Nutritional Deprivation Index

Let us first define an (N × D)-dimensional households' food groups consumption matrix:

$$x = \left[ {\matrix{ {\matrix{ {x_{11}} & \ldots & {x_{1D}} \cr \vdots & \ddots & \vdots \cr } } \hfill \cr {\matrix{ {x_{n1}} & \ldots & {x_{iD}} \cr } } \hfill \cr {\matrix{ \vdots & \ddots & \vdots \cr {x_{N1}} & \ldots & {x_{nD}} \cr } } \hfill \cr } } \right],$$

where the generic element x_ij indicates the amount of food group d consumed by the nth household. That is, each row in x represents the amounts of food groups 1,…, D consumed by the nth household.

Next, for each household n = 1,…, N let us define an (h_n × d)-dimensional matrix of household's minimum consumption requirements for the h_n members of the nth household:

$$S_n = \left[{\matrix{ {\matrix{ {s_{11}} & \ldots & {s_{1D}} \cr } } \hfill \cr {\matrix{ \vdots & \ddots & \vdots \cr } } \hfill \cr {\matrix{ {s_{h_n1}} & \ldots & {s_{h_nD}} \cr } } \hfill \cr } } \right].$$

In this case, each row in S_n corresponds to a distinct member of the nth household whose minimum consumption requirements are based on her/his characteristics (age, sex and/or other factors). Note that the dimension h_n will vary from one household to another based on the number of individuals in the household.

The (N × D) matrix of households' cut-offs Z can then be obtained by placing at each row n of Z the corresponding row sum of the sub-matrix S_n:

$$Z = \left[{\matrix{ {\matrix{ {\mathop \sum \limits_{\,j = 1}^{h_1} s_{\,j1}} & \ldots & {\mathop \sum \limits_{\,j = 1}^{h_1} s_{\,jd}} \cr \vdots & \ddots & \vdots \cr } } \hfill \cr {\matrix{ {\mathop \sum \limits_{\,j = 1}^{h_n} s_{\,j1}} & \ldots & {\mathop \sum \limits_{\,j = 1}^{h_n} s_{\,jd}} \cr } } \hfill \cr {\matrix{ \vdots & \ddots & \vdots \cr {\mathop \sum \limits_{\,j = 1}^{h_N} s_{\,j1}} & \ldots & {\mathop \sum \limits_{\,j = 1}^{h_N} s_{\,jd}} \cr } } \hfill \cr } } \right] = \left[{\matrix{ {\matrix{ {z_{11}} & \ldots & {z_{1D}} \cr \vdots & \ddots & \vdots \cr } } \hfill \cr {\matrix{ {z_{n1}} & \ldots & {z_{nD}} \cr } } \hfill \cr {\matrix{ \vdots & \ddots & \vdots \cr {z_{N1}} & \ldots & {z_{ND}} \cr } } \hfill \cr } } \right].$$

We can then compute the deprivation matrix B in which the generic element b _nd = 1 if x _nd < z _nd and b _nd = 0 otherwise.

Finally, given the vector of weights w = (w ₁, …, w _D) (used to define the importance of each food group for a diverse diet), we can calculate the deprivation score for each household as:

$${\rm ND}{\rm I}_n = \mathop \sum \limits_{d = 1}^D w_db_{nd}\comma \;\;\forall n = 1\comma \;\ldots \comma \;N.$$

The values of the NDI, which fall within the range of $\lsqb 0\comma \;\sum _dw_d\rsqb$, are higher the higher the number of food group deprivations. Note that if each food group is assigned an equal weight (w _d = 1 if normalised), the values of the NDI will fall within the range of [0, D].

Applying the second cut-off (corresponding to a minimum number of deprivations required to be considered malnourished), we obtain a binary version of the NDI, also referred to as the censored deprivation index:

$${\rm ND}{\rm I}_n\lpar k \rpar = \mathop \sum \limits_{d = 1}^D w_d\lpar {b_{nd}I\lpar {n\comma \;k} \rpar } \rpar \comma \;\;\forall n = 1\comma \;\ldots \comma \;N\comma \;$$

where $I\lpar {n\comma \;k} \rpar = I\left[{\left({\sum_db_{nd}} \right)\ge k} \right]$ is an indicator function that assumes the value of 1 if the household n is deprived in at least k food groups and 0 otherwise.

Finally, we can construct the population-level measures. The intensity of deprivation can be obtained as:

$$A = \displaystyle{1 \over Q}\left({\mathop \sum \limits_{n = i}^Q {\rm ND}{\rm I}_n\lpar k \rpar } \right)\comma \;$$

where $Q = \sum _{n = 1}^N I\lpar {n\comma \;k} \rpar$, whereas the incidence (headcount ratio) of households deprived in multiple dimensions as $H = \left( {Q/N} \right)$. The adjusted headcount ratio is then obtained as:

$${M = H \times A = \displaystyle{1 \over N}\mathop \sum \limits_{n = 1}^N {\rm ND}{\rm I}_n\lpar k \rpar}.$$

National Income and Expenditure Survey 2011–2012

The data used in the present study were obtained from the National Income and Expenditure Survey of 2011–2012 (Encuesta de Ingresos y Gastos (EIG) 2011–2012). This was a nationally and sub-nationally representative national household survey conducted by the General Directorate of Statistics, Surveys and Censuses (DGEEC) between August 2011 and July 2012.

The survey collected demographic, socio-economic and expenditure data from a sample of 5417 households, of which 3446 (63 %) were urban and 1971 (37 %) were rural. These households contained a total of 21 130 individuals, implying an average size of a household of 3⋅9 members (one out of four households had six members or more). The survey used a two-stage stratified household design.

Households’ consumption matrices

The use of the NDI requires the construction of the minimum consumption requirement matrix (S_n), households' cut-off matrix (Z) and household consumption matrix (X).

We followed the healthy US-style eating pattern as a basis for the construction of households’ minimum consumption requirements matrix (S_n)⁽¹²⁾. This pattern identifies recommended amounts of foods, in nutrient-dense forms, that an individual should consume from six major food groups (fruits, vegetables, grains, dairy products, protein foods and oils) and their sub-groups in order to meet nutrient and dietary guidelines standards, and also considers a limit on the maximum amount of energy available for other uses, such as added sugars, solid fats, added refined starches, or alcohol. The pattern considers twelve energy levels (from 1000 to 3200 kcal/d (4180 to 13 390 kJ/d)) to meet the needs of an individual across the lifespan.

We used the energy needs estimates provided by the Institute of Medicine to determine the energy level for each member of the household conditional on her age, sex and the level of physical activity⁽¹³⁾ (we restricted our analysis to the sedentary level of physical activity). These estimates are based on the estimated energy requirements equations, using reference heights and reference weights for each age–sex group. In other words, the household-specific minimum consumption requirement matrix is determined by vectors of age-, sex- and activity-specific recommended amounts for each household member. Applying the same procedure to all households then yields matrices S ₁,…, S_N of households' minimum consumption requirements.

The households' cut-off matrix can be calculated as described in the previous section, with each row consisting of D sums of the minimum consumption requirements across all household members, where D represents the number of food groups. Again, recall that we do not consider individual-level cut-offs because the survey only provides consumption data at the aggregated, household-level (see the previous section).

Construction of the household's consumption matrix (X) requires the knowledge of the actual household consumption. Although our data do not provide the actual amounts of foods consumed by the household, they provide a detailed information about the quantities of (as well as the corresponding expenditures on) over 900 unique food items purchased or otherwise acquired by the households over the previous 7 d. For the purposes of our analysis, we only considered food items that the household either purchased or self-produced. Due to significant use of non-standard acquisition units, we did not consider food items that the household received from another household, from a social protection or nutrition programme, or as a gift from church or a non-profit institution, or that either member of the household took from a business. Of the total number of food items acquired by the households over the last 7 d, 90⋅1 % were purchased or self-produced; food items that either member of the household took from a business accounted for 5⋅36 %, food items that the household received from another household accounted for 3⋅82 %, and food items that the household received from a social protection or nutrition programme, or as a gift from church or a non-profit institution, accounted for the remaining <1 %.

Our analysis also excludes alcoholic and non-alcoholic drinks, sweets, spices, condiments and foods consumed away from home. Most foods consumed away from home include dishes whose serving sizes and ingredients are not standardised and thus would require making assumptions about both absolute and relative amounts of their individual food components. However, as these foods represent but a small fraction of households' expenditures in the data (evidencing the fact that eating out is far less common in Paraguay than in developed countries), we have no reason to believe that excluding them significantly impacts the results of the present study.

Using the data, we first classified food items into six food groups, including fruits, vegetables, grains, protein foods, dairy products, and oils^{(Reference Alkire, Foster and Seth8)}. Specifically, the fruits group was constructed by including all the fruit varieties, including fresh, frozen, canned and dried fruits and fruit juices (for example, bananas, grapes, raisins, oranges and orange juice); the vegetables group was constructed by including all the vegetable varieties in fresh, frozen, or canned form; and the proteins group was constructed by including all fish/seafood, meat, poultry, eggs, soya and soya products, nuts and seeds.

Next, we further split vegetables, grains and protein foods groups into food sub-groups. In particular, the vegetables group was classified into five sub-groups, including dark-green vegetables (for example, broccoli, collard greens, kale, spinach), red and orange vegetables (for example, carrots, pumpkin, red peppers, sweet potato, tomatoes), legumes (for example, black beans, garbanzos, green soyabeans, kidney beans, lentils, pinto beans, white beans), starchy vegetables (for example, cassava, green lima beans, green peas, plantains, potatoes) and other vegetables (for example, common lettuce, onion, cucumber, cabbage, celery, mushrooms, green peppers); the grains group was classified into two sub-groups, including whole grains and refined grains; and the protein foods group was classified into three sub-groups, including meats, eggs, soya and soya products, nuts and seeds. Meat and eggs represented by the far the most important constituents of the protein foods group, both in terms of volumes purchased and the relative expenditures. Soya-based products represented a small part of the group (in relative terms) as their consumption remains limited in Paraguay. Given that fruits, dairy products and oils groups remained a single item as before, the final classification included a total of thirteen groups or sub-groups.

For the purposes of our analysis, we converted each food item to its cup- (in case of fruits and vegetables) or ounce- (in case of protein foods) equivalents⁽¹²⁾. For fruits and vegetables, a cup-equivalent corresponds to one cup of vegetable or fruit, one cup of vegetable or fruit juice, two cups of leafy salad greens and 0⋅5 cup of dried fruit or vegetable. For protein foods, 1 ounce-equivalent corresponds to approximately 1 ounce of lean meat, poultry, or fish/seafood, one egg, one tablespoon of peanut butter and 0⋅5 ounce of nuts or seeds. We applied the Food Patterns Equivalents Ingredients Database (FPID) cup-equivalent weights and, where appropriate, the FPID in combination with ARS Food Intakes Converted to Retail Commodities Database (FICRCD) conversion factors to estimate the amount of raw fruits and vegetables to be purchased in order to obtain one cup-equivalent of raw (edible) portion of each food item^{(Reference Bowman, Martin and Friday14,Reference Bowman, Clemens and Friday15)} . The weight/volume of the particular food item can vary significantly depending on whether it is consumed raw or prepared (boiled/cooked). Therefore, for each food item traditionally consumed in a cooked state (such as pumpkin, lentils, meats), we converted the raw amounts to cooked amounts using a yield factor^{(Reference Bognár16,Reference Showell, Williams and Duvall17)} . We used Internet resources to determine the yield factors for the food items that were not available in the manuals. For the meats, we fixed the yield factor at 0⋅8.

Finally, the household's consumption matrix (X) was obtained by adding the household's apparent consumption of food items across each food group (sub-group).

Statistical analyses

Estimates of the NDI (and the related measures) were calculated according to households’ income quintiles and area of residence (rural or urban). Differences among groups were analysed using the χ² test. Where relevant, linear trends across income quintiles and areas were assessed. In all analyses, the data were weighted using the expansion factors provided in the EIG datasets. The analysis was performed in R^TM statistical software, version 3.4.3, using the package ‘survey’^{(Reference Lumley18)}.

The EIG dataset contains detailed information on household income and expenditures. In this study, the monthly per capita household income was used to stratify households into five income quintiles (Q1–Q5). The corresponding income quintile thresholds were as follows: Q1: Paraguyan guarani (Gs.) 0 to 353 992; Q2: Gs. 354 262 to 610 327; Q3: Gs. 610 784 to 930 532; Q4: Gs. 930 913 to 1⋅514⋅103; and Q5: Gs. 1 515 036 and more.

The effects of household income on nutritional deprivation were estimated after statistically controlling for the effects of a number of potentially confounding factors. The factors include household size (0–4, 5–8 and 9 members or more), language spoken by the household head (Spanish, Guaraní, bilingual, or other), education level of female and male household head (no education, primary school education or less, middle school education or less and secondary school education or higher), household's area of residence (rural/urban) and the department (country's basic administrative division). The definition of each variable is provided in the Appendix (Table A1).

The effects of household economic status and other factors on nutritional deprivation were estimated using a multivariate logistic regression procedure. A number of alternative models were estimated to assess the relative significance of various confounding factors included in the analysis, as well as the robustness of the results. Results of multivariate analyses are presented as OR with 95 % CI.

A final note concerns the construction of the dependent variable. For the estimation purposes, the NDI was transformed into a binary variable. Given this transformation is dependent on the value of k (minimum number of food group deprivations), the logistic regression analysis was performed varying the parameter from k = 4 to k = 6. The results of the analysis for k = 5 and k = 6 were qualitatively similar to those for k = 4 presented in the text.