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On the probability of gallstone recurrence
390
Journal of Hepatology, 1987; 4:390-392 Elsevier
HEP 00305
Letter to the Editor
On the probability of gallstone recurrence
S.C. Richardson National Centre for Viral Hepatitis, Athens (Greece)
T h e article by Lanzini et al. [1] on gallstone recur-
T h e c o r r e c t calculations are set o u t in the T a b l e be-
rence rate c o n t a i n s an u n f o r t u n a t e e r r o r which has
low. This uses the d a t a of Lanzini et al's T a b l e 1
a m a j o r effect at least on the c e r t a i n t y with which
(which d o e s not a g r e e with t h e i r Fig. 1) and also cor-
their conclusions can be asserted. T h e m i s t a k e lies in
rects a n o t h e r e r r o r . Since t h e i r ' c o r r e c t e d n u m b e r at
the a r i t h m e t i c of the calculation o f the c o n f i d e n c e
risk' adjusts for the r e c u r r e n c e s in e a c h i n t e r v a l as
limits for r e c u r r e n c e rates, which are s h o w n in t h e i r
well as the losses, t h e y are a p p a r e n t l y e s t i m a t i n g the
Fig. 2. T h e s e are o b v i o u s l y t o o n a r r o w ; it is a sad fact
r e c u r r e n c e rate at the m i d - p o i n t of the i n t e r v a l [2],
that the e s t i m a t i o n of p e r c e n t a g e s with that kind of
r a t h e r than the ' p r o b a b i l i t y of r e c u r r e n c e ' in the in-
precision r e q u i r e s t h o u s a n d s of s u b j e c t s , not just 42,
terval as the labelling of their table states. H o w e v e r ,
as those w h o c o n d u c t political o p i n i o n polls will at-
it is in fact the p r o b a b i l i t i e s that are r e q u i r e d , as t h e s e
test.
are the q u a n t i t i e s which can be m u l t i p l i e d t o g e t h e r to
TABLE 1 LIFE TABLE CALCULATION OF GALLSTONE RECURRENCE PROBABILITY Time (months)
Number starting
Recur
0-6 -12 -18 -24 -30 -36 -42 -48 -54 -60 -66 -72 -78 -84
42 40 32 28 23 21 16 14 10 8 7 4 3 2
0 5 1 1 0 1 1 1 0 1 0 0 0 0
a
Lost
2 3 3 4 2 4 1 3 2 0 3 1 1 1
Number at risk
Prob. of recurrence
Cumul. prob.
Stand. error ~
41.0 38.5 30.5 26.0 22.0 19.0 15.5 12.5 9.0 8.0 5.5 3.5 2.5 1.5
0 0.130 0.033 0.039 0 0.053 0.065 0.080 0 0.125 0 0 0 0
0 0.130 0.158 0.191 0.191 0.233 0.283 0.340 0.340 0.423 0.423 0.423 0.423 0.423
0 0.054 0.059 0.065 0.065 0.075 0.085 0.095 0.095 0.114 0.114 0.114 0.114 0.114
95% conf. interval Ga
R_Wt,
.03 .24 .04 .27 .06.32 .06 .32 .09 .38 .12 .45 .15 .53 .15 .53 .20.65 .20.65 .20.65 .20.65 .20.65
.06 .07 .10 .10 .12 .15 .19 .19 .23 .23 .23 .23 .23
Greenwood's formula; b Rothman-Wilson formula [3].
Correspondence: National Centre for Viral Hepatitis, Athens School of Hygiene, P.O. Box 14085, 115 21 Athens, Greece. 0168-8278/87/$03.50 (~) 1987 Elsevier Science Publishers B.V. (Biomedical Division)
.27 .31 .35 .35 .40 .47 .54 .54 .64 .64 .64 .64 .64
PROBABILITY OF GALLSTONE RECURRENCE give cumulative probabilities of recurrence over more than one time interval. I give the correct probabilities, but the numerical difference is very small - a 5-year recurrence probability of 42% instead of 45%. The very large 95% confidence limits (at least 21 percentage points wide and increasing to 45) shown in my Table affect the conclusions of the analysis in two respects. Firstly, it becomes obvious that a levelling off of recurrence probability after 5 years cannot be claimed, as estimation by then becomes hopelessly imprecise. (After all, there are only 5 person-years of follow-up beyond 5 years.) Secondly, the estimates for the earlier years should not be stated without indicating their imprecision, especially since there might be a tendency to quote the figures of Lanzini et al. as more authoritative than others by virtue of their superior methodology. The above calculations use Greenwood's formula as cited by Lanzini et al., but there are alternatives
391 for calculating standard errors (and hence confidence limits) which appear to be particularly applicable when the number of recurrences is as low as it is here [3]. The final columns of my Table give confidence limits from the Rothman-Wilson formula. The major change is that the limits are asymmetrical about the estimated recurrence probability, although there is no qualitative difference in conclusions.
References 1 Lanzini A, Jazrawi RP, Kupfer RM, Maudzal DP, Joseph AEA, Northfield TC. Gallstone recurrence after medical dissolution - an overestimated threat? J Hepato11986; 3:241 -246. 2 Bradford Hill A. A Short Textbook of Medical Statistics. London: Hodder and Stoughton, 1977. 3 Anderson JR, Bernstein L, Pike MC. Approximate confidence intervals for probabilities of survival and quantiles in life-table analysis. Biometrics 1982; 38: 407-416.
Reply from the authors We thank Dr. Richardson for pointing out the error in the calculation of confidence intervals for gallstone recurrence. As he says, the standard error was greatly underestimated. We have looked again at the question of cumulative recurrence rates calculated by the actuarial method. We think that neither the original table nor Dr. Richardson's is quite correct. The problem lies with the estimation of the number at risk over a time interval. In usual actuarial practice, individuals who are withdrawn from follow-up during an interval each contribute a half person at risk. This is because during the part of the interval for which they were observed, had they died their deaths would have been observed and noted. Thus they contribute to the number at risk. In the present study, however, the re-
currence of a gallstone, which corresponds to a 'death', is only known at the end of the interval when the ultrasound examination is done. Subjects observed for part of the interval do not contribute to the number at risk, the denominator in the recurrence rate, because they cannot contribute to the number of recurrences, the numerator. Thus, in the first period, 0 - 6 months, when there were 42 subjects at the start and only 40 observed for the full 6 months, the number of recurrences which could be observed is 40, the number examined by ultrasound, and so this is the number at risk. The number at risk in any interval is the number who reach the end of the interval. Using this approach, the correct actuarial table is as follows:
Journal of Hepatology, 1987; 4:390-392 Elsevier
HEP 00305
Letter to the Editor
On the probability of gallstone recurrence
S.C. Richardson National Centre for Viral Hepatitis, Athens (Greece)
T h e article by Lanzini et al. [1] on gallstone recur-
T h e c o r r e c t calculations are set o u t in the T a b l e be-
rence rate c o n t a i n s an u n f o r t u n a t e e r r o r which has
low. This uses the d a t a of Lanzini et al's T a b l e 1
a m a j o r effect at least on the c e r t a i n t y with which
(which d o e s not a g r e e with t h e i r Fig. 1) and also cor-
their conclusions can be asserted. T h e m i s t a k e lies in
rects a n o t h e r e r r o r . Since t h e i r ' c o r r e c t e d n u m b e r at
the a r i t h m e t i c of the calculation o f the c o n f i d e n c e
risk' adjusts for the r e c u r r e n c e s in e a c h i n t e r v a l as
limits for r e c u r r e n c e rates, which are s h o w n in t h e i r
well as the losses, t h e y are a p p a r e n t l y e s t i m a t i n g the
Fig. 2. T h e s e are o b v i o u s l y t o o n a r r o w ; it is a sad fact
r e c u r r e n c e rate at the m i d - p o i n t of the i n t e r v a l [2],
that the e s t i m a t i o n of p e r c e n t a g e s with that kind of
r a t h e r than the ' p r o b a b i l i t y of r e c u r r e n c e ' in the in-
precision r e q u i r e s t h o u s a n d s of s u b j e c t s , not just 42,
terval as the labelling of their table states. H o w e v e r ,
as those w h o c o n d u c t political o p i n i o n polls will at-
it is in fact the p r o b a b i l i t i e s that are r e q u i r e d , as t h e s e
test.
are the q u a n t i t i e s which can be m u l t i p l i e d t o g e t h e r to
TABLE 1 LIFE TABLE CALCULATION OF GALLSTONE RECURRENCE PROBABILITY Time (months)
Number starting
Recur
0-6 -12 -18 -24 -30 -36 -42 -48 -54 -60 -66 -72 -78 -84
42 40 32 28 23 21 16 14 10 8 7 4 3 2
0 5 1 1 0 1 1 1 0 1 0 0 0 0
a
Lost
2 3 3 4 2 4 1 3 2 0 3 1 1 1
Number at risk
Prob. of recurrence
Cumul. prob.
Stand. error ~
41.0 38.5 30.5 26.0 22.0 19.0 15.5 12.5 9.0 8.0 5.5 3.5 2.5 1.5
0 0.130 0.033 0.039 0 0.053 0.065 0.080 0 0.125 0 0 0 0
0 0.130 0.158 0.191 0.191 0.233 0.283 0.340 0.340 0.423 0.423 0.423 0.423 0.423
0 0.054 0.059 0.065 0.065 0.075 0.085 0.095 0.095 0.114 0.114 0.114 0.114 0.114
95% conf. interval Ga
R_Wt,
.03 .24 .04 .27 .06.32 .06 .32 .09 .38 .12 .45 .15 .53 .15 .53 .20.65 .20.65 .20.65 .20.65 .20.65
.06 .07 .10 .10 .12 .15 .19 .19 .23 .23 .23 .23 .23
Greenwood's formula; b Rothman-Wilson formula [3].
Correspondence: National Centre for Viral Hepatitis, Athens School of Hygiene, P.O. Box 14085, 115 21 Athens, Greece. 0168-8278/87/$03.50 (~) 1987 Elsevier Science Publishers B.V. (Biomedical Division)
.27 .31 .35 .35 .40 .47 .54 .54 .64 .64 .64 .64 .64
PROBABILITY OF GALLSTONE RECURRENCE give cumulative probabilities of recurrence over more than one time interval. I give the correct probabilities, but the numerical difference is very small - a 5-year recurrence probability of 42% instead of 45%. The very large 95% confidence limits (at least 21 percentage points wide and increasing to 45) shown in my Table affect the conclusions of the analysis in two respects. Firstly, it becomes obvious that a levelling off of recurrence probability after 5 years cannot be claimed, as estimation by then becomes hopelessly imprecise. (After all, there are only 5 person-years of follow-up beyond 5 years.) Secondly, the estimates for the earlier years should not be stated without indicating their imprecision, especially since there might be a tendency to quote the figures of Lanzini et al. as more authoritative than others by virtue of their superior methodology. The above calculations use Greenwood's formula as cited by Lanzini et al., but there are alternatives
391 for calculating standard errors (and hence confidence limits) which appear to be particularly applicable when the number of recurrences is as low as it is here [3]. The final columns of my Table give confidence limits from the Rothman-Wilson formula. The major change is that the limits are asymmetrical about the estimated recurrence probability, although there is no qualitative difference in conclusions.
References 1 Lanzini A, Jazrawi RP, Kupfer RM, Maudzal DP, Joseph AEA, Northfield TC. Gallstone recurrence after medical dissolution - an overestimated threat? J Hepato11986; 3:241 -246. 2 Bradford Hill A. A Short Textbook of Medical Statistics. London: Hodder and Stoughton, 1977. 3 Anderson JR, Bernstein L, Pike MC. Approximate confidence intervals for probabilities of survival and quantiles in life-table analysis. Biometrics 1982; 38: 407-416.
Reply from the authors We thank Dr. Richardson for pointing out the error in the calculation of confidence intervals for gallstone recurrence. As he says, the standard error was greatly underestimated. We have looked again at the question of cumulative recurrence rates calculated by the actuarial method. We think that neither the original table nor Dr. Richardson's is quite correct. The problem lies with the estimation of the number at risk over a time interval. In usual actuarial practice, individuals who are withdrawn from follow-up during an interval each contribute a half person at risk. This is because during the part of the interval for which they were observed, had they died their deaths would have been observed and noted. Thus they contribute to the number at risk. In the present study, however, the re-
currence of a gallstone, which corresponds to a 'death', is only known at the end of the interval when the ultrasound examination is done. Subjects observed for part of the interval do not contribute to the number at risk, the denominator in the recurrence rate, because they cannot contribute to the number of recurrences, the numerator. Thus, in the first period, 0 - 6 months, when there were 42 subjects at the start and only 40 observed for the full 6 months, the number of recurrences which could be observed is 40, the number examined by ultrasound, and so this is the number at risk. The number at risk in any interval is the number who reach the end of the interval. Using this approach, the correct actuarial table is as follows: