Ординатура / Офтальмология / Английские материалы / Principles Of Medical Statistics_Feinstein_2002

TABLE 10.1

Different Ratios Associated with Same Increment and Different Increments Associated with Same Ratio

Conversely, as shown in the last four rows of Table 10.1, a ratio of 4 might seem substantial if it represents an increment from 10% to 40% for therapeutic success, but not if the increase is from 1% to 4%. Similarly, if the ratio of 4 represents the increased chance of death associated with a “risk factor,” an increment of 45 per thousand (representing death rates of .015 and .060) would be much more impressive than an increment of 6 per 10 million (from .000002 to .000008).

For these reasons, the quantitative distinction of a ratio can seldom be interpreted without simultaneous knowledge of the increment, and vice versa. The need to consider two indexes, rather than one, is another reason — beyond the problem of context — why scientists have been reluctant to set specific boundaries (analogous to “P < .05”) for quantitative significance. What would be required is a demarcated zone that includes simultaneous levels of values for a ratio and an increment. The boundary chosen for this zone is not likely to evoke prompt consensus; and furthermore, particularly when the data under contrast are epidemiologic rather than clinical, substantial disagreements may arise between one evaluator and another.

10.4 Distinctions in Dimensional vs. Binary Data

Two mortality rates, .21 vs. .12, can be immediately compared because the units of expression are immediately evident. They are 21 per hundred and 12 per hundred. Two medians or means, 102.3 vs. 98.4, cannot be promptly compared, however, until we know the units of measurement. If they are degrees Fahrenheit, the two central indexes would become 39.1 vs. 36.9 when expressed in degrees Celsius.

Another distinction in dimensional vs. binary data is the connotation of stability for a central index. For dimensional data, the coefficient of stability, (s/n )/ X , will have the same value for the same set of data regardless of units of measurement. In the foregoing example, (s/n )/ X would be the same for each group whether the temperature is measured as Fahrenheit or Celsius. For the proportions (or rates) that summarize binary data, however, the coefficient of stability and the connotation of a contrast will vary according to whether the results are expressed with p or with q = 1 − p. Thus, the increment of

.09 in the foregoing mortality rates will seem more impressive if referred to a “control” mortality rate of .21 than to the equally correct but complementary survival rate of .79.

Binary proportions also have a unique attribute, discussed later in this chapter. They can be compared with a special index, called an odds ratio, which cannot be determined for the medians or means of dimensional data.

For all these reasons, the indexes of contrast are usually expressed (and interpreted) in different ways for dimensional data and for binary data.

10.5 Indexes for Contrasting Dimensional Data

Central indexes for two groups of dimensional data are seldom compared as direct ratios; but direct increments may be difficult to interpret because of the inconsistent effects of units of measurement.

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Consequently, the direct increments are often converted into special expressions that are unit-free. According to the mode of construction, the unit-free expressions are called standardized, proportionate, or common proportionate increments.

10.5.1Standardized Increment

The standardized increment is often used to compare two groups in the psychosocial sciences but is undeservedly neglected in medical publications. Sometimes called the effect size,2 the standardized increment is a type of Z-score formed when the direct increment in the means of two groups, XA and

XB , is divided by their common standard deviation, sp, to produce

XA – XB SI = -------------------

sp

The standa rdized increment is particularly valuable because it is a unit-free expression and it also provides for variance in the data. It therefore becomes, as noted later, a direct connection between a descriptive contrast and the value of t or Z used in stochastic testing.

10.5.1.1 Common Standard Deviation — Calculating a standardized increment requires use of the common standard deviation that will be discussed in greater detail later when we consider stochastic tests. The symbol sp is used, however, because the common standard deviation is found by pooling the group variances of each group. Their sum is

have already been found as sA and sB for each group, the pooled group variance can be calculated as SX XA + SX XB = (nA − 1) s2A + (nB − 1) s2B .

For example, suppose group A has nA = 6, XA = 25.6, and sA = 7, whereas group B has nB = 5, XB

25 .6 – 11.3

--------------------------- = 2.44 5.86

To avoid calculating a common standard deviation, some investigators use the standard deviation of the “control” group rather than sp for the denominator. If sB is the standard deviation of the control group and SI represents standardized increment, the formula becomes

SI = ( XA − XB )/sB

10.5.1.2 Applications of Standardized Increment — As a unit-free ratio that is the counterpart of a Z-score, the standardized increment is an excellent method of comparing two means. Thus, if you recall the distribution of Z-scores from Chapter 6, the value of 2.44 in the previous subsection is relatively large, suggesting an impressive increment. In the social sciences, the standardized increment is frequently used as a conventional index of effect size for comparing results in any two groups.2 The standardized increment has also been used in combining results of different studies in a meta-analysis. 3

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Unfortunately, because the magnitude of descriptive contrasts receives so little attention in medical literature, the standardized increment seldom appears in current publications. The infrequency of its usage thus gives medical investigators little or no experience with which to develop an “intuitive” feeling about boundaries (discussed later) for impressive or unimpressive magnitudes.

10.5.2Conversion to Complex Increment

If both a ratio and increment are to be considered, their combination might allow two groups to be contrasted with a single complex increment. The two combinations that can be constructed are the proportionate increment and the common proportionate increment. They differ in the choice of a denominator for the comparison.

10.5.2.1 Proportionate (Percentage) Increment — The proportionate or percentage increment shows the relative incremental change from a baseline magnitude. For two groups A and B, the calculation is either (A − B)/B or (B − A)/A; and the resulting proportion is customarily multiplied by 100 and expressed as a percentage. [This type of reference to a baseline was used earlier when the standard deviation, as the average change from the mean of a distribution, was divided by the mean to form the coefficient of variation.]

For a contrast of the two means, 11.3 and 25.6, the proportionate increment would be either (11.3 − 25.6)/25.6 = −56% or (25.6 − 11.3)/11.3 = 127%.

If B is clearly the reference value, it is the obvious choice for denominator of the proportionate increment, but if C and D are both “active” treatments, the same problem arises that was noted for the ratio at the end of Section 10.3.2. If the central indexes are 24.8 for Treatment C and 12.7 for Treatment D, the results can be reported proportionately as a rise of 95% [= (24.8 − 12.7)/12.7] from Treatment D, or a fall of 49% [= (12.7 − 24.8)/24.8] from Treatment C. The investigator can legitimately present whichever value seems more “supportive,” and the reader will have to beware that alternative presentations were possible.

10.5.2.2 Redundancy of Simple Ratio and Proportionate Increment — D e s p i t e i t s possible value for certain forms of “intuitive” understanding, the proportionate increment is really a redundant expression, because it can always be converted to the simple ratio. Algebraically, the propor - tionate increment, (A − B )/B = (A/B) − 1, is simply the ratio (A/B) minus 1. Analogously, (B − A)/A = (B/A) − 1. Consequently, the ratio and proportionate increment do not provide distinctively different items of information. A simple ratio can immediately be converted to a proportionate increment by subtracting 1 and then, if desired, multiplying by 100 to form a percentage. Conversely, a proportionate (or percentage) increment can immediately be converted to a ratio after division by 100 and the addition of 1. Thus, a proportionate increment of −15% becomes a ratio of .85.

On the other hand, the direct increment and the standardized increment are unique expressions. They cannot be determined from either the simple ratio or the proportionate increment.

10.5.2.3 Common Proportionate Increment — The problem of choosing a reference denominator can be avoided if the increment of A and B is divided by the mean of the two values. If group sizes are unequal, with nA + nB = N, the common mean will be (nAA + nBB)/N; and the common proportionate increment will be N(A − B)/(nAA + nBB). For the two groups in Section 10.5.1.1, the common mean is [(6)(25.6) + (5)(11.3)]/11 = 19.1, and the common proportionate increment is (25.6 − 11.3)/19.1 = .75. If the group sizes are equal, the mean of the two values will be (A + B)/2, and the common proportionate increment will be (A − B)/[(A + B)/2].

If A is larger than B, the simple ratio will be r = (A/B). Replacing A by rB, the common proportionate increment (for equal group sizes, where nA = nB = N/2) will become 2(r – 1)/(r + 1). It will thus include contributions from the increment and the ratio. If the two means, 25.6 and 11.3, came from groups of equal size, the common proportionate increment would be (25.6 − 11.3)/[(25.6 + 11.3)/2] = 14.3/18.45 = .78. Alternatively, because r = 25.6/11.3 = 2.265, we could calculate (2.265 − 1)/[(2.265 + 1)/2] = 1.265/1.632 = .78.

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A major advantage of the common (or any) proportionate increment is that it is unit-free. If the central indexes are 24 months and 12 months in two equal-sized groups of children, the direct increment is 12 months. If age were expressed in years, however, the direct increment would be 1 year. The proportionate increment in the two groups, however, would be 100% [ = (24 – 12)/12], and the common proportionate increment would be 67% [= (24 – 12)/18], regardless of whether age is measured in months or years.

The main problem with the common proportionate increment is that it is seldom used. Consequently, it is unfamiliar and has no immediately intuitive meaning. The substantive experience that provides “intuition” for most biomedical scientists has usually come from direct increments, simple ratios, and ordinary proportionate increments.

10.6 Indexes for Contrasting Two Proportions

If the compared items are means or medians, only the two individual central indexes are available to form a direct index of contrast. If the two compared items are proportions or rates, however, nine numbers are available. When expressed in the form of p = a/n, any proportion (or rate) can be decomposed into three parts: n, the number of people in the denominator; a, the number of people in the category counted for the numerator; and (n − a), the number of people who are not in that category. When suitably arranged for each group and for the total, the three component parts produce nine numbers.

For example, suppose we want to contrast the two proportions, 33% and 50%, representing the respective rates of success with Treatments A and B. Each proportion comes from a numerator and denominator for which the underlying data might be 6/18 vs. 10/20. These data can be tabulated as the nine numbers shown in Table 10.2.

TABLE 10.2

Table with Nine Numbers Derived from the Four Numbers of Two Proportions, 6/18 vs. 10/20.

10.6.1General Symbols

In general symbols, data showing binary proportions for two groups can be expressed as p1 = a/n1 and p2 = c/n2. These proportions can be arranged to form the classical 2 × 2 or “fourfold” arrangement shown in Table 10.3. The additional five numbers that become available in Table 10.3 are b = n1 − a, d = n2 − c, and the totals a + c, b + d, and N = n1 + n2. Furthermore, in addition to the two stated proportions, p1 and p2, the two complementary proportions, q1 and q2, are also available for comparison. For example, if the stated proportion is a success rate of 33%, its complementary proportion is a failure rate of 67%. Thus, if p1 = a/n1, q1 = (n1 − a)/n1.

TABLE 10.3

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The two proportions, p1 and p2, could be contrasted in the usual two approaches as a direct increment (p2 − p1) or as a simple ratio (p2/p1). We could also, however, contrast the complementary proportions q1 − q2 and q1/q2.

10.6.2Increments

If the index of contrast is an increment, no problems are produced by comparing the complementary proportions, because q2 − q1 = 1 − p2 − (1 − p1) = p1 − p2. The increment for p2 − p1 will therefore be identical but opposite in sign to the increment q2 − q1. For example, with success rates of 55% and 33%, the increment is 22%; with the corresponding failure rates of 45% and 67%, the increment is −22%.

The increment in two proportions is usually expressed directly but can also be cited as a standardized increment.

10.6.2.1Direct Increment — Since p1 − p2 and q1 − q2 will have equal but opposite values, negative signs can be avoided by citing the absolute direct increment as |p1 − p2| . The positive result will represent subtraction of the smaller from the larger value of the pair.

The direct increment is clear and never deceptive, but some people have difficulty interpreting increments cited in decimal proportions such as .17 or .0064.

10.6.2.2Number Needed for One Extra Effect — To ease the interpretation of decimal proportions, Laupacis et al.4 proposed that the increment be inverted into a reciprocal

1/|p1 − p2|

which they called NNT: the number needed to be treated.

For example, if the success rate is .61 for Treatment A and .44 for placebo, the incremental advantage for Treatment A is .61 − .44. = .17, which is 17 persons per hundred. The reciprocal, which is 1/.17 = 5.9, indicates that about six persons must receive Treatment A in order for one to have an incremental benefit that exceeds what would be produced by placebo. If the increment is .27 − .25 = .02, the value of NNT is 1/.02 = 50; and 50 persons must be treated for one to be incrementally benefited.

The result of the NNT is simple, direct, easy to understand, and applicable to many comparisons of proportions or rates. The nomenclature is somewhat confusing, however, because the compared entities are not always the effects of treatment. For example, suppose Disease D occurs at rate .01 in persons exposed to a particular “risk factor” and .0036 in persons who are not exposed. The incremental difference is .0064, and its reciprocal, 1/.0064 = 156, is an excellent way of expressing the contrast. Nevertheless, the result is not a number needed to be treated and is sometimes called NNH — the number needed to harm. To avoid several abbreviations and labels for the same basic idea, the Laupacis index can be called NNE, the number needed for one extra effect — a term applicable for comparing treatments, exposures, or other agents.

The NNE is a particularly valuable and effective way to express the apparent benefit of the therapeutic agents tested in clinical trials, but the expression is not enthusiastically accepted by public-health epidemiologists. One reason, noted later in Chapter 17, is that the available epidemiologic data may not contain the occurrence rates needed to construct and invert the increment. Another reason, discussed later in this chapter, is the very small proportions that become cited as rates of “risk” for etiologic agents. For example, the occurrence rates of disease D may be .005 in persons exposed to a “risk factor” and .001 in persons who are nonexposed. The ratio of .005/.001 is an impressive value of 5, but the direct increment of .004 is relatively unimpressive. Since the NNE of 1/.004 = 250 indicates that 250 persons must be exposed for one to become incrementally diseased, public-health campaigns and other efforts to impugn a “risk factor” are more likely to have an impact if the results are expressed in a “scary” ratio of 5 than in a less frightening NNE of 250.

After pointing out that “perceptions change when data are presented in a way that relates therapeutic efforts to clinical yields,” Naylor et al.5 suggest that “NNTs and similar measures relating efforts to yields dampen enthusiasm for drug therapy by offering a more clinically meaningful view of treatment

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effects.” For an additional innovative expression, the Naylor group proposes the use of TNT, which is “tons needed to treat.” According to Naylor et al., about “10.3 metric tons of cholestyramine were consumed for each definite coronary death prevented in the Lipid Research Clinics Coronary Primary Prevention Trial,” and “about 200,000 doses of gemfibrozil were ingested in the Helsinki Heart Study per fatal or non-fatal myocardial infarction prevented.”

10.6.2.3 Standardized Increment — If two proportions, p1 and p2, are expressed as a standardized increment, the common standard deviation is much simpler to determine than for two means. We first find the common proportion, which is P = (n1p1 + n2p2)/N. [If the groups have equal size, so that n1 = n2, P = (p1 + p2 )/2.] The variance of the common proportion is PQ, where Q = 1 − P. The common standard deviation is then PQ . The standardized increment becomes

p1 – p2

---------------

PQ

Because both P and Q appear in the denominator, the standardized increment will produce the same

absolute magnitude, whether the contrast is between p1 − p2 or q1 − q2.

For smaller proportions, such as the previous comparison of .01 vs. .0036, the common proportion (with equal group sizes) would be P = .0068, with Q = .9932. The standardized increment would be (.01 − .0036)/(.0068)(.9932 ) = .078.

The main problem with the standardized increment is that using PQ in the denominator converts

the result into the counterpart of a ratio. If P and Q are near the meridian value of .5, the value for PQ

Consequently, a tiny incremental change of .001 from .0005 to .0015, in two groups of equal size, will have P = .001 [= (.0015 + .0005)/2] and a standardized increment of .001/.032 = .03, which inflates the actual value of the increment. The NNE of 1000 [= 1/.001] will give a much better idea of the real comparison than the standardized increment.

10.6.3Simple and Complex Ratios

Whenever one proportion is divided by another to form a ratio, the results can be correct but ambiguous or deceptive. Some of the problems, which have already been cited, will be briefly reviewed before we turn to a new complex index, the odds ratio.

10.6.3.1 Ambiguity in Simple Ratios — The simple ratio of two proportions, p1 and p2 , can be an ambiguous index because it can also be formed from the complementary proportions, q1 and q2 , and because p2/p1 does not necessarily equal q2/q1, which is (1 − p2)/(1 − p1). Thus, a success ratio of

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58%/33% = 1.76 (and its reciprocal 1/1.76 = .57) are not the same as the corresponding failure ratio of 42%/67% = .063 (and its reciprocal of 1.60).

10.6.3.2Ambiguity in Proportionate Increments — The proportionate increment for two

proportions, p1 and p2, is a ratio that has two ambiguities. The first occurs when the numerator |p 1 − p2| is divided by a denominator that can be chosen from four possibilities: p1, p2, q1, and q2. The second ambiguity is in the nomenclature that expresses the result. For the two success percentages, 49% and 33%, the proportionate increment can be (49 − 33)/33 = 48% or (33 − 49)/49 = −33%. Expressed for rates of failure, rather than success, the corresponding results would be (51 − 67)/67 = −24% and (67 – 51)/51 = 31%. Aside from ambiguity in numerical constituents, the expressions themselves are potentially confusing, because two percentages are contrasted, but the contrast is expressed with a quite different percentage. Thus, 49% is 16% higher than 33% as a direct increment, but is 48% higher as a proportionate increment.

10.6.3.3Possible Deception in Proportionate Increments — Because investigators will always want to put a “best foot forward” in reporting results and because any of the cited statistical comparisons are correct and “legitimate,” the proportionate increment is frequently (and sometimes deceptively) used to “magnify” the importance of a relatively small direct increment. For example, the mortality rate in a randomized trial may be 21% with placebo and 15% with active treatment. When the results are reported in prominent medical journals (and later in the public media), however, the investi-

gators seldom emphasize the direct increment of 6%. Instead, they usually say that the active treatment lowered the mortality rate by 29% [= (.21 − .15)/.21].

Furthermore, if we examine survival rather than death, the proportionate improvement in survival rates will be (q2 − q1)/q1 = (.85 − .79)/.79 = .06/.79 = .076 = 7.6%. Thus, the same treatment can paradoxically seem to produce an impressive reduction in mortality while simultaneously having relatively little effect

on survival.

The problem is heightened when values of p 1 and p2 are particularly small. For example, suppose the occurrence rate of Disease D is .005 after exposure to Agent X and .001 without exposure. The actual increment is .005 − .001 = .004 and the NNE is 1/.004 = 250. The investigators will seldom use these

unimpressive values, however, for citing their main results. Instead, the indexes of contrast will emphasize the direct ratio of 5, or the proportionate increment of .004/.001, which is a 400% rise from the rate in the unexposed group.

10.6.3.4Relative Difference — Another available but seldom used complex ratio for contrast-

ing two proportions is the relative difference proposed by Sheps.6 If pA is the success rate for treatment A, and pB is the success rate for treatment B, the proportion 1 − pB indicates the “failures” with B who are available to become successes with A. The incremental accomplishment of A can then be propor-

tionately expressed as the relative difference

pA – pB

----------------

1 – pB

For example, if the success rates are pA = .53 and pB = .43, the proportion of failures in B is .57. The relative difference is .10/.57 = .175. In public-health risk results, such as pA = .005 and pB = .001, the relative difference would be .004/.999 = .004.

The relative difference is seldom used in public health statistics because 1 − pB is often close to 1, so that the result is almost the same as the direct increment. For the larger rates that occur in clinical trials, investigators usually prefer to use the direct increment (which would be .10 for the comparison of .53 vs. .43), the NNT (which would be 10), or the proportionate increment (which would be .10/.43 = .23).

For small rates of success increments in clinical trials, the relative difference might be preferred scientifically as more overtly “honest” than the proportionate increment. For example, if the mortality rate is .02 with Treatment A and .04 with Treatment B, the relative difference is (.02 − .04)/.96 = −.021

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or −2.1%. As a proportionate increment, however, the reduction is a much more “impressive” value of (.02 − .04)/.04 = −.50 or −50%.

10.6.4Problems of Interpretation

Because indexes of contrast receive so little formal attention in statistical education, it is not surprising that physicians have great difficulty interpreting published results. In a series of special studies,7–9 physicians were frequently deceived when the same set of results was compared in different ways. Treatment A was regularly preferred over Treatment B when the comparison was reported with a ratio or proportionate increment, but not when the same result was cited with a direct increment or as a number needed to treat.

What is more surprising, perhaps, is that the authors of medical publications are allowed “to get away with” misleading (although accurate) presentations of ratios and proportionate increments, without adequately emphasizing the alternative methods of citing and interpreting the same results. When historians of science review some of the medical and public-health errors of the second half of the 20th century, an interesting point of evaluation will be the vigilance (or lack thereof) with which editors and reviewers guarded the integrity of published literature and carried out the responsibility to prevent even inadvertent deception.

The problem will probably vanish or be greatly reduced in the 21st century when “statistical significance” is decomposed into its component parts, and when quantitative contrasts begin to receive the prime attention now diverted to stochastic inference.10–12

10.7 The Odds Ratio

The odds ratio gets special attention here because it is unique to binary data and is commonly used for contrasting two proportions.

10.7.1Expression of Odds and Odds Ratio

The idea of odds is a classical concept in probability. If the probability that an event will happen is p, the probability that it will not happen is q. The odds of its occurrence are p/q. For example, in a toss of a coin, the chance of getting heads is 1/2 and the chance of tails is 1/2. The odds of getting heads (or tails) are therefore (1/2)/(1/2) = 1 — an “even money” bet. If the chance of failure with Treatment A is 12/18, the chance of success is 6/18, and the odds in favor of failure are (12/18)/(6/18) = (12/6) = 2, or 2 to 1.

Note the way in which the odds were constructed. If a + b = n, the odds in favor of b are (b/n)/(a/n) = b/a. The two probabilities, b/n and a/n, have n as a common denominator, which cancels out to allow the odds to be calculated simply as b/a.

With this tactic in mind, we can now re-examine the results for each treatment in the previous Table 10.2. The odds of success are 6/12 = .5 (or 1 to 2) for Treatment A and 10/10 = 1 (or 1 to 1) for Treatment B. These two sets of odds produce an odds ratio of (6/12)/(10/10) = .5/1 = .5, which expresses the relative odds of success for Treatment A vs. Treatment B.

We also could have formed the odds ratio in another way by asking: what are the odds that a successful patient received Treatment A vs. the odds that a failure received Treatment A? This odds ratio would be expressed as (6/10)/(12/10) = .5, and the result would be the same as before.

10.7.2 Odds Ratio as a Cross-Product Ratio

ratio (a/b)/(c/d) or (a/c)/(b/d). With either algebraic arrangement, the calculation becomes ad/bc. Because the two opposing diagonal terms are multiplied and the two pairs of products are then divided, the odds ratio is sometimes called the cross-product ratio.

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10.7.3Attractions of Odds Ratio

The odds ratio has the statistical appeal of being “independent” of the group sizes expressed in the marginal totals of a 2 × 2 table. As long as the inner four components of the table remain proportionally similar, within each column (or row), the odds ratio will remain the same even if the marginal totals are substantially changed.

For example, suppose the question examined in Table 10.2 was answered from a case-control study, with 48 cases drawn from the “success” group and 110 controls drawn from the “failure” group. If

no longer be calculated from this table, because the persons were “sampled” from the outcome events, rather than from the “exposure“ to treatment. [If calculated, the “success rates” would be .23 (= 18/78) for Treatment A and .375 (= 30/80) for Treatment B, and would be much lower than the correct corresponding values of .33 and .50.] The odds ratio, however, calculated as (18 × 50)/(30 × 60) = .5 in the new table, will be identical to the .5 result obtained in Table 10.2.

This virtue of the odds ratio also leads to its two main epidemiologic attractions. One of them occurs in a multivariable analytic procedure called multiple logistic regression, where the coefficients that emerge for each variable can be interpreted as odds ratios. The most common appeal of the odds ratio, however, is that its result in a simple, inexpensive epidemiologic case-control study is often used as a surrogate for the risk ratio that would be obtained in a complex, expensive cohort study.

10.7.4Approximation of Risk Ratio

The basic algebra that is briefly outlined here will be described and illustrated in greater detail in Chapter 17. Suppose a particular disease occurs at a very low rate, such as p2 = . 0001, in people who have not been exposed to a particular risk factor. Suppose the disease occurs at a higher but still low rate, such as p1 = .0004, in people who have been exposed. The simple ratio, p1/p2, is called the risk ratio. In this instance, it would be .0004/.0001 = 4.

To determine this risk ratio in a cohort study of exposed and nonexposed people, we would have to assemble and follow thousands of persons for a protracted period of time and determine which ones do or do not develop the selected disease. Instead, with an epidemiologic case-control study, we begin at the end. We assemble a group of “cases,” who already have the disease, and a group of “controls,” who do not. We then ask each group about their antecedent exposure to the risk factor. The results are then manipulated as discussed in the next few subsections.

In a cohort study, where we begin with exposed and nonexposed groups, they become the denominator for any calculations of rates of occurrence. Thus, a + b = n1 is the denominator of exposed persons and c + d = n2 is the denominator of unexposed persons.

In a case-control study, however, we assemble diseased and nondiseased groups. They consist of a + c = f1 for the total of diseased cases, and b + d = f2 for nondiseased controls. The values of a + b and c + d can be calculated, but they are not appropriate denominators for determining rates of disease occurrence. If any “rates” are to be calculated, they would have to be a/f1 for rate (or proportion) of exposure in the cases and b/f2 for the corresponding rate in the controls.

10.7.4.2 Advantages of Case-Control Odds Ratio — In a cohort study, the rate of occurrence for the disease in the exposed group would be p1 = a/(a + b) = a/n1. The corresponding rate in the unexposed group would be p2 = c/(c + d) = c/n2. The risk ratio would be p1/p2 = (a/n1)/(c/n2).

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In a case-control study, however, values of n1 and n2 in the exposed and nonexposed groups are not available to allow these calculations of p1 and p2 . Nevertheless, if the cases and controls suitably represent the corresponding members of the cohort population, the odds of exposure should be the same in the two tables. Thus, the odds of a/c in the case group and b/d in the control group should have the same values as their counterpart odds in the cohort. The odds ratio, which is (a/c)/(b/d) = ad/bc, should be the same in both types of research.

In the cohort study, the odds ratio would be (p 1/p2)/(q1/q2) = (p1/p2)[(1 − p2)/(1 − p1)]. It is the desired risk ratio, p1/p2, multiplied by q 2/q1 = (1 − p2)/(l − p1). If the occurrence rates of disease are quite small, however, both 1 − p2 and 1 − p1 will be close to 1. The factor (1 − p2)/(1 − p1) will then be approximately 1. Thus, in the foregoing example, p1 = .004 and p2 = .001. Because 1 − p1 = .996 and 1 − p2 = . 999, the factor .999/.996 = 1.003. If we assume that this factor is essentially 1, the odds ratio will be (p1/p2 ) × ( l) and will approximate the risk ratio.13

A more formal algebraic proof of these relationships is provided in the Appendix to this chapter, and the topic is further discussed in Chapter 17. The main point to be noted now is that if the basic rate of disease is small and if the groups in the fourfold case-control table suitably represent their counterparts in the fourfold cohort-study table, the odds ratio in a relatively small case-control study should approx - imate the risk ratio produced by a much larger and elaborate cohort study.

This mathematical attribute of the odds ratio had made it (and case-control studies) so irresistibly appealing in epidemiologic research that the investigators regularly overlook the scientific criteria needed to accept the odds ratio as a credible scientific substitute for the risk ratio.14

10.8 Proportionate Reduction in System Variance (PRSV)

A descriptive index that is well known to statisticians, but not to most other readers, is the proportionate reduction that occurs in the total variance of a system when it is fitted with a “model.” This index, which is particularly useful for describing the associations considered later in the text, is valuable now because it can be applied to contrasts of both dimensional and binary data, where the two compared groups form the “model.”

For example, consider the two groups of binary data compared in Table 10.2. In the total data, before division into two groups, the system variance, using the formula NPQ, is (16/38) (22/38)(38) = 9.26. After the two-group split, the sum of group variances, using the formula n1p1q1 + n2p2q2, becomes (6/18)(12/18)(18) + (10/20)(10/20)(20) = 9.00. The proportionate reduction in system variance is (9.26 − 9.00)/9.26 = .028.

With dimensional data, as shown in the example in Section 10.5.1.1, the determination of system variance requires first that the grand mean of the system be identified as G = (nA XA + nB XB )/N. In the

cited example, G = [(6)(25.6) + 5(11.3)]/11 = 19.1. Thus, the system variance will be Σ X2A + Σ X2B − NG2 , where Σ X2A = SX XA + nA X2A and Σ X2B = SX XB + nB XB2 . For the cited data, Σ X2A = (5)(7)2 + 6(25.6)2 = 4177.16 and Σ X2B = 4(4)2 + 5(11.3)2 = 702.45. Thus, the system variance is 4177.16 + 702.45 −

11(19.1)2 = 866.7. The sum of the two group variances is SX XA +SX XB = 309, so that the proportionate reduction in system variance is (866.7 − 309)/866.7 = .64.

The proportionate reduction in system variance (PRSV) is seldom considered in ordinary descriptive comparisons of two groups. It has three major advantages, however:

1. With some complex algebra not shown here, it can be demonstrated that the square of the standardized increment (or “effect size”) — discussed in Sections 10.5.1 and 10.6.2.3 — is roughly 4 times the magnitude of the proportionate reduction in system variance, particularly if the two group sizes are reasonably large and relatively equal. Thus, in many situations, (SI)2 4(PRSV) and so SI 2PRSV . For example, for the two proportions 10/20 and 6/18, which have modestly sized groups, PSRV = .028 and .028 = .167 which is about half of the SI (.35) shown in Section 10.6.2.3. (For the very small groups of dimensional data in Section 10.5.1.1, the SI of 2.44 is about three times the PRSV = .64 = .8.)*

* The actual formula, for two groups of dimensional data each having size n, is SI = 2( P RS V n – 1 )/[n(l – PRSV)]. For two groups of binary data, the formula is SI = 2 P RS V .

© 2002 by Chapman & Hall/CRC