Estimation in Stratified Random Sampling

Slides:



Advertisements
Similar presentations
High Resolution studies
Advertisements

Sampling: Theory and Methods
Multistage Sampling.
BUS 220: ELEMENTARY STATISTICS
Lecture 2 ANALYSIS OF VARIANCE: AN INTRODUCTION
Multistage Sampling Module 3 Session 9.
Stratified Sampling Module 3 Session 6.
1 Session 10 Sampling Weights: an appreciation. 2 To provide you with an overview of the role of sampling weights in estimating population parameters.
SADC Course in Statistics Analysis of Variance for comparing means (Session 11)
SADC Course in Statistics Common Non- Parametric Methods for Comparing Two Samples (Session 20)
Basic Sampling Concepts
SADC Course in Statistics Estimating population characteristics with simple random sampling (Session 06)
SADC Course in Statistics Simple Linear Regression (Session 02)
The Poisson distribution
Overview of Sampling Methods II
SADC Course in Statistics Further ideas concerning confidence intervals (Session 06)
SADC Course in Statistics Introduction to Non- Parametric Methods (Session 19)
SADC Course in Statistics Tests for Variances (Session 11)
Assumptions underlying regression analysis
SADC Course in Statistics Basic principles of hypothesis tests (Session 08)
SADC Course in Statistics Meaning and use of confidence intervals (Session 05)
SADC Course in Statistics The binomial distribution (Session 06)
SADC Course in Statistics Sampling weights: an appreciation (Sessions 19)
SADC Course in Statistics Inferences about the regression line (Session 03)
Correlation & the Coefficient of Determination
SADC Course in Statistics Samples and Populations (Session 02)
SADC Course in Statistics Confidence intervals using CAST (Session 07)
SADC Course in Statistics Sample size determinations (Session 11)
SADC Course in Statistics Sampling design using the Paddy game (Sessions 15&16)
SADC Course in Statistics Multi-stage sampling (Sessions 13&14)
SADC Course in Statistics Session 4 & 5 Producing Good Tables.
SADC Course in Statistics Graphical summaries for quantitative data Module I3: Sessions 2 and 3.
SADC Course in Statistics Common complications when analysing survey data Module I3 Sessions 14 to 16.
SADC Course in Statistics Comparing two proportions (Session 14)
SADC Course in Statistics Linking tests to confidence intervals (and other issues) (Session 10)
SADC Course in Statistics Review and further practice (Session 10)
SADC Course in Statistics Revision using CAST (Session 04)
SADC Course in Statistics Introduction to Statistical Inference (Session 03)
SADC Course in Statistics Overview of Sampling Methods I (Session 03)
SADC Course in Statistics General approaches to sample size determinations (Session 12)
SADC Course in Statistics To the Woods discussion (Sessions 10)
SADC Course in Statistics Reporting on the web site Module I4, Sessions 14 and 15.
SADC Course in Statistics Case Study Work (Sessions 16-19)
SADC Course in Statistics Objectives and analysis Module B2, Session 14.
SADC Course in Statistics Revision on tests for means using CAST (Session 17)
Probability Distributions
Chapter 7 Sampling and Sampling Distributions
Biostatistics Unit 5 Samples Needs to be completed. 12/24/13.
Chi-Square and Analysis of Variance (ANOVA)
5-1 Chapter 5 Theory & Problems of Probability & Statistics Murray R. Spiegel Sampling Theory.
7 (a) Under what circumstances is stratified random sampling procedure is considered appropriate?How would you select such samples?Explain by means of.
Factoring Grouping (Bust-the-b) Ex. 3x2 + 14x Ex. 6x2 + 7x + 2.
Chapter 8 Estimation Understandable Statistics Ninth Edition
1 STRATIFIED SAMPLING Stratification: The elements in the population are divided into layers/groups/ strata based on their values on one/several.
Chapter 5 Stratified Random Sampling n Advantages of stratified random sampling n How to select stratified random sample n Estimating population mean and.
Chapter 17 Additional Topics in Sampling
Stratified Simple Random Sampling (Chapter 5, Textbook, Barnett, V
STAT 4060 Design and Analysis of Surveys Exam: 60% Mid Test: 20% Mini Project: 10% Continuous assessment: 10%
SADC Course in Statistics Paddy results: a discussion (Session 17)
Stratified Sampling Lecturer: Chad Jensen. Sampling Methods SRS (simple random sample) SRS (simple random sample) Systematic Systematic Convenience Convenience.
Formalizing the Concepts: STRATIFICATION. These objectives are often contradictory in practice Sampling weights need to be used to analyze the data Sampling.
Chap 20-1 Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chapter 20 Sampling: Additional Topics in Sampling Statistics for Business.
Scot Exec Course Nov/Dec 04 Survey design overview Gillian Raab Professor of Applied Statistics Napier University.
Chapter 18 Additional Topics in Sampling ©. Steps in Sampling Study Step 1: Information Required? Step 2: Relevant Population? Step 3: Sample Selection?
Sampling Design and Analysis MTH 494 Lecture-21 Ossam Chohan Assistant Professor CIIT Abbottabad.
Probability Sampling. Simple Random Sample (SRS) Stratified Random Sampling Cluster Sampling The only way to ensure a representative sample is to obtain.
1. 2 DRAWING SIMPLE RANDOM SAMPLING 1.Use random # table 2.Assign each element a # 3.Use random # table to select elements in a sample.
Meeting-6 SAMPLING DESIGN
STRATIFIED SAMPLING.
2. Stratified Random Sampling.
Presentation transcript:

Estimation in Stratified Random Sampling SADC Course in Statistics Estimation in Stratified Random Sampling (Session 07)

Learning Objectives By the end of this session, you will be able to explain what is meant by stratification, how a stratified sample is drawn, and its advantages explain proportional or Neyman’s allocation of sample sizes to each stratum compute estimates of the population mean and population total from results of a stratified random sample determine measures of precision for the above estimates

Review of stratified sampling We recall first that stratification is done when it is possible to divide the population into groups (strata) so that the within group variance is small, ideally as small as possible. From each stratum, a sample of suitable size is drawn, usually using simple random sampling. The greatest challenge is in defining a suitable stratification variable. It is useful when information is required for each stratum (e.g. each region in a country) as well as for the whole population.

Advantages of stratification Sampling from each stratum guarantees that the overall sample is more representative of the whole population compared to a simple random sample If each stratum is more homogeneous, i.e. less variable than the population as a whole with respect to key responses of interest, then estimates will be more precise Likely to be administratively convenient, e.g. when different sampling procedures need to be applied to different strata (see ELUS example in Practical 2 for large sized estates of >500ha)

Sampling with proportional allocation Suppose there are m strata and a sample of size ni is chosen from the Ni units in stratum i. Then total population size is N =  Ni , while the sample size is n =  ni . Often convenient to choose ni so that This is called proportional allocation

Sampling using Neyman’s allocation If costs of sampling are the same in each stratum, but variability is different (although homogeneous within strata), then sensible to take more samples where there is greater variability, i.e. sample in proportion to the standard deviation. The appropriate value of ni in this case, see below, is called Neyman’s (or optimum) allocation.

Other issues and allocation methods Above assumes within-stratum variances Si are known. A pilot run or a previous study may give estimates. But results from a pilot run may give very poor estimates, since they will often be based on very small sample sizes Also note that Neyman’s allocation may lead to very few units being sampled from some strata – not useful if separate results for each stratum are also needed. Other methods of allocation exists, e.g. incorporating possible differences in sampling costs

Estimating the population mean First carry out computations for each stratum, i.e. find mean and variance for ith stratum. The estimate the population mean is then , with variance

Estimating the population total As with the mean, first find an estimate for the total in ith stratum, i.e. The estimate the population total is then , with variance Note: Use expressions on the previous page in computing these estimates

An example Government agricultural inspectors carry out a survey of cattle ownership in a region divided into 3 administrative areas. Five farms are selected from each area and the number of cattle recorded as shown below. The total number of farms is 636. Area Number of farms No of cattle 1 186 8, 50, 92, 60, 34 2 214 0, 0, 4, 12, 24 3 236 16, 4, 28, 46, 28

Questions to answer Ni 1 - fi Note: fi = ni/Ni in ith stratum. What is the mean number of cattle per farm? What is the total number of cattle in the region? First need to compute some summaries: Area Ni 1 - fi 1 186 48.8 969.2 0.9731 2 214 8.0 104.0 0.9766 3 236 24.4 244.8 0.9788 Note: fi = ni/Ni in ith stratum.

Answers for estimating mean The mean number of cattle per farm is estimated as: = 16547.2/636 = 26.02 i.e. Approximately 26 cows per farm. This has variance: = 25.031 Hence its std. error = 5.0

Answers for estimating total The total number of cattle in the region is estimated as: = 636 x 26.02 = 16547 This has variance: = (636)2 x 25.031 Hence its standard error is 636 x 25.031 = 3181.9

Estimating population proportion As with the mean, first find an estimate for proportion in ith stratum, i.e. pi = ri/ni The estimate the population proportion is then , with variance

References Barnett, V. (1974) Elements of Sampling Theory. Edward Arnold. ISBN 0 340 17387 4 Levy, P.S. and Lemeshow, S. (1999) Sampling and Populations: Methods and Applications (3rd edition) Wiley, New York. ISBN 0-471-15575-6 Lohr, S.L. (1999) Sampling: Design and Analysis. International Thomson Publishing. ISBN 0-534-35361-4

Practical work follows…