Multi-Stage Sampling

In simple random sampling, the first stage in the process is to enumerate (give names or numbers to) the entire population. While this may be feasible for small populations, it is clearly more difficult with larger populations. For example, identifying every individual in a large city or a nation is clearly a non-trivial task. In such circumstances, another variation of random sampling is called for. Multi- stage sampling is a random sampling technique which is based on the process of selecting a sample in two or more successive, contingent stages. Consider, for example, a multi-stage survey of travel patterns for an entire nation. Within an Australian context, the process may proceed in five stages as follows:

(a) First-stage: divide nation into states and sample from total population of states.

(b) Second-stage: divide selected states into Local Government Areas and sample from these Local Government Areas within each selected state.

(c) Third-stage: divide selected Local Government Areas into Census Collectors' Districts and sample Census Collectors' Districts. (d) Fourth-stage: divide selected Census Collectors' Districts into

households and sample households.

(e) Fifth-stage: divide selected households into individuals and sample individuals.

At the end of this process we have a random sample of individuals from the nation (i.e. every individual had an equal chance of being selected at the start of the process) provided that appropriate sampling procedures are used at each of the stages. Thus at the first three stages, it would be necessary to sample states, Local Government Areas and Census Collectors' Districts by means of a selection procedure with probabilities proportional to size (PPS) if all individuals are to have an equal probability of selection. Thus, larger population states would have a higher probability of selection at the initial stage. The PPS sampling procedure can be easily applied to the initial three stages because the population within each state, Local Government Area and Census Collectors' District would generally be known in advance of sampling (from other sources such as National Census statistics).

At the fourth stage, however, a problem arises because, without detailed knowledge of the size of each household in the selected Census Collectors' Districts, it would not be possible to use PPS sampling at this stage. Such detailed knowledge about individual household structure would generally be unavailable.

Without this information, it can easily be seen that if one individual is to be selected from each selected household then an individual in a small household has a higher probability of selection than an individual in a large household. To correct for this it may be necessary to place households in strata in the field by means of filter questions at the start of the interview. The number of households in each stratum would be directly proportional to the household size. When each stratum is filled, no further questions would be asked of that household. Thus if x interviews were conducted in single person households, then 2x interviews should be conducted in two-person households etc., such that each individual has an equal chance of selection, irrespective of household size. Alternatively, households could be selected randomly as if they were all of equal size and then adjustments could be made to the survey results by means of appropriate weighting factors to reflect the distribution of household sizes found in the population.

The fifth stage also requires care in sampling. The interview should not be conducted with whoever opens the door or with anyone who is simply willing to be interviewed. Rather, if individuals are the unit of investigation, random sampling should be performed across all members of the household who are members of the population under investigation (perhaps, for example, there is an age-limit on members of the population). This random sampling may be formalised by printing, on the interview form, instructions to the interviewer for selection of a household member depending on the size of the household. These sampling instructions would be varied randomly, or systematically, from form to form to ensure the desired distribution of household members was obtained in the sample (see Kish, 1965). Examples of such selection grids are provided by Stopher (1985a) for selection of adults aged 18 or over, and are reproduced in Figure 4.6. The interviewer uses each grid on alternating occasions (odd and even grids) and then, depending on the answers to filter questions about the number of adults aged 18 or over and the number of males aged 18 or over, asks to speak with a specified member of the household as indicated in the appropriate selection grid.

O D D

Adults 18 and Over

Males 18 + 1 2 3 4

0 THE WOMAN THE YOUNGER WOMAN THE OLDEST WOMAN THE SECOND YOUNGEST WOMAN 1 THE MAN THE MAN

THE OLDER WOMAN THE YOUNGEST WOMAN 2 THE OLDER MAN THE OLDER MAN THE OLDER MAN 3 THE OLDEST MAN THE YOUNGEST MAN EVEN

Adults 18 and Over

Males 18 + 1 2 3 4 0 THE WOMAN THE OLDER WOMAN THE OLDEST WOMAN THE OLDEST WOMAN 1 THE MAN THE WOMAN THE YOUNGER

WOMAN THE SECOND YOUNGEST WOMAN 2 THE OLDER MAN THE OLDER MAN THE YOUNGEST MAN 3 THE SECOND OLDEST MAN THE SECOND OLDEST MAN

Figure 4.6 Examples of Respondent Selection Grids

(Source: Stopher, 1985a)

Whilst multi-stage sampling may appear to be somewhat complicated from the above description, its major advantage over simple random sampling lies, in fact, in its convenience and economy, especially for surveys of large populations. Thus, in multi-stage sampling, it is not necessary to enumerate all the sampling units in the population. At each stage, the only sampling units which need to be listed are those which belong to the higher level sampling units selected in the previous stage. Thus the expensive and time-consuming compilation of a complete sampling frame list is avoided.

The disadvantage of multi-stage sampling is that the level of accuracy of parameter estimates for a given sample size tends to be less than if a simple random sample had been collected (see Section 4.7 for more details). However, this reduction in accuracy needs to be traded off against the reduction in costs. In many cases, an increase in sample size for multi-stage samples can be paid for by the savings accrued in not having to prepare a full sampling frame.

It should also be noted that at each stage in the multi-stage sampling process, different sampling methods can be applied. Thus stratified sampling and variable fraction sampling can be applied to meet certain objectives. For example, if

certain states must be represented in the final sample (perhaps for political or other reasons outside the scope of the survey) then these states can be segregated into strata by themselves and sampled with certainty. The higher probability of selection of the state must, however, be compensated for by lower probabilities of selection at later stages in the process, such that individuals in all states have an equal probability of selection. This latter criteria is, in fact, the guiding light of multi-stage sampling; virtually anything is allowable at each stage provided that individuals (or whatever the last-stage sampling element happens to be) have an equal chance of selection after all stages have been completed.

Multi-stage sampling can also be used in the design of on-board transit surveys. In such surveys, the sampling unit is the transit passenger, but it would be impractical to have a sampling frame based on the names of all transit passengers. Rather, the sample of transit passengers can be drawn in a four stage process, where each stage takes account of a different dimension in the transit passenger population (see Stopher, 1985b; Fielding, 1985):

Stage 1: Geographically-stratified sampling of routes Stage 2: Sampling of vehicles from the selected routes

Stage 3: Time-stratified sampling of runs on selected vehicles Stage 4: Surveying of all passengers on selected runs