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Procedures for Estimating Growth Rates in Thoracic Aortic Aneurysms
J Clin Epidemiol Vol. 51, No. 9, pp. 747–754, 1998 Copyright 1998 Elsevier Science Inc. All rights reserved.
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Procedures for Estimating Growth Rates in Thoracic Aortic Aneurysms John A. Rizzo,1,* Michael A. Coady,2 and John A. Elefteriades 2 1
Department of Epidemiology and Public Health, Yale University School of Medicine, New Haven, Connecticut; and 2 Section of Cardiothoracic Surgery, Yale University School of Medicine, New Haven, Connecticut ABSTRACT. Thoracic aortic aneurysms (TAAs) are potentially lethal medical conditions often requiring surgical intervention. Reliable information on TAA growth rates and associated risk factors is important for managing this challenging patient population. Unfortunately, a number of studies have employed questionable statistical methods, leading to biased and imprecise estimates. The present study describes these statistical problems in existing studies and delineates procedures for obtaining more reliable results. Using data from the Yale Center for Thoracic Aortic Disease, the study compares TAA growth rate estimates using conventional methods versus the recommended approach of instrumental variables (IV) estimation. The IV approach is designed to mitigate problems of measurement errors inherent in existing estimates of TAA growth. The results demonstrate that IV estimation yields more robust and precise estimates of TAA growth rates and risk factors for TAA growth. For example, the conventional approach yields TAA growth rates that fluctuate substantially—from 0.12 cm/ yr to 0.90 cm/yr—depending on (1) the minimum serial follow-up period for patient inclusion in the study and (2) how subjects with negative measured growth rates are handled. In contrast, growth rate estimates using the IV approach are much more robust, ranging from 0.12 to 0.13 cm/yr. The 95% confidence intervals of estimated TAA growth are much more compact using the IV approach as well. We conclude that the IV estimation procedure yields more reliable estimates of TAA growth than does the conventional approach. j clin epidemiol 51;9:747–754, 1998. 1998 Elsevier Science Inc. KEY WORDS. Thoracic aortic aneurysms, measurement error, risk factor analysis
INTRODUCTION Aneurysms of the thoracic aorta are potentially life-threatening medical conditions often requiring surgical intervention. Such interventions, however, continue to pose serious risks, including death, stroke, and paraplegia. Left untreated, acute dissection or rupture of a thoracic aortic aneurysm is fatal; operating on such patients emergently involves substantial mortality risk as well, with mortality estimates of 15% or more [1–3]. As major medical centers have gained experience in operating on these patients electively, the size criteria for surgical intervention has fallen. A recent study recommends intervention when the ascending aortic aneurysm has attained a size of 5.5 cm, and when the descending aortic aneurysm has attained a size of 6.5 cm; still smaller intervention criteria are recommended for patients with Marfan’s syndrome [1]. As aneurysm size increases, the law of Laplace predicts *
Address for correspondence: Dr. J. A. Rizzo, Yale University School of Medicine, Department of Epidemiology and Public Health, 60 College Street, New Haven, CT 06520-8034. Accepted for publication on 8 April 1998.
that aortic wall tension increases as well [4]. In the case of thoracic aortic aneurysm, disease progression usually consists of self-propagating expansion with possible dissection or rupture. Both aortic size [1] and rate of growth [5] have been shown to be independent risk factors for the serious complications of dissection and rupture. Appropriate preventive care for these patients requires careful monitoring to identify disease progression and determine when surgery is indicated. Accurate information on aortic size and rate of growth is vital to the clinician in assessing patient risk for complication and in developing sound treatment protocols. Unfortunately, substantial scientific uncertainty persists regarding the optimal management of patients with thoracic aortic aneurysms (TAA) [1]. Contributing to this uncertainty are discrepancies in the results of published studies of TAA growth rates and associated risk factors. These discrepancies are due in part to inconsistent statistical methodologies across studies. The present investigation critically evaluates conventional procedures for estimating TAA growth rates and proposes an alternative statistical approach. Using aneurysm
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data from the Yale Center for Thoracic Aortic Disease, we demonstrate that the alternative methodology yields more robust and precise estimates of average TAA growth rates and associated risk factors for higher TAA growth in subgroups of the patient population. The purpose of the study is not to develop a full risk model of factors predicting TAA growth. Rather, we seek to illustrate a more promising statistical approach than the conventional method for ascertaining TAA growth and associated risk factors. In particular, this study addresses the following questions: What is the conventional approach for estimating TAA growth and identifying risk factors? What are the statistical drawbacks to the conventional approach? What is an alternative procedure for estimating TAA growth and identifying risk factors? How does this alternative procedure improve upon the conventional approach? How should we estimate TAA growth and identify risk factors in future research? PROBLEMS IN THE MEASUREMENT OF ANEURYSM SIZE AND GROWTH RATES Obtaining accurate estimates of aneurysm growth rates depends on several key factors. The primary objective is to measure aneurysm size as accurately as possible in order to minimize errors in these measures. Particular strategies employed for handling measurement errors have fundamental implications for the precision and reliability of the resulting TAA growth rate estimates. We contrast the conventional approach with our recommended procedures for estimating TAA growth below. Before turning to these issues, however, it is useful to elaborate upon the idea of measurement error. What is Measurement Error? In the context of aortic aneurysm size, measurement error is the difference between the recorded size of the aneurysm and the true or actual size (unknown). Given measurement errors, measured size, S M, and actual size, SA , are related as follows: S M ⫽ S A ⫹ ⑀,
(1)
where the expected value of the measurement error, ⑀, is zero. We also make the commonly employed assumption that measurement errors across diagnostic tests are independent. An implication of this is that over repeated testing, the measurement errors will tend to cancel out. Using the subscript F to denote the first measured size, and L the last measured size, aneurysm growth rate, GrM, is calculated as: Gr M ⫽ (S ML ⫺ S FM )/T,
(2)
where T ⫽ the time interval between the first and last measured diagnostic studies. When Do Measurement Errors Occur? An error in measurement may occur at the reading of a diagnostic study. The radiologist, for instance, may not always choose precisely the same anatomic location to determine the maximum aneurysm size. This problem will be exacerbated if examinations are read by different radiologists. For example, studies have shown that inter-observer variation in the transverse diameter of abdominal aortic aneurysms is 0.53 cm [6]. Incorrect identification of the limits of the aortic wall or contour may also result in error. A third source of error is incorrect identification of the level of an axial slice in computed tomographic (CT) or magnetic resonance imaging (MRI) scanning. For instance, it is often difficult to determine whether a particular ‘‘cut’’ is above or below the aortic valve. In addition, an oblique slice through the aortic arch on an axial study exaggerates the size of the aorta. Such errors can be minimized but not eliminated by re-reading clinical imaging studies for the purpose of scientific data analysis by an expert team attuned to these issues. A second source of measurement error occurs because aneurysm growth rates, originally based on angiographic measurements, have commonly been evaluated in recent years via alternative noninvasive diagnostic tests, such as echocardiography (ECHO), CT scans, or MRI [7]. These imaging modes apply alternative methods for ascertaining TAA size and growth rates and impose different requirements on the clinician in evaluating them. Some clinicians have argued that echocardiograms underestimate the size of the aorta[8], and others maintain that MRI is the diagnostic imaging method of choice [9]. Nonetheless, comparisons of the actual performance of these alternative imaging modes have found them to be similar in terms of their ability to measure aneurysm size [8]. Thus, when multiple diagnostic imaging modes are used, the potential for measurement errors does increase; however, the magnitude of this source of error appears to be modest. Conventional Approach for Handling TAA Measurement Errors The conventional approach for handling measurement errors is to attempt to minimize them by considering only those studies that have been read by the same clinician for a single diagnostic imaging mode. Other factors equal, it is preferable to track patients using a consistent diagnostic modality, and in fact, a number of studies of aneurysmal progression have done so [5,10–21]. Unfortunately, this intuitively appealing approach involves important drawbacks as well. First, it will in general greatly reduce the available sample size. This is a significant limitation, given that existing studies of TAA growth rates appearing in the litera-
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ture are already based upon modest sample sizes. Moreover, even after restricting the sample to a single imaging mode read by a single clinician, measurement errors will persist. Thus, Dapunt et al. [5] have measured TAA growth rates using a single diagnostic imaging tool (CT scans) read by a well-trained clinician, and still report a number of negative growth rate estimates. Since such negative changes are clinically implausible, persistent measurement errors provide the most likely explanation. The statistical procedures we outline below avoid this problem. The discussion thus far has identified two important limitations to the conventional approach for estimating TAA growth: (1) it reduces sample size and statistical power; and (2) it fails to eliminate measurement errors. This second limitation has lead to at least two further problems: (3) upward-bias in estimated TAA growth rates in studies that have mishandled this type of measurement error; and (4) sensitivity of TAA estimates to time intervals between diagnostic imaging tests. Upward Bias in Growth Rate Estimation How should one handle measurement errors that lead to negative estimated growth rates under the conventional approach? A number of researchers have apparently truncated their samples, eliminating negative growth rate estimates, or censored their samples, scoring negative growth estimates as having 0 growth rates. Thus, Sterpetti et al. [17] reported abdominal aneurysm growth ranging from 0 to 1.8 cm/yr in a cohort of 57 patients. Bengtsson et al. [11] reported a similar range among a cohort of 98 patients followed serially for abdominal aortic aneurysms. Given the sources of measurement error we have discussed, it seems highly unlikely that none of these patients had a negative measured growth rate. Other studies [14,15,21] similarly report lower bounds on measured growth rates equal to 0. Not surprisingly, these studies report the largest mean expansion rates in the literature, ranging from 0.40 cm/yr [11] to a very high mean growth rate of 0.79 cm/yr [20]. Less information is available on the growth of thoracic
than abdominal aneurysms. Table 1 summarizes the available evidence on this issue. As the table indicates, a wide range of TAA growth rates have been reported. Hirose et al. [18] reported a relatively high annual TAA growth rate averaged over all thoracic sites of 0.42 cm/yr. In 1992, Masuda et al. [19] found a mean TAA growth rate of 0.13 cm/ yr, which is consistent with the TAA growth rates based in our Yale series [1,22]. More recently, Hirose and colleagues [20] reported a considerably lower growth rate than in their earlier study [18]. The discrepancy between these two studies may reflect the different statistical methodologies employed. The later study by Hirose et al. [20] estimated TAA growth using regression techniques similar to the approach we are advocating; the earlier study employed the conventional approach [18]. Descending and thoracoabdominal aortic aneurysms may grow faster than ascending and arch aneurysms. Thus, Coady et al. [1] reported growth rates of descending and thoracoabdominal aneurysms of 0.29 cm/yr, which closely approximates the growth rates of 0.32 cm/yr reported by Dapunt et al. [5] for aneurysms in these locations. Although Hirose et al. [18] reported a particularly rapid growth among arch aneurysms of 0.56 cm/yr, we question this finding both because the frequency of arch aneurysms as a percentage of all TAA aneurysms in their series (41.5%) seems implausibly high and because they failed to employ regression techniques in calculating growth rates. The proper handling of statistical outliers would eliminate or decrease the influence of estimated values far above or below the mean, but in such a way as to preserve the symmetric nature of the distribution of measurement errors. Truncating or censoring the data at a single extreme fails to preserve this vital symmetry and results in bias. Even a simple outlier screening rule, such as eliminating observations ⫾3 standard deviations from the mean preserves symmetry and would not introduce a bias in the estimated average growth rate. Alternatively, more sophisticated outlier procedures could be employed as well [23]. Mishandling of statistical outliers confounds efforts to understand differences in aneurysm growth rates that have
TABLE 1. Growth rates and risk factors for expansion in thoracic aortic aneurysms
Study
Sample size
Aneurysm location
Growth rate (cm/yr)
Coady et al. [1]
79
ASC, ARCH, DESC, TA
0.12; all sites 0.29; DESC, TA
Rizzo et al. [22]
53
ASC, ARCH, DESC, TA
0.10; all sites
Hirose et al. [20]
30
ASC, ARCH DESC
0.18a; all sites
Dapunt et al. [5]
67
DESC, TA
0.32; DESC, TA
Hirose et al. [18]
82
ASC, ARCH, DESC, TA
0.42; all sites
Masuda et al. [19]
36
ASC, ARCH DESC
0.13; all sites
Abbreviations ASC ⫽ ascending: ARCH ⫽ arch: DESC ⫽ descending: TA ⫽ thoracoabdominal aneurysm. a These estimates are calculated from a regression equation for TAA expansion estimated in Hirose et al. [20], and assume the average size for an initial TAA measure in their series (4.5 cm).
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appeared in the literature. Thus, high reported growth rates have been explained in terms of ‘‘referral center bias.’’ That is, because studies which report high growth rates were typically conducted in hospitals, they are thought to include patients at higher risk of aortic expansion than do population-based studies [16,24], which report substantially lower growth rates. Some population-based studies, however, have correctly avoided truncating or censoring negative estimated values. Thus, Nevitt et al. [16] report a range in annual growth of ⫺0.8 cm/yr to 3.7 cm/yr. The discrepancy in estimated growth rates between referral-based and population-based studies may thus reflect truncation or censoring bias in referral-based studies. Evidence to support this conclusion is provided by a referral-based study which did not censor or truncate data. That study [12] reported growth rates well in line with those of population-based studies [16,24]. Sensitivity of TAA Growth Rates to Time Intervals between Imaging Studies Another implication of measurement errors is that the shorter the time interval between diagnostic imaging tests, the less reliable the resulting TAA growth rate estimate will be. This may be seen by substituting Eq. (1) into (2), arriving at: Gr M ⫽ (S LA ⫺ S AF )/T ⫹ (⑀L ⫺ ⑀F )/T.
(3)
The shorter the time interval between tests, the larger the error term components will be, increasing the inaccuracy of the resulting growth rate estimates. Another implication of Eq. (3) is that growth rare estimates using the conventional approach are quite sensitive to the time interval chosen between tests. INSTRUMENTAL VARIABLES APPROACH TO ESTIMATING TAA GROWTH Given the problems associated with the conventional approach to estimating TAA growth, it is desirable to consider alternatives. A straightforward alternative to estimating TAA growth is to employ the method of instrumental variables (IV) estimation. What is Instrumental Variables Estimation? In the context of TAA growth rates, the logic of the IV estimation approach is to correlate the change in aneurysm size with a variable which, unlike aneurysm size, has little measurement error. The resulting regression estimates of aneurysm growth will mitigate measurement errors, yielding more stable and less biased estimates of TAA growth. Instrumental variables estimation is a technique for dealing with random measurement errors rather than situations involving systematic bias. Because available evidence suggests
that inter-observer variability in measuring aortic sizes does not involve systematic bias [6], the IV approach seems appropriate for the types of errors incurred when measuring aneurysm sizes. To illustrate the IV approach, we note that the last measured aneurysm size, SML, may be related to the first measured size SMF, as: S ML ⫽ S MFeβ T,
(4)
Where β is a parameter defining the intertemporal growth rate of aneurysm size. Taking the natural logarithm of each side of Eq. (5) and rearranging terms yields: lnS LM ⫺ lnS MF ⫽ β T.
(5)
Equation (5) depicts the relationship assumed for the change in aneurysm size. This change is assumed to grow at an exponential rate over time. The assumption of the exponential relationship is inessential for the IV estimation approach; a linear relationship could also have been specified (we contrast the linear model to the exponential one in the Results section below, finding that they provide very similar estimates). The key feature of IV estimation is not the particular functional relationship chosen, but that the technique relates a term involving measurement error—the left-hand side of Eq. (5), to one involving little or no measurement error—the right-hand side of Eq. (5). The righthand side of Eq. (5) will have little or no measurement error if, as seems reasonable, the dates on which patients’ aneurysm sizes were imaged are reasonably accurate. Equation (5) is estimated by ordinary least squares (OLS) regression analysis [25]. The result of this procedure yields an estimate of β, call it β *, which relates the time interval between diagnostic imaging tests to aneurysm growth. Importantly, the estimate β * will only be correlated with the true variation in aneurysm growth, not with the measurement error terms. This is so because the measurement error terms are random factors that will not correlate with T, the time interval between the first and last imaging studies. Thus, the IV approach allows one to obtain an estimate of aneurysm growth that is purged of measurement errors. As is true with any regression-based method, the β * estimate is itself subject to some uncertainty. The IV approach provides a ‘‘best estimate,’’ β *, and a confidence interval around this estimate. One might argue that IV estimation merely replaces one source of variability under the conventional approach (e.g., measurement errors stemming from inter-observer variability) for another (e.g., uncertainty surrounding the β * estimate). Because our instrument, T, should involve less measurement error than inter-observer variability, it is reasonable to hypothesize that the IV method will yield more precise estimates. Ultimately, however, the practical advantage of the IV estimate is an empirical issue, and a focus of this study.
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Risk Factor Analysis: Conventional versus IV Approaches
The sample consisted of 45 males and 34 females. Chronic dissections were present in 16 patients, and 15 patients had Marfan’s syndrome. Diagnostic imaging studies to ascertain aneurysm size consisted of MRIs (22%), CT scans (34%), and echocardiogram (44%). The IV estimation results presented below do not adjust for diagnostic imaging modality. This implicitly assumes that measurement errors are independent across diagnostic tests, an assumption which may not be valid. We examined this issue in preliminary statistical analysis by including binary variables controlling for the imaging modalities used to measure each patient’s first and last aneurysm size. Including these variables in the analysis had very little effect on any of the other variables in the models. Moreover, imaging modality never had a significant effect on estimated TAA growth. The empirical evidence thus suggests that any correlation in measurement errors across imaging modalities is minor and does not affect the results. We therefore elected to exclude imaging modality from the models reported below for ease of exposition.
It is also important to compare the conventional and IV approaches in terms of risk factor analysis. Under the conventional approach, the growth rate calculated as in Eq. (2) is regressed on a risk factor or vector of risk factors, RISK: Gr M ⫽ β 0 ⫹ β1 RISK.
(6)
Risk factor analysis is conducted under the IV approach by estimating the following equation: lnS ML ⫺ lnS MF ⫽ β 0 T ⫹ β1T ∗ RISK.
(7)
Equation (7) is like Eq. (5) except that it appends a risk factor term. This risk factor term is multiplied by T, because at time T ⫽ 0, first and last measured size must be the same. Another way to understand the rationale for multiplying T and RISK is to note that the effects of any risk factor on increasing aneurysm size are likely to accumulate over time. In contrast, risk factor analysis under the conventional approach is static, for it does not allow the effects of risk factors on aneurysm size to grow over time.
Comparison of TAA Growth Rate Estimates RESULTS
Table 2 shows TAA growth rate estimates under the conventional and IV estimation approaches. Three versions of the conventional approach to estimating TAA growth are presented, each of which differs in terms of how negative estimated growth rates are handled. The first, Conventional Model 1, includes negative growth estimates, neither censoring nor truncating these data points. Conventional Model 2 censors negative growth estimates, re-scaling such observations to equal 0. Conventional Model 3 truncates negative growth estimates, eliminating them from the calculations. Separate IV estimation results are reported for both linear and exponential growth models. Results are shown for different minimum time intervals between tests. Thus the column labelled ‘‘1 month’’ includes patients whose aneurysm sizes were measured at least 1 month apart, while the column labelled ‘‘3 months’’ includes only those patients whose aneurysm measurements
Proper comparison of these two approaches requires analysis on a consistent set of data. We have used data from the Yale Center for Thoracic Aortic Disease for this purpose. The Center maintains an ongoing database of TAA patients monitored serially for progression in their aneurysms. In addition to surgically treated patients, the Yale [1] series consists of individuals followed chronically at Yale-New Haven Hospital for progression in their thoracic aortic aneurysms during the period 1985–1997. At present, the full sample includes data on 225 patients with thoracic aortic aneurysm, with serial imaging studies on 79 aneurysmal patients who did not undergo aortic surgery. The average follow-up period is 35 months. Patients were considered aneurysmal if the maximum diameter of their aortas measured 3.5 cm or greater, an entry criterion consistent with earlier research [1]. The mean age of these 79 patients was 59 years.
TABLE 2. TAA Growth rate estimates: conventional vs. instrumental variables estimation approach a
Minimum time between imaging studies
Estimation approach Conventional model 1: no censoring or truncation Conventional model 2: with censoring Conventional model 3: with truncation b IV: Exponential model IV: Linear model
1 month, n ⴝ 79 (cm/yr) 0.38 0.66 0.90 0.12 0.13
⫾ ⫾ ⫾ ⫾ ⫾
0.64 0.59 0.80 0.06 0.06
3 months, n ⴝ 71 (cm/yr) 0.29 0.37 0.46 0.12 0.13
⫾ ⫾ ⫾ ⫾ ⫾
0.25 0.24 0.30 0.06 0.06
6 months, n ⴝ 62 (cm/yr) 0.12 0.20 0.27 0.12 0.12
⫾ ⫾ ⫾ ⫾ ⫾
0.11 0.08 0.10 0.06 0.06
12 months, n ⴝ 46 (cm/yr) 0.15 0.21 0.27 0.12 0.13
⫾ ⫾ ⫾ ⫾ ⫾
0.10 0.09 0.11 0.06 0.07
Average annual growth rates; ⫾ indicates 95% CIs. b Sample sizes are smaller for the Conventional Model with Truncation (58 with 1-month cutoff; 56 with 3-month; 47 with 6-month; and 35 with 12-month) because negative estimated growth values are eliminated in this case.
a
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were taken at least 3 months apart. Because the conventional estimation approach does not remove measurement errors, TAA growth measures using this approach should be more sensitive to changes in the minimum time interval between tests than would the IV estimation approach. We would also expect the least plausible results under the conventional approach to occur when low cutoffs are specified for the minimum time interval between tests, for that is when the influence of measurement errors is greatest. The results in Table 2 confirm these predictions. Whether we include only patients who are followed for at least 1 month, 3 months, 6 months, or 12 months, the IV approach yields nearly identical estimates of annual TAA growth rates of 0.12 to 0.13 cm/yr. Whether we use the linear or exponential model, the results are very similar. In contrast, the conventional approach leads to widely fluctuating estimates, with implausibly large growth rate estimates when patients who are followed for only brief periods of time (1 month) are included. Moreover, censoring or truncating negative growth estimates leads to substantial upward bias. Using the 1-month minimum time interval between imaging studies, the average annual TAA growth rate is estimated to be 0.38 cm/yr using the Conventional Model without censoring or truncation, 0.66 cm/yr with censoring, and 0.90 cm/yr with truncation. Finally, the IV estimation approach leads to more precise estimates, as measured by the consistently smaller 95% confidence intervals obtained. What can explain the high growth rates calculated using the conventional approach when subjects followed for as little as 1 month are included? To a substantial extent, inappropriate censoring or truncation of negative estimated growth rates is responsible, as we have seen. Mishandling of negative measured growth rates, however, cannot account for the high growth rate under Conventional Model 1, which neither censors nor truncates such observations. An alternative possibility is that the high readings associated with the 1-month cutoff period reflect systematic factors. Perhaps, for example, subjects who are thought to have
unstable aneurysms are imaged more frequently. But if such systematic factors were important, we would have expected them to influence our instrumental variables estimates as well. This is so because the instrumental variables approach mitigates problems associated with random errors, not with systematic ones. Thus if systematic factors were important, we would have expected the IV approach to yield substantially higher estimates as well when subjects followed for only 1 month were included in the study; but that did not happen, as the results in Table 2 attest. Instead, we believe that the substantially higher growth rates under Conventional Model 1, when 1-month follow-ups are included, stem from a failure to treat random measurement error, a problem whose effects are exacerbated when patients followed for brief time periods are included in the analysis (see Eq. (3), above). Comparison of Risk Factor Analyses We also contrasted the conventional and IV approaches in terms of their ability to identify risk factors for TAA growth. To facilitate comparisons, we consider a simple model which includes only one risk factor, the presence of a chronic dissection of the aorta. We hypothesize that chronic dissection will be directly related to growth. Table 3 summarizes the results of these risk factor analyses. Once again, the estimates using the IV approaches yield very consistent results, regardless of the minimum time interval between tests. In contrast, the estimates using the conventional approach are quite unstable, being substantially above the IV estimates for 1-month cutoffs, and below the IV estimates when 6- or 12-month cutoff periods are chosen. DISCUSSION Accurate information on TAA growth rates and associated risk factors are important to clinicians for assessing patient
TABLE 3. Risk factor analysis: conventional vs instrumental variables estimation approach: incremental effect of chronic
dissection on TAA growth a Minimum time between imaging studies
Estimation approach Conventional model no censoring or truncation Conventional model with censoring Conventional model with truncation b IV: Exponential model c IV: Linear model a
1 month n ⴝ 79 (cm/yr) 1.11 0.96 1.06 0.28 0.28
P P P P P
⫽ ⫽ ⫽ ⫽ ⫽
0.0029 0.0049 0.0143 0.0022 0.0023
3 months, n ⴝ 71 (cm/yr) 0.25 0.22 0.21 0.28 0.28
P P P P P
⫽ ⫽ ⫽ ⫽ ⫽
0.0725 0.1055 0.1900 0.0011 0.0012
6 months, n ⴝ 62 (cm/yr) 0.04 0.03 ⫺0.16 0.28 0.27
P P P P P
⫽ ⫽ ⫽ ⫽ ⫽
0.5190 0.9370 0.7382 0.0016 0.0016
12 months, n ⴝ 46 (cm/yr) 0.03 0.01 0.03 0.28 0.27
P P P P P
⫽ ⫽ ⫽ ⫽ ⫽
0.5534 0.7517 0.9461 0.0044 0.0052
Incremental effect of risk factor on TAA growth in cm/yr. Risk factor is presence of chronic dissection. b Sample sizes are smaller for the Conventional Model with Truncation (58 with 1-month cutoff; 56 with 3-month; 47 with 6-month; and 35 with 12-month) because negative estimated growth values are eliminated in this case. c Estimates for exponential model use sample mean of initial size of aneurysm (5.2 cm).
Estimating Growth Rates in Thoracic Aortic Aneurysms
risk for serious complications and developing preventive treatment protocols. Appropriate statistical procedures must be used in a consistent fashion if studies of TAA growth are to provide reliable and comparable evidence. The present study illustrates how instrumental variables estimation is preferable to the conventional approach, and brings to light its advantages by contrasting the two approaches using a consistent set of data. Using the instrumental variables approach, we conclude that TAA grows at an average annual rate of 0.12 cm/yr in the full sample of patients studied, and 0.34 to 0.35 cm/ yr among patients having chronic dissections. These results are quite robust to inclusion criteria: whether the minimum time interval between imaging studies for inclusion is 1 month or 1 year makes little difference for these results. In contrast, the conventional approach yields unstable estimates for overall growth and the effect of chronic dissection which vary considerably by inclusion criteria. We conclude that the method of instrumental variables is preferable to the conventional approach because it leads to more stable and precise estimates of TAA growth rates and associated risk factors for growth. Researchers are beginning to apply instrumental variables to the estimation of TAA growth [15,20]. Although the advantages of IV estimation have been illustrated for the case of thoracic aortic aneurysms, the method would also be useful for the study of growth in abdominal aortic aneurysms as well. Abdominal aortic aneurysms are much more common than thoracic aortic aneurysms and are the thirteenth leading cause of death in the United States [26]. Directions for Future Research While the instrumental variables approach offers an improved statistical methodology for estimating TAA growth rates and identifying risk factors, another important issue that remains to be addressed is that the sample of patients available to the researcher may not be representative of the population of TAA patients. There are at least two sources of this sample selection bias. The first source stems from the fact that TAA patients seen at major referral centers may be different from the general population of TAA patients. This type of selection, known as referral center bias, may lead to upward bias in TAA growth rate estimates, to the extent that TAA patients seen at referral centers have more advanced aortic disease on average than the general population of TAA patients, many of whom may have small, stable aneurysms. Short of obtaining population-based data, there is little that can be done about this source of selection bias. The second source of bias occurs because TAA patients are differentially selected out for graft surgery, based upon their aneurysm sizes, growth rates, the development of symptoms, occurrence of aortic dissection, or other criteria. For example, patients with large aneurysms are differentially chosen for graft surgery. Only individuals with large stable
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aneurysms and few complications tend to remain in cohorts tracked for aortic expansion. Thus, the analyst tends to observe a relatively healthy group of patients with large aneurysms (as the more unstable patients are selected out). Failure to control for such selection effects will lead one to systematically underestimate the true effect of aortic size on TAA growth rates [25]. One would also anticipate that other risk factor estimates would be downward biased as well. Because data on patients selected for surgery are available at most referral centers, one can in principle adjust growth rate estimates and risk factor analyses for these selection effects. Although statistical techniques to control for selection effects are well known, their implementation requires a sufficiently large and clinically detailed data set. As we continue to build our data set at the Yale Center for Thoracic Aortic Disease, we intend to investigate these issues further. An earlier version of this paper was presented at the Birmingham Aortic Surgery Symposium, Birmingham, England, November 1996.
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dominal aortic aneurysm detected by screening. Br J Surg 1993; 80: 718–720. Bernstein EF, Dilley RB, Goldberger LE, et al. Growth rates of small abdominal aortic aneurysms. Surgery 1976; 80: 765– 773. Cronenwett JL, Murphy TF, Zelenock GB, et al. Actuarial analysis of variables associated with rupture of small abdominal aortic aneurysms. Surgery 1985; 98: 472–483. Delin A, Ohlsen H, Swedenborg J. Growth rate of abdominal aortic aneurysms as measured by computed tomography. Br J Surg 1985; 72: 530–532. Nevitt MP, Ballard PJ, Hallet JW Jr. Prognosis of abdominal aortic aneurysm. A population-based study. N Engl J Med 1989; 321: 1009–1014. Sterpetti AV, Schultz RD, Feldhaus RJ, et al. Factors influencing enlargement rate of small abdominal aortic aneurysms. J Surg Res 1987; 43: 211–219. Hirose Y, Hamada S, Takamiya M, et al. Aortic aneurysms: Growth rates measured with CT. Radiology 1992; 185: 249– 252. Masuda Y, Takanasji K, Takasu J, et al. Expansion rate of thoracic aortic aneurysms and influencing factors. Chest 1992; 102: 461–466.
20. Hirose Y, Hamada S, Takamiya M. Predicting the growth of aortic aneurysms: A comparison of linear vs. exponential models. Angiology 1995; 46: 413–419. 21. Littooy FN, Steffan G, Greisler HP, et al. Use of sequential bmode ultrasonography to manage abdominal aortic aneurysms. Arch Surg 1989; 124: 419–421. 22. Rizzo JA, Darr U, Fischer M, Johnson KM, Finkle J, Gusberg R, Kopf G, Abbott TA, Shevchenko I, Elefteriades JA. Multimodality serial follow-up of thoracic aortic aneurysms. Int J Angiol 1997; 6: 153–156. 23. Hampel FR, et al. Robust Statistics: The Approach Based on Influence Functions New York: John Wiley & Sons; 1986. 24. Collin J, Araujo L, Walton J. How fast do very small abdominal aortic aneurysms grow? Eur J Vasc Surg 1989; 3: 15–17. 25. Kennedy P. A Guide to Econometrics. Cambridge, MA: MIT Press; 1992. 26. Goldstein J. Abdominal aortic aneurysms. In: Greenfield LJ, Mulholland MW, Oldham KT, and Zelnock GB, Eds. Surgery: Scientific Principles and Practice. Philadelphia, PA: JB Lippincott; 1993: 1711–1722.
0895-4356/98/$19.00 PII SO895-4356(98)00050-X
Procedures for Estimating Growth Rates in Thoracic Aortic Aneurysms John A. Rizzo,1,* Michael A. Coady,2 and John A. Elefteriades 2 1
Department of Epidemiology and Public Health, Yale University School of Medicine, New Haven, Connecticut; and 2 Section of Cardiothoracic Surgery, Yale University School of Medicine, New Haven, Connecticut ABSTRACT. Thoracic aortic aneurysms (TAAs) are potentially lethal medical conditions often requiring surgical intervention. Reliable information on TAA growth rates and associated risk factors is important for managing this challenging patient population. Unfortunately, a number of studies have employed questionable statistical methods, leading to biased and imprecise estimates. The present study describes these statistical problems in existing studies and delineates procedures for obtaining more reliable results. Using data from the Yale Center for Thoracic Aortic Disease, the study compares TAA growth rate estimates using conventional methods versus the recommended approach of instrumental variables (IV) estimation. The IV approach is designed to mitigate problems of measurement errors inherent in existing estimates of TAA growth. The results demonstrate that IV estimation yields more robust and precise estimates of TAA growth rates and risk factors for TAA growth. For example, the conventional approach yields TAA growth rates that fluctuate substantially—from 0.12 cm/ yr to 0.90 cm/yr—depending on (1) the minimum serial follow-up period for patient inclusion in the study and (2) how subjects with negative measured growth rates are handled. In contrast, growth rate estimates using the IV approach are much more robust, ranging from 0.12 to 0.13 cm/yr. The 95% confidence intervals of estimated TAA growth are much more compact using the IV approach as well. We conclude that the IV estimation procedure yields more reliable estimates of TAA growth than does the conventional approach. j clin epidemiol 51;9:747–754, 1998. 1998 Elsevier Science Inc. KEY WORDS. Thoracic aortic aneurysms, measurement error, risk factor analysis
INTRODUCTION Aneurysms of the thoracic aorta are potentially life-threatening medical conditions often requiring surgical intervention. Such interventions, however, continue to pose serious risks, including death, stroke, and paraplegia. Left untreated, acute dissection or rupture of a thoracic aortic aneurysm is fatal; operating on such patients emergently involves substantial mortality risk as well, with mortality estimates of 15% or more [1–3]. As major medical centers have gained experience in operating on these patients electively, the size criteria for surgical intervention has fallen. A recent study recommends intervention when the ascending aortic aneurysm has attained a size of 5.5 cm, and when the descending aortic aneurysm has attained a size of 6.5 cm; still smaller intervention criteria are recommended for patients with Marfan’s syndrome [1]. As aneurysm size increases, the law of Laplace predicts *
Address for correspondence: Dr. J. A. Rizzo, Yale University School of Medicine, Department of Epidemiology and Public Health, 60 College Street, New Haven, CT 06520-8034. Accepted for publication on 8 April 1998.
that aortic wall tension increases as well [4]. In the case of thoracic aortic aneurysm, disease progression usually consists of self-propagating expansion with possible dissection or rupture. Both aortic size [1] and rate of growth [5] have been shown to be independent risk factors for the serious complications of dissection and rupture. Appropriate preventive care for these patients requires careful monitoring to identify disease progression and determine when surgery is indicated. Accurate information on aortic size and rate of growth is vital to the clinician in assessing patient risk for complication and in developing sound treatment protocols. Unfortunately, substantial scientific uncertainty persists regarding the optimal management of patients with thoracic aortic aneurysms (TAA) [1]. Contributing to this uncertainty are discrepancies in the results of published studies of TAA growth rates and associated risk factors. These discrepancies are due in part to inconsistent statistical methodologies across studies. The present investigation critically evaluates conventional procedures for estimating TAA growth rates and proposes an alternative statistical approach. Using aneurysm
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data from the Yale Center for Thoracic Aortic Disease, we demonstrate that the alternative methodology yields more robust and precise estimates of average TAA growth rates and associated risk factors for higher TAA growth in subgroups of the patient population. The purpose of the study is not to develop a full risk model of factors predicting TAA growth. Rather, we seek to illustrate a more promising statistical approach than the conventional method for ascertaining TAA growth and associated risk factors. In particular, this study addresses the following questions: What is the conventional approach for estimating TAA growth and identifying risk factors? What are the statistical drawbacks to the conventional approach? What is an alternative procedure for estimating TAA growth and identifying risk factors? How does this alternative procedure improve upon the conventional approach? How should we estimate TAA growth and identify risk factors in future research? PROBLEMS IN THE MEASUREMENT OF ANEURYSM SIZE AND GROWTH RATES Obtaining accurate estimates of aneurysm growth rates depends on several key factors. The primary objective is to measure aneurysm size as accurately as possible in order to minimize errors in these measures. Particular strategies employed for handling measurement errors have fundamental implications for the precision and reliability of the resulting TAA growth rate estimates. We contrast the conventional approach with our recommended procedures for estimating TAA growth below. Before turning to these issues, however, it is useful to elaborate upon the idea of measurement error. What is Measurement Error? In the context of aortic aneurysm size, measurement error is the difference between the recorded size of the aneurysm and the true or actual size (unknown). Given measurement errors, measured size, S M, and actual size, SA , are related as follows: S M ⫽ S A ⫹ ⑀,
(1)
where the expected value of the measurement error, ⑀, is zero. We also make the commonly employed assumption that measurement errors across diagnostic tests are independent. An implication of this is that over repeated testing, the measurement errors will tend to cancel out. Using the subscript F to denote the first measured size, and L the last measured size, aneurysm growth rate, GrM, is calculated as: Gr M ⫽ (S ML ⫺ S FM )/T,
(2)
where T ⫽ the time interval between the first and last measured diagnostic studies. When Do Measurement Errors Occur? An error in measurement may occur at the reading of a diagnostic study. The radiologist, for instance, may not always choose precisely the same anatomic location to determine the maximum aneurysm size. This problem will be exacerbated if examinations are read by different radiologists. For example, studies have shown that inter-observer variation in the transverse diameter of abdominal aortic aneurysms is 0.53 cm [6]. Incorrect identification of the limits of the aortic wall or contour may also result in error. A third source of error is incorrect identification of the level of an axial slice in computed tomographic (CT) or magnetic resonance imaging (MRI) scanning. For instance, it is often difficult to determine whether a particular ‘‘cut’’ is above or below the aortic valve. In addition, an oblique slice through the aortic arch on an axial study exaggerates the size of the aorta. Such errors can be minimized but not eliminated by re-reading clinical imaging studies for the purpose of scientific data analysis by an expert team attuned to these issues. A second source of measurement error occurs because aneurysm growth rates, originally based on angiographic measurements, have commonly been evaluated in recent years via alternative noninvasive diagnostic tests, such as echocardiography (ECHO), CT scans, or MRI [7]. These imaging modes apply alternative methods for ascertaining TAA size and growth rates and impose different requirements on the clinician in evaluating them. Some clinicians have argued that echocardiograms underestimate the size of the aorta[8], and others maintain that MRI is the diagnostic imaging method of choice [9]. Nonetheless, comparisons of the actual performance of these alternative imaging modes have found them to be similar in terms of their ability to measure aneurysm size [8]. Thus, when multiple diagnostic imaging modes are used, the potential for measurement errors does increase; however, the magnitude of this source of error appears to be modest. Conventional Approach for Handling TAA Measurement Errors The conventional approach for handling measurement errors is to attempt to minimize them by considering only those studies that have been read by the same clinician for a single diagnostic imaging mode. Other factors equal, it is preferable to track patients using a consistent diagnostic modality, and in fact, a number of studies of aneurysmal progression have done so [5,10–21]. Unfortunately, this intuitively appealing approach involves important drawbacks as well. First, it will in general greatly reduce the available sample size. This is a significant limitation, given that existing studies of TAA growth rates appearing in the litera-
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ture are already based upon modest sample sizes. Moreover, even after restricting the sample to a single imaging mode read by a single clinician, measurement errors will persist. Thus, Dapunt et al. [5] have measured TAA growth rates using a single diagnostic imaging tool (CT scans) read by a well-trained clinician, and still report a number of negative growth rate estimates. Since such negative changes are clinically implausible, persistent measurement errors provide the most likely explanation. The statistical procedures we outline below avoid this problem. The discussion thus far has identified two important limitations to the conventional approach for estimating TAA growth: (1) it reduces sample size and statistical power; and (2) it fails to eliminate measurement errors. This second limitation has lead to at least two further problems: (3) upward-bias in estimated TAA growth rates in studies that have mishandled this type of measurement error; and (4) sensitivity of TAA estimates to time intervals between diagnostic imaging tests. Upward Bias in Growth Rate Estimation How should one handle measurement errors that lead to negative estimated growth rates under the conventional approach? A number of researchers have apparently truncated their samples, eliminating negative growth rate estimates, or censored their samples, scoring negative growth estimates as having 0 growth rates. Thus, Sterpetti et al. [17] reported abdominal aneurysm growth ranging from 0 to 1.8 cm/yr in a cohort of 57 patients. Bengtsson et al. [11] reported a similar range among a cohort of 98 patients followed serially for abdominal aortic aneurysms. Given the sources of measurement error we have discussed, it seems highly unlikely that none of these patients had a negative measured growth rate. Other studies [14,15,21] similarly report lower bounds on measured growth rates equal to 0. Not surprisingly, these studies report the largest mean expansion rates in the literature, ranging from 0.40 cm/yr [11] to a very high mean growth rate of 0.79 cm/yr [20]. Less information is available on the growth of thoracic
than abdominal aneurysms. Table 1 summarizes the available evidence on this issue. As the table indicates, a wide range of TAA growth rates have been reported. Hirose et al. [18] reported a relatively high annual TAA growth rate averaged over all thoracic sites of 0.42 cm/yr. In 1992, Masuda et al. [19] found a mean TAA growth rate of 0.13 cm/ yr, which is consistent with the TAA growth rates based in our Yale series [1,22]. More recently, Hirose and colleagues [20] reported a considerably lower growth rate than in their earlier study [18]. The discrepancy between these two studies may reflect the different statistical methodologies employed. The later study by Hirose et al. [20] estimated TAA growth using regression techniques similar to the approach we are advocating; the earlier study employed the conventional approach [18]. Descending and thoracoabdominal aortic aneurysms may grow faster than ascending and arch aneurysms. Thus, Coady et al. [1] reported growth rates of descending and thoracoabdominal aneurysms of 0.29 cm/yr, which closely approximates the growth rates of 0.32 cm/yr reported by Dapunt et al. [5] for aneurysms in these locations. Although Hirose et al. [18] reported a particularly rapid growth among arch aneurysms of 0.56 cm/yr, we question this finding both because the frequency of arch aneurysms as a percentage of all TAA aneurysms in their series (41.5%) seems implausibly high and because they failed to employ regression techniques in calculating growth rates. The proper handling of statistical outliers would eliminate or decrease the influence of estimated values far above or below the mean, but in such a way as to preserve the symmetric nature of the distribution of measurement errors. Truncating or censoring the data at a single extreme fails to preserve this vital symmetry and results in bias. Even a simple outlier screening rule, such as eliminating observations ⫾3 standard deviations from the mean preserves symmetry and would not introduce a bias in the estimated average growth rate. Alternatively, more sophisticated outlier procedures could be employed as well [23]. Mishandling of statistical outliers confounds efforts to understand differences in aneurysm growth rates that have
TABLE 1. Growth rates and risk factors for expansion in thoracic aortic aneurysms
Study
Sample size
Aneurysm location
Growth rate (cm/yr)
Coady et al. [1]
79
ASC, ARCH, DESC, TA
0.12; all sites 0.29; DESC, TA
Rizzo et al. [22]
53
ASC, ARCH, DESC, TA
0.10; all sites
Hirose et al. [20]
30
ASC, ARCH DESC
0.18a; all sites
Dapunt et al. [5]
67
DESC, TA
0.32; DESC, TA
Hirose et al. [18]
82
ASC, ARCH, DESC, TA
0.42; all sites
Masuda et al. [19]
36
ASC, ARCH DESC
0.13; all sites
Abbreviations ASC ⫽ ascending: ARCH ⫽ arch: DESC ⫽ descending: TA ⫽ thoracoabdominal aneurysm. a These estimates are calculated from a regression equation for TAA expansion estimated in Hirose et al. [20], and assume the average size for an initial TAA measure in their series (4.5 cm).
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appeared in the literature. Thus, high reported growth rates have been explained in terms of ‘‘referral center bias.’’ That is, because studies which report high growth rates were typically conducted in hospitals, they are thought to include patients at higher risk of aortic expansion than do population-based studies [16,24], which report substantially lower growth rates. Some population-based studies, however, have correctly avoided truncating or censoring negative estimated values. Thus, Nevitt et al. [16] report a range in annual growth of ⫺0.8 cm/yr to 3.7 cm/yr. The discrepancy in estimated growth rates between referral-based and population-based studies may thus reflect truncation or censoring bias in referral-based studies. Evidence to support this conclusion is provided by a referral-based study which did not censor or truncate data. That study [12] reported growth rates well in line with those of population-based studies [16,24]. Sensitivity of TAA Growth Rates to Time Intervals between Imaging Studies Another implication of measurement errors is that the shorter the time interval between diagnostic imaging tests, the less reliable the resulting TAA growth rate estimate will be. This may be seen by substituting Eq. (1) into (2), arriving at: Gr M ⫽ (S LA ⫺ S AF )/T ⫹ (⑀L ⫺ ⑀F )/T.
(3)
The shorter the time interval between tests, the larger the error term components will be, increasing the inaccuracy of the resulting growth rate estimates. Another implication of Eq. (3) is that growth rare estimates using the conventional approach are quite sensitive to the time interval chosen between tests. INSTRUMENTAL VARIABLES APPROACH TO ESTIMATING TAA GROWTH Given the problems associated with the conventional approach to estimating TAA growth, it is desirable to consider alternatives. A straightforward alternative to estimating TAA growth is to employ the method of instrumental variables (IV) estimation. What is Instrumental Variables Estimation? In the context of TAA growth rates, the logic of the IV estimation approach is to correlate the change in aneurysm size with a variable which, unlike aneurysm size, has little measurement error. The resulting regression estimates of aneurysm growth will mitigate measurement errors, yielding more stable and less biased estimates of TAA growth. Instrumental variables estimation is a technique for dealing with random measurement errors rather than situations involving systematic bias. Because available evidence suggests
that inter-observer variability in measuring aortic sizes does not involve systematic bias [6], the IV approach seems appropriate for the types of errors incurred when measuring aneurysm sizes. To illustrate the IV approach, we note that the last measured aneurysm size, SML, may be related to the first measured size SMF, as: S ML ⫽ S MFeβ T,
(4)
Where β is a parameter defining the intertemporal growth rate of aneurysm size. Taking the natural logarithm of each side of Eq. (5) and rearranging terms yields: lnS LM ⫺ lnS MF ⫽ β T.
(5)
Equation (5) depicts the relationship assumed for the change in aneurysm size. This change is assumed to grow at an exponential rate over time. The assumption of the exponential relationship is inessential for the IV estimation approach; a linear relationship could also have been specified (we contrast the linear model to the exponential one in the Results section below, finding that they provide very similar estimates). The key feature of IV estimation is not the particular functional relationship chosen, but that the technique relates a term involving measurement error—the left-hand side of Eq. (5), to one involving little or no measurement error—the right-hand side of Eq. (5). The righthand side of Eq. (5) will have little or no measurement error if, as seems reasonable, the dates on which patients’ aneurysm sizes were imaged are reasonably accurate. Equation (5) is estimated by ordinary least squares (OLS) regression analysis [25]. The result of this procedure yields an estimate of β, call it β *, which relates the time interval between diagnostic imaging tests to aneurysm growth. Importantly, the estimate β * will only be correlated with the true variation in aneurysm growth, not with the measurement error terms. This is so because the measurement error terms are random factors that will not correlate with T, the time interval between the first and last imaging studies. Thus, the IV approach allows one to obtain an estimate of aneurysm growth that is purged of measurement errors. As is true with any regression-based method, the β * estimate is itself subject to some uncertainty. The IV approach provides a ‘‘best estimate,’’ β *, and a confidence interval around this estimate. One might argue that IV estimation merely replaces one source of variability under the conventional approach (e.g., measurement errors stemming from inter-observer variability) for another (e.g., uncertainty surrounding the β * estimate). Because our instrument, T, should involve less measurement error than inter-observer variability, it is reasonable to hypothesize that the IV method will yield more precise estimates. Ultimately, however, the practical advantage of the IV estimate is an empirical issue, and a focus of this study.
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Risk Factor Analysis: Conventional versus IV Approaches
The sample consisted of 45 males and 34 females. Chronic dissections were present in 16 patients, and 15 patients had Marfan’s syndrome. Diagnostic imaging studies to ascertain aneurysm size consisted of MRIs (22%), CT scans (34%), and echocardiogram (44%). The IV estimation results presented below do not adjust for diagnostic imaging modality. This implicitly assumes that measurement errors are independent across diagnostic tests, an assumption which may not be valid. We examined this issue in preliminary statistical analysis by including binary variables controlling for the imaging modalities used to measure each patient’s first and last aneurysm size. Including these variables in the analysis had very little effect on any of the other variables in the models. Moreover, imaging modality never had a significant effect on estimated TAA growth. The empirical evidence thus suggests that any correlation in measurement errors across imaging modalities is minor and does not affect the results. We therefore elected to exclude imaging modality from the models reported below for ease of exposition.
It is also important to compare the conventional and IV approaches in terms of risk factor analysis. Under the conventional approach, the growth rate calculated as in Eq. (2) is regressed on a risk factor or vector of risk factors, RISK: Gr M ⫽ β 0 ⫹ β1 RISK.
(6)
Risk factor analysis is conducted under the IV approach by estimating the following equation: lnS ML ⫺ lnS MF ⫽ β 0 T ⫹ β1T ∗ RISK.
(7)
Equation (7) is like Eq. (5) except that it appends a risk factor term. This risk factor term is multiplied by T, because at time T ⫽ 0, first and last measured size must be the same. Another way to understand the rationale for multiplying T and RISK is to note that the effects of any risk factor on increasing aneurysm size are likely to accumulate over time. In contrast, risk factor analysis under the conventional approach is static, for it does not allow the effects of risk factors on aneurysm size to grow over time.
Comparison of TAA Growth Rate Estimates RESULTS
Table 2 shows TAA growth rate estimates under the conventional and IV estimation approaches. Three versions of the conventional approach to estimating TAA growth are presented, each of which differs in terms of how negative estimated growth rates are handled. The first, Conventional Model 1, includes negative growth estimates, neither censoring nor truncating these data points. Conventional Model 2 censors negative growth estimates, re-scaling such observations to equal 0. Conventional Model 3 truncates negative growth estimates, eliminating them from the calculations. Separate IV estimation results are reported for both linear and exponential growth models. Results are shown for different minimum time intervals between tests. Thus the column labelled ‘‘1 month’’ includes patients whose aneurysm sizes were measured at least 1 month apart, while the column labelled ‘‘3 months’’ includes only those patients whose aneurysm measurements
Proper comparison of these two approaches requires analysis on a consistent set of data. We have used data from the Yale Center for Thoracic Aortic Disease for this purpose. The Center maintains an ongoing database of TAA patients monitored serially for progression in their aneurysms. In addition to surgically treated patients, the Yale [1] series consists of individuals followed chronically at Yale-New Haven Hospital for progression in their thoracic aortic aneurysms during the period 1985–1997. At present, the full sample includes data on 225 patients with thoracic aortic aneurysm, with serial imaging studies on 79 aneurysmal patients who did not undergo aortic surgery. The average follow-up period is 35 months. Patients were considered aneurysmal if the maximum diameter of their aortas measured 3.5 cm or greater, an entry criterion consistent with earlier research [1]. The mean age of these 79 patients was 59 years.
TABLE 2. TAA Growth rate estimates: conventional vs. instrumental variables estimation approach a
Minimum time between imaging studies
Estimation approach Conventional model 1: no censoring or truncation Conventional model 2: with censoring Conventional model 3: with truncation b IV: Exponential model IV: Linear model
1 month, n ⴝ 79 (cm/yr) 0.38 0.66 0.90 0.12 0.13
⫾ ⫾ ⫾ ⫾ ⫾
0.64 0.59 0.80 0.06 0.06
3 months, n ⴝ 71 (cm/yr) 0.29 0.37 0.46 0.12 0.13
⫾ ⫾ ⫾ ⫾ ⫾
0.25 0.24 0.30 0.06 0.06
6 months, n ⴝ 62 (cm/yr) 0.12 0.20 0.27 0.12 0.12
⫾ ⫾ ⫾ ⫾ ⫾
0.11 0.08 0.10 0.06 0.06
12 months, n ⴝ 46 (cm/yr) 0.15 0.21 0.27 0.12 0.13
⫾ ⫾ ⫾ ⫾ ⫾
0.10 0.09 0.11 0.06 0.07
Average annual growth rates; ⫾ indicates 95% CIs. b Sample sizes are smaller for the Conventional Model with Truncation (58 with 1-month cutoff; 56 with 3-month; 47 with 6-month; and 35 with 12-month) because negative estimated growth values are eliminated in this case.
a
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were taken at least 3 months apart. Because the conventional estimation approach does not remove measurement errors, TAA growth measures using this approach should be more sensitive to changes in the minimum time interval between tests than would the IV estimation approach. We would also expect the least plausible results under the conventional approach to occur when low cutoffs are specified for the minimum time interval between tests, for that is when the influence of measurement errors is greatest. The results in Table 2 confirm these predictions. Whether we include only patients who are followed for at least 1 month, 3 months, 6 months, or 12 months, the IV approach yields nearly identical estimates of annual TAA growth rates of 0.12 to 0.13 cm/yr. Whether we use the linear or exponential model, the results are very similar. In contrast, the conventional approach leads to widely fluctuating estimates, with implausibly large growth rate estimates when patients who are followed for only brief periods of time (1 month) are included. Moreover, censoring or truncating negative growth estimates leads to substantial upward bias. Using the 1-month minimum time interval between imaging studies, the average annual TAA growth rate is estimated to be 0.38 cm/yr using the Conventional Model without censoring or truncation, 0.66 cm/yr with censoring, and 0.90 cm/yr with truncation. Finally, the IV estimation approach leads to more precise estimates, as measured by the consistently smaller 95% confidence intervals obtained. What can explain the high growth rates calculated using the conventional approach when subjects followed for as little as 1 month are included? To a substantial extent, inappropriate censoring or truncation of negative estimated growth rates is responsible, as we have seen. Mishandling of negative measured growth rates, however, cannot account for the high growth rate under Conventional Model 1, which neither censors nor truncates such observations. An alternative possibility is that the high readings associated with the 1-month cutoff period reflect systematic factors. Perhaps, for example, subjects who are thought to have
unstable aneurysms are imaged more frequently. But if such systematic factors were important, we would have expected them to influence our instrumental variables estimates as well. This is so because the instrumental variables approach mitigates problems associated with random errors, not with systematic ones. Thus if systematic factors were important, we would have expected the IV approach to yield substantially higher estimates as well when subjects followed for only 1 month were included in the study; but that did not happen, as the results in Table 2 attest. Instead, we believe that the substantially higher growth rates under Conventional Model 1, when 1-month follow-ups are included, stem from a failure to treat random measurement error, a problem whose effects are exacerbated when patients followed for brief time periods are included in the analysis (see Eq. (3), above). Comparison of Risk Factor Analyses We also contrasted the conventional and IV approaches in terms of their ability to identify risk factors for TAA growth. To facilitate comparisons, we consider a simple model which includes only one risk factor, the presence of a chronic dissection of the aorta. We hypothesize that chronic dissection will be directly related to growth. Table 3 summarizes the results of these risk factor analyses. Once again, the estimates using the IV approaches yield very consistent results, regardless of the minimum time interval between tests. In contrast, the estimates using the conventional approach are quite unstable, being substantially above the IV estimates for 1-month cutoffs, and below the IV estimates when 6- or 12-month cutoff periods are chosen. DISCUSSION Accurate information on TAA growth rates and associated risk factors are important to clinicians for assessing patient
TABLE 3. Risk factor analysis: conventional vs instrumental variables estimation approach: incremental effect of chronic
dissection on TAA growth a Minimum time between imaging studies
Estimation approach Conventional model no censoring or truncation Conventional model with censoring Conventional model with truncation b IV: Exponential model c IV: Linear model a
1 month n ⴝ 79 (cm/yr) 1.11 0.96 1.06 0.28 0.28
P P P P P
⫽ ⫽ ⫽ ⫽ ⫽
0.0029 0.0049 0.0143 0.0022 0.0023
3 months, n ⴝ 71 (cm/yr) 0.25 0.22 0.21 0.28 0.28
P P P P P
⫽ ⫽ ⫽ ⫽ ⫽
0.0725 0.1055 0.1900 0.0011 0.0012
6 months, n ⴝ 62 (cm/yr) 0.04 0.03 ⫺0.16 0.28 0.27
P P P P P
⫽ ⫽ ⫽ ⫽ ⫽
0.5190 0.9370 0.7382 0.0016 0.0016
12 months, n ⴝ 46 (cm/yr) 0.03 0.01 0.03 0.28 0.27
P P P P P
⫽ ⫽ ⫽ ⫽ ⫽
0.5534 0.7517 0.9461 0.0044 0.0052
Incremental effect of risk factor on TAA growth in cm/yr. Risk factor is presence of chronic dissection. b Sample sizes are smaller for the Conventional Model with Truncation (58 with 1-month cutoff; 56 with 3-month; 47 with 6-month; and 35 with 12-month) because negative estimated growth values are eliminated in this case. c Estimates for exponential model use sample mean of initial size of aneurysm (5.2 cm).
Estimating Growth Rates in Thoracic Aortic Aneurysms
risk for serious complications and developing preventive treatment protocols. Appropriate statistical procedures must be used in a consistent fashion if studies of TAA growth are to provide reliable and comparable evidence. The present study illustrates how instrumental variables estimation is preferable to the conventional approach, and brings to light its advantages by contrasting the two approaches using a consistent set of data. Using the instrumental variables approach, we conclude that TAA grows at an average annual rate of 0.12 cm/yr in the full sample of patients studied, and 0.34 to 0.35 cm/ yr among patients having chronic dissections. These results are quite robust to inclusion criteria: whether the minimum time interval between imaging studies for inclusion is 1 month or 1 year makes little difference for these results. In contrast, the conventional approach yields unstable estimates for overall growth and the effect of chronic dissection which vary considerably by inclusion criteria. We conclude that the method of instrumental variables is preferable to the conventional approach because it leads to more stable and precise estimates of TAA growth rates and associated risk factors for growth. Researchers are beginning to apply instrumental variables to the estimation of TAA growth [15,20]. Although the advantages of IV estimation have been illustrated for the case of thoracic aortic aneurysms, the method would also be useful for the study of growth in abdominal aortic aneurysms as well. Abdominal aortic aneurysms are much more common than thoracic aortic aneurysms and are the thirteenth leading cause of death in the United States [26]. Directions for Future Research While the instrumental variables approach offers an improved statistical methodology for estimating TAA growth rates and identifying risk factors, another important issue that remains to be addressed is that the sample of patients available to the researcher may not be representative of the population of TAA patients. There are at least two sources of this sample selection bias. The first source stems from the fact that TAA patients seen at major referral centers may be different from the general population of TAA patients. This type of selection, known as referral center bias, may lead to upward bias in TAA growth rate estimates, to the extent that TAA patients seen at referral centers have more advanced aortic disease on average than the general population of TAA patients, many of whom may have small, stable aneurysms. Short of obtaining population-based data, there is little that can be done about this source of selection bias. The second source of bias occurs because TAA patients are differentially selected out for graft surgery, based upon their aneurysm sizes, growth rates, the development of symptoms, occurrence of aortic dissection, or other criteria. For example, patients with large aneurysms are differentially chosen for graft surgery. Only individuals with large stable
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aneurysms and few complications tend to remain in cohorts tracked for aortic expansion. Thus, the analyst tends to observe a relatively healthy group of patients with large aneurysms (as the more unstable patients are selected out). Failure to control for such selection effects will lead one to systematically underestimate the true effect of aortic size on TAA growth rates [25]. One would also anticipate that other risk factor estimates would be downward biased as well. Because data on patients selected for surgery are available at most referral centers, one can in principle adjust growth rate estimates and risk factor analyses for these selection effects. Although statistical techniques to control for selection effects are well known, their implementation requires a sufficiently large and clinically detailed data set. As we continue to build our data set at the Yale Center for Thoracic Aortic Disease, we intend to investigate these issues further. An earlier version of this paper was presented at the Birmingham Aortic Surgery Symposium, Birmingham, England, November 1996.
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