WAYNE W. DANIEL, Ph.D., and CAROL E. COOGLER, M.S.
Although the importance of appropriate statistical analyses of the results of physical therapy research is well documented, the theory and methodology of sampling, the procedure by which the data of research are usually obtained, has received much less attention in the physical therapy literature. To help fill this gap, some of the basic concepts and techniques of sampling are discussed.
The importance of appropriate statis tical analyses of the results of physical therapy research is well documented. 1 " 5 Data collection procedures—those procedures which generate the data on which statistical analyses are performed—have received little attention in physical therapy literature. Recently, however, Crocker has made a contribution toward filling this void. 6 Our purpose in this article is to present some basic concepts in the theory and methodology of sampling applicable to physical therapy research. SAMPLING AND STATISTICAL INFERENCE
The subjects who participate in most re search constitute only a part, or subset, called a sample, of some larger collection of similar subjects, called the population. The procedure by which the sample is obtained is called sampling. Dr. Daniel is Professor of Quantitative Methods, School of Business Administration, Georgia State University, Atlanta, GA 30303. Ms Coogler was Associate Professor of Physical Therapy, School of Allied Health Sciences, Georgia State University, at the time this article was written. She is currently Chief Physical Therapist at the Shepherd Spinal Center, West Paces Ferry Hospital, Atlanta, GA 30327.
1326
The Purpose of Sampling
The purpose of sampling is to reach a conclusion about a population on the basis of the information contained in a sample drawn from that population. This process of drawing conclusions about a population from a sample is called statistical inference and may take one of two forms: estimation and hypothesis testing. To be effective, sampling must yield a sample that is representative of the population from which it is drawn. Failure to achieve this goal can be expected to lead to erroneous conclu sions regarding the sampled population. Crocker, for example, states that "a common threat to validity of the results is biased sample selection." 6 The methods discussed in this article, when properly followed, will result in samples that are representative of the popula tions from which they are drawn.
ADVANTAGES OF SAMPLING
The following are some of the advantages of sampling that may be realized by the investi gator who employs this tool of research: 1. Reduced cost. The cost involved in examin ing a part of a population will be less than is PHYSICAL THERAPY
Downloaded from https://academic.oup.com/ptj/article-abstract/55/12/1326/4567579 by University of Texas at Dallas - McDermott Library user on 04 February 2019
Sampling in Physical Therapy Research
KINDS OF SAMPLES
Populations differ with respect to character istics that are relevant to, and affect the outcome of, sampling procedures that are applied to them. A sampling technique that is effective in generating a sample that is repre sentative of one population may not be appropriate for other types of populations. In order to improve the efficiency and effective ness of sampling from different types of populations, several different sampling pro cedures have been developed.
Usually, a simple random sample is selected with the aid of a table of random digits such as that found in the book containing a million random digits published by the Rand Corpora tion. 7 Selection of the items to be included in the sample is made from a frame which is some type of physical representation of the popula tion of interest. A frame, for example, may consist of a printed list of patients; a collection of punched cards or index cards, where each card represents one item in the population; or, if the population of interest consists of house holds in some geographic area, the frame may be a map on which each household in the area is identifiable. The items (subjects, households) in the frame are numbered consecutively from 1 to N, where N is the size of the population. Then, from the table of random digits, n random numbers between 1 and N are selected. The items in the frame whose numbers correspond to the n random numbers constitute the sample of size n. Suppose, for example, that the population of interest consists of N = 1,000 hospitalized patients treated by a physical therapist during the past two years. Suppose also that we wish to draw a sample of size n = 10 from that population. The frame might consist of a computer printout on which the patients' names are listed and numbered from 1 to 1,000. From a table of random digits, the ten numbers 651, 849, 027, 048, 940, 126, 072, 354, 708, 317 might be selected. The sample of size 10 would be composed of patients on the list whose numbers correspond to these random numbers.
Simple Random Samples
In practice, sampling is almost always done An understanding of the basic concepts of sampling theory and methodology is best achieved through a consideration of simple random sampling. Other, more complicated sampling procedures, build on the basic con cepts of the simple random sample. A simple random sample is a sample drawn from a population in such a way that every sample of the given size has the same chance (probability) of being selected. For example, from a population of, size 100, one may draw many samples of size 10. The definition of a simple random sample states that each of these samples has an equal chance of being drawn.
Volume 55 / Number 12, December 1975
without replacement. That is, a given item in the population is allowed to appear only once in a sample. If duplicate numbers are en countered in the process of selecting numbers from the table of random digits they are ignored, and the next usable number that is not a duplicate is chosen. The techniques and concepts of statistical inference—estimation and hypothesis testingare relatively simple and straightforward when the data have been obtained by simple random sampling. Appropriate formulas and the ration ale underlying the procedures may be found in elementary statistics texts. 1327
Downloaded from https://academic.oup.com/ptj/article-abstract/55/12/1326/4567579 by University of Texas at Dallas - McDermott Library user on 04 February 2019
required to examine the population in its entirety. If the population of interest is extremely large, the cost of obtaining data on every item or subject may be prohibitive. Sampling, then, may be the only practical means of arriving at a decision regarding a population. 2. Increased speed. In many instances, time is a critical factor, and sampling may be neces sary in order for results to be available when they are needed. 3. Greater comprehensiveness. Fixed time and cost constraints make sampling necessary for the desired breadth and depth of informa tion to be obtained. 4. Increased accuracy. When a relatively small number of items or subjects is examined, more attention can be focused on the details of collecting and processing the information. The result will be more accurate data on which to base conclusions.
Systematic Sampling
1328
Stratified Random Samples
When estimation is the inferential procedure employed, narrow interval estimates are desir able, and when the statistical analysis takes the form of hypothesis testing, it is desirable to make the chances of rejecting a true hypothesis and failing to reject a false hypothesis small. Usually these goals can be accomplished by taking large samples. Large samples, however, tend to be expensive. Fortunately, there is an alternative procedure that may be used in many situations to reduce both the size of interval estimates and the probabilities of rejecting true hypotheses and failing to reject false hypoth eses. This alternative procedure is called strati
fied random sampling. When stratified random sampling is em ployed, the population of interest is partitioned into nonoverlapping subpopulations called strata. From each of these strata a random sample is drawn, and the results are combined for estimation and hypothesis testing purposes. In order for stratification to be effective, each stratum must be more homogeneous, with respect to the variable of interest, than the population considered as a whole. That is, the variances in the individual strata must be smaller than the overall population variance. Homogeneous strata are obtained through the formation of strata on the basis of a variable that is correlated with the variable of interest. If, for example, one wished to estimate the proportion of physical therapy departments with a high therapist turnover rate, the popula tion of departments might be stratified on the basis of mean salary for therapists. The assumpPHYSICAL THERAPY
Downloaded from https://academic.oup.com/ptj/article-abstract/55/12/1326/4567579 by University of Texas at Dallas - McDermott Library user on 04 February 2019
Selecting a simple random sample from a large population can be a tedious process. The tedium can be avoided, however, through the use of systematic sampling. To draw a system atic sample of size 200 from a population of size 10,000, an item is selected at random from the first 10,000/200 = 50 items in the frame. The remaining 199 items are obtained by taking every 50th item from the initial item drawn. For example, if the item selected at random from the first 50 is item number 35, the second item selected will be the 85th, the third item selected will be the 135th, and so on. An additional advantage of systematic sampling is the fact that this type sample is likely to be more representative of the population than is a simple random sample, since the former is spread more evenly through the population. If the items in the frame from which a systematic sample is drawn are in random order with respect to the variable of interest, the systematic sample will be equivalent to a simple random sample, and the formulas for simple random sampling may be used in testing hypotheses and constructing confidence inter vals. Two situations exist in which a systematic sample cannot be treated as a simple random sample. The first is the case in which the items in the frame are ordered according to the magnitude of the variable of interest. For example, a frame might consist of physical therapy departments listed in order of the number of staff therapists employed, so that departments with fewer therapists appear at the beginning of the frame, and those with more staff therapists appear at the end. When simple random sampling formulas are used with data from this type of population, the sampling variability tends to be overestimated. The second situation in which a systematic sample cannot be treated as a simple random sample occurs when the frame is constructed in such a way that the values of the variable of interest exhibit some cyclical pattern called periodicity. An example of periodic data is the daily occurrence of motor vehicle accidents which tend to peak at the end and drop near the beginning of each week. Simple random sampling formulas used with the data of
systematic samples drawn from this type of population tend to underestimate the sampling variability. When the researcher, wishing to use sys tematic sampling, suspects that, with respect to the variable of interest, the frame from which a sample is to be drawn is either ordered or periodic, special techniques should be em ployed. A frequently used technique is that of drawing several systematic samples and using special formulas rather than the formulas of simple random sampling. These formulas may be found in most sampling textbooks such as those given in the references.
Cluster Sampling
When a sample survey is contemplated, the researcher often finds that no frame of the items (subjects) to be sampled exists. Suppose, for example, that the population of interest consists of all patients receiving physical ther apy in the general hospitals of a certain area during the past year. A list (frame) of such patients probably does not exist. If many hospitals are involved, the expense incurred in compiling such a list may be prohibitive. Another problem that the physical therapy researcher may encounter is that of widely scattered items or subjects. If a survey of patients with a certain type of condition is contemplated, the researcher may find that these patients are scattered over a wide geo graphic area. If the desired information can be obtained only by personal interview, travel costs associated with simple random or strati fied random sampling may be exorbitant if patients are located at great distances from each other. Difficulties such as these may be overcome through the use of cluster sampling. In cluster sampling, the subjects constituting the popula tion of interest are grouped into nonoverlapping subpopulations called clusters. Clusters frequently are geographical units such as census tracts and counties or some other physical entity such as hospitals and classrooms. After clusters have been formed, the sampling process proceeds in two steps. In the first step, a simple Volume 55 / Number 12, December 1975
random sample of clusters is selected from all clusters, and in the second step a random sample of subjects or items is drawn from each of the clusters selected in the first step. Sometimes, in the second step, all of the subjects in the clusters are selected. Informa tion obtained from this two-stage sampling procedure is combined and used as the basis for estimating or testing hypotheses about the parameters of the overall population of interest. When cluster sampling is employed, frames are needed for only those clusters selected in the first step of the sampling procedure. For example, if the population of interest consisted of patients with a certain diagnosis treated at one hundred different hospitals, each hospital could be treated as a cluster. The first "stage of sampling might involve selecting a simple random sample of ten hospitals. For use in the second stage of sampling, the researcher would need to prepare a frame of patients of the desired type for these ten hospitals (clusters) only. The savings in time and money associated with this approach as opposed to simple random or stratified sampling are obvious. The primary purpose of cluster sampling is to save time, reduce costs, and make the sampling process generally more convenient. Such ad vantages of stratified sampling as improved estimates in the form of narrower confidence intervals and reduction of errors in hypothesis testing procedures usually are not realized when cluster sampling is employed. Samples of Convenience
Some readers are likely to become frustrated when they try to reconcile the ideas presented in this article with such statements, so fre quently encountered in published research findings, as "The subjects for this experiment consisted of ten consecutive patients seen in the outpatient clinic," and therapists who find it necessary to use as their subjects "consecutive patients seen in the outpatient clinic" are apt to feel guilty at this apparent violation of the cardinal rules of sampling. They may be able to take little comfort from the rationalization that "everybody does it." In many practical research settings, the sample available for analysis is not random in the strict sense of the word. That is, there is no 1329
Downloaded from https://academic.oup.com/ptj/article-abstract/55/12/1326/4567579 by University of Texas at Dallas - McDermott Library user on 04 February 2019
tion would be that the two variables, mean starting salary and turnover rate, are related. One would expect departments offering higher salaries to have a lower turnover rate. Four strata could be formed, on the basis of salary, according to the four quartiles that could be delineated. If desired, more strata could be formed by making the salary range smaller in each stratum, or if fewer strata were desired, wider ranges could be formed. Stratification is most effective when the variable of interest and the variable on which stratification is based are closely related. If the variable of interest is therapist turnover rate, one would not expect much benefit to accrue from stratification based on the heights of therapists.
1330
SUMMARY AND SUGGESTIONS The application of sampling methodology in physical therapy research has been the subject of this article. Samples of convenience, simple random sampling, and the basic concepts of systematic, stratified, and cluster sampling have been discussed. Only the fundamental ideas have been presented. The reader who wishes to obtain further information on the theory and methodology of sampling is referred to the many available books on the subject. For those who would like a discussion of sampling that is completely devoid of mathematics, the paper back book by Slonim is recommended. 11 Mendenhall and co-workers have written a basic sampling text that is readable and contains only the basic mathematics necessary for the statis tical analysis of the results of sampling. 12 This book makes a useful addition to the library of the researcher who conducts sample surveys. The book by Babbie concentrates on the methodology of surveys, including such topics as questionnaire design, interviewing tech niques, and problems that are likely to be encountered. 13 For a treatment of the subject that places a little more emphasis on theory, yet remains accessible to those with modest preparation in mathematics, the book by Yamane is recommended. 14 Other books on sampling that are helpful, but which are more theoretical and mathematically rigorous, in clude those by Cochran, 15 Deming, 16 Hansen, Hurwitz, and Madow, 17 ' 18 Kish, 19 Raj, 20,21 Sukhatme and Sukhatme, 22 and Yates. 23 REFERENCES
1. Daniel WW, Coogler CE: Some quick and easy statistical tests for physical therapists. Phys Ther 54:135-140, 1974 2. Daniel WW, Coogler CE: Beyond analysis of variance: A comparison of some multiple com parison procedures. Phys Ther 55:144-150, 1975 3. Michels E: Use of the t test (letter to the editor). Phys Ther 50:581-585, 1970 4. Michels E: The 1969 Model t (letter to the editor). Phys Ther 50:751-752, 1970 5. What's the answer? Phys Ther 53:566-568, 1973 6. Crocker LM: Let's reduce the communication gap: Guidelines for preparing a research article. Phys Ther 54:971-976, 1974 7. The Rand Corporation: A Million Random Digits With 100,000 Normal Deviates. Glencoe, Illinois, The Free Press, 195 5 8. Dunn OJ: Basic Statistics: A Primer for the Biomedical Sciences. New York, John Wiley and Sons Inc., 1964, p 12
PHYSICAL THERAPY
Downloaded from https://academic.oup.com/ptj/article-abstract/55/12/1326/4567579 by University of Texas at Dallas - McDermott Library user on 04 February 2019
well-defined population from which subjects are selected in such a way that randomness of selection is assured. The physical therapist who wishes to conduct research with subjects having total hip replacements cannot wait until a large enough pool (population) of such subjects is available and then select from these a random sample for study. For certain research to be possible, subjects must be used as they become available in the clinic or hospital. The therapist, therefore, has little, if any, choice about which subjects compose the sample. Samples composed of subjects of this type are referred to by Dunn as samples of con venience'. 8 If inferences to some population are to be made—and an overwhelming proportion of published research findings are used for this purpose—care must be exercised in deciding to just what population one intends the inferences to apply. Dunn suggests that the researcher in this situation carefully define the sample and then think of the target population (the population about which inferences are made) as being composed of a larger collection of subjects that are similar to those in the sample. 8 Suppose, for example, that the sample of convenience consists of subjects who are char acterized as middle class, urban, between the ages of twenty and forty years, and suffering from no apparent organic or functional disorder other than the one under investigation. The population about which inferences are made may be considered as a large collection of similar subjects who have the same disorder. Remington and Schork point out that the researcher should look for obvious and hidden biases that might cause the sample of conven ience to be unusual in some respect. 9 If, for example, all the subjects in the sample are heavy smokers, the scope of inference may be so limited that the overall usefulness of the results is severely reduced. Armitage also considers the problem of defining the target population when inferences are to be made from samples of convenience. 1 0 He suggests that the target population be thought of as "a hypothetical population which would be generated if an indefinitely large number of observations, showing the same sort of random variation as those at our disposal, could be made." 10
Volume 55 / Number 12, December 1975
York, John Wiley and Sons Inc., 1950 17. Hansen HH, Hurwitz WN, Madow WG: Sample Survey Methods and Theory: Vol I. Methods and Applications. New York, John Wiley and Sons Inc., 1953 18. Hansen HH, Hurwitz WN, Madow WG: Sample Survey Methods and Theory: Vol II. Theory. New York, John Wiley and Sons Inc., 1953 19. Kish L: Survey Sampling. New York, John Wiley and Sons Inc., 1965 20. Raj D: Sampling Theory. New York, McGraw-Hill Book Co., 1968 21. Raj D: The Design of Sample Surveys. New York, McGraw-Hill Book Co., 1972 22. Sukhatme PV, Sukhatme, BV: Sampling Theory of Surveys With Applications 2d ed. Ames, Iowa, The Iowa State University Press, 1970 23. Yates F: Sampling Methods for Census Surveys 3d ed. New York, Hafner Publishing Co., 1960
1331
Downloaded from https://academic.oup.com/ptj/article-abstract/55/12/1326/4567579 by University of Texas at Dallas - McDermott Library user on 04 February 2019
9. Remington RD, Schork MA: Statistics With Applications to the Biological and Health Sci ences. Englewood Cliffs, New Jersey, PrenticeHall, Inc., 1970, p 93 10. Armitage P: Statistical Methods in Medical Re search. Oxford, England, Blackwell Scientific Publications, 1971, p 100 11. Slonim MJ: Sampling. New York, Simon and Schuster Inc., 1960 12. Mendenhall WL, Ott L, Schaeffer RL: Elementary Survey Sampling. Belmont, California, Wadsworth Publishing Company, 1971 13. Babbie ER: Survey Research Methods. Belmont, California, Wadsworth Publishing Co., 1973 14. Yamane T: Elementary Sampling Theory. Engle wood Cliffs, New Jersey, Prentice-Hall, Inc., 1967 15. Cochran WG: Sampling Techniques 2d ed. New York, John Wiley and Sons Inc., 1963 16. Deming WE: Some Theory of Sampling. New
Although the importance of appropriate statistical analyses of the results of physical therapy research is well documented, the theory and methodology of sampling, the procedure by which the data of research are usually obtained, has received much less attention in the physical therapy literature. To help fill this gap, some of the basic concepts and techniques of sampling are discussed.
The importance of appropriate statis tical analyses of the results of physical therapy research is well documented. 1 " 5 Data collection procedures—those procedures which generate the data on which statistical analyses are performed—have received little attention in physical therapy literature. Recently, however, Crocker has made a contribution toward filling this void. 6 Our purpose in this article is to present some basic concepts in the theory and methodology of sampling applicable to physical therapy research. SAMPLING AND STATISTICAL INFERENCE
The subjects who participate in most re search constitute only a part, or subset, called a sample, of some larger collection of similar subjects, called the population. The procedure by which the sample is obtained is called sampling. Dr. Daniel is Professor of Quantitative Methods, School of Business Administration, Georgia State University, Atlanta, GA 30303. Ms Coogler was Associate Professor of Physical Therapy, School of Allied Health Sciences, Georgia State University, at the time this article was written. She is currently Chief Physical Therapist at the Shepherd Spinal Center, West Paces Ferry Hospital, Atlanta, GA 30327.
1326
The Purpose of Sampling
The purpose of sampling is to reach a conclusion about a population on the basis of the information contained in a sample drawn from that population. This process of drawing conclusions about a population from a sample is called statistical inference and may take one of two forms: estimation and hypothesis testing. To be effective, sampling must yield a sample that is representative of the population from which it is drawn. Failure to achieve this goal can be expected to lead to erroneous conclu sions regarding the sampled population. Crocker, for example, states that "a common threat to validity of the results is biased sample selection." 6 The methods discussed in this article, when properly followed, will result in samples that are representative of the popula tions from which they are drawn.
ADVANTAGES OF SAMPLING
The following are some of the advantages of sampling that may be realized by the investi gator who employs this tool of research: 1. Reduced cost. The cost involved in examin ing a part of a population will be less than is PHYSICAL THERAPY
Downloaded from https://academic.oup.com/ptj/article-abstract/55/12/1326/4567579 by University of Texas at Dallas - McDermott Library user on 04 February 2019
Sampling in Physical Therapy Research
KINDS OF SAMPLES
Populations differ with respect to character istics that are relevant to, and affect the outcome of, sampling procedures that are applied to them. A sampling technique that is effective in generating a sample that is repre sentative of one population may not be appropriate for other types of populations. In order to improve the efficiency and effective ness of sampling from different types of populations, several different sampling pro cedures have been developed.
Usually, a simple random sample is selected with the aid of a table of random digits such as that found in the book containing a million random digits published by the Rand Corpora tion. 7 Selection of the items to be included in the sample is made from a frame which is some type of physical representation of the popula tion of interest. A frame, for example, may consist of a printed list of patients; a collection of punched cards or index cards, where each card represents one item in the population; or, if the population of interest consists of house holds in some geographic area, the frame may be a map on which each household in the area is identifiable. The items (subjects, households) in the frame are numbered consecutively from 1 to N, where N is the size of the population. Then, from the table of random digits, n random numbers between 1 and N are selected. The items in the frame whose numbers correspond to the n random numbers constitute the sample of size n. Suppose, for example, that the population of interest consists of N = 1,000 hospitalized patients treated by a physical therapist during the past two years. Suppose also that we wish to draw a sample of size n = 10 from that population. The frame might consist of a computer printout on which the patients' names are listed and numbered from 1 to 1,000. From a table of random digits, the ten numbers 651, 849, 027, 048, 940, 126, 072, 354, 708, 317 might be selected. The sample of size 10 would be composed of patients on the list whose numbers correspond to these random numbers.
Simple Random Samples
In practice, sampling is almost always done An understanding of the basic concepts of sampling theory and methodology is best achieved through a consideration of simple random sampling. Other, more complicated sampling procedures, build on the basic con cepts of the simple random sample. A simple random sample is a sample drawn from a population in such a way that every sample of the given size has the same chance (probability) of being selected. For example, from a population of, size 100, one may draw many samples of size 10. The definition of a simple random sample states that each of these samples has an equal chance of being drawn.
Volume 55 / Number 12, December 1975
without replacement. That is, a given item in the population is allowed to appear only once in a sample. If duplicate numbers are en countered in the process of selecting numbers from the table of random digits they are ignored, and the next usable number that is not a duplicate is chosen. The techniques and concepts of statistical inference—estimation and hypothesis testingare relatively simple and straightforward when the data have been obtained by simple random sampling. Appropriate formulas and the ration ale underlying the procedures may be found in elementary statistics texts. 1327
Downloaded from https://academic.oup.com/ptj/article-abstract/55/12/1326/4567579 by University of Texas at Dallas - McDermott Library user on 04 February 2019
required to examine the population in its entirety. If the population of interest is extremely large, the cost of obtaining data on every item or subject may be prohibitive. Sampling, then, may be the only practical means of arriving at a decision regarding a population. 2. Increased speed. In many instances, time is a critical factor, and sampling may be neces sary in order for results to be available when they are needed. 3. Greater comprehensiveness. Fixed time and cost constraints make sampling necessary for the desired breadth and depth of informa tion to be obtained. 4. Increased accuracy. When a relatively small number of items or subjects is examined, more attention can be focused on the details of collecting and processing the information. The result will be more accurate data on which to base conclusions.
Systematic Sampling
1328
Stratified Random Samples
When estimation is the inferential procedure employed, narrow interval estimates are desir able, and when the statistical analysis takes the form of hypothesis testing, it is desirable to make the chances of rejecting a true hypothesis and failing to reject a false hypothesis small. Usually these goals can be accomplished by taking large samples. Large samples, however, tend to be expensive. Fortunately, there is an alternative procedure that may be used in many situations to reduce both the size of interval estimates and the probabilities of rejecting true hypotheses and failing to reject false hypoth eses. This alternative procedure is called strati
fied random sampling. When stratified random sampling is em ployed, the population of interest is partitioned into nonoverlapping subpopulations called strata. From each of these strata a random sample is drawn, and the results are combined for estimation and hypothesis testing purposes. In order for stratification to be effective, each stratum must be more homogeneous, with respect to the variable of interest, than the population considered as a whole. That is, the variances in the individual strata must be smaller than the overall population variance. Homogeneous strata are obtained through the formation of strata on the basis of a variable that is correlated with the variable of interest. If, for example, one wished to estimate the proportion of physical therapy departments with a high therapist turnover rate, the popula tion of departments might be stratified on the basis of mean salary for therapists. The assumpPHYSICAL THERAPY
Downloaded from https://academic.oup.com/ptj/article-abstract/55/12/1326/4567579 by University of Texas at Dallas - McDermott Library user on 04 February 2019
Selecting a simple random sample from a large population can be a tedious process. The tedium can be avoided, however, through the use of systematic sampling. To draw a system atic sample of size 200 from a population of size 10,000, an item is selected at random from the first 10,000/200 = 50 items in the frame. The remaining 199 items are obtained by taking every 50th item from the initial item drawn. For example, if the item selected at random from the first 50 is item number 35, the second item selected will be the 85th, the third item selected will be the 135th, and so on. An additional advantage of systematic sampling is the fact that this type sample is likely to be more representative of the population than is a simple random sample, since the former is spread more evenly through the population. If the items in the frame from which a systematic sample is drawn are in random order with respect to the variable of interest, the systematic sample will be equivalent to a simple random sample, and the formulas for simple random sampling may be used in testing hypotheses and constructing confidence inter vals. Two situations exist in which a systematic sample cannot be treated as a simple random sample. The first is the case in which the items in the frame are ordered according to the magnitude of the variable of interest. For example, a frame might consist of physical therapy departments listed in order of the number of staff therapists employed, so that departments with fewer therapists appear at the beginning of the frame, and those with more staff therapists appear at the end. When simple random sampling formulas are used with data from this type of population, the sampling variability tends to be overestimated. The second situation in which a systematic sample cannot be treated as a simple random sample occurs when the frame is constructed in such a way that the values of the variable of interest exhibit some cyclical pattern called periodicity. An example of periodic data is the daily occurrence of motor vehicle accidents which tend to peak at the end and drop near the beginning of each week. Simple random sampling formulas used with the data of
systematic samples drawn from this type of population tend to underestimate the sampling variability. When the researcher, wishing to use sys tematic sampling, suspects that, with respect to the variable of interest, the frame from which a sample is to be drawn is either ordered or periodic, special techniques should be em ployed. A frequently used technique is that of drawing several systematic samples and using special formulas rather than the formulas of simple random sampling. These formulas may be found in most sampling textbooks such as those given in the references.
Cluster Sampling
When a sample survey is contemplated, the researcher often finds that no frame of the items (subjects) to be sampled exists. Suppose, for example, that the population of interest consists of all patients receiving physical ther apy in the general hospitals of a certain area during the past year. A list (frame) of such patients probably does not exist. If many hospitals are involved, the expense incurred in compiling such a list may be prohibitive. Another problem that the physical therapy researcher may encounter is that of widely scattered items or subjects. If a survey of patients with a certain type of condition is contemplated, the researcher may find that these patients are scattered over a wide geo graphic area. If the desired information can be obtained only by personal interview, travel costs associated with simple random or strati fied random sampling may be exorbitant if patients are located at great distances from each other. Difficulties such as these may be overcome through the use of cluster sampling. In cluster sampling, the subjects constituting the popula tion of interest are grouped into nonoverlapping subpopulations called clusters. Clusters frequently are geographical units such as census tracts and counties or some other physical entity such as hospitals and classrooms. After clusters have been formed, the sampling process proceeds in two steps. In the first step, a simple Volume 55 / Number 12, December 1975
random sample of clusters is selected from all clusters, and in the second step a random sample of subjects or items is drawn from each of the clusters selected in the first step. Sometimes, in the second step, all of the subjects in the clusters are selected. Informa tion obtained from this two-stage sampling procedure is combined and used as the basis for estimating or testing hypotheses about the parameters of the overall population of interest. When cluster sampling is employed, frames are needed for only those clusters selected in the first step of the sampling procedure. For example, if the population of interest consisted of patients with a certain diagnosis treated at one hundred different hospitals, each hospital could be treated as a cluster. The first "stage of sampling might involve selecting a simple random sample of ten hospitals. For use in the second stage of sampling, the researcher would need to prepare a frame of patients of the desired type for these ten hospitals (clusters) only. The savings in time and money associated with this approach as opposed to simple random or stratified sampling are obvious. The primary purpose of cluster sampling is to save time, reduce costs, and make the sampling process generally more convenient. Such ad vantages of stratified sampling as improved estimates in the form of narrower confidence intervals and reduction of errors in hypothesis testing procedures usually are not realized when cluster sampling is employed. Samples of Convenience
Some readers are likely to become frustrated when they try to reconcile the ideas presented in this article with such statements, so fre quently encountered in published research findings, as "The subjects for this experiment consisted of ten consecutive patients seen in the outpatient clinic," and therapists who find it necessary to use as their subjects "consecutive patients seen in the outpatient clinic" are apt to feel guilty at this apparent violation of the cardinal rules of sampling. They may be able to take little comfort from the rationalization that "everybody does it." In many practical research settings, the sample available for analysis is not random in the strict sense of the word. That is, there is no 1329
Downloaded from https://academic.oup.com/ptj/article-abstract/55/12/1326/4567579 by University of Texas at Dallas - McDermott Library user on 04 February 2019
tion would be that the two variables, mean starting salary and turnover rate, are related. One would expect departments offering higher salaries to have a lower turnover rate. Four strata could be formed, on the basis of salary, according to the four quartiles that could be delineated. If desired, more strata could be formed by making the salary range smaller in each stratum, or if fewer strata were desired, wider ranges could be formed. Stratification is most effective when the variable of interest and the variable on which stratification is based are closely related. If the variable of interest is therapist turnover rate, one would not expect much benefit to accrue from stratification based on the heights of therapists.
1330
SUMMARY AND SUGGESTIONS The application of sampling methodology in physical therapy research has been the subject of this article. Samples of convenience, simple random sampling, and the basic concepts of systematic, stratified, and cluster sampling have been discussed. Only the fundamental ideas have been presented. The reader who wishes to obtain further information on the theory and methodology of sampling is referred to the many available books on the subject. For those who would like a discussion of sampling that is completely devoid of mathematics, the paper back book by Slonim is recommended. 11 Mendenhall and co-workers have written a basic sampling text that is readable and contains only the basic mathematics necessary for the statis tical analysis of the results of sampling. 12 This book makes a useful addition to the library of the researcher who conducts sample surveys. The book by Babbie concentrates on the methodology of surveys, including such topics as questionnaire design, interviewing tech niques, and problems that are likely to be encountered. 13 For a treatment of the subject that places a little more emphasis on theory, yet remains accessible to those with modest preparation in mathematics, the book by Yamane is recommended. 14 Other books on sampling that are helpful, but which are more theoretical and mathematically rigorous, in clude those by Cochran, 15 Deming, 16 Hansen, Hurwitz, and Madow, 17 ' 18 Kish, 19 Raj, 20,21 Sukhatme and Sukhatme, 22 and Yates. 23 REFERENCES
1. Daniel WW, Coogler CE: Some quick and easy statistical tests for physical therapists. Phys Ther 54:135-140, 1974 2. Daniel WW, Coogler CE: Beyond analysis of variance: A comparison of some multiple com parison procedures. Phys Ther 55:144-150, 1975 3. Michels E: Use of the t test (letter to the editor). Phys Ther 50:581-585, 1970 4. Michels E: The 1969 Model t (letter to the editor). Phys Ther 50:751-752, 1970 5. What's the answer? Phys Ther 53:566-568, 1973 6. Crocker LM: Let's reduce the communication gap: Guidelines for preparing a research article. Phys Ther 54:971-976, 1974 7. The Rand Corporation: A Million Random Digits With 100,000 Normal Deviates. Glencoe, Illinois, The Free Press, 195 5 8. Dunn OJ: Basic Statistics: A Primer for the Biomedical Sciences. New York, John Wiley and Sons Inc., 1964, p 12
PHYSICAL THERAPY
Downloaded from https://academic.oup.com/ptj/article-abstract/55/12/1326/4567579 by University of Texas at Dallas - McDermott Library user on 04 February 2019
well-defined population from which subjects are selected in such a way that randomness of selection is assured. The physical therapist who wishes to conduct research with subjects having total hip replacements cannot wait until a large enough pool (population) of such subjects is available and then select from these a random sample for study. For certain research to be possible, subjects must be used as they become available in the clinic or hospital. The therapist, therefore, has little, if any, choice about which subjects compose the sample. Samples composed of subjects of this type are referred to by Dunn as samples of con venience'. 8 If inferences to some population are to be made—and an overwhelming proportion of published research findings are used for this purpose—care must be exercised in deciding to just what population one intends the inferences to apply. Dunn suggests that the researcher in this situation carefully define the sample and then think of the target population (the population about which inferences are made) as being composed of a larger collection of subjects that are similar to those in the sample. 8 Suppose, for example, that the sample of convenience consists of subjects who are char acterized as middle class, urban, between the ages of twenty and forty years, and suffering from no apparent organic or functional disorder other than the one under investigation. The population about which inferences are made may be considered as a large collection of similar subjects who have the same disorder. Remington and Schork point out that the researcher should look for obvious and hidden biases that might cause the sample of conven ience to be unusual in some respect. 9 If, for example, all the subjects in the sample are heavy smokers, the scope of inference may be so limited that the overall usefulness of the results is severely reduced. Armitage also considers the problem of defining the target population when inferences are to be made from samples of convenience. 1 0 He suggests that the target population be thought of as "a hypothetical population which would be generated if an indefinitely large number of observations, showing the same sort of random variation as those at our disposal, could be made." 10
Volume 55 / Number 12, December 1975
York, John Wiley and Sons Inc., 1950 17. Hansen HH, Hurwitz WN, Madow WG: Sample Survey Methods and Theory: Vol I. Methods and Applications. New York, John Wiley and Sons Inc., 1953 18. Hansen HH, Hurwitz WN, Madow WG: Sample Survey Methods and Theory: Vol II. Theory. New York, John Wiley and Sons Inc., 1953 19. Kish L: Survey Sampling. New York, John Wiley and Sons Inc., 1965 20. Raj D: Sampling Theory. New York, McGraw-Hill Book Co., 1968 21. Raj D: The Design of Sample Surveys. New York, McGraw-Hill Book Co., 1972 22. Sukhatme PV, Sukhatme, BV: Sampling Theory of Surveys With Applications 2d ed. Ames, Iowa, The Iowa State University Press, 1970 23. Yates F: Sampling Methods for Census Surveys 3d ed. New York, Hafner Publishing Co., 1960
1331
Downloaded from https://academic.oup.com/ptj/article-abstract/55/12/1326/4567579 by University of Texas at Dallas - McDermott Library user on 04 February 2019
9. Remington RD, Schork MA: Statistics With Applications to the Biological and Health Sci ences. Englewood Cliffs, New Jersey, PrenticeHall, Inc., 1970, p 93 10. Armitage P: Statistical Methods in Medical Re search. Oxford, England, Blackwell Scientific Publications, 1971, p 100 11. Slonim MJ: Sampling. New York, Simon and Schuster Inc., 1960 12. Mendenhall WL, Ott L, Schaeffer RL: Elementary Survey Sampling. Belmont, California, Wadsworth Publishing Company, 1971 13. Babbie ER: Survey Research Methods. Belmont, California, Wadsworth Publishing Co., 1973 14. Yamane T: Elementary Sampling Theory. Engle wood Cliffs, New Jersey, Prentice-Hall, Inc., 1967 15. Cochran WG: Sampling Techniques 2d ed. New York, John Wiley and Sons Inc., 1963 16. Deming WE: Some Theory of Sampling. New
