Ординатура / Офтальмология / Английские материалы / Principles Of Medical Statistics_Feinstein_2002

9.5.4Hyperpropped Proportions

An opposite counterpart of unpropped proportions (in Section 9.2.3) is the hyperpropped proportions that occur when the same denominator is needlessly repeated in a tabulation of relative frequencies for categorical (nominal or ordinal) data.

For example, the relative frequencies of the nominal data in Section 9.1.1.1, can be shown in two ways as follows:

The right-hand arrangement avoids the hyperpropping produced by the needless repetition of the “208” denominator in each row.

9.6Displays of Ordinal Data

A central index for ordinal data can be expressed as a median or with the same plurality or compressed proportions used for nominal data. Sometimes an investigator will cite a mean value, using the ordinal

codes of 1, 2, 3, 4, … as though they were dimensional data; and sometimes, particularly in clinical trials of analgesic agents, the ordinal pain scales are even expressed in standard deviations. A box plot could be used because it relies on quantiles that can be legitimately ranked, but the plot may not be particularly informative if the categorical grades have many frequencies that will be “tied” for the percentile values.

The usual display of ordinal data, therefore, is a bar chart or graph, as shown in Figure 9.12. Bar graphs can also be converted to dot charts, however, if desired for clarity. On the other hand, because the proportions of each category can easily be shown and understood in a simple one-way table, the virtues of a more formal visual display are often dubious.

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9.7Overlapping Multi-Binary Categories

A group of people may have a series of binary attributes, such as fever, chest pain, and sore throat, that are present or absent in different combinations. The proportionate total occurrence can easily be cited for each binary category; and a dot chart is the best way to display the individual proportions, which cannot be shown in a pie graph because they overlap, with a sum that exceeds 100%.

The investigator may also, however, want to display the spectrum of overlapping categories. Venn diagrams are excellent for this purpose. For example, Figure 9.13 shows the overlapping spectrum of different forms of new carditis in 54 recurrent episodes of rheumatic fever.8

FIGURE 9.13

Incidence and types of different manifestations of new carditis in 54 recurrent episodes of rheumatic fever. [Figure taken from Chapter Reference 8.]

Sometimes the investigators (and artists) may get carried away to produce the exotic multi-binary portrait shown in Figure 9.14. For each of 232 “common congenital malformations,”9 17 lesions were identified as present or absent. A line was drawn between each two lesions present in the same patient. The lines get thicker as they are augmented for each identified pair. The resulting picture is probably more remarkable for its artistic imagery than for its statistical communication.

9.8Gaussian Verbal Transformations

The fame of the Gaussian distribution has evoked efforts to display it in diverse ways. Among the most striking are “verbal transformations.”

The first, shown in Figure 9.15, was prepared by the statisticican, W. J. Youden.10 The second, shown in Figure 9.16, was done by John Hollander, a poet who is A. Bartlett Giamatti Professor of English at Yale University. Figure 9.16, which is taken from Hollander’s book11 on “Types of Shape,” shows his poem for the “bell-shaped” Gaussian curve, together with additional exposition. If any of your molecular colleagues uses a new term you do not understand, you can now respond by asking what they know about technopaignia, calligrammes, or a carmina figurata. [You can act particularly superior if your colleagues say that these things are, respectively, an adverse reaction to a new imaging test, the weight gain avoided with non-metabolized sweeteners, or a stand-in for the female lead of an opera by Bizet.]

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FINGER

KIDNEY

EYE

DATEP

SUA

O B D

HIP

EXOS

EAR

FIGURE 9.14

Interrelations of congential defects in 232 cases. HSPD: hypospadias; ANEN: anencephaly; EXOS; exomphalos; SP BIF: spina bifida; SUA: single umbilical artery; IMP AN: imperforate anus; OEATF: oesophageal atresia/tracheal fistula; DH: diaphragmatic hernia; OBD: other brain defects; etc. [Figure derived from Chapter Reference 9.]

FIGURE 9.15

W. J. Youden’s verbal display of the Gaussian distribution. [Figure derived from Chapter Reference 10.]

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Part II

Comparing Two Groups of Data

For almost a century, two groups of data have been compared with various “tests of statistical significance.” The purpose of this part of the text is to describe those tests, but first to clarify and perhaps eliminate the confusion caused by the ambiguous meaning of statistical significance.

The confusion arises because the phrase statistical significance is usually applied to only one of the two evaluations needed for a numerical comparison. The first comparative evaluation refers to the quantitative magnitude of the observed distinction. Is a mean of 8.2 impressively larger than a mean of 6.7? Is a proportion of 25% substantially smaller than a proportion of 40%? This type of quantitative comparison evaluates what was descriptively observed in the data.

The second comparative evaluation looks at the stability of the central indexes. Are the means of 8.2 and 6.7 stable enough to be accepted as adequately representing their groups? Are the two means so fragile that their confidence intervals would have a large overlap? Do the proportions of 25% and 40% come from unstable numerical components, such as 1/4 vs. 2/5? These questions refer not to quantitative magnitudes, but to stability of the central indexes, and the comparison of stability involves stochastic inferences about what might happen if the indexes were perturbed or otherwise rearranged.

Of the two types of comparison, the quantitative decision requires substantive thought about the magnitudes of observed distinctions, whereas the inferential decision usually requires mathematical thought about stochastic probabilities and possibilities. Nevertheless, the inferential activities have been the main focus of attention during the statistical developments of the past century, and almost no emphasis has been given to intellectual mechanisms or criteria for the descriptive contrasts. With this monolithic concept, “statistical significance” has been restricted to the inferential procedures, and the descriptive evaluation has no name and no operational principles.

In statistical inference, the stability of two central indexes is usually contrasted with the t tests, chi-square tests, and other procedures that have become famous in decisions about “statistical significance.” The quantitative contrast of magnitudes, however, is a different and even more fundamental statistical activity, but it is descriptive, not inferential. Lacking the mathematical panache of inference, the descriptive contrasts are hardly discussed in most textbooks and other accounts of statistical reasoning. Descriptive decisions about quantitative significance, however, are constantly made by investigators, reviewers, readers, and policy makers. As a former medical student, Gertrude Stein, once said, “A difference must make a difference to be a difference.” A quantitative comparison may not be worthy of serious attention if the component numbers are too unstable to be statistically significant, but the real significance or importance of a comparative difference depends on the magnitude of what it shows and implies, not on its P value, confidence interval, or other calculation in a mathematical test of inference.

Several other crucial issues are also involved in making decisions about significance. As noted in Chapter 1, these issues are substantive—not an inherent part of the statistical reasoning. They refer to the architectural structure of the comparison and the quality of the raw data and processed data. Although paramount constituents of credibility for the data and the comparison, these substantive scientific issues are beyond the scope of discussion in the next eight chapters, which are concerned with exclusively statistical aspects of comparison.

The first topic will cover the methods of evaluating a quantitative contrast. The subsequent chapters will discuss the principles of forming and testing stochastic hypotheses, the different procedures used for testing different types of data, the mechanisms for displaying two-group contrasts, and the special epidemiologic tactics applied for comparing rates and proportions.

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10

Quantitative Contrasts:

The Magnitude of Distinctions

CONTENTS

10.1Nomenclature for Quantitative and Stochastic Significance

10.2Context of Quantitative Decisions

10.3Simple Indexes of Quantitative Contrast

10.3.1Methods of Simple Expression

10.3.2Orientation of Comparison

10.3.3Customary Scientific Evaluations

10.3.4Role of Two Indexes

10.4Distinctions in Dimensional vs. Binary Data

10.5Indexes for Contrasting Dimensional Data

10.5.1Standardized Increment

10.5.2Conversion to Complex Increment

10.6Indexes for Contrasting Two Proportions

10.6.1General Symbols

10.6.2Increments

10.6.3Simple and Complex Ratios

10.6.4Problems of Interpretation

10.7The Odds Ratio

10.7.1Expression of Odds and Odds Ratio

10.7.2Odds Ratio as a Cross-Product Ratio

10.7.3Attractions of Odds Ratio

10.7.4Approximation of Risk Ratio

10.8Proportionate Reduction in System Variance (PRSV)

10.9Standards for Quantitative Significance

10.9.1Reasons for Absence of Standards

10.9.2Demand for Boundaries

10.9.3Development of Standards

10.10Pragmatic Criteria

10.11Contrasts of Ordinal and Nominal Data

10.12Individual Transitions

10.13Challenges for the Future References

Appendix for Chapter 10

A.10.1 Algebraic Demonstration of Similarity for Cohort Risk Ratio

and Case-Control Odds Ratio

Exercises

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How would you decide that the difference between the two means of 8.2 and 6.4 is “impressive” or “unimpressive”? How would you make the same decision if asked to compare the two proportions, .25 and .40?

One manifestation of the customary inattention to quantitative contrasts is the uncertainty or even discomfort you might feel if asked these questions. What things would you look at or think about? What statistical indexes might you use to do the looking or thinking? For example, suppose you agree that the distinction is quantitatively impressive between the two means 11.3 and 25.6, but not between the two means 18.7 and 19.2. What particular statistical entities did you examine to make your decision? If you were reluctant to decide without getting some additional information, what kind of information did you want?

10.1 Nomenclature for Quantitative and Stochastic Significance

Decisions about quantitative magnitude and numerical stability are different statistical activities, but they are not clearly separated in traditional nomenclature. Both decisions involve thinking about significance for the observed distinction, and two different labels are needed for these two different ideas, but the commonly used term statistical significance is usually applied only to the appraisal of stability. The quantitative decision has no definite name.

In customary usage, statistical significance implies that the distinction is numerically stable and is not likely to arise from the random effects of chance variation. When someone wants to refer to an impressive magnitude, however, a specific title is not available for the quantitative distinction. It is sometimes given such labels as clinical significance, biologic impressiveness, or substantive importance. Lacking specific titles for the different decisions that distinguish quantitative contrasts from numerical stabilities, the two ideas become obfuscated and confused when the term statistical significance is restricted to only one of them. To escape the profound problems of this unsatisfactory nomenclature, the term statistical significance will generally be avoided hereafter in this text.

The term quantitative significance will refer to the magnitude of an impressive or important quantitative distinction. The term stochastic significance will denote the role of chance probabilities in the numerical stability of the observed distinction. [The word stochastic, which is easier to say than the alternative word, probabilistic, refers to random processes and comes from the Greek stochastikos, which means “proceeding by guesswork or conjecture in aiming at a target.” Besides, not all stochastic appraisals involve a consideration of probabilities.]

The rest of this chapter is devoted to principles of quantitative significance in a contrast of two groups. The stochastic strategies for evaluating a two-group contrast are discussed in Chapters 11–16.

10.2 Context of Quantitative Decisions

Quantitative decisions always have two components. One of them is the magnitude of the observed distinction, such as the contrast of 11.3 vs. 25.6, or 19.2 vs. 18.7. The other component is the clinical, biologic, or other substantive context of the comparison. This context is responsible for such labels as clinical significance, biologic impressiveness, or substantive importance to describe a sufficiently large distinction.

For example, suppose .06 and .09 are the proportions of young patients who develop an adverse event after each of two treatments. If the adverse event is a transient, symptomless skin rash, the quantitative distinction will seem unimpressive. If the adverse event is permanent sterility, however, the same distinction will be important. In a different context, suppose the rate of occurrence for a rare but dreadful disease is .000004 after exposure to Agent A and .000003 in the unexposed general population. The incremental difference of .000001 in the two rates, or the ratio of .000004/.000003 = 1.33, may not seem impressive; but a nation that contains 50,000,000 people, half of whom are exposed to the agent, will have 25 (= 100 − 75) extra cases of the disease. As another example, an incremental rise of .0025 or

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0.25% may seem like a trivial change in a proportion, but the change can have a major impact on the stock market if it represents an elevation in the Federal Reserve Board’s prime lending rate.

Another issue in quantitative comparison is the variability of the measurement process. Suppose a patient’s hematocrit rises from 10.2 to 10.4 after a blood transfusion, or suppose the diastolic blood pressure falls from 97 to 94 mm Hg after a particular treatment. Should these changes be ascribed to the effects of therapy, or could variations of 0.2 in hematocrit and 3 mm Hg in diastolic pressure occur simply as ordinary fluctuations in either the method of measurement or the patient’s biologic state?

For all these reasons, quantitative contrasts must always be suitably considered within the context in which they occur. These contexts can include the severity of the observed phenomenon, the method of measurement, the immediately associated clinical or biologic factors, and the external social, regional, or national implications. These contextual features are always a crucial component of the total judgment used for decisions about quantitative significance in groups of people. The context may be less crucial if the compared phenomena refer to events in rodents or in inanimate systems. Even for nonhuman phenomena, however, the decisions may also be affected by important contextual features. An improve - ment of success rate from 90% to 95% may not seem immediately impressive, but may make the world beat a path to your door if you have built a better mousetrap (or a better computer mouse).

Aside from context, however, the other vital component of quantitative decisions is the numerical magnitude itself. Although the final judgment about quantitative significance will come from simultaneous consideration of both the numerical and contextual components, the numerical magnitude alone may sometimes suffice. For example, if a particular agent is suspected of causing a significant change in the potassium or calcium concentration of a rat’s serum, we need not worry (at this point in the research) about the human clinical or populational implications of the change. We do need a way to decide, however, how much of a difference in the potassium or calcium level will be regarded as a “significant change.”

Thus, although quantitative decisions are sometimes made without a concomitant evaluation of context, they always require an appraisal of magnitudes. The rest of this chapter is concerned with the principles that can be used for those appraisals.

10.3 Simple Indexes of Quantitative Contrast

When two groups are descriptively compared, their central indexes are converted into indexes of quantitative contrast.

(The magnitudes found in these contrasts are sometimes called effect size, a term that has two unattractive features. First, the entities being compared may be baseline states, such as age or weight, rather than the “effects” of an intervention, such as treatment. Second, many statisticians reserve size for the number of members in a group, e.g., “sample size.”)

At least two simple indexes of contrast can be created immediately — a direct increment and a ratio — and the emerging results can have at least four different values.

10.3.1Methods of Simple Expression

The four values for the simple indexes of contrast can come from two methods of calculation, each performed in two possible ways.

10.3.1.1 Direct Increments — A direct increment is the subtracted difference between the two values under comparison. If A and B represent the central indexes for Groups A and B, the direct increment can be calculated as A – B, or as B – A. For the two sets of means, 11.3 vs. 25.6 and 19.2 vs. 18.7, the respective increments are 14.3 units (= 25.6 − 11.3) and 0.5 units (= 19.2 − 18.7). Because the order of subtraction could have been reversed, these same values could have been expressed alternatively as −14.3 and −0.5. If A – B yields a positive increment, the negative result of B − A could be called a

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decrement. To avoid two words for this difference, the term direct increment will be used hereafter to represent either the positive or negative result of a subtraction.

10.3.1.2 Ratios — If the two central indexes are divided as A/B or as B/A, the index of contrast is called a ratio. For the two pairs of means under discussion, the respective ratios would be 0.44 (= 11.3/25.6) and 1.03 (= 19.2/18.7) if calculated as A/B, and 2.27 and 0.97 if calculated as B/A.

10.3.2Orientation of Comparison

The four possible simple indexes of contrast for two central indexes, A and B, can promptly be reduced to two, if one of the two groups is used as a “reference” to orient the direction of the calculations. For example, if B represents the result of a control or placebo therapy, and if A represents an active treatment, the respective increment and ratio would be A − B and A/B. This same arrangement would also provide the desired indexes of contrast if A represents the observed value for a group and if B represents the expected value, or if A and B are the values after and before a particular treatment.

Because a reference group can be determined or inferred for almost any comparison of two groups, the decision about orientation is seldom difficult. If we assume that B is the basic or “control” group, B always becomes the subtractor or divisor. With this orientation, the choice of the appropriate simple index can be reduced to two candidates: A − B or A/B.

If a reference group is not obvious, as in a comparison of two standard active treatments, readers should remember that data analysts will usually choose a presentation that puts a “best foot forward” for the results. Thus, if the central indexes are 24.8 for Treatment C and 12.7 for Treatment D, the investigator may decide that a ratio of 1.95 for C/D is more “impressive” than the corresponding .51 for D/C. Both presentations are legitimate and honest, but their results seem quite different if inspected without further thought.

10.3.3Customary Scientific Evaluations

If you had your first exposure to science in activities of physics or chemistry, you may recall that quantitative importance was often shown by the slope of a line on a graph. The effect of substance X on substance Y usually appeared in a plot of points such as in Figure 10.1. The larger the slope of the line that fit those points, the greater is the impact of X on Y.

Y

If this principle is used to compare dimensional data for two groups, the two “one-way graphs” of points (recall Section 5.7.1) can be placed on a single graph if we regard the results as values of Y for an X-variable coded as 0 for group B and 1 for group A. Such a plot is shown in Figure 10.2. If a straight line is fitted to the two sets of points (by methods demonstrated later in Chapter 19), it will connect the two means, YA and YB . With the 0/1 codes used for the X variable, the value of XA − XB will be 1. The slope of the line will be ( YA − YB )/( XA − XB ) = ( YA − YB )/l. Thus, the increment of two means, YA − YB , corresponds to the slope of the line connecting the two groups of data.

The same principle holds true for a comparison of two proportions, pA and pB. Their increment corresponds to the slope of the line connecting them. The individual values of 0 and 1 that are the constituents

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of each proportion are shown as clusters of points in Figure 10.3. The proportions themselves are the “means” of each group, and the slope of the line that connects those means is the increment, pA − pB.

As the “natural” index of scientific comparison for two groups, the increment always takes precedence over the ratio, which is a mathematical artifact. The ratio was developed to help evaluate additional features of increments and is also popular in epidemiologic work (as discussed in Chapter 17), because the ratio allows two groups to be compared even when fundamental data are not available to determine the desired increment. Thus, if only a single index is to be used, it should be the increment, or some appropriately modified arrangement of the increment. Nevertheless, for an enlightened comparison, we usually examine the ratio as well as the increment.

10.3.4Role of Two Indexes

For a contrast to be evaluated effectively, both the increment and the ratio should usually be considered simultaneously. If only one is examined, the result can be misleading.

Table 10.1 contains a set of examples to illustrate this point. The increment of 10 units, shown in the first row, might seem substantial if the contrast is between values of 12 and 2, but trivial if the comparison (as in the second row) is between values of 3952 and 3942. The feature that made the same increment seem “substantial” in one instance and “trivial” in the other was the ratio. In the first case, the associated ratio was 6, and in the second case, 1.003. An increment of 10 derived from the values of 10.2 and .2 in the third row of Table 10.1 might be even more impressive, because the associated ratio is 51.

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