This chapter discusses the traditional methods used to address random error in quantitative research. Before discussing these methods, we should recall the distinction between statistical parameters and estimates as discussed in Chapter 9. The term parameter refers to an error-free numerical constant that statistically describes a numerical characteristic of a population. We cannot in fact know the value of the parameter exactly but can estimate it statistically. However, the calculated statistical estimate will be imperfect due to random and systematic error. Methods presented in this chapter address the problem of random error in statistical estimates but have no influence on the problem of systematic error.

Let us introduce new notation in this chapter that preserves the distinction between parameters and estimates by using Greek letters to denote parameters, while using the same Greek letter with an overhead hat (^{^})] to denote analogous estimators. For example, φ is used to denote a risk ratio parameter, while is used to denote a risk ratio estimator. The only exception to the convention of using Greek characters for parameter is seen in using the Roman letter p to denote the binomial proportion parameter and the symbol to denote its estimator. This exception is adopted to avoid confusion with the constant π and maintain conventions of accepted biostatistical usage.

Understanding the population that is the source of the parameter we want to learn about is fundamental to statistical inference. This topic was introduced in Section 9.2 and bears further comment here. Statistical populations may be real or hypothetical, depending on the object of the research. Real populations are composed of a finite number of potential observations, as describes the situation when conducting a prevalence survey. In contrast, hypothetical populations represent an infinite number of potential observations from which to sample. While imagining a real population is relatively undemanding, conceiving of a hypothetical population is not always self-evident. The difference between a real population and hypothetical population can be made more tangible by considering the Framingham Heart Study. While it is true that the Framingham cohort is real and exists in eastern Massachusetts (United States), most inferences based on these data are used to learn about general physiologic relations between various risk factors (e.g., hypercholesterolemia) and heart disease. Since these generalizations go beyond the present Framingham population, data are seen as representing a sample from a hypothetical “superpopulation” of general causal phenomena.

Regardless of whether a study is based in a real population or hypothetical population, the two traditional methods of inferring statistical parameters are estimation and hypothesis (significance) testing. Examples will introduce their use. It is common for epidemiologists to want to learn about the prevalence of a condition (e.g., smoking) in a population based on the prevalence of the condition in a sample. In a given study, the inference may be “25% of the population smokes” (point estimation). In addition, estimation takes the form of an interval, such as “we are 95% confident that between 20% and 30% of the population smokes” (interval estimation). Finally, the epidemiologist might want to test the claim that the prevalence of smoking has changed. Under such instances, a categorical “yes” or “no” conclusion is sought (hypothesis testing). Our first order of business is to present methods of calculating confidence intervals.

Both incidence proportions (risks) and prevalences are mathematical proportions. Assuming observations are independent, the number of cases in a given sample will follow a binomial distribution with parameters n and p, where n represents the sample size and p represents the incidence proportion or prevalence parameter. The estimator of this parameter, denoted (“p hat”), is simply the sample proportion:

13.1

where A represents the number of cases in the sample and n represents the sample size. For example, a sample of 57 people in which 17 smoke demonstrates = 15/57 = 0.298 for the characteristic of smoking.

Let = 1 − . When ≥ 10, a normal approximation to the binomial can be used to calculate a confidence 95% interval for p as follows:

13.2

where represents the estimated standard error of the proportion:

Other levels of confidence for this and other confidence intervals in this chapter based on normal error distributions are calculated by replacing the 1.96 in formulas with 1.645 for 90% confidence intervals and 2.58 for 99% confidence intervals. For example, a 90% confidence interval for p is given by ± (1.645)(). A 99% confidence interval for p is given by ± (2.58)().

In small samples the above method should be voided in preference for either a quadratic method (Formula 13.3) or exact binomial method. The lower confidence limit (p_LCL) and upper confidence limit (p_UCL) for a 95% confidence interval for p by the quadratic method (Fleiss, 1981, Section 1.4) are

13.3

Coverage of the method used to calculate exact binomial confidence limits is beyond the scope of this book. Fortunately, there exist reliable public domain computer programs that use this method to compute confidence limits. The two that have been emphasized in this book are OpenEpi.com (Dean et al., 2006) and WinPEPI (Abramson, 2011). For example, OpenEpi → Counts → Proportions derives a mid-P exact 95% confidence interval for p of 0.191–0.426.

When estimating an incidence rate based on a person-time denominator, the number of cases in a given sample is assumed to follow a Poisson distribution with expectation μ. Let A represent the observed number of cases in a sample comprising T person-years of population experience. The point estimator of rate parameter λ is

13.4

A Fisher’s exact 95% confidence interval for rate parameter λ is calculated in two steps. First, the lower confidence limit (A_lcl) and the upper confidence limit (A_ucl) for the expected number of cases are determined using the Poisson limits in Appendix 1. Then, these confidence limits are expressed relative to the person-time (T) in the sample:

13.5

The ratio of two incidence proportions is a risk ratio. The ratio of two prevalences is a prevalence ratio. Let φ represent a proportion ratio parameter (either a risk ratio or prevalence ratio). The point estimator of the proportion ratio parameter is

13.6

Let ω represent the rate ratio parameter. The rate ratio estimator is

13.9

where represents the rate in the exposed group (= A₁/T₁) and represents the rate in the nonexposed group (= A₀/T₀). With moderate to large samples, the random error distribution of the natural log of the estimator (ln ) is approximately normal with a standard error of

13.10

The 95% confidence interval for ln ω is calculated in the usual manner:

13.11

The 95% confidence limits for ω is derived by taking the exponents of these limits.

This section considers differences between proportions (risk differences and prevalence differences). To avoid redundancy, only the risk difference will be discussed.

Let δ_risk denote the risk difference parameter. The point estimator of δ_risk is

13.12

where represents the incidence proportion in the exposed group and represents the incidence proportion in the nonexposed group.

The standard error of the risk difference is

13.13

A 95% confidence interval for δ_risk is calculated with the formula

13.14