Document(s) 17 of 27

Flowchart: Steps in the development of a health systems research proposal

NB: Development of a research process is a cyclical process. The double-headed arrows indicate that the process is never linear.

Module 11: SAMPLING

1. Introduction
2. Sampling procedures
   1. Sampling methods for qualitative data
   2. Sampling methods for quantitative data
   3. Bias in sampling
   4. Ethical considerations
3. Sample size

I. INTRODUCTION

WHAT is sampling?

SAMPLING is the process of selecting a number of study units from a defined study population.

Some studies involve only small numbers of people and thus all of them can be included. Often, however, research focuses on such a large population that, for practical reasons, it is only possible to include some of its members in the investigation. We then have to draw a SAMPLE from the total population.

In such cases we must consider the following questions:

• What is the group of people (STUDY POPULATION) we are interested in from which we want to draw a sample?
• How many people do we need in our sample?
• How will these people be selected?

The study population has to be clearly defined (for example, according to age, sex, and residence.) Otherwise we cannot do the sampling. Apart from persons, a study population may consist of villages, institutions, records, etc.

Each study population consists of STUDY UNITS. The way we define our study population and our study unit depends on the problem we want to investigate and on the objectives of the study.

For example:

Representativeness

If researchers want to draw conclusions which are valid for the whole study population, which requires a quantitative study design, they should take care to draw a sample in such a way that it is representative of that population.

For example:

If you intend to interview 100 mothers in order to obtain a complete picture of the weaning practices in District X you would have to select these mothers from a representative sample of villages. It would be unwise to select them from only one or two villages as this might give you a distorted (biased) picture. It would also be unwise to only interview mothers who attend the under-fives clinic, as those who do not attend this clinic may wean their children differently.

When using qualitative research approaches, however, representativeness of the sample is NOT a primary concern. In exploratory studies which aim at getting a rough impression of how certain variables manifest themselves in a study population or at identifying and exploring thus far unknown variables, you may try to select study units which give you the richest possible information: you go for INFORMATION-RICH cases!

For example:

Key informants should never be chosen at random, but purposively from among those who have the best possible knowledge, experience or overview with respect to topic of your study. Moreover they should be willing to share this information with you.

II. SAMPLING METHODS

As the rationale for the use of specific sampling methods in qualitative study designs is very different from the rationale underlying sampling methods in quantitative studies, we will discuss them separately.

Purposeful sampling strategies for qualitative studies

Qualitative research methods are typically used when focusing on a limited number of informants, whom we select strategically so that their in-depth information will give optimal insight into an issue about which little is known. This is called purposeful sampling. There are several possible strategies from which a researcher can choose. Often different strategies are combined, depending on the topic under study, the type of information wanted and the resources of the investigator(s).

This section owes much to Michael Quinn Patton who in his book Qualitative Evaluation and Research Methods (1990: 169-186) discusses a wide range of purposeful sampling techniques.

(1) Extreme case sampling

We have already discussed this type of sampling several times in Modules 9 and 10. Selection of extreme cases, such as good or very poor compliers to treatment, is a powerful and rapid strategy to identify contributing factors to poor compliance. In the same way, selection of well nourished children of the same age will help to identify contributing factors for malnutrition, and systematic comparison of a poorly and a well functioning district health team will give insight into factors that may contribute to the satisfactory functioning of DHTs.

Also ‘thick’ (elaborate) description of single, deviant cases may be useful. In that way AIDS was discovered in California (USA) as a newly emerging disease.

(2) Maximum variation sampling

If a researcher wants to obtain as complete as possible insight in a certain issue in all its variations, maximum variation sampling will be used.

For example, the stigma (see Module 8 Part III) of leprosy, or TB, HIV, epilepsy, is considered a complicating factor in the control of these diseases. In order to obtain insight in how stigma manifests itself in different cultures in males and females, in rural and urban areas, in well-to-do and poor patients, or in educated and illiterate ones, a researcher has to take care that all these groups are included in the sample. To assess whether social distance influences stigma, one could also interview blood relatives (parents or children), spouses, friends, near neighbours of patients and more distant community members. For leprosy and TB patients, it is useful to interview patients on treatment as well as patients declared cured, to assess if any reversal of stigma is experienced when patients’ conditions improve. If a researcher is interested in specific groups and interviews a fixed number per group, this type of sampling is also called quota sampling.

In the study on stigma it would, for example, be possible to select the categories of patients one would like to focus on (male/ female, on treatment and released from treatment) in a systematic way from patient registers in a number of selected clinics. One could e.g., take all, or every second or third patient in the desired categories. The clinics may have been selected purposefully (town/rural, specific ethnic areas, poor/rich areas) but if there is choice the researcher may make lists for each category and select at random (see next section) the desired number of clinics. For identifying social distance variation, selection rules will be much more pragmatic: spouses, relatives and neighbours who are available will be interviewed in a careful, indirect way in order not to hurt the patient’s interests. Still, however interesting data maximum variation sampling may generate, highlighting different factors and different perspectives, it does not provide representative data for the total population.

(3) Homogeneous sampling

Sometimes a researcher would like to have specific information about one particular group only, for example, a group that, for unclear reasons, is more at risk than others:

In country S, death registers indicate that suicide among adolescents is on the increase at an alarming rate. Within that group twice as many boys as girls commit suicide. Researchers may therefore want to concentrate on the boys to identify what factors may be contributing to these suicides, conducting in-depth interviews with parents, other close relatives, teachers and friends of a number of boys who committed suicide.

In focus group discussions (FGDs), we usually select homogeneous groups because participants discuss more freely when they are amongst people of similar social status. (See Module10C.)

(4) Typical case sampling

It is sometimes illustrative to describe in-depth some cases which are ‘typical’ for the group one is interested in. For example, one may describe a ‘typical’ family in a rural village in country A, or a ‘typical’ young school leaver who migrates from the rural areas to town in search of work, or ‘typical’ health problems of miners or malnourished children.

Such descriptions are merely illustrative; they cannot be generalised for the whole group. Typical examples can either be selected with co-operation of key informants who know the study population well, or from a survey that helps to identify the normal distribution and the modus of the characteristics we are interested in.

(5) Critical case sampling

Critical cases are those who ‘can make the difference’ with respect to an intervention you want to introduce or to evaluate.

For example, you have developed a local weaning food that, you hope, is affordable to all mothers. Before propagating it at a larger scale through MCH clinics you first interview and observe some low-income mothers as ‘test cases’. If they manage to produce and use it, this will indicate that it is affordable to the whole group.

(6) Snowball or chain sampling

This approach is particularly suitable for locating key informants or critical cases. You start with one or two information-rich key informants and ask them if they know persons who know a lot about your topic of interest. If a particular person is recommended to you by two or three different people you can be quite sure that he or she will be a valuable key informant.

The same approach can be used if an in-depth interview leads to discoveries, which seem rewarding to follow-up by a number of interviews with an additional group of informants.

For example, in an exploratory study on coping behaviour among AIDS orphans it seemed that child-headed households managed by girls survived better than those managed by boys. The researcher then interviewed more adolescent boys and girls heading households, to see whether this gender difference in ability to cope was real, and how it could be explained.

Patton (1990: 179) labels this kind of additional sampling during the study opportunistic sampling.

Flexible sampling procedures, steered by the data one collects (in relation to the objectives) forms a major opportunity for qualitative researchers to optimally exploit the field situation and explore ‘in-depth’ interesting issues which present themselves. It is exactly the opposite of the random sampling techniques discussed in the next section of this module, which are used in quantitative research to ensure representativeness of the sample for the total population. Still, if qualitative researchers can choose from a group of seemingly similar informants they will also sample at random (see example 2).

* In the earlier version of this module, following epidemiological tradition, all ‘non-random’ sampling methods were categorised under the headings of ‘convenience’ or ‘quota’ sampling. This injustice towards purposeful sampling techniques has been corrected in the present version. In HSR, purposeful and random sampling techniques are used equally.

2. Random sampling strategies to collect quantitative data

If the aim of a study is to measure variables distributed in a population (e.g., diseases) or to test hypotheses about which factors are contributing significantly to a certain problem, we have to be sure that we can generalise the findings obtained from a sample to the total study population. Then, purposeful sampling methods are inadequate, and probability- or random sampling methods have to be used.

Probability sampling requires that a listing of all study units exists or can be compiled. This listing is called the sampling frame.

The following probability sampling methods will be discussed:

• Simple random sampling
• Systematic sampling
• Stratified sampling
• Cluster sampling
• Multistage sampling

(1) Simple random sampling

This is the simplest form of probability sampling. To select a simple random sample you need to:

• Make or search for an existing numbered list of all the units in the population from which you want to draw a sample
• Decide on the size of the sample (this will be discussed in section III)
• Select the required number of sampling units, using a ‘lottery’ method or a table of random numbers (Annex 11.1 explains how to use a table of random numbers.)

For example, a simple random sample of 50 students is to be selected from a school of 250 students. Using a list of all 250 students, each student is given a number (1 to 250), and these numbers are written on small pieces of paper. All the 250 papers are put in a box, after which the box is shaken vigorously, to ensure randomisation. Then, 50 papers are taken out of the box, and the numbers are recorded. The students belonging to these numbers will constitute the sample.

(2) Systematic sampling

For example, a systematic sample is to be selected from 1200 students of a school. The sample size selected is 100. The sampling fraction is:

The sampling interval is therefore 12.

The number of the first student to be included in the sample is chosen randomly, for example by blindly picking one out of twelve pieces of paper, numbered 1 to 12. If number 6 is picked, then every twelfth student will be included in the sample, starting with student number 6, until 100 students are selected: the numbers selected would be 6, 18, 30, 42, etc.

Systematic sampling is usually less time consuming and easier to perform than simple random sampling. However, there is a risk of bias, as the sampling interval may coincide with a systematic variation in the sampling frame. For instance, if we want to select a random sample of days on which to count clinic attendance, systematic sampling with a sampling interval of 7 days would be inappropriate, as all study days would fall on the same day of the week (e.g., Tuesdays only, which might be a market day).

(3) Stratified sampling

The simple random sampling method described above has as disadvantage that small groups in which the researcher is interested may hardly appear in the sample.

Stratified sampling is only possible when we know what proportion of the study population belongs to each group we are interested in.

An advantage of stratified sampling is that we can take a relatively large sample from a small group in our study population. This allows us to get a sample that is big enough to enable us to draw valid conclusions about a relatively small group without having to collect an unnecessarily large (and hence expensive) sample of the other, larger groups. However, in doing so, we are using unequal sampling fractions and it is important to correct for this when generalising our findings to the whole study population.

For example, a survey is conducted on household water supply in a district comprising 20,000 households, of which 20% are urban and 80% rural. It is suspected that in urban areas the access to safe water sources is much more satisfactory. A decision is made to include 100 urban households (out of 4000, which gives a 1 in 40 sample) and 200 rural households (out of 16000, which gives a 1 in 80 sample). Because we know the sampling fraction for both strata, the access to safe water for all the district households can be calculated after the study (by multiplying the findings for the urban households by 40 and those for the rural households by 80, and then calculating statistics for the total sample).

(4) Cluster sampling

It may be difficult or impossible to take a simple random sample of the units of the study population at random, because a complete sampling frame does not exist. Logistical difficulties may also discourage random sampling techniques (e.g., interviewing people who are scattered over a large area may be too time-consuming). However, when a list of groupings of study units is available (e.g., villages or schools) or can be easily compiled, a number of these groupings can be randomly selected.

Clusters are often geographic units (e.g., districts, villages) or organisational units (e.g., clinics, training groups).

For example, in a study of the knowledge, attitudes and practices (KAP) related to family planning in rural communities of a region, a list is made of all the villages. Using this list, a random sample of villages is chosen and all study units in the selected villages are interviewed.

(5) Multi-stage sampling

In very large and diverse populations sampling may be done in two or more stages. This is often the case in community-based studies, in which people are to be interviewed from different villages, and the villages have to be chosen from different areas. This type of sampling is frequently used in HSR.

For example, in a study of utilisation of pit latrines in a district 150 homesteads are to be visited for interviews with family members as well as for observations on types and cleanliness of latrines. The district is composed of 6 wards and each ward has between 6 and 9 villages.

The following four-stage sampling procedure could be performed*:

1. Select 3 wards out of the 6 by simple random sampling.
2. For each ward select 5 villages by simple random sampling (15 villages in total).
3. For each village select 10 households. Since simply choosing households in the centre of the village would produce a biased sample, the following sampling procedure is proposed:
   • Go to the centre of the village.
   • Choose a direction in a random way: spin a bottle on the ground and choose the direction the bottleneck indicates.
   • Walk in the chosen direction and select every (or, depending on the size of the village, every second or every third) household until you have the 10 you need. If you reach the boundary of the village and you still do not have 10 households, return to the centre of the village, walk in the opposite direction and continue to select your sample in the same way until you have 10. If there is nobody in a chosen household, take the next nearest one.
4. Decide beforehand whom to interview (for example the head of the household, if present, or the oldest adult who lives there and who is available).

* This is an adaptation of the method developed by the EPI division in WHO Geneva to measure EPI coverage in districts.

The main advantages of cluster and multi-stage sampling are that:

• A sampling frame of individual units is not required for the whole population. Existing sampling frames of clusters are sufficient. Only within the clusters that are finally selected is there a need to list and sample the individual units (if not using the bottle spinning method).
• The sample is easier to select than a simple random sample of similar size, because the individual units in the sample are physically together in groups, instead of scattered all over the study population.

The main disadvantage of this type of sampling is that:

Compared to simple random sampling, there is a larger probability that the final sample will not be representative of the total study population. The likelihood of the sample not being representative depends mainly on the number of clusters that is selected in the first stage. The larger the number of clusters, the greater is the likelihood that the sample will be representative. Further, the sampling units at community level should be selected randomly (avoid convenience sampling!).

3. Bias in sampling

Module 10 discussed how the use of faulty data collection tools would lead to biased results. Bias can also be introduced as a consequence of improper sampling procedures, which result in the sample not being representative of the study population.

For example, a study was conducted to determine the health needs of a rural population in order to plan primary health care activities. However, a nomadic tribe, which represented one third of the total population, was left out of the study. As a result the study did not give an accurate picture of the health needs of the total population.

There are several possible sources of bias that may arise when sampling. The most well known source is non-response.

Non-response can occur in any interview situation, but it is mostly encountered in large-scale surveys with self-administered questionnaires. Respondents may refuse or forget to fill in the questionnaire. The problem lies in the fact that non-respondents in a sample may exhibit characteristics that differ systematically from the characteristics of respondents.

There are several ways to deal with this problem and reduce the possibility of bias:

• Data collection tools (including written introductions for the interviewers to use with potential respondents) should be pre-tested. If necessary, adjustments should be made to ensure better co-operation.
• If non-response is due to absence of the subjects, follow-up of non-respondents may be considered.
• If non-response is due to refusal to co-operate, an extra, separate study of non-respondents may be considered in order to identify to what extent they differ from respondents.
• Another strategy is to include additional people in the sample, so that non-respondents who were absent during data collection can be replaced. However, this can only be justified if their absence was very unlikely to be related to the topic being studied.

Other sources of bias in sampling may be less obvious, but at least as serious:

• Studying volunteers only. The fact that volunteers are motivated to participate in the study may mean that they are also different from the study population on the factors being studied. Therefore it is better to avoid using non-random selection procedures that introduce such an element of choice.
• Sampling of registered patients only. Patients reporting to a clinic are likely to differ systematically from people seeking alternative treatments.
• Missing cases of short duration. In studies of the prevalence of disease, cases of short duration are more likely to be missed. This may mean missing fatal cases, cases with short illness episodes and mild cases.
• Seasonal bias. It may be that the problem under study, for example, malnutrition, exhibits different characteristics in different seasons of the year. For this reason, data should be collected on the prevalence and distribution of malnutrition in a community during all seasons rather than just at one point in time. When investigating health services’ performance, to take another example, one has to consider the fact that towards the end of the financial year shortages may occur in certain budget items which may affect the quality of services delivered.
• Tarmac bias. Study areas are often selected because they are easily accessible by car. However, these areas are likely to be systematically different from more inaccessible areas.

4. Ethical considerations

If the recommendations from a study will be implemented in the entire study population, one has the ethical obligation to draw a sample from this population in a representative way. If during the research new evidence suggests that the sample was not representative, this should be mentioned in any publication concerning the study, and care must be taken not to draw conclusions or make recommendations that are not justified.

III. SAMPLE SIZE

Having decided how to select our sample, we now have to determine our sample size.

1. Sample size in qualitative studies

There are no fixed rules for sample size in qualitative research. The size of the sample depends on WHAT you try to find out, and from what different informants or perspectives you try to find that out.

For example, if you want to explore how you can involve mothers in your HC catchment area more effectively in early detection and treatment of pneumonia, you might decide to conduct some FGDs to assess mothers’ knowledge, attitudes and practices with respect to pneumonia. You could start with two FGDs among lowly educated mothers and two among mothers with more education (who usually are of higher socio-economic status). If the different data sets reconfirm each other you may stop at this point and start a small scale intervention; otherwise you conduct one or two FGDs more till you reach the point of redundancy*: no new data comes up any more.

If your research objective is more complex e.g., attitudes of males and females towards family planning, and has policy implications for a larger area, your sample will be bigger. You might start with four FGDs, two among males and two among females, subdivided according to socio-economic status. Among male participants you could then select 5-10 users (or spouses of female users) for in-depth interviews on RH history of the couple and reasons for use and non-use of FP. For women, the same procedures can be followed. If possible, you may interview some couples, first separately, then together. Depending on ethnic differences and urban-town differences in attitudes and practices, the ‘clusters’ of FGDs and in-depth interviews can be expanded.

In exploratory studies, the sample size is therefore estimated beforehand as precisely as possible, but not determined.

Patton (1990: 183-186) stresses that richness of the data and analytical capability of the researcher determine the validity and meaningfulness of qualitative data more than sample size. Still, sampling procedures and sample size should always be carefully explained in order to avoid the allusion of haphazardness. Careful analysis of different complementing data sets can result in some plausible generalisations but without ‘proving’ them in a mathematical sense.

* Lincoln and Guba, in Patton (1990: 185)

2. Sample size in quantitative studies

For quantitative studies, calculations can be made which indicate the desirable sample size. The principles of such calculations will be discussed below.

It is a widespread belief among researchers that the bigger the sample, the better the study becomes. This is not necessarily true. In general it is much better to increase the accuracy and richness of data collection (for example by improving the training of interviewers or by better pre-testing of the data collection tools) than to increase sample size after a certain point. Also, it is better to make extra efforts to get a representative sample rather than to get a very large sample.

The following general rules may help to determine the desirable sample size of any given study:

• The desirable sample size depends on the expected variation in the data (of the most important variables): the more varied the data are, the larger the sample size we would need to attain the desired level of accuracy.
• The desirable sample size also depends on the number of cells we will have in the cross-tabulations (see Module 13) which we need to analyse the results. A rough guideline is to have at least 5 to 10 study units per cell.

For example, after conducting FGDs and in-depth interviews in the study on attitudes of men and women towards family planning (see Section III.1 of this module) you might decide to conduct a bigger survey. If your exploratory study revealed that age and education appear to be important factors determining FP use, you will be interested in comparing FP use in groups with different levels of education and of different ages. If you split each of these variables up in three categories, and you select four categories of informants (male users/ spouses of female users; female users; male non-users, female non-users) you would have 12 cells in each table. In order to obtain 5-10 answers per cell you would require 60-120 informants in each research area.

As other variables may have more categories, you may attempt to select 120 informants in rural areas and 120 in urban areas. However, as FP use may not be equally distributed in a population (e.g., 25% users, 75% non-users) your sample will have to be bigger than 120 in order to obtain the desired sample size. HOW big it should be can be calculated. Still, the desirable sample size can not always be achieved for lack of resources such as time, manpower and money. This constraint applies to quantitative as well as qualitative studies.

Sample size calculations

In quantitative studies, researchers will perform sample size calculations before embarking on the project to find the desirable sample size. The formulae for calculating a desired sample size are listed in Annex 11.2. They are divided into two categories, depending on whether the study:

• seeks to measure one single variable (e.g. a mean, a rate or a proportion) in one group with a certain precision, or
• tries to demonstrate a significant difference between two groups.

The formulae can only be used if you have a rough idea about the outcome of the study, which is not always the case. It is always advisable to call upon a statistician or an experienced researcher who can help you in choosing and using the appropriate formulae.

We will look at a few examples to highlight some important issues.

(1) Descriptive studies with one group

Example

In a descriptive study in a certain village we want to measure, with a certain precision, the proportion of children aged 12-23 months who are immunised against measles, using a simple random sample. The following steps should be taken:

1. Estimate how big the proportion might be (say 80%).
2. Choose the margin of error you will allow in the estimate of the proportion (say ± 10%). This means that, if the survey reveals that indeed 80% of the children have been vaccinated, this proportion will probably be between 70 and 90% in the whole study population from which the sample was drawn.
3. Choose the level of confidence at which you want to be able to state that the vaccination coverage in the whole population is indeed between 70 and 90%. You can never be 100% sure. Do you want to be 95% sure? or 99%? A commonly used confidence level in HSR studies is 95%.

   The formula for calculating the sample size for the single proportion expressed as a percentage is presented in Annex 11.2 (1.3).

However, at present it is usually not necessary to calculate the desirable sample size by hand. There are computer programmes to assist us (for example Epi Info). In Annex 11.3 two tables are presented, which will help you to calculate the desired sample size in the most common quantitative HSR studies. Both take a confidence level of 95% (P<0.05) as point of departure. In the measles example, you have to go down the first column till you find the 80% for the estimated vaccination coverage. If you allow a margin of error of 10 + or – the estimated 80%, you look in that column and find a desired sample size of 64. However, if you would allow a margin of error of only + or - 5%, the sample size would increase to 256, and if you would like to be 95% sure that the measles vaccination coverage is between 79 and 81%, you would need to check 6400 children. The smaller your margin of error (also called confidence interval) the bigger your sample size needs to be. If you would like to increase your confidence level from 95 to 99%, the sample would have to increase even more. But then researchers of course start to look at feasibility: do we really need that precision? Most HSR studies will be satisfied with the 95% confidence level.

Note also that in general you need more precision (or a smaller margin of error) if the estimated proportion is very small. This may be the case, for example, for the proportion of HIV* women or the maternal mortality rate in a population.

Table 11.1: Required sample size for studies of HIV prevalence in pregnant women

The table shows that in district A, where HIV is less prevalent, a smaller margin of error is desired and, therefore, the required sample is larger. (Annex 11.4 explains how these sample sizes were calculated).

Table (a) in Annex 11.3 will, however, help you again to identify the required sample sizes in a simple way. The first line of table (a) gives you the required sample sizes for an estimated prevalence of 1%. You would need a sample of 1584 women to be 95% sure that HIV prevalence in District A would be between 0.5 and 1.5% (so 1% plus or minus 0.5%). For district B, with 10% estimated HIV prevalence, and a margin of error plus or minus 5%, the required sample size would be 144.

(2) Comparing two groups for a significant difference

In comparative studies one usually wants to demonstrate that there is a significant difference between two groups. In this type of study the sample size depends primarily on the estimated size of the difference between the two groups that are compared. The larger the difference, the smaller will be the sample that is needed to show this difference. Second, the sample size depends on how large we want the probability to be that we indeed will find a significant difference.

The larger the sample size, the larger will be the probability of finding a significant difference. In the case of many variables, the one with the smallest estimated difference between the groups should be used as the basis for calculating the sample size, as it requires the largest sample. Still, researchers will discuss whether that variable is important enough to maintain if the sample would become too big to handle. They may also decide to measure only some variables in a large sample while measuring variables that require a smaller sample in a sub-sample to save costs and ensure the feasibility of the study.

Example:

In a study, a comparison will be made between the feeding patterns of well-nourished and malnourished children of 12 to 17 months. It is expected that of the well-nourished children 90% are breastfed whereas of the malnourished children approximately 50% are breastfed. The sample size in each group of children needs to be at least 15 to show a significant difference.

However, if 90% of the well-nourished children and 80% of the malnourished children were breast fed, the sample size would need to be at least 175 in each group to show a significant difference. (Annex 11.4 explains how these sample sizes can be calculated).

Table (b) in Annex 11.3 will again help you to identify the required sample size in a simple way. Compare 90% in the column with 50% in the row (or vice versa) and you will find a required sample size of 22. In this case a slightly different formula was used as the one presented in presented in Annex 11.4. Also the calculated sample size for the second example would be bigger (262) when using the table.

Note that it may be useful to conduct sample size calculations for each of the objectives of the study. These calculations may reveal, for instance, that some but not all objectives can be met. Or they may indicate that some variables need only to be measured on a sub-sample.

REFERENCES

All epidemiological and social science research handbooks mentioned in the references in Module 9 are dealing with sampling procedures and sample size. Moreover you can consult:

Swinscow TDV, Revised by MJ Campbell (1998) Statistics at Square One. London: BMJ Publishing Group. (9^th ed.)

Campbell MJ, Machin D (1993) Medical Statistics: A Common Sense Approach. Clichester: John Wiley. (2^nd ed.)

Annex 11.1: How to use a random number table*

1. First, decide how large a number you need. Next, count if it is a one, two or larger digit number. For example, if your sampling frame consists of 10 units, you must choose from numbers 1-10, (inclusive). You must use two digits to ensure that 10 has an equal chance of being included.

   You also use two digits for a sampling frame consisting of 0-99 units.

   If, however, your sampling frame has 0-999 units, then you obviously need to choose from three digits. In this case, you take an extra digit from the table to make up the required three digits. For example, the number in columns 10,11, row 27: 43, would become 431; going down, the next numbers would be 107, 365 etc.

   You would do the same if you needed a four-digit number, for a sampling frame 0-9999 units. In our example of the number on columns 10, 11, 12, row 27 of the table: 431, this would now become 4316, the next down 1075, and so on.

2. Decide beforehand whether you are going to go across the page to the right, ? down the page ?, across the page to the left, ? or up the page. ?
3. Without looking at the table, and using a pencil, pen, stick, or even your finger, pin-point a number.
4. If this number is within the range you need, take it. If not, continue to the next number in the direction you chose before-hand, (across, up or down the page), until you find a number that is within the range you need.

   For example if you need a number between 0-50 and you began at column 21, 22, row 21 you get 74 which is obviously too big. So you could go down (having decided beforehand to go down) to 97, also too big, to 42, which is acceptable, and select it.

*The random number table on the following page has been taken from Hill AB (1977) A Short Textbook of Medical Statistics. London: Hodder and Stoughton, 1977:306-7.

Random sampling numbers

Annex 11.2: Formulae for calculating sample size*

The formulae for calculating required sample size are divided in two categories:

1. For studies trying to measure one variable with a certain precision.
2. For studies seeking to demonstrate a significant difference between two groups.

1. Measuring one variable

In the formulae below the following abbreviations are used:

n - sample size

s - standard deviation

e - required size of standard error
(in the text of the module the term ‘margin of error’ is used for ±2 times the size of the standard error if a precision of 95% is required)

r - rate

p - percentage

1.1 Single mean

In a study the mean weight of newborn babies will be determined. The mean weight is expected to be 3000 grams. Weights are approximately normally distributed and 95% of the birth weights are probably between 2000 and 4000 grams; therefore the standard deviation would be 500 grams. The desired 95% confidence interval is 2950 to 3050 grams, so the standard error would be 25 grams. The required sample size would be:

1.2 Single rate

The maternal mortality rate in a country is expected to be 70 per 10,000 live births. A survey is planned to determine the maternal mortality rate with a 95% confidence interval of 60 to 80 per 10,000 live births. The standard error would therefore be 5/10,000. The required sample size would be:

1.3 Single proportion

The proportion of nurses leaving the health services within three years of graduation is estimated to be 30%. A study that aims to find causes for this, also aims to determine the percentage leaving the service with a confidence interval of 25% to 35%. The standard error would therefore be 2.5%. The required sample size would be:

* Modified from Kirkwood B (1988) Essentials of Medical Statistics. Oxford: Blackwell Scientific Publications, 1988.

1.4 Difference between two means (sample size in each group)

The difference of the mean birth weights in district A and B will be determined. In district A the mean is expected to be 3000 grams with a standard deviation of 500 grams (as in 1.1). In district B the mean is expected to be 3200 grams with a standard deviation of 500 grams. The difference in mean birth weight between districts A and B is therefore expected to be 200 grams. The desired 95% confidence interval of this difference is 100 to 300 grams, giving a standard error of the difference of 50 grams. The required sample size would be:

1.5 Difference between two rates (sample size in each group)

The difference in maternal mortality rates between urban and rural areas will be determined. In the rural areas the maternal mortality rate is expected to be 100 per 10,000 and in the urban areas 50 per 10,000 live births. The difference is therefore 50 per 10,000 live births. The desired 95% confidence interval is 30 to 70 per 10,000 live births giving a standard error of the difference of 10/10,000. The required sample size would be:

1.6 Difference between two proportions (sample size in each group)

The difference in the proportion of nurses leaving the service is determined between two regions. In one region 30% of the nurses are estimated to leave the service within three years of graduation, in the other region 15%, giving a difference of 15%. The desired 95% confidence interval for this difference is 5% to 25%, giving a standard error of 5%. The sample size in each group would be:

2. Significant difference between two groups

In the formulae below the following abbreviations are used:

n, samples size

s, standard deviation

e, required size of standard error

r, rate

p, percentage

u, one-sided percentage point of the normal distribution, corresponding to 100% - the power. The power is the probability of finding a significant result. (e.g. if the power is 75%, u = 0.67).

v, percentage point of the normal distribution, corresponding to the (two-sided) significance level. (e.g. if the significant level is 5% (as usual), v = 1.96.)

2.1 Comparison of two means (sample size in each group)

The birth weights in district A and B will be compared. In district A the mean birth weight is expected to be 3000 grams with a standard deviation of 500 grams. In district B the mean is expected to be 3200 grams with a standard deviation of 500 grams (see 1.4). The required sample size to demonstrate (with a likelihood of 90%) a significant difference between the mean birth weights in district A and B would be:

2.2 Comparison of two rates (sample size in each group)

The maternal mortality rates in urban and rural areas will be compared. In the rural areas the maternal mortality rate is expected to be 100 per 10,000 and in the urban areas 50 per 10,000 live births (compare to 1.5). The required sample size to show (with a likelihood of 90%) a significant difference between the maternal mortality in the urban and rural areas would be:

2.3 Comparison of two proportions (sample size in each group)

The proportion of nurses leaving the health service is compared between two regions. In one region 30% of nurses is estimated to leave the service within three years of graduation, in the other region it is probably 15%.

The required sample size to show with a 90% likelihood that the percentage of nurses is different in these two regions would be:

Annex 11.3: Tables for calculating sample size in the most common HSR studies

(a) Sample size for measuring proportions in one group
Percentage 95% confidence level (+ or – the percentage of allowed error in column heading)

The percentages in the column heading are 2 x standard error (e in formula)

(b) Sample size for comparison of proportions in two groups

       u = 1.28 Power = 90%

       v = 1.96 Significance: P<0.05

(u+v)²= 10.5

Annex 11.4: Explanation of sample size calculations given in the text

1. Prevalence of HIV (p.13)

District A: the estimated HIV⁺ proportion is 1% = 0.01. As the 95% confidence interval is the proportion ± 2 x the standard error, the standard error is 0.25% = 0.0025.

District B: the estimated HIV proportion is 10% = 0.1 Procedures followed are the same as above

2. Feeding patterns in malnourished and well-nourished children (p.14)

Formula used: no. 2.3 in Annex 11.2

If the power is 75%, u = 0.67 and (u + v)² = 6.9;
if the power is 90%, u = 1.28 and (u + v)² = 10.5.
(The power is the probability of finding a significant result).

If the power is increased from 75% to 90%, the sample size is increased 10.5/6.9 (i.e. 1.5 times.)

Trainer’s Notes

Module 11: SAMPLING

Materials

• Calculators
• Paper

Introduction and discussion

• When presenting part I of this module make sure that everyone understands what sampling is and why it is done. Carefully explain the merits of purposeful sampling in small-scale qualitative studies and of random sampling in quantitative studies.
• In the presentation of sampling methods (Part II), use examples from the groups’ own protocols as much as possible. You may do an exercise to show the differences between the different sampling methods with the participants themselves as a group.

  For example, you may sample 6 or 8 persons from your audience using simple random sampling and systematic sampling (from an alphabetical list of participants and facilitators). Ask the participants to name the sampling method applied and discuss the advantages and disadvantages of each method. (As names tend to cluster according to origin, it is likely that the systematic sampling will turn out to be less representative than the simple random sampling.)

• Allow time during and after the presentation for questions and discussion.

Group work, Part I

• Have the working groups choose the appropriate sampling methods for their own projects. The methods should be worked out in as much detail as possible.

Introduction and discussion

• Stress that one does not always have to do calculations to determine the desired sample size. Actually, in many (exploratory) HSR studies one you would not do any calculations, though for the sampling procedures a plan must be worked out which should be adhered to (e.g. selection of extremes from a list of patients, or snowball sampling through different key informants according to criteria set together).
• The formulae for calculating a desired sample size are therefore put in an annex. You are not expected to go into technical details of sample size calculations during your presentation unless the participants are familiar with statistics (all relevant concepts will only be explained during the data analysis workshop). Rather use the tables in Annex 11.3 and make sure participants understand from those tables how sample size and precision (confidence level) go together.

Group work, Part II

• Let each group determine the sample size for the proposal it is working on.
• Participants should be advised to consult experts when they think they will need to calculate sample size but do not know how to go about it. Make sure, for this reason, that there is a statistician present who can be consulted during group work and plenary presentations.
• If a group plans to measure statistical entities such as infant or maternal mortality rates in its study, it should definitely consult a professional with statistical training.

Exercise

• At the end of this group work session each group should examine another group’s chosen sampling procedures and sample size. Ask the groups to look for possible sources of bias and make suggestions for reducing it.

Plenary

• Have each group present their sampling methods and sample size, immediately followed by the comments of the group that examined the sampling methods for bias. A discussion can follow each presentation or be held after all the group presentations.
• Emphasise that, after having incorporated useful suggestions from the plenary discussion, the groups should actually select their samples, as far as possible (e.g. sampling of districts, villages, clinics). This will be useful for the next group work sessions in preparation of the fieldwork (especially Modules 12 and 16).

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