76 J
.A. Bishop et al. Economics Letters 67 2000 75 –85
in Alabama and San Diego, California, which provides an ideal opportunity for comparing the effects of cash and coupons on actual behavior.
In both experiments, substantial proportions of households have nutrient deficits, especially for folacin, vitamin E, and calcium. Other households are well above requirement thresholds and are not
undernourished. To compare relative undernutrition in the treatment and control groups, we apply dominance methods as proposed by Kakwani 1989. Comparing the nutrient distributions rather than
1
summary statistics e.g. the distribution means permits us to focus on undernourished households. Kakwani 1989 has shown that cumulative distributions of nutrients can be used to evaluate
relative undernutrition in the same way that cumulative distributions of incomes have been used to evaluate relative poverty Foster and Shorrocks, 1988. Kakwani’s approach has been applied in India
Kakwani, 1989, Indonesia Ravallion, 1990, 1992, Puerto Rico Bishop et al., 1996, and the United States Bishop et al., 1992. Statistical inference procedures have also been developed for
identifying significant differences between two cumulative distributions. We apply Kakwani’s approach and the statistical inference procedures to data from the cashout experiments.
A dominance approach is well suited to the cashout data for two reasons. First, the classic experimental design of the cashout demonstrations ensures that any differences between the nutrient
distributions of cash and coupon recipients are attributable to the treatment variable. Second, the specific form of the distribution of nutrient requirements in the population does not need to be known;
we need only assume the populations have the same requirement distributions — a reasonable assumption for comparisons of the treatment and control groups in the cashout experiments.
The remainder of the paper is organized as follows. The next section provides a brief description of the methods used in the paper. After describing the data from the experiments, the empirical results
are presented and discussed. A final section offers concluding comments.
2. The dominance approach to comparing relative undernutrition
The starting point for most studies of undernutrition is a set of human dietary requirements for different age and sex groups. By far the most widely recognized of these is the Recommended Dietary
Allowances RDA developed by the Food and Nutrition Board of the National Research Council of the National Academy of Sciences. Yet, as the nutritionists readily acknowledge, the adequate
nutrition level for the human body cannot be established with exact precision; any requirement
2
threshold, such as the RDA, involves an element of arbitrariness. Moreover, in evaluating nutrient distributions it is important to remember that ‘there is no agreed-upon benefit to levels above 100’
of the RDA Emmons, 1986, p. 1690. These concerns can be addressed by evaluating relative undernutrition based on cumulative
1
In their reports to the USDA, Ohls et al. 1992 and Fraker et al. 1992 compare nutrient availability at the means of the treatment and control groups of the two experiments. They also recognize the value of comparing cumulative nutrient
distributions, but consider only a few nutrients and apply no statistical inference procedures to the distributional comparisons.
2
Devaney and Fraker 1986, p. 729 treat the RDA accordingly: ‘The RDA are not dietary ‘requirements’ in the usual sense. Rather, the RDA are intakes of nutrients judged to be adequate for nearly all persons in the United States and are set
such that they exceed the requirements except for food energy for most individuals.’ Butler and Raymond 1996, p. 783 cite several sources in the nutrition literature that propose using two-thirds the RDA as the criterion for adequate nutrition.
J .A. Bishop et al. Economics Letters 67 2000 75 –85
77
distribution functions cdfs of nutrients for two populations, truncated in the neighborhood of the RDA, assuming that the populations have the same distribution of nutrient requirements. Let F x and
1
F x be the cdfs of nutrients for two populations. Widely known procedures developed by nutrition
2
scientists can be used to adjust nutrients for the composition and size of the household, the ages of its members, and the number of meals eaten away from home. Kakwani 1989, p. 539 then establishes
the following proposition:
Proposition 1. if F x F x for all values of x, then the undernutrition in population 1 will always
1 2
be greater than or equal to that in population 2.
Proposition 1 is known as first-order dominance FOD of F by F .
1 2
An analogous theorem on the construction of poverty orderings based on income distributions for two populations is derived by Foster and Shorrocks 1988, who truncate the distributions at the
poverty line. Following Foster and Shorrocks, the distributions of nutrients could be truncated in the neighborhood of the requirement thresholds RDA. Silberberg 1985 and Butler and Raymond
1996 also propose limiting attention to the persons who are at risk of undernutrition. The logic of this approach is made very clear by Butler and Raymond 1996, p. 781, ‘ . . . above some level, extra
nutrients are superfluous. If an individual has attained a nutritionally adequate diet, then whether or not food stamp income allows him or her to consume more nutrients is irrelevant.’
In the construction of dominance tests, it is convenient to compare the inverses of the cdfs in
2 1
Proposition 1, x 5 F p, where p represents a quantile in the nutrient distribution e.g. the bottom
1
20 percent and x is an order statistic, which is the highest nutrient level in the quantile. We approximate the inverse cdf, or quantile function, as the step function formed by the vector of
conditional mean nutrient levels in the sample quintiles. We then conduct pairwise tests of corresponding conditional means for each sample, which permits us to distinguish among three
possible outcomes — dominance, statistical equivalence, and a significant crossing.
To make multiple comparisons we follow the approach of Bishop et al. 1989, who developed a
3
pairwise, joint-testing procedure for a predetermined number of points, k, in the distributions. The points can be the conditional means of the quintiles, deciles, or other quantiles in the two samples.
That is, we test the null hypothesis:
H : m 5 m i 5 1,2, . . . ,k
1
1i 2i
where m and m are the conditional means in populations 1 and 2. An appropriate test statistic for H
1i 2i
is:
1 2
ˆ ˆ
ˆ ˆ
T 5 m 2 m [Q N 1 Q N ] i 5 1,2, . . . ,k
2
i 1i
2i 1i
1 2i
2
where Q and Q
denote asymptotic variances and N and N are the number of observations in
1i 2i
1 2
samples 1 and 2. Bishop et al. 1989, show that for large samples, each of the T is asymptotically
i
standard normal and provide a procedure for estimating Q and Q .
1i 2i
We treat the individual tests at k points of the quantile function as sub-hypotheses of a joint test. To maintain the size of the joint test i.e. to control the type I error, we use critical values for the Student
3
This procedure is closely related to earlier work by Beach and Davidson 1983 and Beach and Richmond 1985.
78 J
.A. Bishop et al. Economics Letters 67 2000 75 –85
Maximum Modulus SMM distribution Miller, 1981. An approximately a -level test of the equality of two vectors of conditional quantile means rejects each of the k sub-hypotheses if
uT u . m k,`,
i a
4
where m k,` is the upper-a critical value of the SMM distribution with ` degrees of freedom. The
a
SMM critical values are obtained from Stoline and Ury 1979. In general, joint tests of the conditional means generate only partial orderings of distributions. If no
pair of conditional means is significantly different, then the two quantile functions are statistically equivalent. If a positive negative and significant difference exists at one or more quantile means and
no negative positive significant differences exist, then we have FOD. Finally, if tests yield both positive and negative significant differences, the two quantile functions intersect and no FOD ranking
is possible. However, truncated dominance may still hold below reasonable requirement thresholds. The next section describes the nutrient data provided by the cashout experiments.
3. Data
