+

SYSTEMATIC SAMPLING

Gladys Grace P. KikoyBS Statistics III

+Description and DefinitionSimple random sampling is the basis of our sampling theory. Sometimes systematic random sampling is better to use. Why? Simplicity-Physical list like a Telephone Book, Old license file on cards. Practicality-Spatial sampling, See next slide. It is much easier to carry out in the field. Precision-Sometimes gain precision over simple random sampling but not always. We will come back to this.

o This method of sampling is at first sight quite different from simple random sampling. Suppose that the N units in the population are numbered 1 to N in some order. To select a sample of n units, we take a unit at random from the first k units and every kth unit thereafter. For instance, if k is 15 and if the first unit drawn is number 13, the subsequent units are numbers 28, 43, 58, and so on. The selection of the first unit determines the whole sample. This type is called an every kth systematic sample.

+Description and Definitiono Systematic random sampling uses the same statistical principles as simple random sampling, that is, p values and confidence intervals are calculated the same way. However, systematic random sampling does not involve separate random selection of each household. For this reason, systematic random sampling is often used to select large samples from a long list of households.o Steps in selecting a systematic random sample:

o Calculate the sampling interval (the number of households in the population divided by the number of households needed for the sample)o Select a random start between 1 and sampling intervalo Repeatedly add sampling interval to select subsequent households

+Description and Definitiono In systematic sampling, only the first unit is selected at random, the rest being selected according to a predetermined pattern. o To select a systematic sample of n units, the first unit is selected with a random start r from 1 to k sample, where k=N/n sample intervals, and after the selection of first sample, every kth unit is included where 1≤ r ≤ k.

+Systematic Random Sampling: Transect ExampleSystematic random sampling sometimes gives a better spatial coverage than simple random sampling. Here is an example.o Think of sampling along a transect of length 100 meters where you start at a random point in first 10 m (7 meters from Excel )and then every 10th meter. The systematic random sample will be 7,17,27,37,47,57,67,77,87,97o I also chose a completely random sample of n=10 using Excel 18, 20, 33, 59, 63, 85, 90, 91, 92, 96.

+Systematic Random Sampling: Grid ExampleSystematic random sampling sometimes gives a better spatial coverage than simple random sampling. o Often have to extend to two dimensions and do grid sampling. Imagine the practical complexity of doing a two dimensional simple random sample compared to sampling on a grid which is relatively simple.

+Systematic Random Sampling: Applications

o Physical file - Telephone directory, license file stubs. Pick the starting point at random and then sample every 50thentry sayo Spatial Sampling - Air, soil, water, vegetation, mineral depositso Aerial Surveys - In many wildlife surveys transects are flown that are regularly spaced to give good coverage. For example, marine mammals like dugong in Australia

+Systematic Random Sampling: Mechanism

o Suppose we have N = 20,000 units in the population and we want n = 500.Take k = 20,000/500 = 40. Then pick a random start between 1 and 40 (say by chance 37) and then take every 40th unit. We thus use units: 37, 77,…, 19,997.o Suppose we have N = 20,000 units in the population and we want n = 500.Take k = 20,000/500 = 40. There are actually only 40 distinct systematic random samples which are: 1. {1, 41, 81,…, 19,961}, 2. {2, 42, 82,…,19,962}, ..., 40. {40,80,…,20,000}.You can thus see that we are simply taking 1of these 40 groups at random.

+Systematic Random Sampling: A Cluster Example

o Systematic random sampling is a special kind of cluster sampling. There are k different clusters and only 1 of these is sampled. In cluster sample notation we have n=1 and M=500 and nM=500 which is what we have been calling n in this section. Notice that therefore we have only 1 cluster and therefore we are going to have trouble with getting a valid variance estimate.

+Systematic Random Sampling: Relation to Cluster Sampling

o There is another way of looking at systematic sampling. With N = nk, the k possible systematic samples are shown in the columns of Table 8.2. It is evident from this table that the population has been divided into k large sampling units, each of which contains n of the original units. The operation of choosing one of these large sampling units at random. Thus systematic sampling amounts to the selection of a single complex sampling unit that constitutes the whole sample. A systematic sample is a simple random sample of one cluster unit from a population of k cluster units.

+

+Systematic Random Sampling: Relation to Cluster Sampling

o A systematic random sample is a cluster sample where only one cluster (out of k) is chosen and all n members of the cluster are sampled. This means that variance estimation is problematic. I recommend that replicated systematic sampling be used as described in the lecture. Otherwise one has to assume that the population is in “random “order and then use the results from simple random sampling.

+Systematic Random Sampling: Comparison to Simple Random Sampling

o Systematic sampling is easier to execute.o Systematic sampling seems to be more precise than simple random sampling.

+Systematic Random Sampling: Comparison to Stratified Random Sampling

o The performance of systematic sampling in relation to that of the stratified or simple random sampling is greatly dependent on the properties of the population. There are populations for which systematic sampling is extremely precise and others for which it is less precise than simple random sampling. For some populations and some values of n, V (yCsy) may even increase when larger sample is taken –a startling departure from good behavior. Thus it is difficult to give general advice about the situations in which systematic sampling is to be recommended. Knowledge of the structure of the population is necessary for its most effective use.

+Systematic Random Sampling: Variance Estimation

o As a systematic random sample is a cluster sample where only one cluster (out of k) is chosen this means that variance estimation is problematic. Approaches to variance estimation are:o Assume that the population is in “random” order and then use the results from simple random sampling.o A much better approach I recommend is that replicated systematic sampling be used. This simply involves taking multiple random starts as I show on the next slide.o Assume Population is arranged in “random order”. Then go back second lecture material for properties of the sample mean under simple random sampling. However, now the results are by assumption and not built into the design as it is not strictly simple random sampling.

+Systematic Random Sampling: Estimation of the Population Mean

o Represent the Population {y1, y2,… ,yN }o Population Mean (Finite Population) –Parameter

o Represent the Sample by {y1, y2,… ,yn }o Sample Mean – Estimate of the Parameter 

+Systematic Random Sampling: Precision

o Systematic Random Sampling may be more or less precise than simple random sampling. We need to review the cluster sampling results we discussed a couple of lectures back.

+Systematic Random Sampling: Examples1. Suppose a supermarket wants to study buying habits of their customers, then using systematic sampling they can choose every 10th or 15th customer entering the supermarket and conduct the study on this sample.This is random sampling with a system. From the sampling frame, a starting point is chosen at random, and choices thereafter are at regular intervals. For example, suppose you want to sample 8 houses from a street of 120 houses. 120/8=15, so every 15th house is chosen after a random starting point between 1 and 15. If the random starting point is 11, then the houses selected are 11, 26, 41, 56, 71, 86, 101, and 116.

+Systematic Random Sampling: ExamplesIf, as more frequently, the population is not evenly divisible (suppose you want to sample 8 houses out of 125, where 125/8=15.625), should you take every 15th house or every 16th house? If you take every 16th house, 8*16=128, so there is a risk that the last house chosen does not exist. On the other hand, if you take every 15th house, 8*15=120, so the last five houses will never be selected. The random starting point should instead be selected as a non integer between 0 and 15.625 (inclusive on one endpoint only) to ensure that every house has equal chance of being selected; the interval should now be non integral (15.625); and each non integer selected should be rounded up to the next integer. If the random starting point is 3.6, then the houses selected are 4, 19, 35, 51, 66, 82, 98, and 113, where there are 3 cyclic intervals of 15 and 5 intervals of 16.

+Systematic Random Sampling: Examples2. Assume one wishes to study dentists’ attitudes toward dental insurance and decides to sample 20 dentists from a list of 100 dentists. One way of doing this is as follows:1. Draw a random number between 1 and 5. Assume the number chosen is 2.2. Include in the sample the dentists numbered 2, 7, 12, 17, 22, —-97. That is, starting with number 2, take every fifth number.

+Systematic Random Sampling: Examples3. A researcher wants to select a systematic random sample of 10 people from a population of 100. If he or she has a list of all 100 people, he would assign each person a number from 1 to 100. The researcher then picks a random number, 6, as the starting number. He or she would then select every tenth person for the sample (because the sampling interval = 100/10 = 10). The final sample would contain those individuals who were assigned the following numbers: 6, 16, 26, 36, 46, 56, 66, 76, 86, 96.