Ординатура / Офтальмология / Английские материалы / Principles Of Medical Statistics_Feinstein_2002

The value of e can be regarded as a sampling variation in which the sample nipi is taken from a “parent” population having the binomial parameter, niP. For this one-group “sample,” a Z-type ratio can be constructed as:

(observed value – parametric value)/variance of observation

The variance of the binomial “parametric” observation will be niPQ = ni(f1/N)(f2/N). Because of the finite “sampling,” however, Mantel–Haenszel believed that the variance should be adjusted by the finite population correction factor, (N − ni)/(N − 1), whose role was discussed in Section 7.6. Accordingly, the variance becomes calculated as [(N – ni)/(N − 1)][ni(f1/N)(f2/N)]. Some further algebra will then show that the variance of nipi − niP = e, in any cell of the 2 × 2 table, will be

The square of the critical Z-type ratio for a cell will be

2 n1 n2 (N – 1)(p1 – p2 )2 e /variance = ---------------------------------------------------

f1 f2

As noted in Chapter 14, p1 − p2 = (ad − bc)/n1n2. Thus, the squared critical ratio becomes

This result is identical to the customary formula for X2 except that Formula [26.3] uses N – 1 rather than N in the numerator. Thus, the square of the critical ratio for a cell in the 2 × 2 table produces essentially the same value as the X2 calculated for the entire table. [According to Mantel–Haenszel, the conventional X 2 formula uses (N − ni)/N, rather than (N − ni)/(N − 1), as the correction factor. The conventional arrangement leads to the customary chi-square formula, (ad − bc)2N/(n1n2f1f2).]

The focus of the adjusted chi-square calculation is usually the a cell, designated as a i in each stratified table. In the first step, the sum of numerators is obtained for the single values of (0 − E)2 in the ai cells. This sum is then divided by the sum of corresponding variances.

26.4.5Alternative Stochastic Approaches and Disputes

The strategy used to calculate the numerators and denominators, and sometimes the entire process, has evoked some of the same kinds of disputes (noted in Chapter 14) for the calculation of chi square in a 2 × 2 table. The disputes involve whether to use or omit Yales correction, whether the effects of the “sampling” should be analyzed with “fixed” or “random” models, and whether (or how) heterogeneous strata should be combined.

26.4.5.1 Determination of Numerators — The sum of the numerators can be determined in two ways. The obvious way is

[Σ (observed ai – exp ected ai )]2

but advocates of Yates correction prefer that it be subtracted from the sum of the absolute increments before they are squared. With the correction, the sum that forms the “adjusted” numerator would be

[(Σ observed ai – exp ected ai )–0.5]2

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The Yates correction was used in the original M-H paper,24 and is maintained in the M-H formulas cited by Breslow and Day,25 Fleiss,26 Kahn and Sempos,27 and Schlesselman,28 but not by Kleinbaum et al.29 or Rothman.30 A vigorous argument is offered in favor of the Yates correction by Breslow and Day and against it by Kleinbaum et al. In the absence of unanimity, probably the best approach is to calculate both ways. If the ultimate (P-value) results agree, there is no problem; if they do not, report both results and let the reader decide. If the adjusted X2 value that emerges is close to the boundary needed for α = .05, honesty in reporting should compel a citation of both sets of results.

26.4.5.2 Determination of Denominators — The denominators of the M-H chi-square can also be calculated in two ways, according to a different type of dispute about variances. In the easy approach, which accepts the Mantel–Haenszel concept, the denominator is determined simply as Σ (variance of ai ). For each stratum, with N members having the marginal totals of n1, n2 and f1, f2, the variance of the ai cell was shown in Formula [26.2]. With the Yates correction, the adjusted chi-square value will be

Without theYates correction, the 0.5 subtraction is omitted in the numerator term. The uncorrected value can be designated as X2M H .

An alternative approach, mentioned previously, uses a “random-effects model” rather than the “fixedeffects model” employed in the customary chi-square calculations. With the “fixed-effects” model, Mantel and Haenszel regard the observed data as the “available universe,” and use N − 1 in the calculations as a “finite population correction factor” for the “sampling process.” With the random-effects model, the strata are regarded as samples from a larger undefined universe, for which variances are calculated according to the DerSimonian–Laird method.31 (The D-L method produces larger variances, thereby reducing the eventual value of the combined X2 result and increasing the size of confidence intervals).

Most calculations in packaged computer programs and in published literature have been done with the fixed-effect M-H model, but the D-L model is becoming increasingly popular, perhaps because it reportedly offers a better management of heterogeneous results in the strata.

26.4.5.3 Example of Calculations — The first stratum (for dusty occupation) in Table 26.12 has 26 as the a cell. Its expected value is (31)(33)/41 = 24.95. The corresponding variance, according to Formula [26.2], is [(33)(8)(31)(10)]/[(41)2(40)] = 1.217. The second stratum, for non-dusty occupation, has 21 as the a cell. Its expected value is (37)(51)/110 = 17.15, and the corresponding variance is [(37)(73)(51)(59)]/[(110)2(109)] = 6.162. Substituting these values into Formula [26.4], we get

Without the Yates correction,

2 (4.9)2

XM H = ------------- = 3.25 7.379

At one degree of freedom, neither result exceeds the 3.84 boundary value of X2 needed for 2P < .05. Note that the adjusted M-H value of X2 exceeds the X2 values of 0.93 and 2.42 in the individual tables.

26.4.5.4 Use of Confidence Intervals — As confidence intervals have increasingly begun to replace P values, the combined results are often expressed with a confidence interval placed around the adjusted value of the odds ratio. For calculational convenience, the latter is regularly expressed as a natural logarithm, for which the standard error is used in the calculation.

Several methods, which were cited earlier (Section 17.5.5.2) for the log odds ratio in one 2 × 2 table, can be expanded to calculate the standard error of the adjusted combined ratio.

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26.4.6Caveats and Precautions

Although often proposed as a mechanism for dealing with “confounders,” the Mantel–Haenszel (or analogous forms of) adjustment can produce a false sense of security if the adjusted variables do not really “confound.” Three other problems which can arise are the results having opposite directions in the individual strata, the numbers in the partitioned strata becoming too small to be meaningful, or the strata being defined by a variable in which an ordinal trend is expected.

26.4.6.1Choice of “Confounders” — To be regarded as a confounder, a variable must exert a confounding effect. For example, if sex does not affect the development of omphalosis but is the only variable used for the Mantel–Haenszel stratification, the results of smoking and nonsmoking would not be adjusted for confounding.

In the example here, occupation was a “legitimate” confounder because it affected both the frequency of smoking and the frequency of omphalosis. In many instances, however, the variables used to “adjust for confounding” may include diverse attributes that affect the exposure or the outcome but not both. The problem is particularly common in multivariable analysis, as discussed elsewhere.10

26.4.6.2Problems of Heterogeneity — The results of any form of combined adjustment can be distorted if the component results, such as risk ratios, odds ratios, or gradients for target rates, go in opposite directions.

In the smoking–occupation–omphalosis example, both of the stratified odds ratios exceeded 1 and so the adjustment was justified. Consider the situation, however, for the case-control study shown in Table 26.13. When stratified, the crude odds ratio of 1.74 becomes an apparently protective value of 0.52 in women but rises to an impressive “causal” value of 11.8 in men.

TABLE 26.13

Hypothetical Case-Control Study

If these data receive a Mantel–Haenszel adjustment, the numerator for the odds ratio would be [(20)(60)/184] + [(36)(63)/137] = 23.08. The denominator would be [(32)(72)/184] + [(6)(32)/137] = 13.92. The Mantel–Haenszel adjusted odds ratio would be

23.08

OM H = ------------ = 1.66 13.92

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This result is consistent with what might have been expected judgmentally. It is lower than the original crude odds ratio of 1.74, and it lies between the odds ratios of 0.52 and 11.8 in the two strata. Nevertheless, a scientific “commonsense” evaluation of the stratified results would suggest that men seem to be at risk but women are not. Because the stratification led to this apparently valuable discovery, the opposing “risk” and “no risk” results should not be combined into a single adjustment, which will distort what is really happening.

Heterogeneity — a problem that plagues any efforts to combine a series of results into a single adjusted value — usually receives two approaches that are reasonable but incompatible. The connois - seurs of substantive issues usually want the results to be substantively “precise,” i.e., kept separate for such detailed distinctions as men vs. women, Stage I vs. Stage III cancer, severely ill vs. moderately ill patients. Connoisseurs of mathematical issues, who usually regard “precision” as an issue in statistical variance rather than biologic or clinical variation, want a combined single result that will have enhanced numerical size and that will avoid the need for inspecting and comparing results in multiple subgroups. In the mathematical approach, heterogeneity depends on the numerical values of the results; in the substantive approach, heterogeneity depends on the biologic (or other substantive) distinctions of the entities being compared. Thus, despite the obvious biologic heterogeneity, a small child, a large dog, and a huge fish might be mathematically regarded as homogeneous if they all weigh the same.

In the mathematical approach, various statistical tests are proposed for appraising heterogeneity of results in the strata that are candidates for combination. No guidelines are offered, however, for a level of heterogeneity that contra-indicates the combined “adjustment.” An advantage claimed for the DerSimonian–Laird random-effects strategy31 is that it improves the mathematical process when heterogeneous results are aggregated.

26.4.6.3 Effect of Small Numbers — When the relationship between clear-cell vaginal carci - noma and in-utero exposure to diethystilbestrol (DES) was explored with the case-control study cited in Section 17.5.4, the original results showed the following:

With 0.5 added to each cell, the odds ratio turned out to be (7.5)(32.5)/(1.5)(0.5) = 325.

From additional data reported by the investigators, however, two additional tables could be noted as follows for mothers who had had pregnancy problems (spontaneous abortions in previous pregnancies or bleeding, cramping, etc. early in current pregnancy):

The odds ratios in these two tables are (7 × 27)/(1 × 5) = 37.8 and (7.5 × 28.5)/(0.5 × 5.5) = 77.7, thus suggesting that pregnancy problems were a strong confounder of the observed relationship.

The results could be divided into the following stratification:

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Calculated with 0.5 added to each cell, the odds ratios are (7.5)(5.5)/(0.5)(0.5) = 165 in the first stratum and (0.5)(27.5)/(0.5)(1.5) = 18.3 in the second. The Mantel–Haenszel adjusted odds ratio is

This value is substantially lower than the unadjusted crude result of 325.

To calculate an M-H adjusted X2, the second stratum will yield a value of 0 for both the observed and expected values in the ai and bi cells. In the other two cells, the observed-minus-expected values will be 0. In the first stratum, the expected value in the a cell is (7 × 7)/12 = 4.08, and the observed- minus-expected value is (7 − 4.08) = 2.82.

The numerator of XM -HC might be calculated as [|2.82 | + | 0 | − 0.5]2 = 5.38, but the denominator is a problem. From Formula [26.2], the variance in the first stratum is (7)(5)(7)(5)/[(11)(12)2] = .773, but the variance in the second stratum will be (1)(27)(28)(0)/[(27)(28)2] = 0. If these two variances are added, the result for X2M Hc would be 5.38/.773 = 6.96. The result is still stochastically significant at 2P < .05, but the adjustment makes X2 much smaller than the original corrected X2c value, which would have been [|(7)(32) − (1)(0)| − (40/2)]2(40)/(8) (32)(33)(7) = 28.15.

On the other hand, a major mathematical assault occurs on scientific “common sense,” when odds ratios, “expected” values, and variances are calculated for 2 × 2 tables with two zero values in cells,

essentially from only the first stratum.

Although possibly justified by diverse forms of mathematical reasoning, the results are too scientifically peculiar to be acceptable. They suggest that the basic design of the study should have been improved. Cases and controls should have been originally matched on the basis of pregnancy problems, rather than leaving the latter variable to be “adjusted” during an analysis afterward.

26.4.6.4 Ordinal Trend in Strata — When stratified results are combined with any of the methods discussed so far, the particular order of the strata is not considered. The standardized or adjusted descriptive summaries, and the corresponding stochastic calculations, will be the same regardless of whether age is divided as young, medium, old or as old, young, medium.

In data for “dose-response curves,” however, we expect the trend in response to go progressively up or down, according to the trend in dose. For example, the odds ratios (or risk ratios) for success with active treatment vs. placebo might be expected to rise in strata with progressively increasing doses of the treatment. An analogous rise in occurrence rates of disease might be anticipated (and might help prove “causation”) with higher magnitudes of exposure to a suspected etiologic agent.

These trends cannot be evaluated with the described arrangements for standardizing or adjusting. A special method of weighting, discussed in Chapter 27, is needed to allow appraisal of “linear trend” in the “dose-response” curves of the ordinal strata.

26.4.7Extension to Cohort Data

Because cohort data can be expressed with rates and risk ratios, the cumbersome use of odds ratios is not necessary. Nevertheless, the M-H adjustment is sometimes applied to cohort data. The tactic is particularly justified when the cohort data contain many “censored” results that preclude a direct calculation and comparison of survival rates. The investigator may also want the stochastic comparison to reflect dynamic results for the entire survival curve rather than for a single time point alone.

Unlike odds-ratio data, for which the M-H adjustment provides both descriptive and stochastic indexes, the adjustment for cohort data is only stochastic and often applies to a comparison of two (or more) survival curves. The customary stochastic index is the log rank statistic, which was discussed in Chapter 22. Because this latter statistic has had so many versions, it is sometimes called the Cox-Mantel test or designated with other eponyms.

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26.5 Matching and “Matched Controls”

All of the stratifications and recombinations that have been discussed so far are but one way of dealing with confounders. Another approach, not discussed here, makes use of multivariable algebraic procedures. A third approach, discussed in the rest of this section, is the use of “matching.”

To match individual persons seems ideal for managing the problem of confounding. In fact, the use of cross-over trials, where the same person receives different treatments, is based on the appealing idea of letting the treated persons be their own “controls.” As noted earlier, however, the idyllic hopes of the cross-over design are destroyed by the reality that severely restricts the clinical circumstances amenable to a cross-over arrangement.

In observational studies, the ideal hope would be to find a “control” whose confounder attributes exactly match those of the compared person. This hope, however, is also regularly thwarted by reality. The main confounders may not be known, or the number of different confounders may be too extensive to permit suitable matchings. Accordingly, although matchings are commonly done, they are arranged mainly for investigative convenience, not for management of confounding. Disputes may then arise about the best way to analyze the “matched” results.

26.5.1Application in Case-Control Studies

In many case-control studies, the cases are chosen as an available group of people who all have the disease under consideration. The controls are selected from a group of people who are believed not to have the disease. Certain criteria may be applied to restrict eligibility, in either the case or control group or in both, to persons having a demarcated range of age, occupations, clinical conditions, or other possibly pertinent attributes.

The selection of a suitable control group is a complex topic about which universal agreement does not exist, and the topic has received abundant discussion elsewhere.1 The main point to be considered now is not the paramount importance of scientific propriety in the matching process, but the pragmatic method often used for choosing controls and for doing the subsequent statistical analysis.

The process sometimes involves getting 2 or more controls for each case, but the most common tactic (which is discussed here) involves a 1-to-1 matching arrangement.

26.5.2Choosing “Matched” Controls

In a case-control study, the investigator usually begins with a group of cases, found via some type of medical roster or registry of persons with the appropriate disease. The controls can then come from three general sources of presumably nondiseased persons: the same medical milieu (e.g., a hospital where the cases were identified); the same large cohort study from which the cases emerged; or the general community (e.g., neighbors, friends, persons found via “random-digit” telephone dialing).

Because age and sex (and sometimes race) are commonly applied for epidemiologic adjustment, these demographic variables are often used to “match” the controls chosen for the cases. The subsequent analysis is easier because the compared groups are demographically similar, but the matching process can also have a major chronologic virtue. If the two groups come from a common roster, such as a list of admissions to a hospital or to a cohort study, the controls can be matched for admission at about the same date as the cases. For example, if the cases have all been hospitalized, the controls can receive a time-age-sex matching if chosen from the chronologic roster of hospital admissions, going upward and downward from the admission date of each case to find the next ensuing or previously admitted person who is similar in sex and in age ±5 years.

26.5.3Formation of Tables

In a 1-to-1 matching, the investigator emerges with N case-control pairs, containing 2N persons. When previous exposure to the suspected etiologic agent is ascertained for each person, each

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case-control pair will have one of four possible results that can be arranged as follows for the N matched pairs:

Table 26.14 shows the agreement matrix for the N matched pairs, and Table 26.15 shows the “unmatched” arrangement for these data, presented in the more customary format of case-control tabulations. In the familiar unmatched arrangement, the odds ratio for the “risk” of exposure will be (n 1 × f2)/(f1 × n2) = [(a + b)(b + d)]/[(a + c)(c + d)]. If you work out all the algebra, this result eventually becomes

It can be compared with the odds ratio for the matched arrangement, which is calculated simply as b/c. The mathematical justification for the latter calculation is the topic for the rest of this discussion.

TABLE 26.14

Agreement Matrix for Matched Case-Control Pairs

TABLE 26.15

Unmatched Arrangement of Data in Table 26.14

26.5.4Justification of Matched Odds Ratio

If the unmatched odds ratio is 1, ad = bc; so ad − bc = 0, thus reducing expression [26.5] to b/c. For the matched and unmatched ratios to be equal, however, b/c must also equal 1. Therefore b = c, and ad

matched odds ratio is 1, and the process needs a mathematical justification that is provided by the Mantel–Haenszel procedure.

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26.5.4.1 Stratification of Matched Pairs — If we consider each of the N matched pairs as an individual stratum in an “unmatched” arrangement, each of the strata will have one of the four possible arrangements shown in Table 26.16. Because each pair contains 2 persons, the matched pairs in Table 26.14, when displayed in Table 26.16, will contain 2a persons in Arrangement I, 2b in Arrangement II, 2c in Arrangement III, and 2d in Arrangement IV.

TABLE 26.16

Arrangements of Strata of Matched Pair Results in Case-Control Study

26.5.4.2Formation of Adjusted Odds Ratio — For the M-H summary odds ratio, each stra-

tum adds its own “intrinsic” aidi/ni term to the adjusted numerator and its own bici/ni term to the adjusted denominator. In arrangements I, III, and IV of Table 26.16, the aidi terms will be 0. In arrangements I,

II, and IV, the bici terms will be 0. The only nonzero values will occur in arrangement II, where aidi/ni = (1)(1)/(1 + 1) = 1/2, and in arrangement III, where bici /ni = (1)(1)/(1 + 1) = 1/2. The number of strata

will be a for arrangement I, b for II, c for III, and d for IV. Because b strata will have aidi/ni values of 1/2, with all other strata having corresponding values of 0, Σ aidi/ni for the M-H adjusted numerator will

be b/2. For the M-H denominator, c strata will have bici/ni values of 1/2 and all other strata will have corresponding values of 0. Thus, Σ bici/ni for the M-H adjusted denominator will be c/2. The M-H adjusted odds ratio will then be (b/2)/(c/2) = b/c.

26.5.4.3Formation of Adjusted Variance — In arrangements I and IV of Table 26.16, the

variance will be 0, for reasons discussed earlier. In arrangements II and III, the variance will be nonzero

and identical. The four marginal totals in each arrangement will be 1, with ni = 2. Thus, according to Formula [26.2], the variance will be (1)(1)(1)(1)/[(22)(1)] = 1/4 for each arrangement of the b strata

(with arrangement II) and the c strata (with arrangement III). Thus, Σ (variance of a i ) will be b(1/4) + c(1/4) = (b + c)/4. The expected value in the ai cell of arrangements II and III will be 1/2[= (1)(1)/(2)]. The observed-expected value in the ai cell will be 1 – 1/2 = 1/2 in arrangement II and 0 − 1/2 in arrangement III. Thus, the value of Σ (observed − expected) will be b(1/2) + c(−1/2) = (b − c)/2.

26.5.5Calculation of “Matched” X2

When the (b − c)/2 value is squared for the uncorrected calculation, we get X2M-H = [(b – c)2 /4]/(b + c)/4 = (b − c)2/(b + c), which is the same result as the formula previously cited for the uncorrected McNemar chi-square test for a matched 2 × 2 table. Co nsequently, when uncorrected, the McNemar and the Mantel–Haenszel procedures give identical results (and thus help justify one another) for determining X2 in the matched analysis of a 2 × 2 table.

The 0.5 subtraction to “correct” the M-H results, however, becomes very tricky to apply to the matched arrangements II and III in Table 26.13, because the observed values are either 1 or 0 and the expected values are 0.5. Accordingly, if Yates correction is desired, the best stochastic approach for the matched 2 × 2 table is simply to calculate a corrected McNemar chi-square value as

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26.5.6Scientific Peculiarities

Despite the apparent mathematical merits, the procedure just described for getting a matched odds ratio has at least four scientific peculiarities:

26.5.6.1Nonconfounding “Confounders” — The Mantel–Haenszel adjustment is intended to deal with confounding variables, but the matching in most case-control studies is an act of demographic convenience, not a precaution against confounding. For reasons discussed elsewhere,1 age and sex (and perhaps race) are seldom the main confounding variables in case-control studies of etiology (or therapy).

26.5.6.2Neglect of Bulk of Data — In a study where most of the data are in the matched a and d cells of Table 26.14, it seems odd to place the main emphasis on the “disagreement” in the relatively

sparse b and c cells. This problem can be illustrated as follows. Suppose the matched data for 250 pairs

tically nonsignificant X2 = 1.3. Although the investigator might want to use the matched data to claim a

“significantly” elevated risk, the claim would be contradicted by the unmatched results. The main findings in this study would seem to be the predominant similarity of exposure or nonexposure in 222 (= 150 + 72) of the matched pairs, rather than the dissimilarity noted in 28 pairs. To draw a conclusion from only the latter results would ignore what is found in the great bulk of all the other data.

26.5.6.3Results with Variance of 0 — It seems intuitively unacceptable to dismiss arrangements I and IV of Table 26.16 as having 0 variance. Although quite correct mathematically, the dismissal

runs counter to the ingrained idea that all samples have variations. For example, a proportion such as 0/2 has 0 variance according to the formula pq/n (for the proportion) or the formula npq (for the numerator). Nevertheless, despite the 0 variance, most evaluators would regard such a proportion as

unstable. The stability might be determined by resorting to binomial distributions, but these distributions are not used in the McNemar or M-H formulas; and the Yates corrections are but arbitrary attempts to substitute for the actual distributions.

26.5.6.4Ambiguous Results — Because the odds ratios and the X2 values can become ambiguous, yielding substantially different results when calculated with the matched and unmatched methods,32 an investigator can always do things both ways and then report the “desired” result. For this reason, reviewers, editors, and readers should always demand that both sets of results be presented. If the conclusions disagree, the decision about which set to accept will usually be controversial, depending on the professional ancestry of the person who does the deciding. For persons with a mathematical background, the preferred choice will usually be the statistical appeal of the matched analysis. For persons with a scientific background, the preferred choice — for reasons stated in the first two points here — will be the unmatched analysis. As long as both sets of results are presented, however, you can make your own choice.

26.6 Additional Statistical Procedures

If you feel mentally tired and somewhat confused after all the foregoing discussion, you are in good company. Most health professionals and scientific workers become distinctly uncomfortable (if not overtly oppressed) when confronted with the extensive mathematical reasoning used for the cited concepts. Complex as they may be, however, the concepts just discussed are but a few of the diverse mathematical strategies that have been developed for the statistical analysis of “risk” in case-control and cohort studies.

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The Mantel–Haenszel and many other methods can be used to adjust the risk ratio in cohort studies as well as the odds ratio in case-control studies. The matched odds ratio can be calculated in special ways when the controls are matched as ≥ 2 to 1, rather than 1 to 1. Special confidence intervals can be calculated for these multiple matchings, as well as for 1 to 1 matchings. And all of the mathematical strategies have inspired multiple ideas and multiple pages of statistical literature.

As a reward for having come this far in the chapter, however, you will be spared an account of all the additional ideas and operational tactics. If you ever want (or need) to find out what they are, you can look them up (and try to understand them) from the accounts and cited references given in several textbooks.25–28 After discovering that the authors do not always agree on the best way to do things, you can feel particularly grateful to have escaped the risk of the exposure here.

References

1. Feinstein, 1985; 2. Chan, 1988; 3. Carson, 1994; 4. Yerushalmy, 1951; 5. Abramson, 1995; 6. Kitagawa, 1966; 7. Anderson, 1998; 8. Sorlie, 1999; 9. Royal College of General Practitioners, 1974; 10. Feinstein, 1996; 11. Concato, 1992; 12. Roos, 1989; 13. Morris, 1988; 14. Armitage, 1971; 15. Vandenbroucke, 1982; 16. Mulder, 1983; 17. Liddell, 1984; 18. Ulm, 1990; 19. Lee, 1994; 20. Greenland, 1991; 21. Kupper, 1978; 22. Decoufle, 1980; 23. Cochran, 1954; 24. Mantel, 1959; 25. Breslow, 1980; 26. Fleiss, 1981; 27. Kahn, 1989; 28. Schlesselman, 1982; 29. Kleinbaum, 1982; 30. Rothman, 1986; 31. DerSimonian, 1986; 32. Feinstein, 1987b.

Exercises

26.1.Use your calculator to confirm the statement in the text that the direct standardized rates in Table 26.5 are 15.02% for OPEN and 14.38% for TURP. Show at least some of the component calculations.

26.2.If you did an indirect rather than direct standardized adjustment for the data in Table 26.5, what would be the standardized mortality ratios for TURP and OPEN? What would be the indirectly standardized mortality rates?

26.3.The investigators whose work was contradicted have vigorously disputed the conclusions by Concato et al.11 What do you think is the focus of the counter-attack by the original investigators?

26.4.The five-year survival rate for breast cancer is reported to be 71% at the prestigious Almamammy Medical Center and 59% at the underfunded Caritas Municipal Hospital. The nationally reported fiveyear rate of survival for breast cancer is 63%. We also know the following data:

What could you do to standardize the two sets of results for comparison? What are the standardized values?

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