Small area estimation of food insecurity prevalence for the state of uttar pradesh in India

Abstract

The 2 ${}^{\text{nd}}$ Sustainable Development Goal (SDG) of the United Nations attempt to eliminate the potential hunger and food insecurity issues by 2030, but in the plight of COVID19 pandemic it has become far more critical and persistent issue globally as well as in India. The nation-wide socio-economic surveys of National Sample Survey Office (NSSO) in India are designed to produce reliable and representative estimates of important food insecurity parameters at state and national level for both rural and urban sectors separately but these surveys cannot be used directly to generate reliable district level estimates. Whereas, efficient and representative disaggregated level estimates for the extent (or incidence) of food insecurity prevalence has direct impact on strategizing effective policy plans and monitoring progress towards eliminating food insecurity. In this backdrop, the paper outlines small area estimation approach to estimate the incidence of food insecurity across the districts of rural Uttar Pradesh in India by linking data from the 2011–12 Household Consumer Expenditure Survey of NSSO, and the 2011 Indian Population Census. A spatial map has been generated showing spatial disparity for the incidence of food insecurity in Uttar Pradesh. These disaggregated level estimates are relevant and purposeful for SDG indicator 2.1.2 – severity of food insecurity. The estimates and map of food insecurity incidences are expected to deliver invaluable information to the policy-analysts and decision-makers.

Keywords

Food insecurity sustainable development goal small area estimation precise representative rural districts

1. Introduction

Food security exists when all people, always, have physical and economic access to sufficient, safe, and nutritious food that meets their dietary needs and food preferences for an active and healthy life (FAO, 2010). Inversely food insecurity is the situation when people do not have adequate physical, social or economic access to food. United Nations’ (UN) Sustainable Development Goal-2 is to eliminate all sorts of hunger, malnutrition issue and achieve food security. Government of India has also set it as one of the highest priorities to meet UN goal by 2030 aiming at ‘Leaving No One Behind’. The major data source to produce the estimates of food insecurity in India is the Household Consumer Expenditure Survey (HCES) of National Sample Survey Office (NSSO), Ministry of Statistics and Program Implementation. The HCES is designed to generate representative estimates for various indicators at macro level i.e., state and national level for both rural and urban sectors and combinedly. However, such macro level estimates can not reflect the heterogeneities that are available at local, micro or regional levels In spite of high importance, the estimates of food insecurity indicators are not available for lower administrative units e.g. district or lower level in the country. Lack of sufficient and representative food insecurity measures at the local or disaggregated level often put constraints for the policy planners, government and public agencies in designing targeted interventions and policy developments related to beating the inequality and disparity in food insecurity. Hence, for aiming inclusive developments with zero food insecurity it is necessary to obtain statistical summaries of relevant paramter of interest for smaller domains or small areas. The small areas are created by cross classifying demographic and geographic variables such as small geographic areas (e.g. districts) or small demographic groups (e.g. age-sex groups, social groups) or a cross classification of both. The sample sizes for such small domains in the survey data are usually very small, negligible or even zero. The SAE methodology provides a viable and cost effective solution to obtain precise estimates for small areas, see for example, Chandra et al. (2011) and Rao and Molina (2015). The SAE methods basically invokes the idea of borrowing strength from data of other areas, other time periods or both to faciliate the estimation process.

The SAE methods are generally based on model-based methods. The idea is to use statistical models to link the variable of interest with auxiliary information, e.g. Census and Administrative data, for the small areas to define model-based estimators for these areas. Based on the level of auxiliary information available, the models used in SAE are categorized as area level or unit level. Area-level modelling is typically used when unit-level data are unavailable, or, as is often the case, where model covariates (e.g. census variables) are only available in aggregate form. The Fay-Herriot model is a widely used area level model in SAE (Fay & Herriot, 1979). This model is an area level linear mixed model (Chandra, 2013; Chandra et al., 2015; Chandra & Chandra, 2015 & 2020). Standard SAE methods based on linear mixed models for continuous data can produce inefficient and sometime invalid estimates when the variable of interest is binary. If the variable of interest is binary and the target of inference is a small area proportion, then the generalized linear mixed model with logit link function, also referred as the logistic linear mixed model (LLMM) is generally used. An empirical plug-in predictor (EPP) under a LLMM is commonly used for the estimation of small area proportions, see for example, Chandra et al. (2012), Rao and Molina (2015) and references therein An alternative to EPP is the empirical best predictor (EBP, Jiang, 2003). This predictor does not have a closed form and can only be computed via numerical approximation. This is generally not straightforward, however, and so national statistical agencies favour computation of an approximation like the EPP. In this context, when only area level data are available, an area level version of a LLMM is used for SAE, see for example, Johnson et al. (2010), Chandra et al. (2011), Chandra et al. (2017), Chandra et al. (2018) and references therein Unlike the Fay-Herriot model, this approach implicitly assumes simple random sampling with replacement within each area and ignores the survey weights. Unfortunately, this has the potential to seriously bias the estimates if the small area samples are seriously unbalanced with respect to key population charcteristics, and consequently use of the survey weights appears to be inevitable for if one wishes to generate representative small area estimates. Recently, Chandra et al. (2019) introduce an approach to model the survey weighted estimates as binomial proportions, with an “effective sample size” chosen to match the binomial variance to the sampling variance of the estimates. This article considers Chandra et al. (2019) approach to model survey weighted small area proportions under a LLMM and attempts to produce the district level estimates of proportion of food insecurity (also refers as food insecurity prevalence or incidence of food insecurity) for rural areas of Uttar Pradesh. Note that if unit level data is available, the three suitable SAE methods based on unit-level model are the ELL (Elbers et al., 2003), the empirical Bayes (Best) prediction method (Molina & Rao, 2010) and the M-Quantile method (Tzavids et al., 2008). Das and Haslett (2019) described the performances of these methods in terms of their underlying model assumptions.

Rest of the paper is organized as follows. Next Section describes the data from the 2011–12 HCES of the NSSO and the 2011 Population Census that will be used to estimate the district-wise proportion of household food insecurity for rural areas of the State of Uttar Pradesh in India. The target variable of interest, the auxiliary variables and model specifications for SAE analysis are illustrated in Section 3. Section 4 presents a brief overview of SAE methodology. The empirical results and a map showing district-level inequalities in the distribution of food insecure households in Uttar Pradesh along with various diagnostic measures are reported in Section 5. Finally, Section 6 provides concluding remarks.

2. Data description and study area

The state of Uttar Pradesh inhabits around 16.16 percent of India’s population and is the most populous state in the country. It covers 243,290 square km, equal to 6.88% of the total area of the country. The analysis in the paper has been focused to rural areas because about 78% of the population of the State live in rural areas according to 2011 Population Census. The sources of the study and auxiliary variables used in SAE application are 2011–12 HCES of the NSSO in India and the 2011 Population Census respectively. Data obtained from these sources has been used to estimate the district level incidences of food insecurity in rural Uttar Pradesh. In 2011–12 HCES the stratified multi-stage random sampling design was used with districts being the strata, villages as first stage units and households as second stage units. In the 2011–12 HCES, a total of 5916 households were surveyed from the 71 districts of Uttar Pradesh. The district sample sizes ranged from 32 to 128 with an average of 83. It is evident that these district level sample sizes are relatively small, with average sampling fraction of 0.0002 (Table 1). Consequently, it is difficult to generate reliable district level direct survey estimates with associated standard errors from this survey. We address this small sample size problem in the 2011–12 HCES data by implementing the SAE methodology and using auxiliary information from the 2011 Population Census to strengthen the limited sample data from the districts to produce district level estimates.

Table 1
Summary of sample size, sample count (i.e., number of food insecure households) and sampling fraction in 2011 HCES data

Features	Minimum	Maximum	Average	Total
Sample size	32	128	83	5915
Sample count	10	111	53	3778
Sampling fraction	0.00015	0.00032	0.00023	0.01647

3. Variables and model specifications

The target variable $Y$ at the unit (household) level in the published NSSO survey data file is binary, corresponding to whether a household is food insecure (household consuming less than 2400 Kcal per day) or not. Average dietary energy intake per person per day in rural India is 2400 kilocalorie (Kcal), as defined by the Ministry of Health and Family Welfare, Government of India. The target is to estimate the proportion of rural households that are not getting satisfactory proportion of calories consistently (i.e., food insecurity prevalence or also referred as the incidence of food insecurity) at district level.

The auxiliary variables are taken from the 2011 Population Census of India. These auxiliary variables are only available as counts at district level, and so SAE methods based on area level small area models must be employed to derive the small area estimates. There are approximately 20–25 such auxiliary variables that are available for use in SAE analysis. We therefore carried out an exploratory data analysis to choose few auxiliary variables to determine appropriate covariates for SAE modelling. We also employed Principal Component Analysis (PCA) to derive composite scores for some selected groups of variables. In particular, we did PCA separately on two groups of variables, all measured at district level and identified as S1 and S2 below. The first group (S1) consisted of the proportions of main worker by gender, proportions of main cultivator by gender and proportions of main agricultural labourer by gender. The first principal component (S11) for this first group explained 44% of the variability in the S1 group, while adding the second component (S12) increased explained variability to 69%. The second group (S2) consisted of proportions of marginal cultivator by gender and proportions of marginal agriculture labourers by gender. The first principal component (S21) for this second group explained 52% of the variability in the S2 group, while adding the second component (S22) increased explained variability to 90%.

We fitted a generalised linear model using direct survey estimates of proportions of food insecure households as the response variable and the four principal component scores S11, S12, S21, S22 and some selected auxiliary variables from the 2011 Population Census as potential covariates. The final selected model included five covariates namely proportional scheduled caste population (SC), literacy rate (Lit), proportion of working population (WP), index for main worker population (S11) and index for marginal worker population (S21), with Akaike Information Criterion (AIC) value of 636.34. For this model, null deviance is 430.88 on 70 degrees of freedom and including the five independent variables has decreased the deviance to 294.72 on 65 degrees of freedom, a significant reduction in deviance. The residual deviance has reduced by 136.16 with a loss of five degrees of freedom. We use Hosmer Lemeshow goodness of fit test to examine the fitted model (i.e., model fits depend on the difference between the model and the observed data). The $p$ -value of Hosmer Lemeshow goodness is 0.9987. This indicates that model appears to fit well because we have no significant difference between the model and the observed data (i.e., the $p$ -value is above 0.05). In this fitted model it can be noted that SC, Lit, WP, S11 influence proportion of food insecure households positively, while S21 has a slightly negative effect. Further, the coefficients of SC ( $-$ 1.3741), Lit ( $-$ 1.10334), WP ( $-$ 5.0617), S11 ( $-$ 0.385) and S21 (0.3123) are significant ( $p<$ 0.001). This final model was then used to produce district wise estimates of food insecure households.

4. Small area estimation methodology

Let us assume that a finite population $U$ of size $N$ consists of $D$ non-overlapping and mutually exclusive small areas (or areas), and a sample $s$ of size $n$ is drawn from this population using a probability sampling method. We use a subscript $d$ to index quantities belonging to small area $d(d=1,\ldots,D)$ . Let $U_{d}$ and $s_{d}$ be the population and sample of sizes $N_{d}$ and $n_{d}$ in area $d$ , respectively such that $U=\bigcup\nolimits_{d=1}^{D}{U_{d}}$ , $N=\sum\nolimits_{d=1}^{D}{N_{d}}$ , $s=\bigcup\nolimits_{d=1}^{D}{s_{d}}$ and $n=\sum\nolimits_{d=1}^{D}{n_{d}}$ . We use subscript $s$ and $r$ respectively to denote quantities related to sample and non-sample parts of the population. Let $y_{di}$ denotes the value of the variable of interest for unit $i(i=1,\ldots,N_{d})$ in area $d$ . The variable of interest, with values $y_{di}$ , is binary (e.g., $y_{di}=1$ if household $i$ in area $d$ is food insecure and 0 otherwise), and the aim is to estimate the small area population count, $y_{d}=\sum\nolimits_{i\in U_{d}}{y_{di}}$ , or equivalently the small area proportion, $P_{d}=N_{d}^{-1}y_{d}$ , in area $d$ . The standard direct estimator (denoted by Direct) for $P_{d}$ is $\hat{p}_{d}^{\textit{Direct}}=\sum\nolimits_{i\in s_{d}}{\tilde{w}_{di}y_{di}}$ , where $\tilde{w}_{di}={w_{di}}\mathord{\left/{\vphantom{{w_{di}}{\sum\nolimits_{i\in s% _{d}}{w_{di}}}}}\right.\kern-1.2pt}{\sum\nolimits_{i\in s_{d}}{w_{di}}}$ with $\sum\nolimits_{i\in s_{d}}{\tilde{w}_{di}}=1$ and $w_{di}$ is the survey weight for unit $i$ in area $d$ . The estimate of variance of Direct is approximated as $v(\hat{p}_{d}^{\textit{Direct}})\approx\sum\nolimits_{i\in s_{d}}{\tilde{w}_{% di}(\tilde{w}_{di}-1)(y_{di}-\hat{p}_{d}^{\textit{Direct}})^{2}}$ . Under simple random sampling (SRS), this estimator is $p_{d}=n_{d}^{-1}y_{sd}$ , with $v(p_{d})\approx n_{d}^{-1}p_{d}(1-p_{d})$ , where $y_{sd}=\sum\nolimits_{i\in s_{d}}{y_{di}}$ denotes the sample count in area $d$ . Similarly, $y_{rd}=\sum\nolimits_{i\in s_{r}}{y_{di}}$ denotes the non-sample count in area $d$ . If the sampling design is informative, this SRS-based version of Direct may be biased.

If we ignore the sampling design, the sample count $y_{sd}$ in area (district) $d$ can be assumed to follow a Binomial distribution with parameters $n_{d}$ and $\pi_{d}$ , i.e. $y_{sd}|u_{d}\sim\text{Bin}(n_{d},\pi_{d})$ . Similarly, for the non-sample count, $y_{rd}|u_{d}\sim\text{Bin}(N_{d}-n_{d},\pi_{d})$ . Further, $y_{sd}$ and $y_{rd}$ are assumed to be independent binomial variables with $\pi_{d}$ being a common success probability. This leads to $E({y_{sd}|u_{d}})=n_{d}\pi_{d}$ and $E({y_{rd}|u_{d}})=({N_{d}-n_{d}})\pi_{d}$ . Let $x_{d}$ be the $k$ -vector of covariates for area $d$ from available data sources. Following Johnson et al. (2010) and Chandra et al. (2011), the model linking the probability $\pi_{d}$ with the covariates $x_{d}$ is the logistic linear mixed model (LLMM) of form

$\displaystyle\textit{logit}(\pi_{d})=\ln\{{\pi_{d}(1-\pi_{d})^{-1}}\}=\eta{}_{% d}={\rm{\bf x}}_{d}^{T}{\rm{\bf\beta}}+u_{d},$ (1)

with $\pi_{d}=\exp({\rm{\bf x}}_{d}^{T}{\rm{\bf\beta}}+u_{d})\{{1+\exp({\rm{\bf x}}_% {d}^{T}{\rm{\bf\beta}}+u_{d})}\}^{-1}$ . Here ${\rm{\bf\beta}}$ is the $k$ -vector of regression coefficients, often known as fixed effect parameters, and $u_{d}$ is the area-specific random effect that capture the area dissimilarities. We assume that $u_{d}$ is independent and normally distributed with mean zero and variance $\sigma_{u}^{2}$ . We can express the total population counts $y_{d}$ as $y_{d}=y_{sd}+y_{rd}$ , where the first term $y_{sd}$ , the sample count is known whereas the second term $y_{rd}$ , the non-sample count, is unknown. Under model Eq. (1), a plug-in empirical predictor (EPP) of $y_{d}$ in area $d$ is

$\displaystyle\hat{y}_{d}^{\textit{EPP}}=y_{sd}+({N_{d}-n_{d}})[{\exp({\rm{\bf x% }}_{d}^{T}{\rm{\bf\hat{\beta}}}+\hat{u}_{d})({1+\exp({\rm{\bf x}}_{d}^{T}{\rm{% \bf\hat{\beta}}}+\hat{u}_{d})})^{-1}}].$ (2)

An estimate of the corresponding proportion in area $d$ is $\hat{p}_{d}^{\textit{EPP}}=N_{d}^{-1}\hat{y}_{d}^{\text{EPP}}$ . It is obvious that in order to compute the small area estimates by Eq. (2), we require estimates of the unknown parameters $\beta$ and $u=(u_{1},\ldots,u_{D})^{T}$ . We use an iterative procedure that combines the Penalized Quasi-Likelihood estimation of $\beta$ and ${\rm{\bf u}}$ with REML estimation of $\sigma_{u}^{2}$ to estimate unknown parameters. Following Chandra et al. (2019), an approximate MSE estimate of Eq. (2) is:

$\displaystyle\textit{mse}(\hat{p}_{d}^{\textit{EPP}})=M_{1}(\hat{\sigma}_{u}^{% 2})+M_{2}(\hat{\sigma}_{u}^{2})+2M_{3}(\hat{\sigma}_{u}^{2}).$ (3)

The theoretical development is given in Chandr et al. (2019). Let define by $A=\{{\textit{diag}(N_{d}^{-1})}\}\hat{V}_{r}$ , $B=\{{\textit{diag}(N_{d}^{-1})}\}\hat{V}_{rd}X-A\hat{T}\hat{V}_{s}X$ and $\hat{T}=({\hat{\sigma}_{u}^{2}I_{D}+\hat{V}_{s}})^{-1}$ where $\hat{V}_{s}=\textit{diag}\{{n_{d}\hat{p}_{d}^{\textit{EPP}}(1-\hat{p}_{d}^{% \textit{EPP}})}\}$ and $\hat{V}_{r}=\textit{diag}\{{(N_{d}-n_{d})\hat{p}_{d}^{\textit{EPP}}(1-\hat{p}_% {d}^{\textit{EPP}})}\}$ . Here $I_{D}$ is an identity matrix of order $D$ and $X=(x_{1}^{T},\ldots,x_{D}^{T})^{T}$ is a $D\times k$ matrix. We further write $\hat{T}_{11}=\{{X^{T}\hat{V}_{s}X-X^{T}\hat{V}_{s}\hat{T}\hat{V}_{s}X}\}^{-1}$ and $\hat{T}_{22}=\hat{T}+\hat{T}\hat{V}_{s}X\hat{T}_{11}X^{T}\hat{V}_{s}^{T}\hat{T}$ . With these notations, assuming model Eq. (1) holds, the various components of MSE estimate are:

$\displaystyle M_{1}(\hat{\sigma}_{u}^{2})=A\hat{T}A^{T},M_{2}(\hat{\sigma}_{u}% ^{2})=B\hat{T}_{11}B^{T},\,\text{and}\,M_{3}(\hat{\sigma}_{u}^{2})=\textit{% trace}({\hat{\nabla}_{i}\hat{\Sigma}{\hat{\nabla}}^{\prime}_{j}v(\hat{\sigma}_% {u}^{2})})$

with $\hat{\Sigma}=\hat{V}_{sd}+\hat{\phi}I_{D}\hat{V}_{sd}\hat{V}_{sd}^{T}$ . Let $\Delta=A\hat{T}$ and $\hat{\nabla}_{i}=\left.{{\partial(\Delta_{i})}\mathord{\left/{\vphantom{{% \partial(\Delta_{i})}{\partial\phi}}}\right.\kern-1.2pt}{\partial\phi}}\right|% _{\phi=\hat{\phi}}=\left.{{\partial(A_{i}\hat{T})}\mathord{\left/{\vphantom{{% \partial(A_{i}\hat{T})}{\partial\sigma_{u}^{2}}}}\right.\kern-1.2pt}{\partial% \sigma_{u}^{2}}}\right|_{\sigma_{u}^{2}=\hat{\sigma}_{u}^{2}}$ , where $A_{i}$ is the $i^{\text{th}}$ row of the matrix $A$ . Here $v(\hat{\sigma}_{u}^{2})$ is the asymptotic covariance matrix of the estimate of variance component $\hat{\sigma}_{u}^{2}$ . This also depends upon whether we use ML or REML estimate for of variance component $\hat{\sigma}_{u}^{2}$ . We use REML estimates for $\hat{\sigma}_{u}^{2}$ and then $v(\hat{\sigma}_{u}^{2})=2({(\hat{\sigma}_{u}^{2})^{-2}(D-2t_{1})+(\hat{\sigma}% _{u}^{2})^{-4}t_{11}})^{-1}$ with $t_{1}=(\hat{\sigma}_{u}^{2})^{-1}\textit{trace}(\hat{T}_{22})$ and $t_{11}=\textit{trace}(\hat{T}_{22}\hat{T}_{22})$ .

The model Eq. (1) is based on unweighted sample counts, and hence it assumes that sampling within areas is non-informative given the values of the contextual variables and the random area effects. The small area predictor based on Eq. (2) therefore ignores the complex survey design used in NSSO data. The sampling design used in 2011 HCES is informative. Using the effective sample size rather the actual sample size allows for the survey weights under complex sampling. Furthermore, the precision of an estimate from a complex sample can be higher than for a simple random sample, because of the better use of population data through a representative sample drawn using a suitable sampling design. Following Chandra et al. (2019), we model the survey weighted probability estimate for an area as a binomial proportion, with an “effective sample size” that equates the resulting binomial variance to the actual sampling variance of the survey weighted direct estimate for the area. Hence, in our analysis we replaced the “actual sample size” and the “actual sample count” with the “effective sample size” and the “effective sample count” respectively.

5. Results and discussions

Firstly, the sampling design in HCES sample data were examined whether informative or not. The sampling design is called informative if the distribution in the sample is different from the distribution in the population. In this case, sampling design used in survey data collected must be incorporated in making the valid analytical inference about the population. Such sampling design is also referred as non-ignorable designs. For this purpose, the effective sample sizes and the effective sample counts for the 2011 HCES data were computed to illustrate whether the 2011 HCES data is informative sample or not. Refer Chandra et al. (2019) for details about calculation of the effective sample sizes and the effective sample counts. Figure 1 plots the effective sample sizes against the observed sample sizes. The effective sample counts and observed sample counts are shown in Fig. 2. It is evident from Fig. 1 that the effective sample size is smaller than the observed sample sizes in almost all the districts. Similarly, in Fig. 2 the effective sample counts is lower than the observed sample counts. This indicates that the sampling design results in a loss in information, when compared with simple random sampling, in all the districts. Figure 3 presents the district-wise survey weighted and unweighted direct estimates of proportion of household food insecurity. It can be seen from Fig. 3 that the unweighted direct estimates underestimate the proportion of food insecure households, in majority of the districts. These examples are evident that the sampling design is informative and therefore must be accounted in SAE. Hence, the SAE analysis reported in paper uses effective sample sizes and effective sample counts in replace of observed sample sizes and observed sample counts respectively to incorporate the sampling design of HCES data.

Figure 1.

Effective sample size versus observed sample size in 2011 HCES data.

Figure 2.

Effective sample count versus observed sample count in 2011 HCES data.

Figure 3.

District-wise survey weighted direct estimates versus unweighted direct estimates of proportion of food insecure households.

5.1 Diagnostics measures

The estimates of food insecurity prevalence (i.e., incidence of food insecurity) at district level for rural areas in the state of Uttar Pradesh is generated from the EPP method described in Sections 4 using 5 significant covariates described in Section 3. We now describe some important diagnostics to examine the assumptions of the underlying models, and to validate the empirical performances of the EPP method. Generally, two types of diagnostics measures are advised in SAE applications. These are (i) the model diagnostics, and (ii) the diagnostics for the small area estimates. See Chandra et al. (2011). The model diagnostics are applied to verify model assumptions. The other diagnostics are used to validate reliability of the model-based small area estimates of incidence of food insecurity generated by the EPP method. In model Eq. (1) the random district specific effects are assumed to have a normal distribution with mean zero and fixed variance. If the model assumptions are satisfied, then the district level residuals are expected to be randomly distributed around zero. Histogram and normal probability (q-q) plot can be used to examine the normality assumption. Figure 4 shows the histogram (left plot), the normal probability (q-q) plot (centre plot) and the distribution of the district-level residuals (right plot). We also use the Shapiro-Wilk test (implemented using the shapiro.test() function in R) to examine the normality of the district random effects. The Shapiro-Wilk test with $p$ -value lower than 0.05 indicate that the data deviate from normality. Here, the value of Shapiro-Wilk test statistics is 0.988 with 71 degree of freedom and $p$ -value 0.746. From the Fig. 4, the district level residuals appear to be randomly distributed around zero. Further, histogram and the q-q plot also provide evidence in support of the normality assumption. The Shapiro-Wilk $p$ -value is larger than 0.05 and hence, the district random effects are likely to be normally distributed.

Figure 4.

Histograms (left plot), normal q-q plots (centre plot) and distributions of the district-level residuals (right plot).

We consider three commonly used diagnostics measures for assessing the validity and the reliability of the model-based small area estimates: the bias diagnostic, the percent coefficient of variation (CV) diagnostic and the 95 percent confidence interval diagnostic. The first diagnostics assesses the validity and last two assess the reliability or improved precision of the model based small area estimates. In addition, we implemented a calibration diagnostic where the model-based estimates are aggregated to higher level and compared with direct survey estimates at this level. See for example, Chandra et al. (2011). Note that here direct estimates are defined as the survey weighted direct estimates.

The bias diagnostic is based on following idea. The direct estimates are unbiased estimates of the population values of interest (i.e., true values), their regression on the true values should be linear and correspond to the identity line. If model-based small area estimates are close to these true values the regression of the direct estimates on these model-based estimates should be similar. We therefore plot direct estimates (y-axis) vs. model-based small area estimates (x-axis) and we looked for divergence of the fitted least squares regression line from the line of equality. In Fig. 5 we provide a bias diagnostic plot, defined by plotting direct survey estimates ( $Y$ axis) against corresponding small area estimates generated by the EPP ( $X$ -axis) and testing for divergence of the fitted least squares regression line (shown by dashed line) from the line of equality, i.e., $Y=X$ line (shown by solid line). The bias diagnostic plot in Fig. 5 clearly indicates that the model-based EPP estimates are less extreme when compared to the direct survey estimates, demonstrating the typical SAE outcome of shrinking more extreme values towards the average. The value of $R^{2}$ for the fitted (OLS) regression line between the direct survey estimates and the EPP estimates is 95.6 per cent. In general, these different bias diagnostics all show that the estimates generated by all the SAE methods appear to be consistent with the direct survey estimates.

We now illustrate the second set of diagnostics to assess the extent to which the EPP estimates improve in precision compared to the direct estimates. The percent coefficient of variation (CV) is the estimated sampling standard error as a percentage of the estimate. District level estimates with large CVs are considered unreliable. Table 2 provides a summary of CVs of the direct and the EPP estimates Fig. 6 presents the district-wise values of CV for the direct and EPP methods. In one of the 71 districts, smaller CV (2.16%) of direct estimate is due to extreme value of proportion. Sample size and sample count for this district are 64 and 58 respectively while and direct estimate of proportion of food insecurity is 0.967. Note that the effective sample size and effective sample count for these districts are 25 and 24 respectively. In Table 2, we therefore presented the summary based on 70 districts (excluding one district extreme value of proportion). In further discussion we refer summary based on 70 districts only. The CVs of the direct estimates are larger than the EPP estimates. Table 2 and Fig. 6 show that direct survey estimates of incidence food insecurity are unstable with CVs that vary from 5.53 to 45.52 % with average of 14.59 %. In contrast, the CV values of EPP range from 5.12 to 24.29% with average of 10.65%. The relative performance of the EPP as compared to the direct survey estimates improve with decreasing district specific observed sample sizes. That is, the estimates computed from the EPP are more reliable and provide a better indication of food insecurity incidence in Uttar Pradesh. The district-wise plot of the 95 percent confidence intervals (CIs) generated by direct and EPP methods are displayed in Fig. 7. The width of CIs are given Fig. 8. Figures 7 and 8 show that the 95% CIs for the direct estimates are wider than the 95% CIs for the EPP.

Table 2

Summary of area distributions of percentage coefficients of variation (CV, %) for the direct and EPP methods applied to HCES data

Values	Summary of 71 districts		Summary of 70 districts
	Direct	EPP	Direct	EPP
Minimum	2.16	5.12	5.53	5.12
Q1	8.97	7.90	9.06	7.99
Mean	14.41	10.60	14.59	10.65
Median	12.31	9.56	12.38	9.56
Q3	12.31	9.56	12.38	9.56
Maximum	45.52	24.29	45.52	24.29

Figure 5.

Bias diagnostic plot with $y=x$ line (solid) and regression line (dotted) for proportion of food insecurity for rural areas in Uttar Pradesh: EPP versus direct estimates.

We inspect the aggregation or calibration property of the model-based district-level estimates generated by EPP at higher (e.g. State or Region) level. Let $\hat{P}_{d}$ and $N_{d}$ denote the estimate of proportion of household food insecurity and population size for district $d$ . The state-level estimate of the proportion of food insecure households is calculated as $\hat{P}={\sum\nolimits_{d=1}^{D}{N_{d}\hat{P}_{d}}}\mathord{\left/{\vphantom{{% \sum\nolimits_{d=1}^{D}{N_{d}\hat{P}_{d}}}{\sum\nolimits_{d=1}^{D}{N_{d}}}}}% \right.\kern-1.2pt}{\sum\nolimits_{d=1}^{D}{N_{d}}}$ . The state of Uttar Pradesh is divided into Central, Eastern, Western and Southern regions, and calibration properties can also be examined for these regions. State and regional level estimates of the proportion of household food insecurity generated by the EPP method is reported in Table 3. Comparing these with the corresponding direct estimates we see that the EPP estimates are very close to the direct survey estimates as state level as well in each of the four regions.

Table 3

Aggregated level estimates of incidence of food insecurity generated by direct and EPP method in different regions in Uttar Pradesh

Estimator	State	Central	Eastern	Southern	Western
Direct	0.644	0.557	0.698	0.431	0.649
EPP	0.646	0.565	0.695	0.455	0.650

Figure 6.

District-wise percentage coefficient of variation (CV, %) for the direct (dotted line, $\circ$ ) and EPP (solid line, $\bullet$ ) estimates for the food insecurity prevalence in Uttar Pradesh.

Figure 7.

District-wise 95 percentage nominal confidence interval (95% CI) for the direct (solid line) and EPP (thin line) methods. Direct (dotted point) and EPP estimates (dash point) for the food insecurity prevalence in Uttar Pradesh are shown in the 95% CI.

Figure 8.

District-wise width of 95 percentage nominal confidence interval for the direct (solid line) and EPP (thin line) methods. Direct estimate (dotted point) and EPP estimate (dash point) are plotted in the 95% CI.

Figure 9.

Map showing location of state of Uttar Pradesh in India.

Table 4

Direct and model-based (EPP) estimates along with 95% confidence interval (95% CI) and percentage coefficient of variation (CV) of the incidence of food insecurity by District in rural areas of Uttar Pradesh

District	Direct				EPP
	Estimate	95 % CI		CV	Estimate	95 % CI		CV
		Lower	Upper			Lower	Upper
Saharanpur	0.641	0.488	0.794	11.90	0.637	0.514	0.760	9.65
Muzaffarnagar	0.696	0.575	0.817	8.72	0.685	0.578	0.792	7.79
Bijnor	0.667	0.523	0.811	10.81	0.675	0.559	0.791	8.59
Moradabad	0.664	0.545	0.783	8.98	0.679	0.576	0.782	7.58
Rampur	0.681	0.512	0.850	12.44	0.714	0.590	0.838	8.69
Jyotiba Phule Nr	0.700	0.537	0.863	11.67	0.685	0.558	0.812	9.26
Meerut	0.452	0.275	0.629	19.62	0.505	0.367	0.643	13.63
Baghpat	0.480	0.191	0.769	30.12	0.538	0.363	0.713	16.29
Ghaziabad	0.843	0.727	0.959	6.90	0.762	0.647	0.877	7.55
Gautam B. Nr	0.486	0.232	0.740	26.17	0.553	0.389	0.717	14.87
Bulandshahr	0.611	0.480	0.742	10.70	0.595	0.482	0.708	9.46
Aligarh	0.516	0.337	0.695	17.37	0.551	0.414	0.688	12.42
Hathras	0.356	0.165	0.547	26.82	0.429	0.287	0.571	16.53
Mathura	0.685	0.514	0.856	12.45	0.666	0.539	0.793	9.56
Agra	0.844	0.743	0.945	5.97	0.786	0.687	0.885	6.28
Firozabad	0.698	0.519	0.877	12.83	0.679	0.550	0.808	9.51
Etah	0.777	0.643	0.911	8.59	0.718	0.593	0.843	8.73
Mainpuri	0.967	0.925	1.009	2.16	0.811	0.698	0.924	6.96
Budaun	0.701	0.543	0.859	11.29	0.705	0.579	0.831	8.94
Bareilly	0.585	0.427	0.743	13.50	0.627	0.500	0.754	10.12
Pilibhit	0.842	0.733	0.951	6.46	0.776	0.659	0.893	7.54
Shahjahanpur	0.673	0.502	0.844	12.72	0.668	0.540	0.796	9.56
Kheri	0.465	0.306	0.624	17.12	0.506	0.376	0.636	12.82
Sitapur	0.483	0.345	0.621	14.25	0.519	0.400	0.638	11.50
Hardoi	0.626	0.496	0.756	10.35	0.627	0.512	0.742	9.18
Unnao	0.608	0.452	0.764	12.85	0.588	0.462	0.714	10.68
Lucknow	0.639	0.472	0.806	13.04	0.646	0.515	0.777	10.16
Rae Bareli	0.647	0.527	0.767	9.30	0.637	0.531	0.743	8.32
Farrukhabad	0.649	0.443	0.855	15.89	0.665	0.524	0.806	10.63
Kannauj	0.776	0.625	0.927	9.71	0.700	0.563	0.837	9.76
Etawah	0.279	0.105	0.453	31.22	0.401	0.252	0.550	18.63
Auraiya	0.659	0.495	0.823	12.45	0.647	0.514	0.780	10.31
Kanpur Dehat	0.483	0.265	0.701	22.60	0.506	0.354	0.658	14.99
Kanpur Nagar	0.646	0.459	0.833	14.51	0.600	0.458	0.742	11.83
Jalaun	0.550	0.368	0.732	16.57	0.529	0.392	0.666	12.96
Jhansi	0.217	0.083	0.351	30.96	0.272	0.152	0.392	22.12
Lalitpur	0.271	0.059	0.483	39.19	0.355	0.189	0.521	23.35
Hamirpur	0.399	0.095	0.703	38.16	0.427	0.244	0.610	21.44
Mahoba	0.282	0.025	0.539	45.52	0.364	0.187	0.541	24.29
Banda	0.727	0.539	0.915	12.96	0.678	0.542	0.814	10.04
Chitrakoot	0.432	0.165	0.699	30.95	0.481	0.310	0.652	17.82
Fatehpur	0.489	0.345	0.633	14.76	0.486	0.364	0.608	12.52
Pratapgarh	0.887	0.789	0.985	5.53	0.828	0.743	0.913	5.12
Kaushambi	0.896	0.769	1.023	7.10	0.800	0.697	0.903	6.43
Allahabad	0.621	0.468	0.774	12.31	0.623	0.505	0.741	9.48
BaraBanki	0.674	0.499	0.849	12.95	0.637	0.507	0.767	10.20
Faizabad	0.584	0.356	0.812	19.49	0.610	0.463	0.757	12.05
Ambedkar Nr	0.701	0.578	0.824	8.74	0.690	0.585	0.795	7.63
Sultanpur	0.650	0.529	0.771	9.35	0.662	0.558	0.766	7.88
Bahraich	0.740	0.583	0.897	10.60	0.739	0.622	0.856	7.93
Shrawasti	0.819	0.672	0.966	8.96	0.768	0.642	0.894	8.19
Balrampur	0.616	0.405	0.827	17.14	0.644	0.496	0.792	11.49
Gonda	0.866	0.770	0.962	5.56	0.810	0.720	0.900	5.56
Siddharthnagar	0.839	0.735	0.943	6.17	0.800	0.705	0.895	5.93
Basti	0.839	0.737	0.941	6.05	0.791	0.695	0.887	6.05
Sant Kabir Nr	0.823	0.697	0.949	7.68	0.799	0.699	0.899	6.23

Table 4, continued
District	Direct				EPP
	Estimate	95 % CI		CV	Estimate	95 % CI		CV
		Lower	Upper			Lower	Upper
Mahrajganj	0.708	0.558	0.858	10.60	0.707	0.590	0.824	8.27
Gorakhpur	0.787	0.690	0.884	6.19	0.783	0.692	0.874	5.81
Kushinagar	0.696	0.573	0.819	8.87	0.727	0.620	0.834	7.37
Deoria	0.790	0.664	0.916	7.98	0.788	0.687	0.889	6.40
Azamgarh	0.593	0.470	0.716	10.41	0.626	0.518	0.734	8.65
Mau	0.634	0.457	0.811	13.98	0.665	0.531	0.799	10.10
Ballia	0.414	0.242	0.586	20.77	0.517	0.377	0.657	13.58
Jaunpur	0.650	0.522	0.778	9.84	0.661	0.550	0.772	8.37
Ghazipur	0.609	0.482	0.736	10.43	0.629	0.520	0.738	8.63
Chandauli	0.650	0.476	0.824	13.40	0.660	0.531	0.789	9.75
Varanasi	0.596	0.456	0.736	11.71	0.610	0.492	0.728	9.66
Bhadohi	0.827	0.686	0.968	8.50	0.785	0.679	0.891	6.76
Mirzapur	0.588	0.453	0.723	11.50	0.604	0.489	0.719	9.54
Sonbhadra	0.629	0.466	0.792	12.94	0.632	0.501	0.763	10.36
Kanshiram Nr	0.395	0.173	0.617	28.14	0.528	0.358	0.698	16.06

Nr: Nagar.

Figure 10.

EPP estimates showing the spatial distribution of incidence of food insecurity by District in Uttar Pradesh.

5.2 Spatial distribution of food insecurity prevalence

Figure 9 shows the location of state of Uttar Pradesh in India. In Fig. 10 we present a map showing the estimated proportion of food insecurity in different districts in rural areas of Uttar Pradesh produced by the EPP method. This map provides the district-wise degree of inequality with respect to distribution of extent of food insecurity in rural areas of Uttar Pradesh. This map is supplemented by the results set out in Table 4, where we report the district-wise estimates along with CVs and 95% confidence intervals generated by direct and EPP. The results indicate an east-west divide in the distribution of food insecurity. For example, in the western part of Uttar Pradesh there are many districts with low level of incidence of food insecurity. Similarly, in the eastern part and in the Bundelkhand region (north-east) we see districts with high incidence of food insecurity. This should prove useful for policy planners and administrators aiming to take effective financial and administrative decisions.

6. Concluding remarks

In this paper we first summarise a plug-in empirical predictor (EPP) for small area proportions under an area level logistic linear mixed model. Then the EPP method in the 2011–12 HCES data collected by the NSSO of India has been applied to estimate the incidence of food insecurity and to produce a spatial map of the different districts of rural areas of the state of Uttar Pradesh in India. The auxiliary variables used in this analysis were taken from the 2011 India’s Population Census. The effective sample sizes in place of the observed sample sizes were used to account for the sampling design information of the 2011–12 HCES. The use of survey information through effective sample size leads to more representative and realistic estimates of the incidence of food insecurity. The empirical results were also evaluated through several diagnostic measures and revealed that the model-based SAE method defined by EPP provide significant gains in efficiency for generating district level estimates of proportions of food insecurity. Spatial map produced from the estimates generated by the EPP method provides an evidence of inequality in distribution of incidence food insecurity across different districts of the State of Uttar Pradesh in India.

Availability of reliable district level estimates can definitely be useful for various Organizations and Ministries in Government of India as well as International organizations for their policy research and strategic planning. These estimates will also be useful for budget allocation and to target welfare interventions by identifying the districts/regions with high food insecurity incidence. This application clearly demonstrates the advantage of using SAE technique to cope up the small sample size problem in producing the cost effective and reliable disaggregate level estimates and confidence intervals from existing survey data by combining auxiliary information from different published sources with direct survey estimates.

Footnotes

Acknowledgments

This paper is a tribute to Dr. Hukum Chandra who was a passionate researcher in the field of Small Area Estimation and left for heavenly abode after writing this paper with me (Dr. Bhanu Verma). I am highly grateful to Dr. Girish Chandra, Indian Council of Forestry Research and Education, Dehradun, India for communicating this paper.

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Small area estimation of food insecurity prevalence for the state of uttar pradesh in India

Abstract

Keywords

1. Introduction

2. Data description and study area

Table 1 Summary of sample size, sample count (i.e., number of food insecure households) and sampling fraction in 2011 HCES data

4. Small area estimation methodology

6. Concluding remarks

Footnotes

Acknowledgments

References

Table 1
Summary of sample size, sample count (i.e., number of food insecure households) and sampling fraction in 2011 HCES data