a Department of Mathematics, Tulane University, New Orleans, LA 70118, USA. E-mail: renj{at}ultral.math.tulane.edu
b University of Nijmegen, Department of Medical Statistics, PO Box 9101, 6500 HB Nijmegen, The Netherlands. E-mail: N.Peer{at}MIE.KUN.NL
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Abstract |
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Methods The study group consisted of cancers diagnosed in women who participated in a serial screening programme with a mean screening interval of 2 years. To study the effectiveness of the screening, a comparison is made between the distribution of age at which the tumour could be detected when biennial mammographic screening is the only detection method, and the distribution of age at which the tumour would be detected by either biennial mammographic screening or the development of symptoms. Some recently developed statistic methods, such as bootstrap, the maximum likelihood distribution estimator for doubly censored data and the EM algorithm, are used in estimation of these distributions.
Results The hypothesis tests and confidence intervals show that the difference between the two distributions was statistically significant for women younger than 50 and 5070 years old, but not for women over 70 years.
Conclusions The statistical analysis indicates that for women younger than 50, and 5070 years of age, a screening mammogram every other year is not frequent enough to detect primary breast cancer, but for women over 70 years, it might be sufficient.
Keywords Bootstrap, breast cancer, doubly censored data, screening mammograms
Accepted 13 March 2000
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Introduction |
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This study analyses the effectiveness of screening mammograms with a mean screening interval of 2 years in the detection of primary breast cancer. Some recently developed statistical methods are employed to handle doubly censored observations encountered in the data set used in this research. The results of our analysis, limited to the observational data currently available, indicate that a screening mammogram every other year is not frequent enough to detect primary breast cancer in women under 50.
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Data and Methods |
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From the introduction of the mediolateral oblique projection in 1981 to the end of 1990, nearly 30 000 women in Nijmegen, The Netherlands, were invited for screening mammograms every other year. The percentages of these women who accepted invitations were 70% among those under 50, 60% among those aged 5070 years, and 20% among those over 70.
For cancers diagnosed between screening examinations, i.e. the interval cancers, previous screening mammograms were re-examined by one radiologist to determine whether, in retrospect, a tumour could be identified and measured. The review showed that in several cases, a tumour nucleus shadow could be identified in the previous negative mammogram. In other interval cancers, although there were visible tumours radiographically at the time of the diagnosis, no suspect lesions could be found in the previous negative mammogram. An analogous review of earlier screening mammograms was also carried out for the screen-detected cancers.
In this study, the patients were eligible if at least two mammographic examinations with the mediolateral oblique projection were available. Radiographically occult cancers were excluded (16% of the interval cancers in the Nijmegen programme).6 They were invisible on a mammogram and showed no signs of microcalcification, thus could never be detected by mammographic screening. The resultant study group consists of 289 cancers with age at diagnosis ranging from 41 to 84 years. Among these 289 cancers, 132 were interval cancers not diagnosed at the time of screening. Of these 132 interval cancers, 79 were visible on the mammogram at the time of diagnosis, while 53 of them were palpable, but not visible on any of the mammograms (some indirect signs like microcalcification were shown on the mammogram, but no tumour shadow). The percentages of these 289 women who actually had their screening mammograms once every 2 years were 83.3% among those under 50, 84.9% among those aged 5070, and 78.6% among those older than 70 (for interval cancers, a patient with only one previous screening examination is not considered as having had her biennial screening regularly if the time between screening and diagnosis is more than 2 years).
Statistical analysis
Consider the following two variables (in years):
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and
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where the index S is for screening and SS for screening or symptoms. One should note that AS only includes the screen-detected cancers, while ASS includes screen-detected cancers and interval cancers. Let FS and FSS be the distribution functions of AS and ASS, respectively. A comparison between FS and FSS may be conducted to study the effectiveness of the screening mammograms in the detection of primary breast cancer. If mS and mSS are the medians of FS and FSS, respectively, then a big difference between mS and mSS implies that the age at which the tumour could be detected using biennial mammographic screening as the only detection method is much later than the age at which the tumour developed. One may note that mS > mSS should always hold based on the above definitions of AS and ASS, and relevant inference in this study is the size of the difference between mS and mSS. A statistically significant difference between mS and mSS, say, (mS mSS) > for some
> 0, would indicate that the screening was not done frequently enough to detect the cancer.
Among 289 cancers in the study group, 45 women had tumour volumes observed at their first screening mammograms. Some of these women, though not treated as interval cancers, had positive mammograms at their first screenings because of the retrospective examination due to interval cancers, while the rest of them continued their biennial mammograms after their positive first screenings. If these women had started their screening earlier and continued having mammograms every 2 years, the tumour could have been (at least very likely) observed before their first mammograms. Thus we assume that AS for these 45 individuals was less than the age at the first screening mammogram, and so they yield 45 left censored observations. A left censored observation is an available quantity which is larger than the desired, but unavailable, quantity (AS in the case considered here). Similarly, 132 of the 289 cancers did not have tumour volume observed at the last available screening mammogram, yielding 132 right censored observations (for each individual, AS was greater than the age at the last screening examination). During the serial screening mammograms, 112 were observed tumour growth yielding uncensored observations (for each individual, AS was actually observed). All these observations together consist of a doubly censored data set, which is used to estimate the distribution of AS. Intuitively, the use of the left censored or right censored observation in this problem takes into account that, for some individuals, AS is, with positive probability, less than or greater than the observation we have. Thus, it gives a more accurate estimation for FS. The estimator for FS, denoted as S, is called the non-parametric maximum likelihood estimator and its computation and asymptotic properties are studied in recent statistics literature.7
One may notice that for those 132 right censored observations in the doubly censored model of AS, we assume that AS could have been observed sometime after the last screening if the screening had been conducted continuously every 2 years. These 132 cases are all these interval cancers. However, this does not mean that we assume AS is definitely observable at the next regular screening. The reasons for this are that not all observed interval cancers occurred before the next scheduled screening (some patients did not have their screenings on schedule), and that, as mentioned earlier, among all interval cancers, 53 did not show any tumour nucleus shadow on the mammograms at the time of diagnosis. These might have not been detected by the screening even if the next screening mammograms were conducted on time in these cases. Thus it is not appropriate to assume that all interval cancers can be observed at the next scheduled screening.
Another point worth mentioning is that since the screening interval was fixed as 2 years, there is no information available on the exact time at which the tumour developed between a negative screening mammogram and the subsequent positive mammogram. The definition of AS is not equivalent to the time at which the tumour developed, thus AS is not subject to any censoring between a negative screening mammogram and the subsequent positive mammogram, and there were in total 112 such cases in our study for AS.
To estimate the distribution of ASS, the 45 left censored observations above are still used as left censored observations, but the rest of 244 cancers, yielding uncensored observations (for each individual, ASS was actually observed), were either detected at screening or clinically diagnosed based on symptoms. These observations together consist of a left censored data set, and are used to compute the reversed Kaplan-Meier estimator7 for FSS, denoted as SS.
To compare the distributions FS and FSS, the following hypothesis test is conducted for women under 50, women aged 5070, and women over 70, respectively:
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Our goal is seeking sufficient evidence to conclude the alternative hypothesis H1 with some > 0 for different age groups. For this test, the test statistic is computed by
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where n is the total number of observations used to compute S and
SS, and SE is the standard error of
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We know9 that for large n, T is asymptotically standard normal, SE can be estimated by the bootstrap method10,11 (1000 bootstrap samples are used in this study), and the rejection region for 95% significance level is T > 1.645.
It should be pointed out that the complexity of this current analysis is due to the doubly censored data naturally encountered for the estimation of the distribution of AS. In recent years, new statistical methods89 have been developed precisely for this type of data.
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Results |
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Discussion and Conclusions |
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As summarized earlier, some recent studies13,10 have demonstrated a statistically significant morality reduction for women under 50, but so far no trials with an interval of 2 years have demonstrated the same fact. This current analysis indicates that biennial mammographic screening is not frequent enough to detect primary breast cancer in women under 50.
However, it must be noted that due to the limits of the observational data we have, cost-benefit and other risk factors (e.g. radiation exposure) are not taken into account in our analysis, and further study should be conducted to determine how frequently younger women should have their screening mammograms in order to detect primary breast cancer effectively.
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Acknowledgments |
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References |
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