Fundamental misunderstanding of the relation between energy density (kcal/g) and energy cost ($/kcal)

Dear Sir: We are writing to comment on the recent article by Drewnowski (1) regarding the relative prices of food groups. First, we commend the thorough data collection methods, which ensured a comprehensive investigation of the research question. However, we have significant concerns regarding certain elements of the data analysis, interpretation, and conclusions put forth in the article. Most important, we state as fact that there is always an association between any number (X) and its reciprocal (1/X), and that this association is necessarily negative. This is relevant to the relation between energy density (kcal/g) and energy cost ($/kcal), because the relation is influenced by the presence of kcal in both variables and on opposite sides of the divisor (2). For example, 3 statistically independent and unrelated exponentially distributed random variables x, y, and z (representing energy, cost, and mass, respectively) could be used to create ratios r = x/z and q = y/x. It can be shown that, where 1/λi is the mean of the exponential random variable i, the conditional expectation of r given a value of q is the expectation of (r|q) = (λxq + λy)/λzq, which has a negative first derivative ( 0) and appears as the familiar hyperbolic shape reported in numerous studies of the relation between energy density and energy cost (proof available on request). However, the unconditional medians of r and q will vary independently depending on the distributional parameters of x and y. We select the exponential distribution because of the relative ease of analytic computations, but this same principle is true for more general distributions. In Drewnowski's recent article, “x”, “y” and “z”, used to create “x/y” and “y/z”, were each randomly generated variables specified with a uniform distribution and identical range. Given the distributions of each variable, individual realizations of x/y are systematically related to y/z. However, the analysis merely illustrated that, as expected, the unconditional means and medians of x/y and y/z have no particular implied relation. Rather, his analysis served more to show that, empirically, the distribution of kcal/g is associated with food category: fruit and vegetables have lower mean and median energy density than do other food groups and thus also have high $/kcal due to the tautological inverse relation between kcal and 1/kcal and the relatively smaller variance of $ and g (2). We also wish to highlight the article's empirical finding that the relative price of foods depended on the specific outcome price measure used (1). For example, fruit and vegetables had the highest $/kcal but had among the lowest $/100g and a relatively moderate to high $/serving. These differences raise the unanswered question as to the relevance of these metrics for how consumers perceive food cost and evaluate food choices. The hypothesis that $/kcal is a key influence on consumer behavior is inconsistent with much in the literature about food preferences (3) and leads to many improbable outcomes. For example, in Drewnowski's article, legumes had among the lowest $/kcal, whereas meat had the highest $/100g and $/serving and among the highest $/kcal. However, Americans’ intake of legumes is extremely low, whereas meat intake is far more adequate (4). Furthermore, if consumers perceived food cost in terms of $/kcal, marketers would avoid labeling foods as “reduced calorie” or similar, because it would indicate more expensive calories. Likewise, providing menu calorie information would actually promote energy-dense options because this would clearly show the “value” of these foods; yet, evidence suggests such information instead discourages intake of high-calorie foods and promotes low-calorie menu items (5), which cost more per kcal. When evaluating food groups by any price measure, it is imperative to acknowledge the dangers of comparing “apples to oranges” or, in this case, “apples to potato chips,” as there are many differences between food groups that influence food choice besides nutritive value and price (6). Thus, to assume consumers choose potato chips over apples due to their cheap calories requires a huge leap in logic and cannot be tested by a simple correlation analysis of food attributes (energy density and energy cost). An unintended consequence of focusing primarily on $/kcal is the creation of a belief among consumers and health professionals that a healthy diet is unaffordable to persons with low socioeconomic status, which does a disservice to this population. Indeed, there exists a wide range of cost among fruit and vegetables (7, 8), and longitudinal research indicates that food expenditure does not necessarily increase as individuals increase diet quality (9), which suggests that people are able to select healthful options within their budget. Of greater concern may be the variable availability of produce in some areas (10), which further speaks to the need to address the complexity of these issues. We fully appreciate the importance to public health of policies aimed at reallocating subsidies away from nutrient-poor foods toward nutrient-rich fruit and vegetables. However, we believe that the argument for such policies should not be based on the correlation between variables with untested effect on consumer behavior but rather on the urgent need to promote intake patterns to reduce morbidity, mortality, and related health expenditures.