Ratios can be used to depict the value of one variable divided by the value of another.  A proportion, often expressed as a percentage, is a kind of ratio that can be used to represent the value of a single variable for one class divided by the value for all classes combined.

Whenever multiple ratios are involved—either across many individuals in a group or over numerous days of intake for each individual—analysts can use different ways to summarize them, and these different calculations can lead to different answers.  This is because the calculations involve both summation and division, and an elementary principle of mathematics dictates that the order of these operations matters. The mathematical properties of ratios are the same, whether one is considering simple ratios, proportions, or percentages.

In survey analyses involving multiple dietary recalls per person, consideration of which kind of summary ratio to use must be made at both the group and individual levels.

#### Group-level Ratios

At the group level, two different, but equally correct, answers can be given in response to the question “What proportion of the calcium that is consumed comes from milk?”  This is because the question can have two different meanings:

• How much of all the calcium consumed by the group comes from milk?” (Ratio of Means) or
• What is the group’s daily contribution of milk to calcium intake?”  (Mean Ratio)

Whenever ratios involve the division of one variable by another—both of them, by definition, free to vary—these two ratios can be different from one another.

Ratio of Means

The ratio of means is used to answer questions such as, “How much of all the calcium consumed by the group comes from milk?” It is calculated by summing the amount of calcium from milk for all persons and then dividing that by the sum of the calcium from all foods for all persons.   The answer would be the same if both the numerator and denominator were divided by a constant, such as the sample size.   Therefore, it can also be calculated by dividing the group’s mean amount of calcium from milk by the group’s mean total calcium, and for this reason it can be thought of as a ratio of means.

The ratio of means yields information about the diet of the population as a whole because both the numerator and the denominator are computed for the whole population before the ratio is derived.  That is, the whole population has only one aggregate value and the distribution of the ratio among members of the population is not available.  However, the ratio of means can be obtained for various subgroups in the population, if comparisons are warranted.  The ratio of means has been employed to identify important sources of nutrients in the US diet as a whole and to examine diet quality using to the Healthy Eating Index-2005.

Mean Ratio

The mean ratio is used to answer questions such as, “What is the group’s daily contribution of milk to calcium intake?”  It is determined by first calculating the proportion of calcium from milk for each person and then taking an arithmetic mean of all the proportions.  Often, the mean ratio is similar to the ratio of means; however, sometimes they are quite different, depending on the variability in the ratio, variation in the denominator, and the correlation between the ratio and the denominator.

The mean ratio requires that a ratio be calculated for each person before averaging .  When the ratio itself varies among the population, its distribution can be examined, and the ratio can be studied in relation to other variables.  Also, the distribution of ratios provides other summary statistics, such as the median, the ratio at other percentiles, and the proportion of the population above or below a certain cut-off, in addition to the mean ratio.  Such statistics have been used in tracking progress toward meeting national health objectives.

When the intent is to say something about how the intake varies among the population, or how the ratio relates to other factors, deriving the ratio for each person before summarizing (as with the mean ratio) is the method of choice.  However, it should be remembered that the generalizability of such an approach is subject to whatever time period limitations the individual ratios impose.  For example, if the individual ratios each represent only a single day, then the mean ratio can only be used to make inferences “for a given day,” and relating a single day’s ratio to some other factor is rarely of interest.

Means should be examined along with their standard errors to get an indication of the variation about the mean.  Special statistical procedures are required to get appropriate standard errors when using data from a complex sample such as the NHANES.  In addition, appropriate sample weights should be applied, if the data are being used to represent the population as a whole.  See “Module 13: Estimate Variance, Analyze Subgroups, and Calculate Degrees of Freedom” for further information.

#### Individual-Level Ratios

If the mean ratio is being used at the group level, then an individual-level ratio is needed for each person.  If you are using only one observation per person—such as a single 24-hour recall—then there is only one value for the numerator and one for the denominator and, therefore, only one way to derive the individual-level ratio.  If data were available for each person’s intake on every day over an extended period, then the individual’s daily ratios would need to be summarized.

As with group-level ratios, two different questions could be posed: “How much of all the calcium consumed by this person, over time, has come from milk?” or “What is the person’s daily contribution of milk to calcium intake?”  And again, because the ratios would involve the division of one variable by another, these two ratios could be different from one another.  Although long-term intake observations are not available in NHANES, available data can be modeled to represent usual intake and, in that case, decisions about which individual-level ratio to use must be made.  That topic will be covered in the Advanced Dietary Analysis course.  The example of the single day’s ratio for each person will be used in the next section, to demonstrate the difference between the group-level ratio of means and mean ratio.

IMPORTANT NOTE

Summary ratios can be calculated in multiple ways, at both the group and individual levels, depending on the question of interest. At the group level, the ratio of means is the simplest calculation, and if it is used, individual-level ratios do not need to be calculated. If the mean ratio is used, then an individual-level ratio is needed. If single day ratios are being used, it should be noted that the mean ratio represents “a given day” rather than usual intake. If usual intake estimates are being modeled, then the individual ratio can be either the ratio of usual intake or the usual ratio of intake (see Advanced Dietary Analysis Course).