The Bayesian Approach to Decision Making and Analysis in Nutrition Research and Practice

Published:October 01, 2019DOI:https://doi.org/10.1016/j.jand.2019.07.009

      Abstract

      This is part of a series of monographs on research design and analysis. The purpose of this article is to describe the purposes of and approach to conducting Bayesian decision making and analysis. Bayesian decision making involves basing decisions on the probability of a successful outcome, where this probability is informed by both prior information and new evidence the decision maker obtains. The statistical analysis that underlies the calculation of these probabilities is Bayesian analysis. In recent years, the Bayesian approach has been applied more commonly in both nutrition research and clinical decision making, and registered dietitian nutritionists would benefit from gaining a deeper understanding of this approach. This article provides a background of Bayesian decision making and analysis, and then presents applications of the approach in two different areas—medical diagnoses and nutrition policy research. It concludes with a description of how Bayesian decision making may be used in everyday life to allow each of us to appropriately weigh established beliefs and prior knowledge with new data and information in order to make well-informed and wise decisions.

      Keywords

      The Continuing Professional Education (CPE) quiz for this article is available for free to Academy members through the MyCDRGo app (available for iOS and Android devices) and through www.jandonline.org (click on “CPE” in the menu and then “Academy Journal CPE Articles”). Log in with your Academy of Nutrition and Dietetics or Commission on Dietetic Registration username and password, click “Journal Article Quiz” on the next page, then click the “Additional Journal CPE quizzes” button to view a list of available quizzes. Non-members may take CPE quizzes by sending a request to [email protected] . There is a fee of $45 per quiz (includes quiz and copy of article) for non-member Journal CPE. CPE quizzes are valid for 1 year after the issue date in which the articles are published.
      Research Question: What are the origins and background of Bayesian decision making and analysis, and how has it been applied in medical diagnoses, nutrition policy research, and everyday life?
      Key Findings: Bayesian decision makers base decisions on the probability of an outcome, using Bayesian analysis to account for both prior information and new evidence. This approach is used in making medical diagnoses, where diagnoses about a patient’s condition are based on both prior information about the patient and the results of diagnostic tests. Nutrition policy researchers use Bayesian analyses to draw conclusions about topics, such as nutrition program participation rates or the prevalence of nutrition-related disease in developing countries.

      What Is Bayesian Decision Making and Analysis?

      Bayesian decision making is the process in which a decision is made based on the probability of a successful outcome, where this probability is informed by both prior information and new evidence that the decision maker obtains. Bayesian analysis is the statistical analysis that underlies the calculation of these probabilities. This article describes the basics of Bayesian decision making and analysis, as applied to nutrition research and practice.
      Bayesian decision making and analysis are based on Bayes’ Theorem, a mathematical formula for updating prior probabilities based on new information or evidence. The theorem was developed originally by Thomas Bayes (1701-1761), a Presbyterian minister and mathematician in England. Bayes published only one mathematical work during his lifetime, unrelated to Bayes’ Theorem, but at the time of his death had been working on a manuscript on probability theory that was left to a friend, Richard Price.
      • Bayes T.
      An essay towards solving a problem in the doctrine of chances.
      Through Price’s efforts, Bayes’ paper, “An Essay Towards Solving a Problem in the Doctrine of Chances,” was published posthumously. Bayes’ Theorem was later generalized by Pierre LaPlace (1749-1827).
      In its basic form, Bayes’ Theorem provides a formula for calculating the probability of an event—such as the presence of a disease or nutritional condition—given some new piece of information or evidence that has been collected about the likelihood of that event. The probability, known as the posterior probability, is equal to the prior probability of that event (the belief about the likelihood of that event before the new information or evidence was obtained) multiplied by a fraction that represents the information that the new evidence provides about the likelihood of the event (referred to as the likelihood). This formula can be written:
      Prob{Event|Evidence}=Prob{Event}Prob{Evidence|Event}}/Prob{Evidence}


      In the case of Bayesian decision making for medical diagnoses, described later, the “event” is the actual presence of a disease in a patient and the “evidence” is the result of a diagnostic test for the presence of the disease in the patient.
      In its more general form, Bayes’ Theorem gives the full posterior distribution of a given parameter rather than just a single probability (and the probability can be calculated from the full distribution). This posterior distribution is equal to the prior distribution modified by the new information (the likelihood). For example, suppose a nutrition policy analyst wants to examine state-level participation in the Supplemental Nutrition Assistance Program (SNAP). The prior distribution might be based on the national proportion of eligible people who participate in SNAP, which was 79% in 2011,
      • Cunnyngham K.E.
      • Sukasih A.
      • Castner L.A.
      Empirical Bayes Shrinkage Estimates of State Supplemental Nutrition Assistance Program Rates in 2009-2011 for All Eligible People and the Working Poor.
      with some variance around the national rate to account for differences across states. New evidence on the rate for a given state could be obtained by surveying a small sample from that state. The sample might be too small to provide a precise estimate on its own but could be used in combination with the prior distribution to do so. Under Bayes’ Theorem, the prior distribution based on the national estimate would be updated using the new evidence from the state survey (the state proportion along with its standard error) to form the posterior distribution for that particular state. The mean value of this posterior distribution would be the Bayesian estimate of the SNAP participation rate among eligible households in that state.
      The logic of Bayesian decision making and analysis is straightforward. If a decision maker or researcher wishes to estimate an unknown parameter, such as the probability an individual has a disease or a state’s SNAP participation rate, the Bayesian framework assumes they start with some information. This information determines the prior probability or distribution. But the decision makers want better information, so they collect some new evidence about the unknown parameter. Bayesian analysis combines the new evidence with the prior information to yield a better estimate of the probability or distribution (the posterior). If the new information was unreliable or not especially highly accurate, then the prior information will be weighted more heavily in determining the posterior probability or distribution. However, if the new evidence is highly reliable and accurate, it will be weighted more heavily.
      Consider the state SNAP participation rate example. The prior distribution based on national estimate in this hypothetical example suggests the state will have a 79% participation rate (Figure 1). This is the starting point. But suppose that in the small sample collected in state B, only 63% of eligible people participated in SNAP. That sample provides new evidence that will contribute to the likelihood through a series of calculations. Ultimately, the Bayesian posterior estimate of the state’s participation rate will use information from both state B’s participation rate and the national participation rate. Because the new evidence was based on a small sample, the prior distribution based on the national estimate will be more heavily weighted in the calculations and the posterior estimate will likely fall somewhere closer to the prior of 79% (for example, the posterior estimate for state B might be 75%, as shown in Figure 1). If the new evidence had been based on a larger and more reliable sample, the posterior estimate would fall closer to the percentage of SNAP participants (63%) based on the state sample. This would be the case in state A, for example, where the Bayesian posterior estimate of the participation rate might be 67%. State C, with a medium-sized sample, would fall in between states A and B.
      Figure thumbnail gr1
      Figure 1Hypothetical example of calculating Bayesian estimates of state Supplemental Nutrition Assistance Program participation rates by combining information from the national estimate and an estimate based only on data from the state. Note: The size of each circle represents the precision of the estimate.
      Decision making and analysis using a Bayesian framework can be contrasted with the more traditional framework for statistical analysis, often referred to as the frequentist approach. Frequentist statistical analysis does not use prior information about the parameter being estimated, but instead relies entirely on new evidence from data collected specifically for the purpose of estimating that parameter. More fundamentally, the frequentist approach operates under the assumption that the parameter being estimated has a single true value, about which we have no prior information. The Bayesian approach, by contrast, assumes that unknown parameters have distributions of values, and we do know something about these distributions based on prior information. So rather than disregard prior knowledge, it is considered together with new knowledge.
      For example, suppose that a researcher is estimating the effect of a specific dietary program designed for weight loss on participants’ body weight. If using a Bayesian approach, the researcher would form a prior belief about the distribution of the program’s effect, most likely based on previous research on the program or similar kinds of programs. The frequentist approach, by contrast, assumes that this program has a single true effect, waiting to be discovered, and the analysis would not formally take into account any prior beliefs about this effect in producing an estimate of the program effect. This does not mean that researchers using frequentist statistical analysis entirely ignore evidence from previous studies. They may use pilot study data with smaller sample sizes or foundational randomized clinical trials. They may use such evidence to help put the results of their study into context or conduct meta-analyses summarizing the results of an area of research as a whole.
      The two approaches to analysis usually lead to differences in statistical inference—that is, what sorts of statements are made to summarize the results of the analysis and contribute to decision making based on the analysis. Frequentist analysis results in point estimates of parameter values, standard errors and CIs for these point estimates, and P values arising from hypothesis tests. For example, a frequentist analysis of the weight loss program might estimate that the program has an effect of –10 lb (a 10-lb weight loss), a standard error of 5.5, and a P value of 0.07. In the frequentist framework, the P value represents the probability that the current data could have occurred under the specified model and assumption that the null hypothesis is true, and nothing more. The P value does not, for example, tell us the probability that the null hypothesis is false. The misinterpretation of P values is widespread, according to a recent statement on this issue by the American Statistical Association.
      • Wasserstein R.
      • Lazar N.
      The ASA's statement on p-values: Context, process, and purpose.
      Under a Bayesian framework, the researcher starts with prior beliefs about or estimates of the effects of the weight loss program, then collects data in the study to provide new evidence. The Bayesian analysis combines the new data with prior beliefs to estimate the posterior distribution of the effect of the weight loss program. Statistical inference in the framework is then based on this estimated posterior distribution. Based on this distribution, a researcher can calculate the mean value of the distribution, which can be interpreted as a single estimate of the parameter, similar to the point estimate in frequentist analysis. In this example, this would be an estimate of the effect of the weight loss program. However, the researcher can also make inferences about the effectiveness of the program in probabilistic terms. For example, the researcher could use the posterior distribution to estimate the probability that the program leads to any weight loss or the probability that it leads to “substantial” weight loss—such as a weight loss of ≥5 lb. Proponents of Bayesian analysis view its ability to produce these types of probabilistic statements in a straightforward way as a key advantage, because they are helpful to decision makers in determining a course of action.
      A key limitation of the Bayesian approach is that it includes a subjective element in the sense that it requires the researcher to choose the prior distribution to include in the analysis. The researcher may choose to base this prior distribution on evidence from prior research, but even in that case must select the specific research studies upon which to base the prior distribution. Bayesian estimates of the posterior distribution, or probability statements based on that distribution, will likely be affected by this choice of a prior distribution. To address this issue, the researcher may assess the sensitivity of their estimates to the selection of different prior distributions, such as those based on highly specific information and those that reflect more uncertainty (such as a “non-informative” prior).
      The role of Bayesian analysis in decision making and policy analysis is discussed in the sections that follow. The next section discusses Bayesian decision making within the context of medical diagnoses. Following that, the article presents examples of Bayesian analysis in nutrition policy making. It concludes with a discussion of Bayesian decision making in everyday life.

      Bayesian Decision Making and Medical Diagnosis

      Registered dietitian nutritionists (RDNs) implement nutrition assessment to gather data in order to make an accurate diagnosis of nutrition-related problems in alignment with the Nutrition Care Process. Bayes’ Theorem is a useful approach in this diagnostic process.
      • Ashby D.
      • Smith A.
      Evidence-based medicine as Bayesian decision-making.
      • Sox H.
      • Higgins M.
      • Owens D.
      Medical Decision Making.
      In medical decision making, one implements Bayes’ Theorem by estimating the initial chances of a disease or condition for a person. This estimate is made based on the existing scientific evidence and clinical experience. Demographics and personal characteristics may be considered in this estimation process. Once the RDN estimates this prevalence it is designated as the prior, or pretest, probability of the disease, whether it is vitamin or mineral deficiency, protein-energy malnutrition, or a condition such as hypertension or prediabetes. For example, what is the initial probability that the presenting person has an iron or calcium deficiency? As mentioned previously, the best evidence and clinical experience would guide one’s judgment. For instance, the pretest probability estimate for iron deficiency would be higher in a menstruating young female than a young man. The protein-energy malnutrition estimate is likely to be higher for someone with a history of anorexia nervosa than one without this history.
      Once the pretest probability is estimated for a given condition then assessment data are collected—this is the new evidence or likelihood described above. The goal of using Bayes’ Theorem is to estimate the disease or medical condition probability by using the accurate and evidence-based parameters to modify the perception of pretest probability. This can be presented as:
      Pretest probability+assessment information=post-test probability


      Here, the concepts are the same as described in the first section. The pretest probability is the prior probability, the assessment information is the potentially probability-modifying new information, and the post-test probability is the posterior probability. The ultimate goal is to use a series of assessments to be confident in ruling in or out a disease or condition. In other words, the decision maker wants to collect enough new information from these assessments so that the likelihood is highly reliable and any recommended treatment is based more on what is appropriate for that particular patient rather than heavily based on the prior probability, which captures average tendencies for others who are like that particular patient. For instance, the desire is to have a high degree of certainty that someone has iron deficiency or not, alcohol use disorder or not, or coronary artery disease or not. Effective treatment will be predicated on an accurate diagnosis.
      Gleason and colleagues
      • Gleason P.M.
      • Harris J.
      • Sheean P.M.
      • Boushey C.J.
      • Bruemmer B.
      Publishing nutrition research: Validity, reliability, and diagnostic test assessment in nutrition-related research.
      present procedures for determining the accuracy of diagnostic testing and nutrition assessment. Only aspects related to the Bayesian decision-making process will be explored here. Two calculated values are particularly important in determining the accuracy of a test or assessment parameter, the sensitivity and specificity. Sensitivity is the extent to which a diagnostic test correctly identifies those who have a particular condition or disease. If a person has a condition or disease, a sensitive test will identify him or her as having the condition or disease nearly always (the test will be positive). Specificity is the extent to which a diagnostic test correctly identifies those who do not have a particular condition or disease. If a person does not have the condition or disease, a specific test will nearly always be negative. The ideal test will have a both high sensitivity and specificity, yielding few false positives or negatives. Parameters called likelihood ratios can be derived from the sensitivity and specificity of a test or assessment parameter.
      • Gleason P.M.
      • Harris J.
      • Sheean P.M.
      • Boushey C.J.
      • Bruemmer B.
      Publishing nutrition research: Validity, reliability, and diagnostic test assessment in nutrition-related research.
      The likelihood ratio (LR) for diagnostic tests is the likelihood that a given test result would be expected in a patient with the target disorder compared to the likelihood that that same result would be expected in a patient without the target disorder. A positive LR (LR+) is the likelihood that a positive test would be expected in a patient with the condition of interest compared to the likelihood of a positive test in a patient without the condition of interest. A negative LR (LR–) is the likelihood that a negative test would be expected in a patient with the condition of interest compared to the likelihood of a negative test in a patient without the condition of interest. These likelihood ratios are easily calculated using the following formulas:
      LR+=sensitivity/1-specificity
      (formula 1.1)


      LR-=1-sensitivity/specificity
      (formula 1.2)


      LR of 1 for a test or assessment parameter indicates that the diagnostic test provides no valuable additional information in determining the presence of the disease or condition. The higher the LR+ the better the test is at ruling in a disease or condition. The lower the LR, the better the test is in ruling out a disease or condition. An LR+ that is 10 or higher or LR– that is 0.1 or lower indicate a highly valuable and informative test in either ruling in or out a disease or condition. LRs for many diagnostic tests and parameters can be acquired by a thorough review of the scientific literature, using a database such as PubMed. Particular LRs are characteristic of the specific test or diagnostic parameter based on the characteristics of the sample used in determining the LRs. For instance, LRs can be different for elderly and young people for a given diagnostic test. Also, for a given test, the published LR will vary based on the different levels of the specific test. An illustration of this is found in the Table for serum ferritin (μg/L), a measure of the amount of stored iron in the liver and a parameter for diagnosing iron deficiency anemia. The Table shows values of LR+ in a diagnostic test for iron deficiency anemia in elderly people, given different test results (ie, different values of serum ferritin).
      • Guyatt G.H.
      • Oxman A.D.
      • Ali M.
      • Willan A.
      • McIlroy W.
      • Patterson C.
      Laboratory diagnosis of iron-deficiency anemia: An overview.
      As one can see in the Table, only when the diagnostic test results in a serum ferritin value of ≤15 μg/L does the resulting LR+ value meet the above criteria of ≥10.
      TableSerum ferritin cutoffs and corresponding positive likelihood ratios
      • Guyatt G.H.
      • Oxman A.D.
      • Ali M.
      • Willan A.
      • McIlroy W.
      • Patterson C.
      Laboratory diagnosis of iron-deficiency anemia: An overview.
      Serum ferritin cutoffs (μg/L)Positive likelihood ratios for iron deficiency anemia
      ≥1000.08
      45 to <1000.54
      35 to <451.83
      25 to <352.54
      15 to <258.83
      ≤1551.85
      In applying Bayes’ Theorem the LR’s represent the new information derived from diagnostic tests that may potentially modify the pretest probability estimate. In order to use LRs in determining the post-test probability of a disease or condition, the pretest probability must be converted to pretest odds by using the following formula:
      Pretest odds=pretest probability/(1-pretest probability)
      (formula 1.3)


      The odds then can be multiplied by the LR to determine the post-test odds. The post-test odds can then be converted to the post-test probability using this formula:
      Post-test probability=post-test odds/(1+post-test odds)
      (formula 1.4)


      This calculation will lead to the probability of the disease or condition, given the added information of the assessment test or parameter. If the laboratory test is highly accurate, the probability should increase substantially if there is a positive test result. In this situation, one can be more confident that a person has a disease or condition. The word substantially is defined contextually in that if the pretest probability is already high, then the difference between the pretest probability and post-test probability will be less than if the pretest probability is lower. As one can see, this modification of probability based on diagnostic test information represents the application of Bayes’ Theorem.
      If more than one test or assessment parameter is used, additional tests can be factored in by multiplying a sequence of LRs by the pretest odds:
      Post-test odds=pretest-odds×LR1(test 1)×LR2 (test 2)× . . .
      (formula 1.5)


      One hopes that by adding more tests, the accuracy of the diagnosis can be improved. However, it is important that the tests be somewhat independent of one another and be the most accurate ones.

      Example of Bayesian Decision Making in Medical Diagnosis

      Suppose Marge, an 81-year-old white woman, contacts your practice wanting some nutrition advice. As her RDN, you know that Marge has type 2 diabetes mellitus, has taken metformin for 15 years, and is a vegan. She is reporting fatigue, tingling in her hands and feet, forgetfulness, dizziness, and muscle weakness. When you examine her, you notice she is unusually pale. After having done a thorough anthropometric, clinical, and dietary assessment, you suspect possible anemia. You ask her primary care physician to order a comprehensive battery of laboratory tests, including serum ferritin and holotranscobalamin B-12 (holoTC). When you receive the results, you notice her serum ferritin is 105 μg/L and holoTC is 13 pmol/L. The published LR+ for the levels of these two measurements are 0.08 μg/L and 14 pmol/L, respectively for diagnosing either iron or vitamin B-12 deficiencies.
      • Guyatt G.H.
      • Oxman A.D.
      • Ali M.
      • Willan A.
      • McIlroy W.
      • Patterson C.
      Laboratory diagnosis of iron-deficiency anemia: An overview.
      • Valente E.
      • Scott J.M.
      • Ueland P.M.
      • Cunningham C.
      • Casey M.
      • Molloy A.M.
      Diagnostic accuracy of holotranscobalamin, methylmalonic acid, serum cobalamin, and other indicators of tissue vitamin B12 status in the elderly.
      What is the probability that she has iron deficiency anemia? How about vitamin B-12 anemia? The prevalence of iron deficiency anemia is approximately 13.8% in white older women.
      • Le C.H.
      The prevalence of anemia and moderate-severe anemia in the US population (NHANES 2003-2012).
      The prevalence of vitamin B-12 deficiency among white women older than 70 years of age is approximately 3.3%.
      • Green R.
      • Allen L.H.
      • Bjørke-Monsen A.L.
      • et al.
      Vitamin B12 deficiency.
      Note that taking metformin increases the risk of B-12 deficiency and the prevalence is near 9% for those on metformin for longer than 13 years.
      • Aroda V.R.
      • Edelstein S.L.
      • Goldberg R.B.
      • et al.
      Long-term metformin use and vitamin B12 deficiency in the Diabetes Prevention Program Outcomes Study.
      Therefore, the pretest probabilities based on the best prevalence evidence for iron deficiency anemia and B-12 deficiency are 13.8% and 9%, respectively.
      Let us focus on iron deficiency anemia first. This choice was logical to consider, given the profile and the shared symptoms of iron deficiency anemia and those associated with B-12. The pretest odds is calculated this way (formula 1.3):
      Pretest odds = 0.138/(1-0.138)=0.16


      The post-test odds is calculated in the following way using the LR+ of 0.08 based on her serum ferritin level:
      Post-test odds = 0.16×0.08=0.0128


      The post-test probability is (formula 1.4):
      Post-test probability=0.0128/(1+0.0128)=0.0126


      With the inclusion of the LR+ related to the serum ferritin value, the post-test probability is now 1.3%. The post-test probability decreased substantially from 13.8% to 1.3%. It is very unlikely Marge has iron deficiency anemia.
      What about vitamin B-12 deficiency? As previously stated, the pretest probability for older women with diabetes who are using metformin is 9%. The pretest odds (formula 1.3) is:
      Pretest odds=0.09/(1-0.09)=0.10


      Using the LR of 14 based on the holoTC test yields a post-test odds of:
      Post-test odds = 0.10×14=1.40


      The post-test probability (formula 1.4) is:
      Post-test probability=1.40/(1+1.40)=0.58


      Therefore, the probability of a B-12 deficiency has increased to 58%. This is quite an increase, but not yet entirely convincing, given that Marge is almost as likely (42% chance) to not have B-12 deficiency, as she is likely (58% chance) to have this deficiency.
      Suppose serum methyl malonic acid (MMA) was measured during the laboratory analysis and yielded a value of 0.50 μmol/L. Vitamin B-12 is a required coenzyme in the enzymatic metabolism of MMA. When missing, the level of MMA increases. The LR for this level of MMA is 2.
      • Sox H.
      • Higgins M.
      • Owens D.
      Medical Decision Making.
      This is a case in which the pretest odds can be multiplied by the LR for both holoTC and MMA. So using the previous formula (formula 1.5):
      0.10×14×2=2.80


      Converting this to post-test probability yields (formula 1.4):
      Post-test probability=2.80/(1+2.80)=0.74


      A post-test probability of 74% is much more convincing that Marge has vitamin B-12 deficiency. Suppose after the laboratory tests, the RDN asks Marge if she uses vitamin B-12–fortified foods while eating the vegan diet. If Marge answers no, the RDN can be even more convinced and begin treating Marge with high doses of sublingual B-12 supplements.

      Why a Bayesian Approach?

      It is important in clinical nutrition to avoid making too quick a diagnosis of a condition. An RDN can be biased based on history of treating individuals of a particular disease or condition. If one has encountered a much larger number of people with iron deficiency anemia than those with B-12 deficiency anemia, there might be an instinct to overlook B-12 and in a biased fashion assume iron deficiency. Frequent encounters with people having a particular disease state can prompt an RDN to see this disease first before any others when diagnosing. Also without a proper evidence base an RDN may assume that a state is more or less prevalent than it really is.

      Is This Approach Practical?

      A Bayesian process may seem to be quite laborious in environments that are pressure-packed. However, the time that can be wasted with misdiagnosis can be much greater than the time it takes to use a systematic process.

      Bayesian Analysis in Nutrition Policy Making

      A Bayesian approach to statistical analysis and decision making may be appealing to nutrition policy makers for several reasons. First, many policy decisions have the potential to be highly consequential, and so policy makers may wish to use both the evidence from the data collected for a single study at a single place and time and incorporate prior information to inform their decision making. A Bayesian approach allows the analysis to make use of prior knowledge or beliefs about the specific question being studied, as well as the new evidence collected specifically for the study.
      Second, Bayesian analysis can be translated in terms that should be highly relevant to policy makers. These policy makers are likely to be most interested in whether the policy or program “works,” in the sense that it leads to favorable changes in a particular outcome of interest (eg, a weight loss program actually leads to substantial weight loss). The results of Bayesian analysis can be easily translated into probabilities of these favorable outcomes. It is less straightforward to translate frequentist statistical analysis—with its framework of null and alternative hypotheses, P values, and statistical significance—into policy-friendly terms involving the probability that the true effect falls into a particular range of values.
      Finally, the framework in which Bayesian analysis is conducted lends itself to a flexible presentation of results, which allows policy makers to use their own judgments about what constitutes a sufficient level of evidence for them to make a policy decision. In the case of the weight loss program, for example, the policy maker could decide what constitutes a “substantial” enough weight loss to be meaningful (5 lb? 10 lb?), as well as the preferred magnitude of the probability that the program led to this weight loss to suggest that the program should be implemented more widely. Alternatively, the policy could calculate the Bayesian credible interval. The credible interval is the interval of values containing 95% of the posterior distribution—that is, we can say that there is a 95% chance that the true value falls within this interval.
      Frequentist statistical analysis can also produce useful statistical results such as the estimate of a program’s effect size and the CI of this estimate, which gives a policy maker an estimate of the program’s effect and an indication of the precision of that estimate. However, the interpretation of the CI in frequentist statistical analysis aligns less well with the way most people think about the world. As emphasized by Greenland and colleagues,
      • Greenland S.
      • Senn S.J.
      • Rothman K.J.
      • et al.
      Statistical tests, P values, confidence intervals, and power: A guide to misinterpretations.
      it does not imply that there is a 95% chance that the true value falls within this interval; rather, “the 95% refers only to how often 95% confidence intervals computed from very many studies would contain the true size if all the assumptions used to compute the intervals were correct.”
      This section presents two examples of analyses that used a Bayesian approach. The first example is the analysis of state SNAP participation rates introduced above. The second is an analysis of trends in the prevalence of vitamin A deficiency and its consequences in a set of developing countries around the world.

      State SNAP Participation Rates

      SNAP provides nutrition assistance to low-income households by increasing their food purchasing power. A state’s SNAP participation rate is the percentage of eligible individuals in that state who receive SNAP benefits in a given month. Policy makers may be interested in knowing state SNAP participation rates for several reasons. First, the participation rate is a measure of how well the program is reaching its target population of eligible households. A low participation rate in a state may indicate that there are households in need that are not receiving benefits that might help them provide more nutritious diets for household members. Variation in this rate may suggest states in which there are program implementation issues, or factors that discourage eligible households from applying for benefits, as well as states with policies and conditions leading to high participation rates. Finally, estimating participation rates may help state and national SNAP offices more accurately project future program budgets.
      Each year, the US Department of Agriculture collects administrative data on the number of people receiving SNAP in each state, but does not necessarily have complete data on the number who are eligible for benefits. To estimate SNAP participation rates, information on participation must be combined with estimates of the numbers of eligible people. The Current Population Survey (CPS), for example, provides detailed income information and other relevant data on large nationally representative samples, which can be used to estimate the number of eligible individuals nationally and, thus, national SNAP participation rates. Because CPS samples are so large overall, they can also provide representative data on the number of eligible individuals in states. The CPS samples at the state level in any given year, however, tend to be small. Thus, estimates of the SNAP participation rate in a given state would be highly imprecise if based only on data from that state (especially in smaller states). As a result, policy makers examining such estimates would be less certain that states with high estimated participation rates actually had policies and factors leading more eligible households to participate, worrying instead that these high rates could have been the result of estimation error.
      Cunnyngham and colleagues
      • Cunnyngham K.E.
      • Sukasih A.
      • Castner L.A.
      Empirical Bayes Shrinkage Estimates of State Supplemental Nutrition Assistance Program Rates in 2009-2011 for All Eligible People and the Working Poor.
      addressed this challenge by using a Bayesian approach to estimating state SNAP participation rates by combining prior information with new evidence. In this context, the researchers’ prior information on the SNAP participation rate in a given state (that is, the prior distribution of the state participation rate) was based on CPS data on SNAP participation rates in other states with similar characteristics (eg, similar demographic characteristics and economic conditions). For example, for a rural, Western state like Wyoming, the prior distribution would have been based on the estimated participation rates in similar states (eg, Idaho, Montana, North Dakota, South Dakota). The Bayesian likelihood portion of the calculation of a state’s SNAP participation rate (the new information) came from the direct evidence based on the CPS and administrative data collected from that state itself (from those sample members living in Wyoming in the above example). In larger states with larger CPS samples, the Bayesian estimate of the participation rate weighted this direct evidence more heavily because the estimated participation rates based only on data from the state was more precise. In smaller states with smaller CPS samples, this direct evidence was less heavily weighted and the prior estimates based on participation rates in other, similar states received a bit more weight.
      The authors used this approach to produce estimates of SNAP participation rates for every state and the District of Columbia in a given year (2011). They also ranked states from the highest to lowest estimated participation rates, including measures of statistical uncertainty. As a result, policy makers could use these estimates to identify possible policies and conditions in states that were associated with greater SNAP participation. It should be noted that while some models based on a frequentist approach would calculate SNAP participation rates for a given state using only data from that state, other models (such as a generalized linear model with random state effects) could also be used to incorporate information from other states in estimating a given state’s effect. The strength of the Bayesian approach, however, is that this use of additional or prior information is explicit in the estimation framework.

      Vitamin A Deficiency and Mortality among Children across the World

      In this study, Stevens and colleagues
      • Stevens G.A.
      • Bennett J.E.
      • Hennocq Q.
      • et al.
      Trends in mortality effects of vitamin A deficiency in children in 138 low-income and middle-income countries between 1991 and 2013: A pooled analysis of population-based surveys.
      were interested in estimating trends in the prevalence of vitamin A deficiency among children in 83 low- and middle-income countries. The motivation for this study was that vitamin A deficiency is a relatively common nutritional problem among children in developing countries, and is a risk factor for blindness and mortality from diarrhea and measles among children from 6 to 59 months. Moreover, there is a potentially effective policy remedy, vitamin A supplementation, which is recommended by the World Health Organization.
      WHO
      Global prevalence of vitamin A deficiency in populations at risk 1995-2005. WHO Global Database on Vitamin A Deficiency.
      However, to most effectively plan for and implement these policies, accurate information is needed on where vitamin A deficiency is most common and what its trends look like in different countries and regions.
      The researchers obtained available data and published research from 134 data sources covering these countries during the 1991 to 2013 period. The challenge to examining country-specific trends was that for any given year and country there may have been relatively few data upon which to base an estimate. If the researchers had based the estimate of the prevalence of vitamin A deficiency for a given year and country relying on the data from that year and country alone, these estimates would have been very imprecise. This imprecision may have made it appear as though there were dramatic changes in prevalence from year to year in a country, or big differences in vitamin A deficiency between similar countries in the same region.
      To address these issues in a straightforward way, the researchers used a Bayesian model. To estimate the prevalence of vitamin A deficiency in a given country and year, this approach took advantage of both the data for that country and year, as well as other data on vitamin A deficiency prevalence to inform the priors for the estimate. In the Bayesian framework described here, the data for that country and year provided the likelihood portion of the Bayesian prevalence estimate. The prior distribution was based on data on the prevalence of vitamin A deficiency in other years and other countries. Stevens and colleagues describe how this worked
      • Stevens G.A.
      • Bennett J.E.
      • Hennocq Q.
      • et al.
      Trends in mortality effects of vitamin A deficiency in children in 138 low-income and middle-income countries between 1991 and 2013: A pooled analysis of population-based surveys.
      :In the . . . model, estimates for each country-year were informed by data from that country-year itself, if available, and by data from other years in the same country and in other countries, especially those in the same region with data from similar time periods. The . . . model shares information to a greater degree where data are non-existent or weakly informative (i.e., have large uncertainty), and to a lesser degree in countries or regions and in years with more data.
      To make sure the prior distribution was as accurate as possible for a given country and year, the model included covariates as well. In other words, the prior was informed most strongly by available data from countries and years most similar to the country and year being estimated. These covariates included national income, maternal education, proportion of the population in urban areas, mean weight-for-age z score, and a measure of the availability of calories and animal-source foods.
      • Stevens G.A.
      • Bennett J.E.
      • Hennocq Q.
      • et al.
      Trends in mortality effects of vitamin A deficiency in children in 138 low-income and middle-income countries between 1991 and 2013: A pooled analysis of population-based surveys.
      Based on the posterior distributions of vitamin A deficiency, the researchers presented three types of estimates. For each country or group of countries in a given year, they presented the estimated prevalence of vitamin A deficiency, as well as the Bayesian 95% credible interval for the estimate. The credible interval is the interval of values containing 95% of the posterior distribution—that is, we can say that there is a 95% chance that the true value falls within this interval. Finally, they estimated the prevalence of vitamin A deficiency at the beginning (1991) and end (2013) of their period of study, and presented the posterior probability that this prevalence declined over this period.
      • Stevens G.A.
      • Bennett J.E.
      • Hennocq Q.
      • et al.
      Trends in mortality effects of vitamin A deficiency in children in 138 low-income and middle-income countries between 1991 and 2013: A pooled analysis of population-based surveys.
      Using this Bayesian approach, Stevens and colleagues
      • Stevens G.A.
      • Bennett J.E.
      • Hennocq Q.
      • et al.
      Trends in mortality effects of vitamin A deficiency in children in 138 low-income and middle-income countries between 1991 and 2013: A pooled analysis of population-based surveys.
      found that the prevalence of vitamin A deficiency declined from 39% to 29% across the full set of 83 low- and middle-income countries, and the posterior probability of a true decrease in prevalence was 81%. Prevalence rates varied greatly by region, however. By the end of the period, prevalence rates were 44% and 48% in South Asia and Sub-Saharan Africa, respectively, and ≤11% in the other three regions examined. In further analysis, they concluded that the vast majority of children’s deaths worldwide from diarrhea and measles due to vitamin A deficiency were in these two regions. These findings have clear policy implications—the researchers suggest that the findings “should be used to reconsider, and possibly revise, the list of priority countries for high-dose vitamin A supplementation.”
      • Stevens G.A.
      • Bennett J.E.
      • Hennocq Q.
      • et al.
      Trends in mortality effects of vitamin A deficiency in children in 138 low-income and middle-income countries between 1991 and 2013: A pooled analysis of population-based surveys.
      The analysis relied on a Bayesian approach to estimate the prevalence of vitamin A deficiency in each country in each year of the study. This approach combined prior evidence (based on data from other countries or years) with new information (based on data from that country and year) to estimate this prevalence. If the researchers had used only data from that country and year (in some cases based on small samples), it may not have been possible to make such policy recommendations. For example, this approach may have found that the estimated prevalence of vitamin A deficiency in South Asia and Sub-Saharan Africa was not significantly different from the estimated prevalence in the other regions (even if the point estimates of the prevalence rates were higher in South Asia and Sub-Saharan Africa). Without the clear finding of a statistically significant difference, policy makers may have been reluctant to take action based on this study. Without considering this additional information (the prior information, in a Bayesian framework) a significant argument for public health intervention may have been lost.

      A Bayesian Approach to Everyday Decision Making

      As has been presented, Bayes’ Theorem and a Bayesian approach involves probability statements. However, the principles underlying the Theorem can be used to guide systematic decision making and belief shifting in daily life.
      • Bain R.
      Are our brains Bayesian?.
      • Achtziger A.
      • Alos-Ferrer C.
      • Hugelschafer S.
      • Steinhauser M.
      The neural basis of belief updating and rational decision making.
      • Dvorsky G.
      How Bayes’ Rule Can Make You A Better Thinker. @io9.
      • de Lange F.P.
      • Heilbron M.
      • Kok P.
      How do expectations shape perception?.
      • Gigerenzer G.
      • Garcia-Retamero R.
      Cassandra's regret: The psychology of not wanting to know.
      The idea that one can state an initial position about a decision or belief but then remain flexible in thinking with the acquisition of additional information and evidence is helpful in making better decisions and having more useful beliefs. This is termed Bayesian updating in the scientific literature.
      • Buser T.
      • Gerhards L.
      • van der Weele J.
      Responsiveness to feedback as a personal trait.
      Bayes’ Theorem in principle can be very valuable to an RDN.
      Each RDN holds a personal set of beliefs about a variety of issues related as well as unrelated to the profession. Suppose an RDN is a vegan and has arrived at that position through processing scientific, political, and ethical information. On a scale from 0 to 10, with 0 being an unapologetic carnivore and 10 a completely committed vegan, this RDN is a 10. She writes articles about veganism and in her practice encourages even to the point of pushing a vegan diet. She has a choice of whether she will be flexible in her thinking or dogmatic in her dietary approach. In one of her graduate courses she is assigned to conduct a review of scientific literature. She conducts a search on the hazards of eating animal products but stumbles upon credible evidence verifying the value of dairy products, including information on the benefits of dairy products in influencing metabolic syndrome biomarkers in those with type 2 diabetes mellitus.
      • Imamura F.
      • Fretts A.
      • Marklund M.
      • et al.
      Fatty acid biomarkers of dairy fat consumption and incidence of type 2 diabetes: A pooled analysis of prospective cohort studies.
      What will she do with this information incongruent with her beliefs? A Bayesian approach would dictate that prior information or beliefs are important but malleable and that beliefs should have a flexibility based on additional information. If not using a Bayesian approach, she would base her beliefs only on the new information she collected (alternatively, she might disregard any new information and continue with her original beliefs). If the RDN were following a Bayesian approach, she would consciously acknowledge her prior position, but then modify her beliefs based on additional information. Having started at a 10 on veganism she would be willing to adjust those beliefs based on what she has learned in her research, perhaps shifting to a 5, and let that shift be reflected in her writing and practice.
      It would be valuable to teach and encourage a Bayesian approach in dietetics education. Embracing this approach would encourage an RDN to use all of the available information—whether prior or new—and reduce the risk of an RDN engaging in quackery and dogmatic non-evidence-based approaches. It is noteworthy that a Bayesian approach in this situation takes into consideration prior beliefs with the new information modifying them, not completely discounting or trumping the prior beliefs. In this situation, the RDN would not necessarily completely abandon veganism, given the information and data on the value of plant-based diets as well as her ethical considerations. The new information is considered in the context of the prior beliefs, with the prior beliefs providing a valuable starting point.
      It is vital that RDNs maintain a flexibility in thinking so belief modifications can be made as the situation and evidence arises. Also, it is important that dietetics and nutrition-related scientists maintain a Bayesian approach in conducting their research. A scientist who is flexible in thinking is less likely to introduce bias into their research designs and studies.
      In addition, the Bayesian approach can be applied to everyday decisions. A private practice RDN may be contemplating the acquisition of professional liability insurance. Initially, he minimizes his need to obtain it. He believes that it is very unlikely that he will engage in any behavior that could result in litigation because of his ethical behavior and professionalism. Taking a Bayesian approach, he consciously assesses his position on the insurance. Next, he conducts a search of the literature to determine the probability of litigious events among RDNs. He also examines reported cases of RDNs being sued to determine the financial, personal, and professional risks. Upon examination of all the evidence, he decides the risk is not worth taking and acquires the insurance.
      A senior-level foodservice RDN has worked in the past as a military institutional foodservice manager. Her leadership style had been authoritarian with a strong “top-down” approach, giving orders and expecting them to be followed. Upon retirement from the military, she has acquired a top foodservice management leadership position in a University-based medical center. She could very well use an authoritarian leadership style based on past experience. If she chose a Bayesian approach, she would study the most current research regarding the most effective style of leadership in private sector situations. In addition, she would consult with leaders in the medical center to discuss with them the most effective approach to leadership in that setting. Also, she might consult with members of her Academy of Nutrition and Dietetics Dietetic Practice Group. As a result of gathering this information, she might choose to use a more collaborative leadership approach rather than an authoritarian one, especially with her direct foodservice supervisor reports. So she had an initial view about leadership, but by gathering reliable information she modified her approach believing that it would be more effective in the new setting.
      Even though an individual may not be strong in mathematics, Bayes’ Theorem can be applied to improve dietetics practice and promote personal transformation. This approach can help guide the individual to be more flexible in their thinking and less biased in their decision making and beliefs.

      Conclusions

      Bayesian analysis and decision making is an approach to drawing evidence-based conclusions about a particular hypothesis on the basis of both prior information relevant to that hypothesis and new evidence collected specifically to address it. Mathematically, the approach is based on Bayes’ Theorem, which dates back to the 18th century. Bayesian analysis and decision making can be applied in a variety of different contexts. In this article, we have described its use in medical diagnoses, nutrition policy making, and everyday decision making. RDNs and others should consider using these methods in these and other settings. See Figure 2 for a list of terms useful in understanding Bayesian decision making and analysis in nutrition research and practice. For a more thorough treatment of Bayesian analysis, refer to A Student’s Guide to Bayesian Statistics by Lambert.
      • Lambert B.
      A Student's Guide to Bayesian Statistics.
      Figure 2Glossary: the Bayesian approach to decision making and analysis in nutrition research and practice.
      The following terms are useful in understanding Bayesian decision making and analysis in nutrition research and practice.
      Bayesian analysis: Statistical analysis used to calculate Bayesian posterior distributions and probabilities based on prior information and new evidence.
      Bayesian decision making: Process in which a decision is made based on the probability of a successful outcome, where this probability is informed by both prior information and new evidence that the decision maker obtains.
      Bayesian updating: Process in which an initial position or set of beliefs based on prior information is revised on the basis of additional information and evidence considered in conjunction with the prior information.
      Bayes’ Theorem: Mathematical formula originally developed by Thomas Bayes and used for updating prior probabilities on the basis of new information or evidence.
      Credible interval: Extent to which several different measures of a concept agree with each other and with a test measure of that concept; considered a type of relative validity.
      Distribution: Set of possible values of a random variable (such as a parameter being estimated) with an associated probability that the variable takes on each of these values.
      False negative: Result of a test for the presence of a disease or condition indicating that the disease or condition is not present for a given subject when in fact the disease or condition is present.
      False positive: Result of a test for the presence of a disease or condition indicating that the disease or condition is present for a given subject when in fact the disease or condition is not present.
      Frequentist statistical analysis: Approach to analysis that relies solely on patterns and frequencies in the data set being examined and does not make use of prior information about the parameter being estimated.
      Likelihood: In Bayes Theorem, refers to new information collected to update prior estimates of the probability of an event or the prior distribution of a parameter.
      Negative likelihood ratio (LR–): Measure of the usefulness of a diagnostic test for the presence of a particular condition or disease; indicates the odds of the test yielding a false negative among those with the condition or disease relative to yielding a true negative among those without the condition or disease; lower values of the LR– indicate that the test is effective at ruling out the condition or disease.
      Odds: The odds of an event is the ratio of the probability that the event occurs to the probability that the event does not occur.
      Positive likelihood ratio (LR+): Measure of the usefulness of a diagnostic test for the presence of a particular condition or disease; indicates the odds of the test yielding a true positive among those with the condition or disease relative to yielding a false positive among those without the condition or disease; higher values of the LR+ indicate that the test is effective at establishing the condition or disease.
      Posterior distribution: The estimated distribution of a parameter based on prior information and new data providing additional evidence about the distribution; in Bayesian statistics, the posterior distribution is estimated based on the prior distribution and likelihood.
      Posterior probability: The estimated probability of an event based on prior information and new data providing additional evidence about the probability; in Bayesian statistics, the posterior probability is estimated based on the prior probability and likelihood.
      Post-test probability: Another term for the posterior probability.
      Pre-test probability: Another term for the prior probability.
      Prevalence: Measure of how common a condition or disease is in the population of interest; the percentage of all individuals who have the condition or disease.
      Prior distribution: The estimated distribution of a parameter based only on prior information such as prior research on the same topic; in Bayesian statistics, the prior distribution is combined with the likelihood (based on new evidence) to estimate the posterior distribution.
      Prior probability: The estimated probability of an event based only on prior information such as estimates from prior research on the same probability; in Bayesian statistics, the prior probability is combined with the likelihood (based on new evidence about the probability) to estimate the posterior probability.
      P value: The probability of an estimate at least as extreme as the observed result in a sample under the assumption that the null hypothesis is true; when the P value is less than a predetermined critical value (such as 0.05) then the researcher rejects the null hypothesis.
      Sensitivity: Extent to which a diagnostic test correctly identifies those who have a particular condition or disease; if a person has a condition or disease, a sensitive test will identify them as having the condition or disease nearly always.
      Specificity: Extent to which a diagnostic test correctly identifies those who do not have a particular condition or disease; for those without the condition or disease, a specific test will nearly always be negative.
      Statistical inference: Process of drawing conclusions about characteristics of a population (or population parameters) on the basis of analysis of data from a sample drawn from that population.

      Acknowledgements

      The authors thank Carol Boushey, PhD, MPH, RD; Elizabeth Metallinos-Katsaras, PhD, RD; Mary Naglak, PhD, MMSc, RD, LDN; and Linda Snetselaar, PhD, RD, LDN, FAND, for insightful comments on a draft of this manuscript. We also thank all members of the Journal of the Academy of Nutrition and Dietetics STATS team for useful discussions.

      Author Contributions

      P. M. Gleason and J. E. Harris contributed equally to writing the initial draft of the paper and on subsequent revisions after receiving comments.

      References

        • Bayes T.
        An essay towards solving a problem in the doctrine of chances.
        Philos Trans R Soc Lond. 1763; 53: 370-418
        http://rstl.royalsocietypublishing.org/
        Date accessed: October 25, 2018
        • Cunnyngham K.E.
        • Sukasih A.
        • Castner L.A.
        Empirical Bayes Shrinkage Estimates of State Supplemental Nutrition Assistance Program Rates in 2009-2011 for All Eligible People and the Working Poor.
        Mathematica Policy Research, Washington, DC2014
        • Wasserstein R.
        • Lazar N.
        The ASA's statement on p-values: Context, process, and purpose.
        Am Stat. 2016; 70: 129-133
        • Ashby D.
        • Smith A.
        Evidence-based medicine as Bayesian decision-making.
        Statist Med. 2000; 19: 3291-3305
        • Sox H.
        • Higgins M.
        • Owens D.
        Medical Decision Making.
        Second ed. Wiley and Sons, Hoboken, NJ2013
        • Gleason P.M.
        • Harris J.
        • Sheean P.M.
        • Boushey C.J.
        • Bruemmer B.
        Publishing nutrition research: Validity, reliability, and diagnostic test assessment in nutrition-related research.
        J Am Diet Assoc. 2010; 110: 409-419
        • Guyatt G.H.
        • Oxman A.D.
        • Ali M.
        • Willan A.
        • McIlroy W.
        • Patterson C.
        Laboratory diagnosis of iron-deficiency anemia: An overview.
        J Gen Intern Med. 1992; 7: 145-153
        • Valente E.
        • Scott J.M.
        • Ueland P.M.
        • Cunningham C.
        • Casey M.
        • Molloy A.M.
        Diagnostic accuracy of holotranscobalamin, methylmalonic acid, serum cobalamin, and other indicators of tissue vitamin B12 status in the elderly.
        Clin Chem. 2011; 57: 856-863
        • Le C.H.
        The prevalence of anemia and moderate-severe anemia in the US population (NHANES 2003-2012).
        PLoS One. 2016; 11e0166635
        • Green R.
        • Allen L.H.
        • Bjørke-Monsen A.L.
        • et al.
        Vitamin B12 deficiency.
        Nat Rev Dis Primers. 2017; 3: 17040
        • Aroda V.R.
        • Edelstein S.L.
        • Goldberg R.B.
        • et al.
        Long-term metformin use and vitamin B12 deficiency in the Diabetes Prevention Program Outcomes Study.
        J Clin Endocrinol Metab. 2016; 101: 1754-1761
        • Greenland S.
        • Senn S.J.
        • Rothman K.J.
        • et al.
        Statistical tests, P values, confidence intervals, and power: A guide to misinterpretations.
        Eur J Epidemiol. 2016; 31: 337-350
        • Stevens G.A.
        • Bennett J.E.
        • Hennocq Q.
        • et al.
        Trends in mortality effects of vitamin A deficiency in children in 138 low-income and middle-income countries between 1991 and 2013: A pooled analysis of population-based surveys.
        Lancet. 2015; 3: e528-e536
        • WHO
        Global prevalence of vitamin A deficiency in populations at risk 1995-2005. WHO Global Database on Vitamin A Deficiency.
        World Health Organization, Geneva, Switzerland2009
        • Bain R.
        Are our brains Bayesian?.
        Significance. 2016; 13: 14-19
        • Achtziger A.
        • Alos-Ferrer C.
        • Hugelschafer S.
        • Steinhauser M.
        The neural basis of belief updating and rational decision making.
        Soc Cogn Affect Neurosci. 2014; 9: 55-62
        • Dvorsky G.
        How Bayes’ Rule Can Make You A Better Thinker. @io9.
        • de Lange F.P.
        • Heilbron M.
        • Kok P.
        How do expectations shape perception?.
        Trends Cogn Sci. 2018; 22: 764-779
        • Gigerenzer G.
        • Garcia-Retamero R.
        Cassandra's regret: The psychology of not wanting to know.
        Psychol Rev. 2017; 124: 179-196
        • Buser T.
        • Gerhards L.
        • van der Weele J.
        Responsiveness to feedback as a personal trait.
        J Risk Uncertain. 2018; 56: 165-192
        • Imamura F.
        • Fretts A.
        • Marklund M.
        • et al.
        Fatty acid biomarkers of dairy fat consumption and incidence of type 2 diabetes: A pooled analysis of prospective cohort studies.
        PLoS Med. 2018; 15e1002670
        • Lambert B.
        A Student's Guide to Bayesian Statistics.
        Sage Publications Ltd, Los Angeles, CA2018

      Biography

      P. M. Gleason is a senior fellow, Mathematica Policy Research, Geneva, NY.
      J. E. Harris is a professor and chair of the Department of Nutrition, West Chester University of Pennsylvania, West Chester.