Lesson 4: Displaying Public Health Data

Section 3: Graphs

A graph (used here interchangeably with chart) displays numeric data in visual form. It can display patterns, trends, aberrations, similarities, and differences in the data that may not be evident in tables. As such, a graph can be an essential tool for analyzing and trying to make sense of data. In addition, a graph is often an effective way to present data to others less familiar with the data.

In epidemiology, most graphs have two scales or axes, one horizontal and one vertical, that intersect at a right angle. The horizontal axis is known as the x-axis and generally shows values of the independent (or x) variable, such as time or age group. The vertical axis is the y-axis and shows the dependent (or y) variable, which, in epidemiology, is usually a frequency measure such as number of cases or rate of disease. Each axis should be labeled to show what it represents (both the name of the variable and the units in which it is measured) and marked by a scale of measurement along the line.

In constructing a useful graph, the guidelines for categorizing data for tables by types of data also apply. For example, the number of reported measles cases by year of report is technically a nominal variable, but because of the large number of cases when aggregated over the United States, we can treat this variable as a continuous one. As such, a line graph is appropriate to display these data.

Arithmetic-scale line graphs

An arithmetic-scale line graph (such as Figure 4.1) shows patterns or trends over some variable, often time. In epidemiology, this type of graph is used to show long series of data and to compare several series. It is the method of choice for plotting rates over time.

In an arithmetic-scale line graph, a set distance along any axis represents the same quantity anywhere on that axis. In Figure 4.2, for example, the space between tick marks along the y-axis (vertical axis) represents an increase of 10,000 (10 × 1,000) cases anywhere along the axis — a continuous variable.

Furthermore, the distance between any two tick marks on the x-axis (horizontal axis) represents a period of time of one year. This represents an example of a discrete variable. Thus an arithmetic-scale line graph is one in which equal distances along either the x- or y- axis portray equal values.

Arithmetic-scale line graphs can display numbers, rates, proportions, or other quantitative measures on the y-axis. Generally, the x-axis for these graphs is used to portray the time period of data occurrence, collection, or reporting (e.g., days, weeks, months, or years). Thus, these graphs are primarily used to portray an overall trend over time, rather than an analysis of particular observations (single data points). For example, Figure 4.2 shows prevalence (of neural tube defects) per 100,000 births.

Figure 4.3 shows another example of an arithmetic-scale line graph. Here the y-axis is a calculated variable, median age at death of people born with Down's syndrome from 1983–1997. Here also, we see the value of showing two data series on one graph; we can compare the mortality risk for males and females.

Semilogarithmic-scale line graphs

In some cases, the range of data observed may be so large that proper construction of an arithmetic-scale graph is problematic. For example, in the United States, vaccination policies have greatly reduced the incidence of mumps; however, outbreaks can still occur in unvaccinated populations. To portray these competing forces, an arithmetic graph is insufficient without an inset amplifying the problem years (Figure 4.4).

An alternative approach to this problem of incompatible scales is to use a logarithmic transformation for the y-axis. Termed a “semi-log” graph, this technique is useful for displaying a variable with a wide range of values (as illustrated in Figure 4.5). The x-axis uses the usual arithmetic-scale, but the y-axis is measured on a logarithmic rather than an arithmetic scale. As a result, the distance from 1 to 10 on the y- axis is the same as the distance from 10 to 100 or 100 to 1,000.

Another use for the semi-log graph is when you are interested in portraying the relative rate of change of several series, rather than the absolute value. Figure 4.5 shows this application. Note several aspects of this graph:

• The y-axis includes four cycles of the order of magnitude, each a multiple of ten (e.g., 0.1 to 1, 1 to 10, etc.) — each a constant multiple.
• Within a cycle, the ten tick-marks are spaced so that spaces become smaller as the value increases. Notice that the absolute distance from 1.0 to 2.0 is wider than the distance from 2.0 to 3.0, which is, in turn, wider than the distance from 8.0 to 9.0. This results from the fact that we are graphing the logarithmic transformation of numbers, which, in fact, shrinks them as they become larger. We can still compare series, however, since the shrinking process preserves the relative change between series.

Consider the data shown in Table 4.16. Two hypothetical countries begin with a population of 1,000,000. The population of Country A grows by 100,000 persons each year. The population of Country B grows by 10% each year. Figure 4.6 displays data from Country A on the left, and Country B on the right. Arithmetic-scale line graphs are above semilog-scale line graphs of the same data. Look at the left side of the figure. Because the population of Country A grows by a constant number of persons each year, the data on the arithmetic-scale line graph fall on a straight line. However, because the percentage growth in Country A declines each year, the curve on the semilog-scale line graph flattens. On the right side of the figure the population of Country B curves upward on the arithmetic-scale line graph but is a straight line on the semilog graph. In summary, a straight line on an arithmetic-scale line graph represents a constant change in the number or amount. A straight line on a semilog-scale line graph represents a constant percent change from a constant rate.

Table 4.16 Hypothetical Population Growth in Two Countries

Semilog graph paper is available commercially, and most include at least three cycles.

Histograms

A histogram is a graph of the frequency distribution of a continuous variable, based on class intervals. It uses adjoining columns to represent the number of observations for each class interval in the distribution. The area of each column is proportional to the number of observations in that interval. Figures 4.7a and 4.7b show two versions of a histogram of frequency distributions with equal class intervals. Since all class intervals are equal in this histogram, the eight of each column is in proportion to the number of observations it depicts.

Figures 4.7a, 4.7b, and 4.7c are examples of a particular type of histogram that is commonly used in field epidemiology — the epidemic curve. An epidemic curve is a histogram that displays the number of cases of disease during an outbreak or epidemic by times of onset. The y-axis represents the number of cases; the x-axis represents date and/or time of onset of illness. Figure 4.7a is a perfectly acceptable epidemic curve, but some epidemiologists prefer drawing the histogram as stacks of squares, with each square representing one case (Figure 4.7b). Additional information may be added to the histogram. The rendition of the epidemic curve shown in Figure 4.7c shades the individual boxes in each time period to denote which cases have been confirmed with culture results. Other information such as gender or presence of a related risk factor could be portrayed in this fashion.

Conventionally, the numbers on the x-axis are centered between the tick marks of the appropriate interval. The interval of time should be appropriate for the disease in question, the duration of the outbreak, and the purpose of the graph. If the purpose is to show the temporal relationship between time of exposure and onset of disease, then a widely accepted rule of thumb is to use intervals approximately one-fourth (or between one-eighth and one-third) of the incubation period of the disease shown. The incubation period for salmonellosis is usually 12–36 hours, so the x-axis of this epidemic curve has 12-hour intervals.

The most common choice for the x-axis variable in field epidemiology is calendar time, as shown in Figures 4.7a–c. However, age, cholesterol level or another continuous-scale variable may be used on the x-axis of an epidemic curve.

In Figure 4.8, which shows a frequency distribution of adults with diagnosed diabetes in the United States, the x-axis displays a measure of body mass — weight (in kilograms) divided by height (in meters) squared. The choice of variable for the x-axis of an epidemic curve is clearly dependent on the point of the display. Figures 4.7a, 4.7b, or 4.7c are constructed to show the natural course of the epidemic over time; Figure 4.8 conveys the burden of the problem of overweight and obesity.

Six bars are captioned from under-weight to extreme obese. Percentages of population decrease after the overweight category to the extremely obese.

The component of most interest should always be put at the bottom because the upper component usually has a jagged baseline that may make comparison difficult. Consider the data on pneumoconiosis in Figure 4.9a. The graph clearly displays a gradual decline in deaths from all pneumoconiosis between 1972 and 1999. It appears that deaths from asbestosis (top subgroup in Figure 4.9a) went against the overall trend, by increasing over the same period. However, Figure 4.9b makes this point more clearly by placing asbestosis along the baseline.

This graph is similar to the one above except the variables in the stacks are arranged in a different order. It dramatizes the increase of asbestosis deaths over time.

Some histograms, particularly those that are drawn as stacks of squares, include a box that indicates how many cases are represented by each square. While a square usually represents one case in a relatively small outbreak, a square may represent five or ten cases in a relatively large outbreak.

Population pyramid

A population pyramid displays the count or percentage of a population by age and sex. It does so by using two histograms — most often one for females and one for males, each by age group — turned sideways so the bars are horizontal, and placed base to base (Figures 4.10 and 4.11). Notice the overall pyramidal shape of the population distribution of a developing country with many births, relatively high infant mortality, and relatively low life expectancy (Figure 4.10). Compare that with the shape of the population distribution of a more developed country with fewer births, lower infant mortality, and higher life expectancy (Figure 4.11).

While population pyramids are used most often to display the distribution of a national population, they can also be used to display other data such as disease or a health characteristic by age and sex. For example, smoking prevalence by age and sex is shown in Figure 4.12. This pyramid clearly shows that, at every age, females are less likely to be current smokers than males.

Frequency polygons

A frequency polygon, like a histogram, is the graph of a frequency distribution. In a frequency polygon, the number of observations within an interval is marked with a single point placed at the midpoint of the interval. Each point is then connected to the next with a straight line. Figure 4.13 shows an example of a frequency polygon over the outline of a histogram for the same data. This graph makes it easy to identify the peak of the epidemic (4 weeks).

A frequency polygon contains the same area under the line as does a histogram of the same data. Indeed, the data that were displayed as a histogram in Figure 4.9a are displayed as a frequency polygon in Figure 4.14.

A frequency polygon differs from an arithmetic-scale line graph in several ways. A frequency polygon (or histogram) is used to display the entire frequency distribution (counts) of a continuous variable. An arithmetic-scale line graph is used to plot a series of observed data points (counts or rates), usually over time. A frequency polygon must be closed at both ends because the area under the curve is representative of the data; an arithmetic-scale line graph simply plots the data points. Compare the pneumoconiosis mortality data displayed as a frequency polygon in Figure 4.14 and as a line graph in Figure 4.15.

Cumulative frequency and survival curves

As its name implies, a cumulative frequency curve plots the cumulative frequency rather than the actual frequency distribution of a variable. This type of graph is useful for identifying medians, quartiles, and other percentiles. The x-axis records the class intervals, while the y-axis shows the cumulative frequency either on an absolute scale (e.g., number of cases) or, more commonly, as percentages from 0% to 100%. The median (50% or half-way point) can be found by drawing a horizontal line from the 50% tick mark on the y-axis to the cumulative frequency curve, then drawing a vertical line from that spot down to the x-axis. Figure 4.16 is a cumulative frequency graph showing the number of days until smallpox vaccination scab separation among persons who had never received smallpox vaccination previously (primary vaccinees) and among persons who had been previously vaccinated (revaccinees). The median number of days until scab separation was 19 days among revaccinees, and 22 days among primary vaccinees.

A survival curve can be used with follow-up studies to display the proportion of one or more groups still alive at different time periods. Similar to the axes of the cumulative frequency curve, the x-axis records the time periods, and the y-axis shows percentages, from 0% to 100%, still alive.

The most striking difference is in the plotted curves themselves. While a cumulative frequency starts at zero in the lower left corner of the graph and approaches 100% in the upper right corner, a survival curve begins at 100% in the upper left corner and proceeds toward the lower right corner as members of the group die. The survival curve in Figure 4.17 shows the difference in survival in the early 1900s, mid-1900s, and late 1900s. The survival curve for 1900–1902 shows a rapid decline in survival during the first few years of life, followed by a relatively steady decline. In contrast, the curve for 1949–1951 is shifted right, showing substantially better survival among the young. The curve for 1997 shows improved survival among the older population.

Note that the smallpox scab separation data plotted as a cumulative frequency graph in Figure 4.16 can be plotted as a smallpox scab survival curve, as shown in Figure 4.18.

References (This Section)

12. Schmid CF, Schmid SE. Handbook of graphic presentation. New York: John Wiley & Sons, 1954.
13. Cleveland WS. The elements of graphing data. Summit, NJ: Hobart Press, 1994.
14. Brookmeyer R, Curriero FC. Survival curve estimation with partial non-random exposure information. Statistics in Medicine 2002;21:2671–83.