Analysis of ratios of means

A procedure is presented for the analysis of the ratio of two means obtained from independent samples. The use and interpretation of this ratio is discussed along with formulas for calculating confidence intervals and significance tests. Comparison is made between this ratio approach and the more commonly used mean differences (subtraction) technique.


Introduction:

Testing the equality of two means obtained from independent samples usually involves subtracting one mean from the other to obtain the difference between them. Dividing the difference by its standard error allows reference to a t-distribution to evaluate the significance of that difference. More generally, a confidence interval can be placed around the obtained difference value. If this interval, constructed to be wide enough to contain the population difference with a certain level of confidence, includes zero, the conclusion of no significant difference is suggested. (More accurately the means may be considered significantly different but with less confidence than that used to form the interval.)

As an alternative to this testing procedure the ratio of the two means could be estimated. Differences found by subtraction measure the inequality of two means in an absolute sense, i.e ., the number of units apart ("units" being expressed in the same metric as the variable on which the means were calculated). However, the ratio measures a relative difference: the magnitude of one mean relative to or as a proportion of another. Under certain circumstances calculation of the ratio and associated confidence intervals can lead to informative alternate interpretations of data.

Consider the following situation: A survey was conducted among households to determine preference for a new formulation of grape drink. (More extensive details of the survey are given later in this paper.) Analysis of the survey results showed the average household consumption of the new formulation to be 16.28 cans over an eight-week period. Average consumption of a control formulation (the product currently on the market) was 14.89 cans over the same eightweek period.

Assume the researcher analyzing these data is interested in making a comparison between the new formulation and the control product. The most common comparison approach involves subtracting one average from the other: 16.28 – 14.89. The researcher can claim that users of the new formulation consumed 1.4 more cans than users of the control product. A test of the statistical significance of this difference could be computed along with calculating a confidence interval around this difference. In fact, the 95% confidence bound would be from .51 cans to 2.27 cans.

However, other researchers might go one step further with these data by considering the ratio of the two averages: 16.28/14.89 = 1.093. Consumers of the new formulation drank, on average, 9.3% more juice than their control counter parts. Further, these researchers could assume that a simple statistical relationship exists between the subtraction and ratio approach. They would then calculate confidence bounds around the value of 9.3% using the confidence interval information from the subtraction analysis. These bounds would be (.51/14.89) to (2.27/14.89) or 3.4% to 15.24%.

Unfortunately, the relationship between the subtraction and ratio approach is not that simple. Although the ratio bounds given above may at times be close to those obtained from the appropriate formula (given later), the above methods will not, in general, yield correct bounds.

The purpose of this paper is to consider the viability of using ratios of means as an effective alternative to subtraction. Along the way, it will be shown under what conditions the ratio can be used and how associated statistical considerations, like confidence intervals, can be handled correctly.


Scale Considerations:

Ratios of means can have clear meaning and interpretation only when estimated from variables measured on a ratio scale. Ratio scales are continuous scales with an absolute zero point (a score or value of zero indicates the complete absence of the quality), and where a unit change has a constant meaning over the entire range of the scale. Quantities or amounts of things are examples of ratio scales: income, age, number of beers consumed in a week. Each has an absolute zero point, e.g.. zero income means no earned money. All are continuous, e.g., a person or object can be any age from zero to infinity and, depending on the fineness of measurement, can take any value in between. Lastly, a unit change has constant meaning. Defining a unit as one can of beer the difference in the quantity between two and three cans is equal to the difference in the quantity between sixteen and seventeen cans (although perhaps not to the consumer). Only under these conditions will the mathematical operation of division yield meaningful results and will the ratio of two means be of value.

Note: Attitude rating scales and the like do not possess an absolute zero point. The scale is arbitrarily coded and fixed. Adding and subtracting constants from these scales will not affect mean differences but will affect ratios. Further, a unit change typically does not have constant meaning over the entire range of the scale.


Estimation of the Ratio:

The ratio of two means can be calculated by dividing the larger mean by the smaller. Denoting the ratio as R:

where the subscripts "L" and "S" indicate the larger and smaller means, respectively. (The "^" over R indicates that the ratio is a sample estimate of the population ratio.) Placing the larger mean in the numerator is a convention taken here to facilitate notation and interpretation. In general, where each mean is placed in the ratio is up to the researcher and can be considered arbitrary. It has no effect on statistical techniques involving the ratio. However, when one group's mean is to serve as a control, to which many other means are to be compared, placing that mean in the denominator will make comparison across groups easier. All group means would then be expressed relative to the control.

The value of can be interpreted as a proportion: one mean is larger, relative to another, by a certain proportion. Further, equality of means would lead to an value of one. From a conceptual point of view, statistically determining the equality of means is tantamount to testing whether the obtained ratio of the means differs significantly from one. (This parallels the practice of testing differences between means through subtraction. In this instance, a difference of zero is the statistical focal point.)


Calculating Confidence Intervals:

Although it is possible to test the equality of means expressed as a ratio, it is more informative to calculate the ratio along with a confidence interval.*

To obtain the values of this interval, statistical theory shows that the roots of a complex quadratic equation must be solved for. (Sources and references for the quadratic equations and the formulas to follow can be supplied upon request.) However, a simpler approach can be taken by approximation by first calculating the value C:

The value of t in the above formula, which is squared and shown as t², is taken from a t-table corresponding to the amount of confidence desired and with (nL + nS - 2) degrees of freedom. (nL and ns are sample sizes for the groups with largest and smallest means, respectively.) is the variance estimate associated with the group having the smaller mean.

The value of C itself is difficult to interpret and at this point should simply be viewed as the means to the end of calculating confidence bounds. (The denominator of C is roughly akin to the value of the lower bound of a confidence interval drawn around . It could then be roughly interpreted as the ratio of to its lower bound. The larger C is for a given value of the further the lower bound is from . More details will be given later.)

Once C is obtained the confidence interval is then calculated as:


Note: C is included on the left side of the above equation as a correction factor. The bounds obtained from this formula are, in fact, those for but are drawn around . The resulting interval clearly is symmetric is symmetric around and non-symmetric for when C is not equal to one.

*See Addendum on significance testing.


Example:

As mentioned in the introduction, an experiment was performed to assess preference for a new formulation of grape drink. Two panels of one-hundred households each were constructed, balanced and blocked (see Research on Research Report #13) on demographic and previous drink consumption characteristics. (Households were randomly assigned to panels within blocks.) Each household received a small initial stock of several cans of one of two formulations. Panel A received the new formulation; Panel B received the standard control formulation. All cans were purchased by the household at a reduced price.

The measurement of interest in this experiment was total household consumption over a period of eight weeks. The comparison of consumption between panels would lead to a reasonably effective measure of preference as price (although slightly below market value), consumption patterns and other relevant factors were experimentally controlled.

The statistics in Table One summarize the findings of the experiment. All numbers below are expressed in the metric of number of cans.


Table One
Panels consist of one-hundred households

  Mean Std Dev*
Panel A (new formulation) 16.28 3.1862
Panel B (control) 14.89 3.1862


Expressing the inequality of the two means in Table One as a ratio yields a value of 16.28 / 14.89 = 1.093. Panel A households consumed, on average, 9.3% more grape drink than Panel B households over the eightweek period. To construct confidence intervals around the ratio estimate of 1.093, C is calculated first:


* The standard deviation are equal because they are from a pooled estimate of variability obtained from an analysis of variance performed on these data. The analysis of variance allows the complexities and subtleties of the experiment (balancing and blocking) to be taken into account. As such, the precision of the experiment is enhanced. The formulas given above are still applicable.)

** A t value of 1.96 is used in the above calculation. This sets the bounds of the interval wide enough to encompass the population ratio value with 95% confidence with 198 degrees of freedom.

or from 1.033 to 1.157. To complete the interpretation, consumption of the new formulation appears to be between 3% and 15.7% greater than consumption of the original formulation. The significance of the difference between the means, with at least 95% confidence, also is indicated since the confidence interval does not include the value one, which represents equality of the means.

Comparison with ( L - s):

The estimation and interpretation of ratios yields alternative or complementary views of data. Side-by-side comparison makes this clear. The ratio of the means gives a value of 1.093; 9.3% greater consumption of the new formulation. Expressed through subtraction, Panel A households consumed, on average, about (16.28-14.89)= 1.4 more cans of grape drink than Panel B households over the eight-week period. Algebraically, the relationship between the ratio result and subtraction result is:

As Such, the ratio can be viewed as a simple re-expression of mean differences.

The bounds around the ratio, can be compared to those obtained by subtraction. Bounds for an interval around with a predetermined level of confidence can be defined as:

where t is taken from a t-distribution with (nL+ nS - 2) degrees of freedom for a given level of confidence. Using 1.96 to define an interval with 95% confidence, the bounds are:

or from .51 to 2.27. Consumption of the new formulation is between half a can to two and a quarter cans greater than consumption of the original formulation. (Again, the significance of the difference between the means can be noted, with at least 95% confidence, since the confidence interval does not include the value zero which represents equallty.)

Although a simple algebraic relationship exists between the ratio results and the subtraction results there is, in general, no simple relationship between their confidence bounds. However, in certain circumstances knowledge of one set of bounds can lead to an approximation of the other. This approximation. which shows the similarity of information obtained from both approaches, is a function of the size of C.

For small C values, less than 1.2, the approximation is obtained as follows, using the data from Table One: dividing the lower bound value of the subtraction interval by the larger mean, .51/16.28, yields a value which when added to one approximates the lower bound of the ratio interval: 1.031. Dividing the upper bound value of the subtraction interval by the smaller mean, 2.27/14.89, yields a value which when added to one approximates the upper bound of the ratio interval: 1.152. As C increases in size the approximations become quite poor.

The bounds found by use of this approximation can be compared to those calculated in the preface. Although the values are close, the great differences in the logic of the calculations suggests this closeness will not always occur. This is especially true when values of C become larger. As such, the method mentioned in the preface should not be used.

Note: The size of C is directly influenced by the precision with which the means are measured. Examining the denominator of C:

shows it to be of a form analagous to that used in calculating the lower bound of the confidence interval around the mean . Values of C will become larger as this denominator becomes smaller or as the lower bound of this interval gets further from the mean. For a given value of , C changes as elements of the term

change. In developing any confidence interval, the distance from the lower bound to the statistic on which it is based is a function of the variability of that statistic ( ), the size of the sample used (ns), and the degree of confidence desired (t²). Variability and sample size directly affect precision. That is, as variability is reduced and/or the sample size is increased precision improves. As precision improves the distance from the lower bound to the statistic gets smaller; and the lower bound value are more alike. Referring back to C, as precision improves C gets smaller. In general, for reasonably large samples (two-hundred or more), C should be small enough so that the approximations cited above will hold.


Conclusion:

The ratio of two means has been suggested as an alternative to examining differences among means. When the measurement scale is of a ratio nature, the mean ratio and its confidence interval can yield useful information and complement statistics more routinely estimated, such as mean differences obtained by subtraction.


Addendum: Statistical Tests

Two approaches now appear viable for testing the equality of the means obtained from independent samples: ratio or subtraction. However, it can be shown that for all practical intents the same statistical test can be used for both for assessing the equality of means. The appropriate statistical test is the t-test. Although the t-test differs in its general form for ratios and differences by subtraction, they are equivalent when testing the hypothesis of equality of means.

To test the hypothesis of zero difference, the test of equality by subtraction, the t-statistic is, in its general form:


represents the hypothesized difference between the population means. The hypothesis of equal means assumes this value is zero. (" ") and "n" represent the variance and sample size, respectively, associated with each mean.

To test the hypothesis of a ratio value of one the general form of the t-statistic is:


Under the hypothesis of equality, as expressed by an R value of one, this t-statistic is the same as that for the difference (XL - Xs) when = 0. As such, under the hypothesis of equality of means, the t-tests for the ratio of means and differences between means are identical.