This article is concerned with two questions of interest in nutritional epidemiology. The first is biological: is there a relationship between dietary fat intake and breast cancer? The second is methodological: how do various dietary assessment instruments compare in their power to detect diet–disease relationships? To address these questions we analyzed an important data set that was subject to biased sampling via truncation. This article addresses statistical issues involved in the analysis of such truncated data. Substantive results of the analysis may be found in Freedman et al. (2006)
Case–control studies, international comparisons, and laboratory experiments in animals generally support a positive association between fat consumption and the incidence of breast cancer (Howe et al., 1990
). Conversely, a pooled analysis of several prospective studies, which are free of some of the biases that potentially affect case–control studies, has not found such an association (Hunter et al., 1996
). A major problem with studies relating diet and disease is that of dietary measurement error. Both case–control and prospective studies employ self-reporting techniques for measuring dietary intake. For reasons of logistics and cost the most commonly used instrument is the food frequency questionnaire (FFQ) (Willet, 1990
). Little is known about the nature and the extent of measurement error in FFQ-reported dietary fat intake, and there has been much discussion about whether such error could have led to the failure of the prospective studies to find a fat–breast cancer relationship (Prentice, 1996
; Kipnis et al., 2001
Other dietary assessment instruments are available, however, including multiple-day food records (FR) (see, e.g., Patterson et al., 1999
). These instruments are more expensive to administer, but are often thought to be better measures of dietary intake than the FFQ, particularly because the respondent is required only to write down all that is eaten over a relatively short period, whereas completion of a FFQ requires estimation of average long-term diet, a much more difficult cognitive task. Moreover, the FR has already been used in a few large-scale studies (e.g., Bingham et al., 2003
; Prentice et al., 2006
). Nevertheless, the superiority of the FR over the FFQ for studies of long-term diet and chronic disease is far from certain, because the FR measures only short-term intake, where long-term intake is of primary interest.
Recent calibration studies using reliable biomarker measures of protein and energy intake as reference instruments have indicated that measurement error in FFQ-reported protein and energy is substantial (Kipnis et al., 2003
), and that for these two variables other dietary instruments, such as the FR or multiple 24-hour recalls, may have less measurement error than the FFQ and thus better predict true intake (Day et al., 2001
; Schatzkin et al., 2003
). No reliable reference instrument exists for dietary fat intake, however, so that direct estimation and comparison of measurement error in reported fat intake is not possible.
Recently, Bingham et al. (2003)
reported the results of an indirect comparison of two instruments, a FFQ and a quantitative 7-day diary (a variant of the FR), both completed by a cohort of 13,070 women. They tested the fat–breast cancer hypothesis in this cohort and found that a statistically significant positive association between saturated fat intake and breast cancer incidence could be demonstrated using the 7-day diary, but not using the FFQ. These results suggested not only that the fat–breast cancer association exists, but that the 7-day diary may have more power than the FFQ to detect this association, thus providing indirect evidence that the 7-day diary may predict fat intake better than the FFQ.
One might object to this last conclusion, inasmuch as the measured association with breast cancer is distorted by measurement error and the positive association detected by the diary might be spurious; in order to properly compare the instruments, one must adjust the observed relationships for measurement error, but this is not possible without a reliable biomarker for fat intake. However, these objections may be overcome by noting that dietary measurement error predominantly causes simple attenuation of the estimated fat-intake regression coefficient, so that one may assume that tests of the null hypothesis, unadjusted for measurement error, are valid. Empirical evidence for this is available from a large biomarker study (Kipnis et al., 2003
). In such a case, one may indeed compare the strength of observed relationships between fat and breast cancer for two instruments, and ascribe differences (if significant) to differences in the attenuation caused by the measurement error in the instruments. This reasoning underlies our current investigation.
The analysis of Bingham et al. (2003)
was based on a relatively small number of breast cancer cases (168). Seeking to corroborate these results in a larger study, researchers at the National Cancer Institute and the Fred Hutchinson Cancer Research Center planned a similar comparison within the control group of the dietary modification (DM) arm of the Women’s Health Initiative (WHI) Clinical Trial (Hays et al., 2003
). The DM study is a randomized controlled trial of a low-fat diet that is high in fruits, vegetables, and grains. General eligibility criteria for the trial are provided in detail elsewhere (Hays et al., 2003
). Women who participated in this trial completed both a FFQ and a 4-day FR on entry, allowing a comparison between these instruments similar to the comparison of Bingham et al.
The data were truncated in that approximately 42% of the women screened were excluded from the trial after the first visit because they reported consuming a diet with less than 32% energy from fat, as estimated by the FFQ. Percent energy from fat is defined as 100 × (fat in kilo-calories)/(energy in kilo-calories). The purpose of this screening was to enroll into the study women having relatively high fat intake and thereby increase the difference in percent energy from fat between women in the dietary intervention and control groups.
The control group we analyzed comprised approximately 30,000 women who received general advice on diet and health, but no intensive dietary counseling. At the time of the analysis, after a median follow up of 7 years, 603 of these women had been diagnosed with invasive breast cancer. Staff at the WHI Clinical Coordinating Center selected two women with no diagnosis of breast cancer for every case, resulting in a sample of 603 cases and 1206 noncases. This was done because the cost of analyzing the food records of all women in the control group would have been prohibitive. Thus, not only is the sample truncated, but it is in the form of a nested case–control study.
The main goal of this article is to describe a simple methodology for estimating risks in the full (nontruncated) population using truncated data such as ours, both for the FFQ (on which the truncation is based) and for the FR. It is important for us to provide inference for the full population for two reasons. Firstly, the truncated population is of little biological interest because it is instrument specific. Secondly, the truncation, being based on one of the instruments, affects risk estimates differently for the two instruments and restricting inference to the truncated population would preclude a fair comparison of these two instruments.
The method uses as its motivation work by Gail, Wieand, and Piantadosi (1984)
, and requires little more than ordinary linear logistic regression. It posits a risk model in which the truncation variable is included, and then employs a new residualization method to approximate the marginal model when the truncation variable is not included. We will demonstrate that the approximation gives estimates with very small bias, and with good power.
The second goal of this article is to describe a simple methodology that allows comparison of the instruments in terms of the local power each would have in a nontruncated prospective study, again using data from the truncated sample. To do so, we develop estimates of the standard errors of the estimators that would have been obtained in a (hypothetical) nontruncated sample. We show that under some fairly mild assumptions, one can obtain an approximate comparison of the instruments without having to model the often high-dimensional distribution of all the covariates.
An outline of the article is as follows. Section 2 describes the potential for bias caused by truncated sampling and our method for addressing this problem. Section 3 describes the methods used to compare the two instruments when data are from a truncated sample. Section 4 presents results of simulations demonstrating that our approximate truncation-adjusted methods work well. Section 5 provides an analysis of the WHI study data. We find that various types of fat intake appear to be risk factors for breast cancer when using the FR to assess diet, but that no such effect is found using the FFQ. We also find that the estimated local power of the FR to detect a fat–breast cancer relationship is higher than that of the FFQ and that the difference in local powers is marginally significant.
The first part of Section 6 points out some of the difficult asymptotic issues that arise in the analysis of nest case–control studies, as described in detail by Arratia, Goldstein, and Langholz (2005)
. The second part of Section 6 briefly describes two likelihood-based alternatives, comparing them with our methodology. Section 7 presents further discussion of our method.