Ординатура / Офтальмология / Английские материалы / Principles Of Medical Statistics_Feinstein_2002

20.5.

20.5.1.

(a)Value of X2 for “agreement” seems to be [(19 × 12) − (5 × 1)]2 (37)/(24 × 13 × 20 × 17) = 17.345, which is an ordinary chi-square calculation. This is an inappropriate approach to these data.

(b)Value of X2 for “change” seems to be (5 − 1)2/(5 + 1) = 16/6 = 2.67. This is the McNemar

formula and would be appropriate for indexing disagreement, but these small numbers probably need a correction factor so that X2M = (|5 − 1| − 1)2/6 = 9/6 = 1.5, not 2.67.

(c)Better citations would be p o = (19 + 12)/37 = 31/37 = .84, or kappa. Because pe = [(20 ×

24) + (13 × 17)]/372 = .512, kappa = (.84 − .512)/(l − .512) = .672.

20.5.2. McNemar index of bias here is (5−1)/(5+1) = 4/6 = .67, suggesting that the therapists are more likely to make “satisfactory” ratings than the patients. Because of the small numbers, this index would have to be checked stochastically. The McNemar chi-square values in 20.5.1.(b), however, are too small (even without the correction factor) to exceed the boundary of 3.84 (= 1.962) needed for stochastic significance of X2 at 1 d.f.

Chapter 21

21.1. If we start with 10,000 school-aged children, of whom 4% are physically abused, the fourfold table will show

Of the 400 abused children, 96% (384) are detected by the physical exam; of the 9600 nonabused, 8% (768) have false-positive exams. Consequently, nosologic sensitivity = 96% (384/400), nosologic specificity = 92% (8832/9600), and diagnostic sensitivity (positive predictive accuracy) = 33% (384/1152).

21.3. Results from nomogram table show:

Usefulness of test cannot be determined without knowing its negative predictive accuracy. The nomogram technique, however, does not seem easy to use, and many clinicians might prefer to do a direct calculation, if a suitable program were available to ease the work. For example, recall that posterior odds = prior odds × LR. For P(D) = .02, prior odds = .02/.98 = .0196. For LR = 10, posterior odds = 10 × .0196 =

.196. Because probability = odds/(1 + odds), posterior probability = .196/(1 + .196) = .16. This is close to the 17% estimated from the nomogram. For a working formula, convert P(D) to prior odds of P(D)/[1 – P(D)]; multiply it by LR to yield posterior odds, and convert the latter to posterior probability as LR[P(D)/{1 – P(D)}]/(1 + LR[P(D)/{(1 – P(D)}]). The latter algebra simplifies to P(D T) = LR[P(D)]/{l – P(D) + LR[P(D)]}. Thus, for LR = 20 and P(D) = 20%, P(D T) = [20(.20)]/[1 – .2 + (20)(.20)] = .83, as shown in the foregoing table.

© 2002 by Chapman & Hall/CRC

21.5. Prevalence depends on total group size, N, which is not used when likelihood ratios are calculated from the components T and S in N = T + S.

21.7. Individual answers. 21.9. Individual answers.

Chapter 22

22.1.

22.1.1. Calculations for fixed-interval method:

22.1.2. Calculations for direct method:

22.1.3. The distinctions clearly illustrate the differences between analyses in which the censored people do or do not contribute to the intervening denominators. At the end of the fifth year of follow-up in this cohort of 126 people, we clinically know only two things for sure: 56 people have died and 60 people have actually been followed for at least 5 years. Everyone else in the cohort is either lost or in the stage of censored suspended animation called withdrawn alive. Assuming that the 12 (= 4 + 6 + 0 + 2) lost people are irretrievably lost, we still have 54 people (= 15 + 11 + 15 + 7 + 6) who are percolating through the follow-up process at various stages of continuing observation before 5 years. The cumulative survival rate of 0.067 in the direct method is based on the two clearly known items of clinical information at 5 years. The cumulative survival rate of 0.443 in the fixed-interval method is based on the interval contributions made by all the censored people. In fact, if you take 33 as half the number of the 66 (= 12 + 54) censored people and add 33 to the direct numerator and denominator at 5 years, the cumulative survival becomes 0.398; and the two methods agree more closely.

Rather than dispute the merits of the two methods, we might focus our intellectual energy on the basic scientific policy here. Why are 5-year survival rates being reported at all when the 5-year status is yet to be determined for almost as many people (54) as the 60 for whom the status is known? Why not restrict the results to what was known at a 1-year or at most a 2-year period, for which the fate of the unknown group has much less of an impact? A clinician predicting a patient’s 5-year chances of survival from these data might be excessively cautious in giving the direct estimate of 0.067. On the other hand, the clinician would have a hard time pragmatically defending an actuarial estimate of 0.443, when the only real facts are that four people of a potential 60 have actually survived for 5 years.

© 2002 by Chapman & Hall/CRC

Chapter 23

23.1. In 1936, telephones were not yet relatively inexpensive and ubiquitous in the U.S. Owners of telephones were particularly likely to be in higher socioeconomic classes and, therefore, Republicans. In addition, readers of the “upscale” literary magazine were likely to be college graduates, a status which, in 1936, was also commonly held by Republicans. The use of these two “sampling frames,” despite the random selection of telephone owners, produced an overwhelmingly Republican response.

23.3.

23.3.1. Authors did not indicate why they made no recommendations. Perhaps they were worried about “β error” because the confidence interval around the observed result goes from –0.120 to +.070. [Observed mortality rates were 15/100 = .150 for propranolol and 12/95 = .126 for placebo, so that result was lower in placebo by .126 − .150 = −.024. Standard error under H0 is (27 )(168 ) ⁄[195(100 )(95 )] = .049 and two-tailed 95% C.I. would be −.024 ± (1.96)(.049) = −

.024 ± .096, which extends from −0.12 to +.072. Thus, the result might be regarded as a stochastic variation for the possibility that propranolol was 7.2% better. The authors also state that their population was unusually “mild” in prognosis. [Perhaps propranolol works better in patients with worse prognoses.]

23.3.2. The most direct, simple approach is as follows. If mortality rate with propranolol is 10% below that of placebo, the true propranolol mortality rate is 12.6% − 10% = 2.6%. In the 100 patients treated with propranolol, we would expect 2.6 deaths and 97.4 survivors. Applying a chisquare goodness-of-fit test to this “model,” we would have

2 = (15 – 2.6 )2 (85 – 97.4)2 = + = <

X ------------------------- + ---------------------------- 59.14 1.58 60.72 and P .001

2.697.4

The chance is therefore tiny that the observed results arose by chance from such a population. In a more “conventional” way, we can use the formula

where δ − do would be .10 − (−.024) = .124, and

[(.15)(.85) ⁄100 ] + [(12 ⁄95)(83 ⁄95) ⁄95 ] = .04936

so that ZH = .124/.04936 = 2.51, P < .025. We can reject the alternative hypothesis that propranolol mortality was 10% below that of placebo.

23.5. Figure E.23.5 shows that the values of do (labeled PC − PT) ranged from −.20 to +.20, with negative values favoring treatment. Without knowing the actual values of P C and PT, we cannot decide how large δ should be for quantitative significance. Most investigators would probably accept do = .10 as quantitatively significant, regardless of magnitudes for PC and PT. Using the principle that δ = θ PC and letting θ = .5, we would have δ ≥ .1 for PC ≥ .2, δ ≥ .05 for PC ≥ .1, and δ ≥ .025 for PC ≥ .05. With

© 2002 by Chapman & Hall/CRC

these boundaries, most of the “positive” trials that favored treatment in Figure E.23.5 seemed to have dO > δ and certainly had upper confidence limits that exceeded δ , although the lower confidence interval included 0. Accordingly, the main defect in these trials was capacity, not power. A recommendation to use the Neyman-Pearson approach would needlessly inflate the sample sizes. Conversely, the top eight negative trials, favoring the control group would seem to have their quantitative “nonsignificance” confirmed by upper confidence limits that were < δ . The other “negative” trials in Figure E.23.5 seemed to have upper confidence limits that included δ and lower limits that included 0. The investigator would thus have to concede both the original null and the alternative hypothesis. An appropriate sample size for the latter trials would depend on which hypothesis the investigator wanted to reject, i.e., to confirm a big or small difference.

Thus, although a Neyman-Pearson calculation would probably have been unnecessary for any of the cited trials, Freiman et al. recommended that the “determination of sample size” should be done as cited in two references9,l5 that presented the Neyman-Pearson approach.

Chapter 24

24.1. For the big difference, set Zα at 1.645 for a one-tailed Po < .05. With π A = .08 and π B = .10, π =

.09 and so

n Š [1/(.02)2][1.645]2 [2(.09)(.91)] = 1068

and 2n would be 2136.

For the tiny difference, Zα can also be set at 1.645 for a one-tailed PE < .05. The calculation will be

n Š [1/(.02 − .005)2] [1.645]2[(.08)(.92) + (.10)(.90)] = 1968

and 2n would be 3935 — which is larger than the 2958 calculated previously in Exercise 23.2.

The main reason why sample size for the big difference did not become much smaller than before is the shift from δ = .03 to δ = .02. These values are quite small for a “big” difference, and the “gain” of having only one Zα term in the numerator of the calculation is offset by the loss of having to divide by the square of a smaller value of δ .

24.3. Using the data employed in Exercise 14.4, we can assume that the rate of total myocardial infarction is .02 per person in the placebo group, and will be reduced by 25% to a value of .015 in the aspirin group, so that δ = .005. The anticipated value for π will be (.02 + .015)/2 = .0175. If we choose a twotailed α = .05, so that Zα = 1.96, and a one-tailed β = .2, so that Zβ = 0.84, the Neyman-Pearson calculation becomes

n Š [1/(.005) 2][1.96 + 0.84]2[2(.0175)(.9825)] = 10783.9

This result is close to the actual sample size of about 11,000 in each group.

According to the original grant request, which was kindly sent to me by Dr. Charles Hennekens, the investigators expected that aspirin would produce “a 20% decrease in cardiovascular mortality” and a “10% decrease in total mortality” during “the proposed study period of four and one-half years.” The

Zβ = 1.645. With these assumptions, n Š [1/(.009749) 2] [1.645 + 1.645]2 [2(.043871) (.956129)] = 9554.2. The increase to about 11,000 per group was presumably done for “insurance.”

An important thing to note in the published “final report” of the U.S. study (see Exercise 10.1) is that neither of the planned reductions was obtained. The total cardiovascular death rate and the total death rate in the two groups were almost the same (with relative risks of .96 and .96, respectively). The significant reduction in myocardial infarction (which had not been previously designated as a principal endpoint) was the main finding that led to premature termination of the trial.

© 2002 by Chapman & Hall/CRC

Chapter 25

25.1. Individual answers.

25.3. The stochastic calculations rely on the idea that each comparison is “independent.” Thus, if treatments A, B, and C are each given independently and if the effects of the agents are not interrelated, the three comparisons are independent for A vs. B, A vs. C, and B vs. C. In a person’s body, however, the values of different chemical constituents are usually interrelated by underlying “homeostatic” mechanisms. Consequently, the different chemical values are not independent and cannot be evaluated with the [1 − (1 − α )k] formula used for independent stochastic comparisons. Another possible (but less likely) explanation is that the originally designated normal zones were expanded, after a plethora of “false positives” were found, to encompass a much larger range that would include more values in healthy people.

25.5. Individual answers.

Chapter 26

26.1. Stratum proportions in the total population are W1 = 101/252, W2 = 134/252, and W3 = 17/252. Direct adjusted rate for OPEN would be (W1)(6/64) + (W2)(10/59) + (W3)(1/3) = .1502. Analogous calculation for TURP yields .1438. [To avoid problems in rounding, the original fractions should be used for each calculation.]

26.3. Concato et al. used two different taxonomic classifications (Kaplan–Feinstein and Charlson et al.) for severity of co-morbidity, and three different statistical methods (standardization, composite-staging, and Cox regression) for adjusting the “crude” results. With each of the six approaches, the adjusted risk ratio was about 1. Nevertheless, despite the agreement found with these diverse methods, the relatively small groups in the Concato study led to 95% confidence intervals that reached substantially higher (and lower) values around the adjusted ratios. For example, with the Cox regression method in the Concato study the 95% confidence limits for the adjusted risk ratio of 0.91 extended from 0.47 to 1.75. The original investigators contended that this wide spread made the Concato result unreliable, and besides, it included the higher RR values, ranging from 1.27 to 1.45, that had been found in the previous claimsdata study.

26.5.

26.5.1.No. Crude death rate is “standardized” by the division of numerator deaths and denominator population.

26.5.2.(1) Guyana rates may be truly lower in each age-specific stratum; (2) many of the deaths in Guyana may not be officially reported; (3) Guyana rates may be the same or higher than U.S. in each age-specific stratum, but Guyana may have a much younger population.

26.5.3.The third conjecture in 26.5.2 is supported by the additional data. Despite the high infant mortality, if we subtract infant deaths from births, we find that Guyana increments its youth at

Chapter 27

27.1.

27.1.1. By ordinary linear regression for the two variables in Figure 27.2. The tactic seems inappropriate. If a Pearson correlation coefficient is not justified, the results do not warrant use of an ordinary regression model.

© 2002 by Chapman & Hall/CRC