Sampling and Estimation in Agricultural Surveys

Transcription

1 GS Training and Outreach Workshop on Agricultural Surveys Training Seminar: Sampling and Estimation in Cristiano Ferraz 24 October 2016

2 Download a free copy of the Handbook at:

3 Objective: To provide the participants the opportunity to get in touch with key-concepts and practical aspects of designing a sample to generate agriculture estimates.

4 Overview: : Challenging features Typical Frames in Ag-Surveys Single Frame Surveys Multiple Frame Designs Dual Frame Survey

5 : Challenging Features Covers a large spectrum of subjects There is a great variety of variables of interest

6 : Challenging Features Often a multi-subject/multi-purpose survey Suffers influence from nature and culture

7 : Challenging Features Require periodicity

8 Typical Frames in Agriculture: List Frames Area Frames Dual Frames Multiple Frames

9 What is a Sampling Frame? A Sampling Frame can be defined as a reference system composed by a set of materials, devices or coordinates that identifies and provides access to sampling units, so that a sample can be selected and its elements can be reached. Sampling FRAME

10 What is a List Frame? This type of frame is recognized by the main Sampling characteristic of listing its components. Examples of FRAME list frames include: a list of farmers from a country or region; a list of associates from a cooperative association; a list of beneficiaries of a type of government policy program, etc.

11 What is an Area Frame? Area frames are used to geographically cover a Sampling target population. FRAME Typical area frames use technological devices to identify and to provide access (coordinates) to well defined segments of lands.

12 What is a Master Sampling Frame? Sampling FRAME Household Survey Sampling FRAME Grain Survey Sampling FRAME Agricultural Survey Sampling FRAME Livestock Survey

13 What is a Master Sampling Frame? MASTER SAMPLING FRAME Household Survey Grain Survey Agricultural Survey Livestock Survey A Master Sampling Frame is a unique Sampling Frame System from which samples for different surveys can be selected, each one using its own probability sample design. Used in this way, Master Frames can be an efficient tool to integrate surveys.

14 What is a Master Sampling Frame? MASTER SAMPLING FRAME Agricultural Survey: T1 Agricultural Survey: T2 Agricultural Survey: T3 Agricultural Survey: T4 A Master Sampling Frame can also be used to select samples for the same survey at different points in time. Used in this way, Master Frames provide the sampling support to longitudinal, and panel type surveys.

15 Challenge: Master Sampling Frames for Agriculture Surveys must satisfy the needs of three statistical units: the farm or agricultural holding; the household; and the land. While in many cases there is a one-to-one relationship between the agricultural holding, the household, and the land parcel, it is not always that this happens.

16

17 Single Frame Surveys: Basic fundamental concepts: Population or target population Subpopulation Frame and Sampled Population Sampling unit Observation unit Reporting unit

18 Sistema de Produção de Gado de Leite do Agreste Meridional de Pernambuco

19 Sistema de Produção de Gado de Leite do Agreste Meridional de Pernambuco Population or Target Population:

20 Sistema de Produção de Gado de Leite do Agreste Meridional de Pernambuco Subpopulation: Multi-purpose aspects of agricultural surveys may require estimates for subpopulations of interest. These are specific subsets of elementary units for which inferences are required. For example, inference for the subpopulation of milk producers that have received technical support from local governmental agencies could be necessary.

21 Sistema de Produção de Gado de Leite do Agreste Meridional de Pernambuco Frame and Sampled Population: FRAME: 95% coverage level Target population: set of all milk producers from the Agreste Meridional de Pernambuco - AMPE Frame: List of all milk producers from AMPE that sells their milk to a given Industry

22 Sistema de Produção de Gado de Leite do Agreste Meridional de Pernambuco Sampling Unit, Observation Unit and Reporting Unit:

23 Survey error = sampling error + non-sampling error Sample Surveys Census Sampling error Non-sampling error Non-sampling error

24

25

26 Design-based inference for finite populations Suppose that NN is the size of the target population, and let UU be the set of indices uniquely identified: UU = {1,2,, NN}. Let SS UU be a sample of nn from UU. Let yy kk be the value of the variable of interest yy for unit kk of the target population UU. The inclusion of kk in the sample is indicated by the following random variable: 1, iiii kk SS II kk = II kk SS = 0, ooooooooooooooooo

27 II kk ~BBBBBBBBBBBBBBBBBB(ππ kk ) EE(II kk ) = ππ kk VVVVVV(II kk ) = ππ kk (1 ππ kk ) In general, when sampling from finite populations, CCCCCC(II kk, II ll ) = ππ kkll ππ kk ππ ll

28 The randomization role: Probability sampling designs determine the exact distribution of II kk, providing the sample inclusion probabilities: ππ kk = PP II kk = 1 ; ππ kkll = PP II kk II ll = 1. Probability sampling designs require that all ππ kk >0.

29 Design-based inference for finite populations Probability sampling designs determine the exact distribution of II kk, providing the sample inclusion probabilities: ππ kk = PP II kk = 1 ; ππ kkll = PP II kk II ll = 1. Probability sampling designs require that all ππ kk >0. First-order inclusion probability

30 Design-based inference for finite populations Probability sampling designs determine the exact distribution of II kk, providing the sample inclusion probabilities: ππ kk = PP II kk = 1 ; ππ kkll = PP II kk II ll = 1. Probability sampling designs require that all ππ kk >0. Second-order inclusion probability

31 Parameter and estimator: Given a probability sampling design, a unifying result, due to Horvitz and Thompson (1952) ensures unbiased estimation of parameters such as means, totals and percentages. Lets focus on the problem of estimating a population total (parameter): YY = kk UU The Horvitz-Thompson estimator for YY is given by: yy kk YY = yy kk ππ kk kk SS

32 The variance of the Horvitz-Thompson estimator can be written as: VVVVVV pp YY = kk UU ll UU (ππ kkkk ππ kk ππ ll ) yy kk ππ kk yy ll ππ ll In addition, an unbiased estimate of this variance may be obtained using: VVVVVV pp YY = kk SS (ππ kkkk ππ kk ππ ll ) ll SS ππ kkkk yy kk ππ kk yy ll ππ ll

33 Frame and sample design: An important characteristic of frames is the nature of its sampling unit. On one hand, it is possible to identify either LIST or AREA frames. On the other, it is possible to identify: Type A: Frames with sampling units as elements of the population; Type B: Frames with sampling units as sets of elements of the population. Availability of type A frames allows for direct element sampling designs to be used.

34 Suppose a type A frame is available: Simple Random Sampling Systematic Sampling Probability Proportional to Size Design PPS Multivariate Probability Proportional to Size Design MPPS Stratified Sampling

35 Suppose a type A frame is available: Simple Random Sampling Systematic Sampling Probability Proportional to Size Design PPS Multivariate Probability Proportional to Size Design MPPS Stratified Sampling These designs need auxiliary information

36 Simple Random Sampling Samples selected from a population of size N according to a simple random sampling design have a pre-assigned size n, and are such that the probability of selecting a given sample s is PP ss = NN 1 nn In a simple random sample, the first and second order inclusion probabilities are ππ kk = nn NN aaaaaa ππ nn(nn 1) kkkk = NN(NN 1)

37 Systematic Sampling Suppose that a sample of size n is to be selected from a population of size N using a systematic sampling design. First, a sample interval, given by aa = NN nn is calculated. Suppose that a is an integer number. Then, a sample of one is randomly selected from the first a elements identified by the frame. Thereafter, every a-th element of the frame is also included in the sample.

38 In systematic sampling, the inclusion probabilities are ππ kk = nn NN aaaaaa ππ kkkk = nn NN, iiii kk aaaaaa ll aaaaaa iiii ttttt ssssssssssss 0, ooooooooooooooooo

39 Probability Proportional to Size Sampling (PPS) In the previous examples, each population unit had the same chance of being selected, regardless of the method of selection or the population unit s actual size. If a measure of size (relevance) can be attached to each unit, a probabilityproportional-to-size (PPS) sample can be drawn.

40 The following example is used to illustrate PPS sampling: Name Measure of Size Accumulated Measure

41 The following example is used to illustrate Systematic PPS sampling: Name Measure of Size Accumulated Measure

42 Multivariate probability-proportional-to-size (MPPS) The same example as that described for the PPS, with two available measurements of size, follows: Name Measure 1 Measure 2 Improved Accumulated of Size of Size Size Measure Measure

43 Multivariate probability-proportional-to-size (MPPS) Suppose that there are J 2 variables of interest (items), each having at least one auxiliary variable that can be used as a measurement of size. Let xx jjjj be the value of the size measure j for element k in a given f frame. Let also XX jj = kk ff xx jjjj be the total of the auxiliary variable j over frame f.

44 Multivariate probability-proportional-to-size (MPPS) In addition, let nn jj be the sample size needed for the variable of interest j. Then, the inclusion probability under an MPPS design is given by ππ kk ff = mmmmmm 1, mmmmmm nn jj xx jjjj XX jj, jj = 1,2 JJ The remaining steps for selecting the sample are identical to PPS sampling.

45 Stratified Sampling In stratified sampling, the population is first divided into subgroups called strata, in a process called stratification. Then, elements are sampled from each stratum (subgroup) on the basis of a given probability sample design, such as simple random sampling. Stratification can be used for several purposes, but each requires some information on the sample units. Sometimes, stratification is used when estimates are to be made for subpopulations of interest, such as geographic or administrative areas or rare items.

46 Suppose a type B frame is available: Cluster Sampling Two-stage Sampling

47 Cluster Sampling The main characteristic of cluster sampling is that the sampling unit is a cluster of units. To select a cluster sample, a simple random sample of clusters is taken and each unit in the selected clusters is investigated. Systematic can also be used to select a cluster sample.

48 Two-stage Sampling Two-stage sampling is the sampling procedure that results when each selected cluster is subsampled for population elements. Suppose that 50 farms clustered into 15 villages are to be surveyed. Suppose further that it is decided to select five villages at random, obtain a listing of all farms within each selected village, and then select two farms from within each village. In this case, each farm has a chance of appearing in the sample at least once with each of the other farms, and the overall sample size and survey workload can thus be controlled.

49 Cochran (1977) suggests a survey planning according to the following general topics: I. Identification of the goals of the survey; II. Definition of the target population; III. Definition of the variables of interest and the data to be collected; IV. Identification of the desired degree of precision; V. Selection of the data collection instrument; VI. Identification of a frame; VII. Design of the sample;

50 Cochran (1977) suggests a survey planning according to the following general topics: VIII. Pre-test; IX. of the sample and collection of the data / organization of the fieldwork; X. Data description and analysis; XI. Summary of the obtained information and recommendations for future surveys.

51 Cochran (1977) suggests a survey planning according to the following general topics: I. Identification of the goals of the survey; II. Definition of the target population; III. Definition of the variables of interest and the data to be collected; IV. Identification of the desired degree of precision; V. Selection of the data collection instrument; VI. Identification of a frame; VII. Design of the sample;

52 Sampling Design The choice of sample design depends on the type of the frame and the availability of auxiliary information. Example:

53 Multiple Frame Design Population

54

55 Multiple Frame Design Population Frame 1

56 Multiple Frame Design Population Frame 1 Frame 2

57 Multiple Frame Design Population Frame 1 Frame 3 Frame 2

58 Multiple Frame Design Population Frame 1 Frame 3 Frame 4 Frame 2

59 Multiple Frame Design Population Frame 1 S1 Frame 3 Frame 4 Frame 2

60 Multiple Frame Design Population Frame 1 S1 S2 Frame 3 Frame 4 Frame 2

61 Multiple Frame Design Population Frame 1 Frame 4 S1 S2 Frame 2 Frame 3 S3

62 Multiple Frame Design Population Frame 1 S1 Frame 4 S4 S2 Frame 2 Frame 3 S3

63 Dual Frame Design Very flexible approach Can accommodate a variety of estimators Accommodates the advantages of area and list frames Compromise solution for dealing with disadvantages of area and list frames

64

65

66

67

68 Hartley s Estimator

69 Hartley s Estimator

70 Hartley s Estimator

71 Hartley s Estimator

72 Dual Frame Assumptions: 1. Completness 2. Identifiability Population

73 Dual Frame Assumptions: 1. Completness 2. Identifiability Population Area Frame provides full coverage

74 Dual Frame Assumptions: 1. Completness 2. Identifiability Population Area Frame Sample List Frame

75 Dual Frame Assumptions: 1. Completness 2. Identifiability Area Frame Sampled Elements Identified at List Frame Population Area Frame Sample List Frame

76 Thank You Cristiano Ferraz Universidade Federal de Pernambuco Departamento de Estatística CAST Computational Agriculture Statistics Laboratory