Overview Sampling is choosing which subjects to measure in a research project. Usually the subjects are people, but subjects could also be organizations, cities, even nations. Measurement is typically accomplished through survey instruments, but could also be by observation, archival record, or other method. Regardless, sampling will determine how much and how well the researcher may generalize his or her findings. A bad sample may well render findings meaningless. Sampling in conjunction with survey research is not only one of the most popular approaches to data collection in the social sciences, random sampling is also the foundation assumption for much of inferential statistics and significance testing. Most national and other large samples by governmental agencies and polling firms use complex sampling designs, such as multistage or stratified sampling. Thes compex designs mean that significance tests must be computed differently. That is, most significance tests in statistical software are somewhat inaccurate when applied to complex samples. Specialized software, discussed in the section on complex samples, should be used to obtain the highest level of accuracy. See also the section on Survey Research.

Contents Key concepts and terms Types of random sampling Sampling problems Assumptions Frequently asked questions Complex samples Bibliography




Key Concepts and Terms

• Population: The population is the set of people or entities to which findings are to be generalized. The population must be defined explicitly before a sample is taken. Care must be taken not to generalize beyond the population. Doing so is a common error in social science writing.

• Enumerations or censuses are collections of data from every person or entity in the population. If data are an enumeration, the researcher does not have to be concerned with significance, which is a form of estimate of sampling error.

• Random sampling is data collection in which every person in the population has a chance of being selected which is known in advance. Normally this is an equal chance of being selected. If data are a random sample, the researcher must report not only the magnitude of relationships uncovered but also their significance level (the chance the findings are due to the chance of sampling). There are various types of random sampling, discussed below.

• The sampling frame is the list of ultimate sampling entities, which may be people, households, organizations, or other units of analysis. The list of registered students may be the sampling frame for a survey of the student body at a university. Problems can arise in sampling frame bias. Telephone directories are often used as sampling frames, for instance, but tend to under-represent the poor (who have fewer or no phones) and the wealthy (who have unlisted numbers). Random digit dialing (RDD) reaches unlisted numbers but not those with no phones, while overrepresenting households owning multiple phones. In multi-stage sampling, discussed below, there will be one sampling frame per stage (ex., a list of the 50 states, lists of Census tracts for sampled states, lists of Census blocks for sampled tracts, and finally a list or residences for sample blocks).

• Significance is the percent chance that a relationship found in the data is just due to an unlucky sample, and if we took another sample we might find nothing. That is, significance is the chance of a Type I error: the chance of concluding we have a relationship when we do not. Social scientists often use the .05 level as a cutoff: if there is 5% or less chance that a relationship is just due to chance, we conclude the relationship is real (technically, we fail to accept the null hypothesis that the strength of the relationship is not different from zero).

  Significance testing is not appropriate for non-random samples or for enumerations/censuses. We would like to make similar inferences for non-random samples, but that is impossible. Any relationship, not matter how small, is a true relationship (barring measurement error) for an enumeration.

  • Confidence intervals are directly related to coefficients of significance. For a given variable in a given sample, one could compute the standard error, which, assuming a normal distribution, has a 95% confidence interval of plus or minus 1.96 times the standard error. If a very large number of samples were taken, and a (possibly different) estimated mean and corresponding 95% confidence interval was constructed from each sample, then 95% of these confidence intervals would contain the true population value, assuming random sampling. The formula for calculating the confidence interval, significance levels, and standard errors, depends on the type of random sampling, whether simple or complex, discussed below.

  • The design effect (D) is a coefficient which reflects how sampling design affects the computation of significance levels compared to simple random sampling (discussed below). A design effect coefficient of 1.0 means the sampling design is equivalent to simple random sampling. A design effect greater than 1.0 means the sampling design reduces precision of estimate compared to simple random sampling (cluster sampling, for instance, reduces precision). A design effect less than 1.0 means the sampling design increases precision compared to simple random sampling (stratified sampling, for instance, increases precision). Unfortunately, most computer programs generate significance coefficients and confidence intervals based on the assumption of formulas for simple random sampling. Formulas for all types are found in Kalton (1983).

• Types of non-random sampling is widely used as a case selection method in qualitative research, or for quantitative studies of a preliminary and exploratory nature where random sampling is too costly, or where it is the only feasible alternative. . Non-random samples are often “convenience samples,” using subjects at hand. Non-random samples, like random samples, also raise the issue of findings being merely an artifact of the chance of sampling, but the rules of statistical significance testing do not apply. That is, there is no way to assess the validity of results of non-random samples.
  • Availability sampling is where the researcher selects subjects on the basis of availability. Also called haphazard sampling, examples include interviewing people who emerge from an event or location, interviewing a captive audience such as one's students, and mail-in surveys printed in magazines and newspapers.

  • Quota sampling is availability sampling, but with the constraint that proportionality by strata be preserved. Thus the interviewer(s) will be told to interview so many white male Protestants, so many Hispanic female Catholics, and so on, to improve the representativeness of the sample. Maximum variation sampling is a variant of quota sampling, in which the researcher purposively and non-randomly tries to select a set of cases which exhibit maximal differences on variables of interest. Further variations include extreme or deviant case sampling or typical case sampling.

  • Expert sampling, also called judgment sampling, is where the researcher interviews a panel of individuals known to be expert in a field. Expertise is any special knowledge, not necessarily formal training. Depending on the topic of study, experts may be policy issue academics or devotees to a popular culture fad. Critical case sampling is a variant of expert sampling, in which the sample is a set of cases or individuals identified by experts as being particularly significant (ex., award winners in a given field).

  • Chain referral sampling , also called snowball sampling or network sampling, is where the researcher starts with a subject who displays qualities of interest (ex., being a private militia member), then obtains referred subjects from the first subject, then additional referred subjects from the second set, and so on. (Note chain referral sampling is different from chain sampling in quality control applications. Chain sampling in this sense refers to basing acceptance of a given production lot based not only on a sample of that lot, but also on samples of the previous 10 or so lots.)

• Types of random sampling There are several types of random sampling, each of which affects how significance is computed.
  • Simple random sampling Simple random sampling is where the researcher has a list which approximates listing all members of the population, then draws from that list using a random number generator or possibly takes every nth subject (called interval sampling). Conventional significance testing is appropriate for simple random samples.

    Simple random sampling is common when the sampling frame is small. Starting at an arbitrary point in a table of random numbers, such as found in most statistics books, sets of random numbers are read and associated with members of the sampling frame. The first three digits might be 712, for instance, and thus the 712th person in the sampling frame might be selected. If the sampling frame had over 999 people, then four-digit random sequences would be used. Selections outside the range of members of the frame would be ignored. Using pseudo random number generating software (used to construct tables of random numbers found in books), such as that built into SPSS, a simpler, more modern method is simply to request such software generate n random digits between 1 and N, where n is sample size and N is population size.

    • Simple random sampling with replacement (SWSWR) is where selections are replaced back into the sampling frame such that repeat selections are possible. For a large sample, such repeats are rare and SWSWR becomes equivalent to SRSWOR, discussed below.

    • Simple random sampling without replacement (SRSWOR) does not allow the same random selection to be made more than once. Confidence intervals are slightly (usually trivially) smaller (more precise) for SRSWOR samples compared to simple random samples. Most computer programs use formulas which assume SRSWOR sampling.

  • Equal Probability Systematic sampling also involves the direct selection of subjects or other primary sampling units from the sampling frame. The researcher starts at a random point and selects every nth subject in the sampling frame. The random starting point equals the sampling interval, n, times a random number between 0 and 1, plus 1, rounded down. In systematic sampling there is the danger of order bias: the sampling frame list may arrange subjects in a pattern, and if the periodicity of systematic sampling matches the periodicity of that pattern, the result may be the systematic over- or under-representation of some stratum of the population. If, however, it can be assumed that the sampling frame list is randomly ordered, systematic sampling is mathematically equivalent to and equally precise as simple random sampling. If the list is stratified (ex., all females listed, then all males), systematic sampling is mathematically equivalent to stratified sampling an is more precise than simple random sampling.
    • Repeated systematic sampling is a variant which seeks to avoid the possibility of systematic biases due to periodicity in the sampling frame. This is done by taking several smaller systematic samples, each with a different random starting point, rather than using one pass through the data as in ordinary systematic sampling. Repeated systematic sample has the side benefit that the variability in the subsample means for a given variable is a measure of the variance of that estimate in the entire sample.

  • Stratified simple random sampling is simple random sampling of each stratum of the population. For instance, in a study of college students, a simple random sample may be drawn from each class (freshman, sophomore, junior, senior) in proportion to class size. This guarantees the resulting sample will be proportionate to known sizes in the population. One may simultaneously stratify for additional variables, such as gender, with separate simple random samples of freshman women, freshmen men, sophomore women, etc. The finer the stratification, the more precision compared to unstratified simple random sampling. That is, confidence intervals will be narrower for stratified sampling than for simple random sampling of the same population. The more heterogenous the means of the strata are on a variable of interest, the more stratified sampling will provide a gain in precision compared to simple random sampling. Stratified sampling, therefore, is preferred to simple random sampling.

  • Multistage stratified random sampling is where the researcher draws simple random samples from successively more homogenous groups (“strata”) until the individual subject level is reached. For instance, the researcher may sample occupations, then sample companies within occupations, then sample individual workers. The purpose of stratified random sampling is to increase research precision by ensuring that key populations of subjects are represented in the sample (ex., people in certain job categories). The greater the heterogeneity of the strata and the finer the stratification (that is, the smaller the strata involved) depending on the topic of study., the more the precision of the results. For instance, stratifying by gender at the highest level might well introduce bias in measuring opinions about an item known to be gender-related, whereas stratifying by state would be less likely to introduce a bias since there are more categories (more states than genders) and there is less likely to be a correlation with the opinion item. At each stage, stratified sampling is used to further increase precision. Because the variance of individuals from their group mean in each strata is less than the population variance, standard errors are reduced. This means conventional significance tests, based on population variances, will be too conservative – there will be too many Type I errors, where the researcher wrongly accepts the null hypothesis.

  • Disproportionate stratified sampling is the case where disproportionate numbers of subjects are drawn from some strata compared to others. A typical use of disproportionate stratified sampling is oversampling of certain subpopulations (ex., African Americans) to allow separate statistical analysis of adequate precision. When significance and confidence levels are computed for the entire sample under disproportionate stratified sampling, cases must be weighted to restore the original proportions. This weighting process generally reduces the precision benefits of stratified sampling. That is, disproportionate stratified samples tend to be less precise than proportionate stratified samples of the same population. Standard error estimates based on disproportionate stratified samples may be either more or less precise than those based on simple random samples, whereas those based on proportionate stratified samples are more precise. The more heterogeneous the means of the strata are on a variable of interest, the more the positive effect on precision of stratification compared to the negative effect of weighting, and the more likely the disproportionate stratified sample will be more precise than one based on a simple random sample.

  • Cluster sampling is where the researcher draws simple random samples from certain aggregational units of interest (ex., from certain census block groups). When the strata are geographic units, this method is sometimes called area sampling.

  • Multistage cluster sampling is where the researcher draws simple random samples from successively smaller aggregations until the individual subject level is reached. For instance, the researcher may sample counties, then sample census block groups within counties, then sample individuals within the block groups. Probability proportional to size sampling (pps) is a related variant in which each of the hierarchical levels prior to the ultimate level is sampled according to the number of ultimate units (ex., people or households) it contains. Technically, cluster sampling is where all subjects at the lowest hierarchical level (ex., all students in a school) are sampled for each primary sampling unit (PSU's, which are the second-lowest hierarchical level, such as schools or Census blocks), whereas multistage sampling is where only a random sample of lowest hierarchical level subjects are selected. Where stratification reduces standard error, clustering increases it. That is, there is an accumulation of chances of sampling error at each stage. Conventional significance tests will be too liberal – there will be too many Type II errors, where the researcher wrongly sustains an hypothesized relationship.

    Warnings on Use of Cluster Sampled Data

    Clustering will produce correlated observations, which violates the assumption of independently sampled cases - an assumption of many statistical techniques. Multi-level modeling is an example of a technique which is appropriate for clustered samples (see Goldstein, 1995). Nonetheless, it should be noted that it is common practice to treat data from cluster sampling as if it were randomly sampled data.

    Overall, multi-stage or cluster sampling is usually less precise than simple random sampling, which in turn is less precise than one-stage stratified sampling. Warning: Since multistage sampling is the most prevalent form for large, national surveys, and since most computer programs use standard error algorithms based on the assumption of simple random samples, the standard errors reported in the literature often underestimate sampling error, leading to too many Type II errors (false positives). See the discussion below regarding estimation and software for complex samples.

  
  

• Dealing with sampling problems
  • Dealing with mismatched sampling frames. Often, the sampling frame does not match the primary sampling unit. For instance, the sampling frame may be a list of residences, but the ultimate sampling units are individuals. The interviewer needs instruction on just which individual within a residence to select as a subject. This is handled by a selection grid (see Kish, 1965). The selection grid may be of a variety of forms. For instance, in a survey of taxpayers, for simplicity assume that a residence can have a maximum of three taxpayers. Let the researcher number the taxpayers from 1 to 3 by age and use the selection grid below:

    		If no. of taxpayers in residence is
    		1	2	3
    Survey form	% of all forms	Interview taxpayer #
    A	1/3	1	1	1
    B	1/6	1	1	2
    C	1/6	1	2	2
    D	1/3	1	2	3

    In this example, there are four different forms of the survey (A, B, C, D), printed and randomly distributed in the proportions in the table above. Each survey form would have one of the rows in the table. For instance, in form B, if there were two taxpayers in the residence then the interviewer would interview taxpayer number 1. For forms C or D, taxpayer number 2 would be interviewed. For form A, taxpayer number 1 would be interviewed. Kalton (1983: 61) presents the table for the assumption of a maximum of four selectable subjects per residence. Similar tables can be constructed for any assumption. All such selection grid tables equalize the probability of any appropriate individual being chosen for inclusion in the sample.

    A similar, simpler approach is the "last birthday" method, whereby the researcher asks the number of adults in the household, then interviews the sole adult in one-adult households; every other time interviews the adult with the most recent birthday in two-adult households, and every other time the other adult; every third time in three-adult households, etc. Asking about birthdays rather than ages may be less sensitive. Some evidence suggests that in telephone surveys, these methods may require additional time screening and interview time, and may generate too many callbacks or refusals. In face-to-face interviews there is greater subject tolerance for such screening questions.

  • Increasing the response rate. Response rates for face-to-face interviews are typically in the 75% range, and for mail surveys 10% is considered good in some marketing surveys. However, "good" depends of purpose and resources of the polling organization. The Office of Management and Budget's minimal standards require a 75% expected response rate, and federal data collected with response rates under 50% should not be used except if a special OMB waiver is granted. The OMB guidelines point up the salient point, that high response rate, like random sampling, is essential to reliable statistical inference. Response rate can be increased by
    1. Having legitimating sponsorship for the survey, geared to sponsors who are highly-regarded in the community being surveyed
    2. Having a good, short explanation justifying the survey
    3. Notifying individuals in advance that a survey is coming
    4. Keeping the survey instrument short, and letting prospective subjects know this.
    5. Assure confidentiality and anonymity.
    6. Offer to reschedule to a time at the subject's convenience.
    7. Make call-backs (four is typical) where needed. In mail surveys, provide a new copy of the instrument each time.
    8. Start the instrument with non-threatening questions which arouse interest.
    9. Offer to provide final results later on.
    10. In mail surveys, use certified and express mail. In e-mail surveys, use priority e-mail. Postcard and e-mail reminders also help. In mail surveys, plan to wait about six weeks for all responses to drift in.
    11. Offer token remuneration (ex., $1 enclosed in mail surveys; $5 in face-to-face interviews; or use pens, keychains, calendars, or other tokens to make respondents feel like they are getting something in return). Linking the return of the survey to entrance into a lottery for a substantial prize (ex., $1,000) can also boost response rates.

  • Analyzing non-response. The researcher must assume that non-respondents differ significantly from respondents with regard to variables of interest. There are three general strategies for assessing the effect of non-response. These strategies are not mutually exclusive.
    1. Population comparison. Survey averages can be compared with known population averages, particularly feasible for demographic variables such as gender, age distribution, income, occupation, and other variables covered by the Census. In educational studies, population data may be available on the population distribution of gender, age, class, and test performance. The researcher seeks to identify variables where the sample mean deviates from the population mean, and speculates (preferably on the basis of prior literature) on the impact of such bias on the dependent variables of interest.

    2. Intensive postsampling. An intensive effort may be made to interview a sample of non-respondents, using the postsample to assess nonresponse. Adding the postsample to the sample also reduces the bias of the sample from that estimated by this method, which is costly and therefore rare. If non-respondents are sampled, the researcher may also wish to ask instrumentation assessment questions to probe why non-response occurred in the initial sample, thus throwing light on the nature of non-response. If the non-respondent post-sample is by a different mode (ex., telephone vs. initial web survey), the researcher may wish to ask question about the initial modality (ex., does the non-responder have Internet access? an email address? check his/her email regularly? have problems loading the survey web page? etc.)

    3. Wave extrapolation. The researcher codes the initial response set and each of the four call-back response sets, computes the mean of key variables on each of the five sets, and then sees if consistent extrapolation is possible for any variable. For instance, each successive set might show fewer women respondents, suggesting the non-response set would be expected to have an even lower proportion of women yet. This method carries little marginal expense and therefore is highly recommended, particularly in supplementation to population comparison.

    4. Imputing responses. Not all respondents answer all items in a questionnaire. Andrew Gelman, Gary King, and Chuanhai Liu (1998) have presented a method of analyzing a series of independent cross-sectional surveys in which some questions are not answered in some surveys and some respondents do not answer some of the questions posed. The method is also applicable to a single survey in which different questions are asked, or different sampling methods used, in different strata or clusters. The method involves multiply-imputing the missing items and questions by adding to existing methods of imputation designed for single surveys a hierarchical regression model that allows covariates at the individual and survey levels. Information from survey weights is exploited by including in the analysis the variables on which the weights were based, and then re-weighting individual responses (observed and imputed) to estimate population quantities. They also develop diagnostics for checking the fit of the imputation model based on comparing imputed to non-imputed data. They illustrate with the example that motivated this project --- a study of pre-election public opinion polls, in which not all the questions of interest are asked in all the surveys, so that it is infeasible to impute each survey separately. See Gelman, King, and Liu (1998).

  • Weighting. There are a variety of reasons for weighting and, indeed, weighting can be a multi-step process as the researcher sequentially adjusts for probability of selection, non-response, stratification, and other purposes. Some researchers object to weighting on the grounds that it requires the artificial replication of cases, usually makes little or no difference to conclusions, and may convey a false confidence in the data while forestalling candid discussion of the biases of the sample. Supporters of weighting argue that weighting is a reasonable form of approximation to adjust an existing sample for known biases, and such correction is better than the alternative of no correction. The WEIGHT parameter in SAS and the WEIGHT BY command in SPSS are used for weighting, as discussed in their respective manuals.
    • Weighting for non-response. Sometimes it is possible to compare respondents and nonrespondents on the basis of gender or some other attribute. For instance, in a random sample of a city, census data gives a good estimate of the true proportion of males and females. In a mail survey, the true proportion of males and females may be estimated from first names. If in such situations one finds the observed distribution does not conform to the true population, one may wish to weight responses to adjust accordingly. For instance, if too few women are in the respondent pool, one might wish to weight their responses more than the male responses. For instance, if the true proportion by gender is 50-50, and if one got 40 females and 60 males, then one could weight each female response by 1.5. This, in effect, gives 60 females and 60 males. However, to avoid artificially increasing sample size from 100 to 120, one needs further weighting to scale back to 100. This could be achieved by further weighting both females and males by 5/6. This logic can be extended to the case of item non-responses for subjects who are in the respondent pool. Reliable non-response adjustment requires a high response rate (ex., > 70%).

      Bourque and Clark (1992: 60) state, "It has been our experience that the use of weights does not substantially change estimates of the sample mean unless nonrespondents are appreciably different from respondents and there is a substantial proportion of nonrespondents."

    • Weighting for post-stratification adjustment. The same logic and same weighting strategy applies if the underrepresentation of a given strata (ex., black males age 18-25) is due to non-response or due to disproportionate stratified sampling. Either way the objective is to weight the existing cases in a way which increases the representation in the adjusted sample of the strata that are underrepresented in the raw data.

    • Weighting to account for the probability of selection. In a random sample of people, each individual should have an equal chance of being selected. However, the reality of the sampling procedure may be such that each household, not individual, has an equal chance. Individuals within households with more people have a lower chance of being selected: they are underrepresented and should be weighted more. A weighting adjustment can be made if the researcher thought to have an item asking for the number of eligible people in the household. Summing the responses to this item and dividing by the number of households surveyed, assuming one survey per household, gives the average number of individuals per household, say 2.5. The weight for any surveyed individual in the sample is then the number of people in that household divided by this average. For instance, if a given household had 5 eligible individuals, the weight for that case would be 5/2.5 = 2.

  • Dealing with missing data. Non-response to individual survey items raises the question of whether to delete cases with missing data or to interpolate responses. Specialized software exists to aid in dealing with this problem. an example is the “Missing Values Analysis” module which can be added to base SPSS. This topic is dealt with in the separate page on missing data.

  • Sample size often needs to be estimated in the design phase of research. The size of the sample will need to be larger if one or more of the following applies:
    1. the weaker the relationships to be detected
    2. the more stringent the significance level to be applied
    3. the more control variables one will use
    4. the smaller the number of cases in the smallest class of any variable
    5. the greater the variance of one's variables.

    Combinations of these factors create complexities which, in combination with lack of knowledge about the population to be sampled, usually make sample size estimation just that -- an arbitrary estimate. Note that needed sample size does not depend at all on the size of the population to be sampled. Even in the most complex analyses, samples over 1,500 are very rarely needed. Specialized software, such as the “SamplePower” module which can be added to base SPSS, exists to help the researcher calculate needed sample size. See http://www.spss.com/samplepower/. Other such software as well as online sample size calculators may be found on the web. There are also rules-of-thumb and various manual methods.

    
    

Assumptions

• Significance testing is only appropriate for random samples.
  Random sampling is assumed for inferential statistics (significance testing). "Inferential" refers to the fact that conclusions are drawn about relationships in the data based on inference from knowledge of the sampling distribution. Significance tests are based on a sampling theory which requires that every case have a chance of being selected known in advance of sample selection, usually an equal chance. Statistical inference assesses the significance of estimates made using random samples. For enumerations and censuses, such inference is not needed since estimates are exact. Sampling error is irrelevant and therefore inferential statistics dealing with sampling error are irrelevant. Significance tests are sometimes applied arbitrarily to non-random samples but there is no existing method of assessing the validity of such estimates, though analysis of non-response may shed some light. The following is typical of a disclaimer footnote in research based on a nonrandom sample:
  

  "Because some authors (ex., Oakes, 1986) note the use of inferential statistics is warranted for nonprobability samples if the sample seems to represent the population, and in deference to the widespread social science practice of reporting significance levels for nonprobability samples as a convenient if arbitrary assessment criterion, significance levels have been reported in the tables included in this article." See Michael Oakes (1986). Statistical inference: A commentary for social and behavioral sciences. NY: Wiley.




Frequently Asked Questions

• How can one estimate sample size in advance?
  As mentioned above, sample size calculation depends on a number of complex factors. In practice, researchers use specialized software designed for these calculations. One rule of thumb is based on standard error as used in normal curve tests: Sample size = ss = s*z/T)², where s is the standard error of the variable with the largest variance (perhaps estimated in a pretest sample), z is the number of standard units corresponding to the desired proportion of cases (z = 1.96 for two-tailed tests at the .05 significance level), and T is the tolerated variation in the sample. For instance, this formula might be used to compute the necessary sample size such that the variable with the largest standard deviation will have a sample mean within t=2 years of the true population mean, with .05 significance.

  Another rough rule of thumb is based on chi-square testing needs, may be followed:

  1. Determine desired significance and difference levels. The researcher must first select the desired level of significance (typically .05) and the smallest difference he or she wiches to be detected as significant. For instance, in a study of gender and presidential support, one might want a 10% gender difference to be found significant at the .05 level.

  2. Specifying expected and least-difference tables. Researchers then must create two tables. This requires estimating the marginal frequencies (the number of men and women, and of presidential supporters and non-supporters, for example). Expected cell frequencies are then calculated as usual for chi-square. Then the researcher creates a least difference table as, for example, placing 10% more cases than expected on the diagonal (ex., on the male-non-supporters, female-supporters diagonal).

  3. Solving for n. The researcher then solves for n, sample size, using the chi-square formula, which is chi-square = SUM{(Observed - Expected)²/Expected}. For instance, in a 2-by-2 table, let the upper-left cell be .20n, the upper right .30n, the lower-left .20n, and the lower-right .30n. Let the least-difference cells be .25n, .25n, .15n, and .35n respectively. Degrees of freedom is (rows-1)*(columns-1), which is 1 for a 2-by-2 table. For 1 degree of freedom, at the .05 significance level, the critical value of chi-square is 3.841. Therefore chi-square = 3.841 - SUM{[(.25n-.20n)²/.20n] + [(.25n-.30n)²/.30n] + [(.15n-.20n)²/.20n] + [(.35n-.30n)²/.30n]. Solving for n, n = 3.841/.0416 = 92.3. Therefore, a sample size of 93 is the minimum sample size needed to detect a 10% difference at the .05 significance level, by chi-square.

  However, as mentioned, these are simple rule-of-thumb methods and more satisfactory estimates require taking more complex factors into account.

• How is standard error computed for dichotomous responses?
  
  For yes-no and other dichotomous items, where the object is to arrive at percentages which are within plus or minus a certain percentage (ex., 5%) of the unknown population percent, the formula is [Py*Pn]/se, where se is the standard error term. The "se" term is estimated as (siglevel/z)-squared, where siglevel is our chosen significance level (.05) and z is the corresponding number of standard units in the normal curve (1.96 for the .05) level. Also Py and Pn are the percentages estimated to answer yes and no on the dichotomy. If we lack any knowledge at all, we estimate these percentages as both being .5. The formula then becomes .25/.0006507, or 384. That is, if we sample 384 persons, we can be confident that the percentages we arrive at on a dichotomous item will be within 5% of the true but unknown population percentage.

• How do significance tests need to be adjusted for multi-stage, cluster, or other complex sampling designs, as compared to simple random samples?
  This is a central focus of the reading by Lee, Forthofer, and Lorimar (1989). These authors set forth a number of different strategies for variance estimation for complex samples, including replicated sampling: Multiple independent subsamples are drawn using an identical sampling design such as repeated systematic sampling (discussed above). The sampling variance of the total sample is estimated from the variability of the subsample estimates. Let U be a parameter such as the response to a survey item. Each of t subsamples will generate an estimate of the mean of U, designated u₁, u₂, ..., u_t. One can also pool all the subsamples to get the sample mean, ü. One can then enter these values into a formula which gives the sampling variance of ü, whose square root is interpreted as an estimate of the standard error:
  
  v(ü) = SUM (u_i - ü)²/t(t-1)
  For descriptive statistics a minimum of 4 and a recommended number of 10 subsamples should be drawn. For multivariate analysis, the number of subsamples should be 20 or more. Note that the same logic and formula apply if U is a percentage, an odds ratio, an intercept, a regression coefficient, or any other parameter being estimated.

  SPSS distributes an add-on module called Complex Samples for purposes of significance testing with multistage and other complex samples. In turn, it has the following sub-modules:

  Complex Samples Descriptives (CSDESCRIPTIVES)—
  Complex Sample Tabulate (CSTABULATE)—
  Complex Samples General Linear Models (CSGLM)—
  Complex Ordinals Selection (CSORDINAL)—
  Complex Samples Logistic Regression (CSLOGISTIC)—
  Complex Samples Cox Regression (CSCOXREG)
  

  The replication method is implemented by WesVar Complex Samples software, distributed by SPSS, Inc. Note Wesvar is limited in the statistical procedures for which it provides adjusted significance tests. Other software handling estimation and variance estimateion under stratified and other unequal probability sampling methods include SUDAAN (from the Research Triangle Institute) and VPLX (from the Bureau of the Census).

  Lee, Forthofer, and Lorimar also discuss three other methods of variance estimation: (1) balanced repeated replication, used primarily in paired selection designs; (2) repeated replication through jackknifing, which is based on pseudo-replication involving serially leaving cases out in successive subsamples ; and (3) the Taylor series method, which is less computationally intensive but cannot handle certain types of estimates, such as estimates of medians or percentiles.

  It should be noted that in practice, most social science researchers utilize the default significance tests generated by SPSS and other leading statistical packages, whose defaults are based on the assumption of simple random sampling. This is justified on the argument that in most cases, conclusions are not affected. However, ignoring complex sample design and using simple random sampling methods runs the risk of biased estimates (Landis et al., 1982; Kish 1992; Korn and Graubard, 1995).

  Resampling is an alternative inductive approach to significance testing, now becoming more popular in part because of the complexity and difficulty of applying traditional significance tests to complex samples.

  * Landis, Richard J., Lepkowski, James M., Eklund, Stephen A., and Stehouwer, Sharon A., (1982). A Statistical Methodology for Analyzing Data From a Complex Survey: The First National Health and Nutrition Examination Survey, Vital and Health Statistics, National Center for Health Statistics, Series 2-No. 92, DHHS Pub. No. 82-1366.
  * Kish, L. (1992). Weighting for unequal Pi, Journal of Official Statistics, 8, 183-200.
  * Korn, E.L. and Graubard, B.I. (1995). Examples of differing weighted and unweighted estimates from a sample survey. The American Statistician, 49, 291-295.




Bibliography

• Bourque, Linda B. and Virginia A. Clark (1992). Processing data: The survey example. Thousand Oaks, CA: Sage Publications. Quantitative Applications in the Social Sciences Series No. 85. A non-technical work which covers designing forms, pretesting, training data collectors, data entry, and data preparation, including handling nonresponse, missing data, and outliers.
• Brogan, D. (1998). Software for sample survey data, misuse of standard packages. Pp. p. 4167-4174. in P. Armitage and T. Colton, eds., Encyclopedia of Biostatistics, Volume 5. New York: Wiley. Brogan demonstrates that biased point estimates, inappropriate standard errors and confidence intervals, and misleading tests of significance can result from using standard statistical software packages which do not adjust for complex sampling designs.
• Czaja, Ron and Johnny Blair (2005). Designing surveys: A guide to decisions and procedures. Second ed.. Thousand Oaks, CA: Sage Publications. Covers item construction, survey design, testing, sampling, and reducing sources of error.
• Fink, Arlene (2002). How to sample in surveys, Vol. 7. Thousand Oaks, CA: Sage Publications. Covers probability and non-probability sampling methods, calculating response rates, dealing with sampling error, and also treats determining sample size and statistical power. Journal of the American Statistical Association, March, 1998. You can retrieve the paper from File Transfer Protocol (FTP) to ftp wizard.ucr.edu, then login as user: anonymous, password: , and cd to pub/polmeth/working_papers97, and retrieve gelma97c.zip.
• Goldstein, H. (1995). Multilevel statistical models. London, Edward Arnold; New York, Halstead Press.
• Gurr, Ted (1972). Polimetrics. Englewood Cliffs, NJ: Prentice-Hall. Gurr argues for a rule of thumb for assessing the appropriateness of imputation for multivariate analysis, holding that if one divides one's data into a set without missing data and a set with imputed data, then if both sets have the same range of data but the set with imputed data displays weaker relationships, this suggests that imputation contains unacceptable levels of error (Gurr, 1972: 87).
• Henry, Gary T. (1990). Practical sampling. Thousand Oaks, CA: Sage Publications. A very accessible introductory treatment.
• Kalton, Graham (1983). Introduction to survey sampling. Quantitative Applications in the Social Sciences Series, No. 35. Thousand Oaks, CA: Sage Publications. This work is a recognized authority.
• Kalton, Graham and D. Kasprzyk (1982). Imputing for missing survey responses. Proceedings of the Section on Survey Research Methods, American Statistical Association: 22-31.
• Kish, L. (1965). Survey sampling. NY: John Wiley. Cited with regard to selection grids.
• Lee, Eun Sol, Ronald N. Forthofer, and Ronald J. Lorimor (1989). Analyzing complex survey data. Thousand Oaks, CA: Sage Publications. Quantitative Applications in the Social Sciences Series No. 71. Good coverage of what statistical adjustments must be made for stratified, multistage, clustered, and other complex sampling designs as compared to simple random sampling.

Copyright 1998, 2008 by G. David Garson
Last update 3/29/2008,