TABLE 1. Types of Variables

TABLE 2. Frequency Distributions

x = Σx/n = x1 + x2 + … xn/n

where: n = the sample size; x1, x2, and so forth = the individual data. The mean for the population is found by replacing n with N, which signifies the population size.

The median is defined as the middle value when the numbers are arranged in either ascending or descending order. When the distribution of data contains extreme values, the median often is a better indicator of the central tendency. The mode, on the other hand, is the value that occurs most often. All of these measures of central tendency indicate the overall shape of the distribution curve. When the mean, median, and mode all share the same value, then the resulting curve is said to be perfectly symmetric. Conversely, if the curve is skewed to the left or right, the mean, median, and mode values also are skewed to the left or right, respectively (Fig. 1).

Population-weighted variance:

σ² = 1/N Σ(xi - μ)²

Sample-weighted variance:

s² = 1/n - 1 Σ(xi - x)²

STANDARD DEVIATION

By taking the positive squared root of the variance, the standard deviation (SD) in the data set may be calculated as follows:

σ = √1/N Σ(xi - μ)²

1. Within ±1 SD from the mean value for the data, 68% of the data is contained under the bell-shaped curve,
   
2. Within ±2 SD from the mean valued for the data, 95% of the data is contained under the bell-shaped curve,
   
3. Within ±3 SD from the mean value for the data, nearly 99.5% of all the data is contained under the bell-shaped curve.
   
4. The area under the normal curve is equivalent to the probability of randomly selecting a value within that range and is equal to unity, or 1.
   

  95% CI = sample mean ± 1.96 standard error
  or
  95% CI = x ± 1.96 standard error
  where: standard error = standard deviation = SD
  √sample size √n

z = x - μ/σx

When both μ and σ are known for a given population and the sample size is large (n > 30), it is possible to understand the pattern that the distribution of the sample mean will take. It is assumed that it has a normal distribution with a mean = μ and a standard error (σx) = σ/N. The critical region is defined as that region or area under the curve that includes values that will result in the rejection of the null hypothesis.

When the SD is unknown, a t-statistic is used. The t-statistic uses a ratio to assess the significance of the difference between the mean of two groups of data, such as observed or treatment group and a control. In general, the t-statistic is found using the following formula:

  x - μ/σx
  where: μ = population mean under the null hypothesis (H_o).

Last, with respect to the t-test, there is the concept of degrees of freedom (df). Thus, to calculate the number of degrees of freedom for problems using the t-distribution with a sample size n, 1 is subtracted from n (i.e., n - 1) to give the number of degrees of freedom (df) for a one-sample mean population.

Overall, both the z- and t-distributions and the calculation of their corresponding test statistic form the basis of calculating whether the two sets of data are significantly different from each other. This works for two sets of data, such as two treatment groups. However, among many sample means (e.g., from multiple treatment groups), the t-test is no longer valid. Rather, the more advanced statistical concepts of analysis of variance must be explored.

Perhaps the most commonly used nonparametric test is the chi-square (ξ²) test, which measures the difference between the distribution of observed and expected values. The ξ² test also can be used to examine the association between more than two variables, so long as the variables are nominal and the expected frequency in any one cell is greater than five. The basic formula for determining the Þgc² statistic is as follows:

ξ² (df) = Σ of all cells (observed - expected)²/expected

where: observed = the actual values; expected = values that would be anticipated given the calculation of the value for that cell based on the multiplication of the row and column values for that particular cell divided by the grand total for all rows and columns. The corresponding degrees of freedom (df) are found by subtracting one from each of the number of rows and columns and then multiplying them together, that is, df = (r - 1)(c - 1). If the calculated ξ² value is greater than the given value for the ξ² distribution with the exact same number of degrees of freedom, the null hypothesis must be rejected and it must be concluded that there is an association.

Other important nonparametric tests include the following: the binomial test, Fisher's exact test, the McNemar test, the Mann-Whitney U test, the Kolmogorov-Smirnov test, the sign test, and the Wilcoxon test. (The reader is advised to explore these in more detail by consulting the bibliography.)

The sensitivity of a test is the proportion of those with the disease (a + c) who also test positive (a) to the diagnostic test (a/[a + c]), whereas specificity is the proportion of those without the disease (b + d) who test negative (d) to the diagnostic test (d/[b + d]) (Table 3). Similarly, the predictive value of a positive test is given by (a + c)/(a + b), that is, all those with the disease (a + c) divided by all those testing positive to the disease (a + b). Conversely, the predictive value of a negative test is given by (b + d)/(c + d), that is, all those without the disease (b + d) divided by those testing positive to the disease (c + d) (see Table 3).

TABLE 3. Results of a Screening Test

ROC curves originally were developed to measure the signal-to-noise ratio of early radio transmitters and have been adapted in the field of epidemiology to measure the true-positive values detected using the diagnostic test versus the false-negative values also picked up using the same diagnostic test. Thus, the ROC curve is a graph of the sensitivity (true-positive rate) to the false-positive rate. As shown in Figure 2, the closer the ROC curve is to the upper left-hand corner of the graph, the greater the accuracy of the test, since the true-positive rate is 1 and the false-positive rate is 0. Thus, as the values for a positive test become greater, the point on the curve corresponding to sensitivity and specificity (denoted by point A) decreases and moves to the left (i.e., lower sensitivity and higher specificity). Conversely, if fewer data are needed for a positive test, the point on the curve corresponding to sensitivity and specificity (denoted by point B) moves up and to the right (i.e., higher sensitivity and lower specificity). Last, the diagonal line drawn from the origin in Figure 2 represents a test with a positive or negative result purely by chance alone.

PROSPECTIVE (COHORT OR LONGITUDINAL) STUDIES

Prospective studies start with a group or groups of patients free of the specific disease under investigation, yet each group is exposed to various known or unknown risk factors suspected or implicated in the development of the disease. The various cohorts then are followed longitudinally, that is, over time, to see whether certain cohorts develop the disease at different rates than other cohorts. Because there is no control group per se, all cohorts must be similarly constructed to minimize any potential sources of bias during the analysis period later on. In general, the more homogenous the cohorts the better, except with the exposure to possible risk factors under investigation, since any observed difference in disease severity will be more likely to result from variations in exposure rather than potential confounding variables. In general, prospective studies require a long follow-up period, often an individual's complete lifetime, and are thus costly to administer. Moreover, if the disease is rare, they require numerous enrollees. This said, prospective studies do carefully document the progression and onset of the disease in relation to other known health status indicators and risk factors.

RETROSPECTIVE (CASE-CONTROL) STUDIES

Retrospective studies are similar to prospective studies in that both a cohort of cases and a cohort of controls is identified, and then the strength of the relationship for a possible causal factor is determined using various means of determining measures of exposure to a given event or phenomenon often using personal exposure histories. As can be imagined, there is great room for recall bias on the part of the patients who may experience difficulty in fully remembering all of the necessary information to construct an epidemiologic association. Equally, because retrospective studies require knowledge of the patient's disease status before the intervention, it is difficult for the investigator to be absolutely convinced that the definition of a case actually matches the disease under investigation. A slight change in the definition of what is regarded as a case could drastically alter the interpretation of the data. Also, there is the concern that a control group has been properly and systematically identified, and it often is a problem to ensure that controls are equally matched for all clinically relevant variables. Despite these limitations, the retrospective study has the distinct advantage that it can be performed quickly at relatively low cost and is more amenable to studying rarer diseases, since cases can be collected from a much larger population. Case-control studies are particularly useful for obtaining pilot data to undertake more elaborate studies in the future.

CROSS-SECTIONAL STUDIES

Because the population is only sampled at one particular interval in time, that is, in a snapshot fashion, cross-sectional studies rely on the clear definition of cases (diseased) and controls (nondiseased). Cross-sectional studies are best suited to measure the prevalence of a particular disease or condition at a given point in time, such as the prevalence of visual acuity less than 3/60 (blindness according to the World Health Organization) in a given population at a point in time. The main limitation is that they provide only a one-time or snapshot view of the disease or condition, and this may not adequately reflect the true situation at that time or over a period of time.

CASE SERIES

A case series typically is a report of some clinically interesting finding or feature in a patient or group of patients. Because there are no control groups associated with the case series reports, the value of the case series study is lower than other studies involving an attempt to define a control group.

TABLE 4. Calculation of Relative Risk and Odds Ratios

The effect size refers to the difference between both the control and intervention groups, which need to be detected. Thus, to calculate the necessary sample size, one must know the smallest difference between the two treatments that would miss detection, or not be detected at all. Overall, the effect size may be thought of as the degree to which the phenomenon is present in the population; it measures the degree of difference between the intervention and control group.

Power refers to the probability of detecting a real effect if it is present. Recall that to calculate the power, the following equation must be used, namely, 1 - β. In general, power values of .8 (80%) or higher are selected when conducting a clinical trial. Thus, with a power of 0.8 (80%), the probability of a type II error is 0.2 (20%). Table 5 presents a range of sample size calculations to attain desired power and effect sizes.

TABLE 5. Sample Size Calculations

ANALYSIS OF VARIANCE

1. Daly LE, Bourke GJ, McGilvaray J: Interpretation and Uses of Medical Statistics, pp 139–156, 245–248. 4th ed. Oxford, UK: Blackwell Scientific, 1991

2. Armitage P, Berry G: Statistical Methods in Medical Research, pp 207–228. 3rd ed. Oxford, UK: Blackwell Scientific, 1994

NONPARAMETRIC STATISTICS

1a. Bland M: An Introduction to Medical Statistics, pp 216–218. Oxford, Oxford Medical Publications, 1993

1b. Campbell MJ, Machin D: Medical Statistics: A Comprehensive Approach, pp 83, 84, 150–152. 2nd ed. Chichester, UK: John Wiley & Sons, 1993