matrix

X-squared = 8.3383, df = 1, p-value = 0.003882

Based on the outbreak data, carry out an appropriate test and report the results for testing the hypothesis that people who drank water were more likely to get diarrhea than those who did not.

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