Block level estimates of cotton crop production for Sae in Amreli district

Introduction

Sample surveys is the only tool to gather information on various variables, where ultimate units of observation were selected randomly. The stratified multi stage complicated sampling design can help with this. It has been noted that various sampling strategies can deliver data at a higher degree of aggregation at the national, state, or district level. Recently, the government has changed its planning priorities from the macro to the micro levels, emphasizing the need for trustworthy data at the lower levels of aggregation, such as tehsil, block, and village panchayat, to ensure the most effective and efficient use of scarce resources. The problem of the development of reliable estimation at a lower level has been attempted by several researchers. These techniques are known as small area estimation. (Sharma, 2013) The concept of a small area is not new. It has been derived from large scale surveys where survey statisticians are interested in estimating the characteristics of sub populations, known as domain, as well as population parameters like mean or total. The fact that domain studies use information from large scale surveys instead of planning surveys particularly because they demand so many resources to estimate domain characteristics makes them unique. Sometimes the typical direct sampling approach may not adequately reflect such domains in the sample when the domain sizes are very tiny, and as a result, estimations of these domain parameters may not be accurate. Such types of small domains are also called small areas. According to Brackstone (1987), "Any small domain may be regarded as a small area for which direct design based estimates cannot be reliably produced from the current sample survey program".

The estimates of crop productivity/production are often provided at the district level. Aside from those that are available at the block and district levels, there are other factors relating to agricultural productivity. These factors include the crop's understory, whether it is irrigated or not, how much fertilizer is used, and how much rain falls, etc. So, an initial attempt was undertaken to create an appropriate multiple regression model between crop production and associated factors (also known as predictor or independent variables) at the district level. Then, using the data from the fitted regression model at the district level, procedures have been devised to scale down the district level crop production estimates at the block level. (Sharma, 2013). In agriculture, it is common practice that crop estimation surveys are designed to provide reliable estimates at the aggregated administrative levels like states or districts. Now there is a growing for reliable estimates for smaller area like community development blocks or gram panchayt or even for individuals villages. This need has arisen due to plan agricultural development for increasing production rapidly considering the resources and need of individual farmers and their environments. Since community development blocks have become the unit of planning, reliable data needed for formulating scientific plans and assessing their progress must be available. The planners should also have an opportunity to examine the achievements as a result of various plan efforts made in the area.

The key indicator of the achievements of such plan efforts is agricultural production. (Sharma, 2013) Cotton, the king of apparel fibre, has played a key role in the development of human civilization. Even today, it occupies an outstanding position in the textile industry despite pressure of manmade fibres and blended fabrics. Gujarat is one of the leading cotton producing state in the country. Gujarat contributes substantially to the national cotton area (24%) and productions (37%). Gujarat ranks first in cotton production occupying nearly 24% cropped area of Gujarat, cotton contributes nearly 1/3 to the State Gross Agricultural Product. In the last 10 years, greater than 500 Bt Cotton hybrids notified for commercial cultivation in central zone. Bt cotton area has grown by greater than 85% in last ten years unseating food crops. (Anon., 2023).

Materials and Methods

In the present study districts were considered as broad area and villages were considered as small area. The sampling unit (Individuals farmers) were selected as multistage sampling method. Purposively selected districts were considered as first stage unit. Whereas, talukas within district, villages within taluka and farmers within villages were selected randomly as second, third and last stage unit, respectively.

Broad Area Ratio Estimator With No Auxiliary Data

The broad area ratio estimates of cotton growing crop or the smaller area is defined as (1) Where, Y = Broad area ratio estimate of each small area. X = Direct estimate of cotton crop growing regions in broad area. N = Total number of cottons growing farmers in the broad area. n = Total number of cottons growing farmers in the small area SRSWOR By Using Mean Per Unit Method SRSWOR by using the mean per unit method is the point estimate of the population mean, which is defined as the mean per unit method and is obtained by using the following formula.

Where, y i is the yield of i th cotton growing farmer in the given year.

Variance of the sample mean

Where is the sampling variance given by Furthermore, an unbiased estimator of , when the population variance is not known is, (5) Where, is the estimator of ,

Confidence limits on mean

If the sample size is sufficiently large so that the distribution of can be approximated by a normal distribution, with confidence coefficient 100 (1 -/2) may be taken as, (6) For large sample size (n) is the value of the standard normal variate corresponding to the value 100 (1 -α/2) of the normal probability integral. For smaller sample sizes, the confidence interval may be constructed using a t-variate for (n-1) degrees of freedom, where in is the 100 (1 -α/2) th percentile point in the t-table for (n-1) degrees of freedom.

Multiple Linear Regression At District Level

In India, estimates for crop production/yield at district level were made available through CCEs. The district level estimates were then aggregated at state and national level. There was lack of estimates of below this level, e.g., Block level which was an important level for policy planning for rural area therefore describe scale down approach using multiple linear regression at

to obtain the block level estimates from district level crop production estimates. Availability of auxiliary variables that were related to cotton crop production/yield at both district and block level.

Where crop production in i th year (i = 1, 2, …, n) value of j th auxiliary variables in the i th year (i = 1, 2, …, n)

unknown parameters error term which follows iid ~ N (0, σ 2 )

The set of auxiliary variables may include area under the crop, farm harvest price, fertilizer consumption, rainfall etc. Let the fitted model, using least square (l.s.) technique, be denoted as,

Where, least square estimate of estimated value of Yi for corresponding values of Decompose the sum of squares due to regression i.e., SS R (β 1, β 2 …. β p | β 0 ) to define a weight that determines the relative contribution of each auxiliary variables included in the model follow:

Using these weights, an estimator of crop production (Y q ) for q th block, (q = 1,2 ... Q) is constructed as

Where, = ; is obtained through the fitted model (8).

A is the area under the crop in a given year is the value of j th predictor at village level in a given year weight Wj depends on the set of data on Y and X ij used to fit model (7). To find out a stable value of Wj one can use iteration technique by fitting model (7) first with n years data and then with (n + 1), (n + 2) ... years data till we get a stable value of Wj. Consequently, the estimate also be stable one. The estimator was an unbiased estimator of Y q if was constant quantity for given block and expected value of is under assumption of the model (7).

The variance of Y q is obtained as

The variance of is easily available by fitting the model ( 7), which is equal to σ 2 , the estimated error variance. The estimator Y q can also be referred to as synthetic estimator because Y q also borrows the strength from which is an estimate at the district level; whereas the coefficient of in () is the weighted value of the predictors at the block level.

Scalling Of Village Estimates To District Total Production

It is obvious that in general reported at district level through crop cutting experiments in a given year. Thus, a new estimator of Y q is proposed as,

The question arises that how to choose a q . The two alternatives choice of a q are suggested below: Choice I: The simplest choice of a q is to take a q = a, i.e., the constant for each taluka. It can easily be shown that,

Thus, a new estimator of Y q is given by,

Choice II: Another choice of a q could be to minimize the sum of squared differences between and . To do so, by minimizing the sum of squared differences between and subject to the condition. Using a lagrange multiplier λ, to impose the desired constraint, by minimizing the function, (18) Differentiating φ with reference to a q and equating it to zero and solving for a q , following equation was obtained, (19) Thus, another new estimator of Y q is given by, (20) Note that the scalled estimator are obtained by adjusting the original estimator Y q by adding a factor which is ratio of the difference between the actual crop production at district level, Y and the sum of the original villages estimates, , to number of villages in district.

Efficiency Of Small Area Estimation Against Larger Area Estimation

In order to find out the relative performance of the estimators, the variance of these estimators is derived. The variance of Y q is already given in (4). However, an alternative expression for V (Y q ) is given by,

The MSE of and are obtained as

In order to study the relative efficiency of the estimators, the MSE of these three estimators are compared.

The percent of standard error (% SE) of the estimates was calculated as (25)

The overall average error in as compared to was calculated as

Results

BARE Estimates For Total Numbers Of Farmers, Cotton Crop Production And Cotton Crop Yield.

Small area estimation techniques which include Broad area ratio estimation to know the percentage of farmers involved in cotton growing activity, total production of cotton crop, yield of cotton crop. BARE estimates for total numbers of cotton growing farmers were 80.45, 77.16 and 79.33 percent for Rajula, Babara and overall Amreli district. BARE estimates of total production of cotton for Amreli district were found to be 19329.10 tonnes.

It was found that the expected production was over-estimated as compared to actual production for Amreli district in the year 2023. Minimum standard error of small area estimates of cotton yield for Amreli district found in Rajula (55.57) village. Based on this it concluded that the estimates of cotton yield for various villages within district were found to be have lesser standard error which make them more efficient.

Correlation Between Cotton Production And Other Variables For Amreli District

The association of cotton production for Amreli district showed positive correlations with key variable like area coverage (r = 0.726), fertilizer consumption (r = 0.737), rainfall (r = 0.467) and farm harvest price (r = 0.607) found to be positive and significant. It suggests that an increase in cotton cultivation area, fertilizer usage, rainfall, and farm harvest price significantly and positively contribute towards higher cotton production. Conversely, relative humidity in the morning (r = -0.181), relative humidity in the evening (r = -0.052), maximum temperature (r = -0.448), and minimum temperature (r = -0.374) exhibits negative and non significant correlations.

Multiple Regression Model At District Level

The multiple linear regression analysis was employed to know the functional relationship between cotton production (Y) and different variables like total area under the cotton crop (X 1 ), fertilizer consumption (X 2 ), rainfall (X 3 ), relative humidity in the morning (X 4 ), relative humidity in the evening (X 5 ), maximum temperature (X 6 ), minimum temperature (X 7 ), and farm harvest price (X 8 ) of Amreli district by using least square technique and the results were presented in Table 5 showed values of regression coefficients, their standard error and coefficients of determinants (R 2 ) for the districts included in the study.

Regression equation for Amreli district Y = 89440.1 + 1.841 X 1 + 0.006 X 2 + 5.896* X 3 -439.091 X 4 + 448.044 X 5 -2288.119** X 6 + 71.106 X 7 + 0.259 X 8 The results indicated that certain factors do not exhibit a significant relationship with cotton production. These included total area under the cotton crop (X 1 ), fertilizer consumption (X 2 ), relative humidity in the morning (X 4 ), relative humidity in the evening (X 5 ), minimum temperature (X 7 ) and farm harvest price (X 8 ).

Interpretation of results of regression analysis for Amreli district

The larger areas exhibit an anticipated count 1.841 units higher compared to smaller areas, with other variables held constant If the fertilizer consumption (X 2 ) of the cotton crop increase by one year, the difference in the counts would be expected to be increase by 0.006 units, while holding the other variables in the model constant. Each unit increase in rainfall (X 3 ) during the cotton crop's growth period is met with a substantial boost of 5.896 units in the expected counts, while other variables remain steady. A rise in morning humidity (X 4 ) per unit growth period is associated with a decrease of -439.091 units in expected counts, holding other variables constant. Conversely, an increase in evening humidity (X 5 ) per growth period sees a robust surge of 448.044 units in expected counts, all other factors remaining stable. The temperature's maximum (X 6 ) rise per growth period introduce a considerable decrease of -2288.119 units in expected counts, with other variables unchanged Conversely, an elevation in minimum temperature (X 7 ) per growth period corresponded to an increase of 71.106 units in expected counts, keeping other variables constant. An upward shift in farm harvest price (X 8 ) throughout the growth period of the cotton crop was associated with a modest increase of 0.259 units in the expected counts. This relationship holds under the assumption that all other variables remain constant. The details of standard errors, coefficient of determination (R 2 ) and regression coefficients were presented in table no. 4.14. Rainfall (X 3 ) was found to be a statistically significant and favourable factor that increases cotton yield. On the other hand, the effect of maximum temperature (X 7 ) was not only negative but also quite large. The high coefficient of determination (R 2 ) of 90.60% indicated that the variables included in the model had significant and sufficient explanatory power and can be used to interpret variations in district-level cotton production. Using the analysis of variance, the individual contribution of independent variables viz., total area under the cotton crop (X 1 ), fertilizer consumption (X 2 ), rainfall (X 3 ), relative humidity in the morning (X 4 ), relative humidity in the evening (X 5 ), maximum temperature (X 6 ), minimum temperature (X 7 ) and farm harvest price (X 8 ) towards the regression sum of square and their weights (W j ) calculated based on regression sum of square were presented in table no. 6. The result presented in table no. 6 revealed that the maximum weights were observed in X 2 (0.1415) followed by X 8 (0.1415), X 5 (0.1411), X 4 (0.1320), X 3 (0.1310), X 7 (0.1102), X 6 (0.1102) and X 1 (0.0916) Efficiency Of Small Area Estimation Against The Larger Area Estimation By using Weights (W j ) and districts data on the variables viz., X 1 , X 2, X 3, X 4, X 5, X 6 , X 7 and X 8 , three different Synthetic estimators ( , , ) were determined. Based on these estimators the block (village) level estimates of cotton production, their percent standard error and percent relative efficiencies were calculated for different districts for the year 2023. The overall average error in percentage were also calculated for all the districts and the results were presented in table no. 8.Result presented in table no. 8 showed that the block (village) level synthetic estimates obtained from three different estimators were subject to maximum of almost 5% standard error. The percent standard error for block estimates varied between 2.64 to 2.79 percent in case followed by 3.79 to 4.37 percent in case and 4.37 to 4.93 percent in case of . It showed that the range of percent standard error for block estimate is smaller for as compared to the other estimators. An overall average error was also found to be 772.96 in case of was smaller as compared to other estimators. Therefore, it concluded that was best scaled improved estimator for estimating the block level estimates as compared to other estimator.

Conclusion

In the present study, the percentage of cotton growing farmers were 79.33%. BARE estimate of cotton production was found 19329.10 tonnes over estimates the actual production. BARE estimates cotton yield of Rajula (55.57%) had lower standard errors. The correlation analysis showed a significant and positive association with factors such as total area under cotton cultivation, fertilizer consumption and rainfall. MLR revealed a positive and significant relationship between rainfall and cotton production. An empirical study for cotton production suggested that among scaled estimator, was more stable.

References

1. Ahmed NDG, Jalikatti VN. Small area estimation techniques in wheat production -An empirical study. Int Res J Agric Econ Stat. 2014;5(2):231-234.
2. Brackstone GJ. Small area data: Policy Issues and Technical challenges. In: Small Area Statistics. 1987;3:20.
3. Montgomery DC, Peck EA. Introduction to Linear Regression Analysis. Canada: John Wiley and Sons; 1982.
4. Sharma M. Some methods for estimation of small area statistics in agriculture. 2013.