Learn — Population, sampling & variables

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Population and sample — sampling frame, accessible population and sample size

A population is the entire set of people or units that meet the criteria the researcher sets, and a sample is the subset actually studied, chosen so that what is found in it can be said of the whole. Sampling exists because a whole population can seldom be studied, and its quality is judged by one thing — how representative the sample is of the population the study wants to describe.

The target population is the whole group the researcher wants to generalise to — every staff nurse in the state; the accessible population is the part of it she can actually reach — the nurses of three hospitals; the sampling frame is the list of the accessible population from which the sample is drawn — the nursing register, the duty roster, the outpatient register; and the eligibility (inclusion and exclusion) criteria define who belongs.
A sample is described by statistics and a population by parameters; a sample is representative when its key characteristics match the population's, and sampling bias is the systematic over- or under-representation of some segment; the two families of methods are probability (random) sampling, which allows sampling error to be estimated, and non-probability sampling, which does not.
In quantitative research the sample size is calculated before the study by power analysis from the expected effect size, the level of significance (usually 0.05), the power wanted (usually 0.80) and the variability of the outcome; the formula for estimating a proportion, n = Z²pq ÷ d² with Z = 1.96 for 95 per cent confidence and d the allowable error, is the one examinations quote, and thirty per group is the common floor for the parametric tests.
A larger sample gives more precision and a smaller sampling error, and the size is raised for a weak expected effect, a heterogeneous population, planned subgroup analyses and the attrition expected; in qualitative research the size is set by data saturation, not by calculation, and is often between ten and thirty.
Sampling error is the difference between the sample's estimate and the true population value that arises by chance from which units happened to be selected — it is reduced by a larger sample but never removed; non-sampling error — a faulty frame, non-response, mismeasurement — is not cured by size at all.
The sampling plan runs in steps — define the population, set the criteria, identify the frame, choose the method, decide the size, recruit — and the report gives the response rate, the attrition and a comparison of the sample with the population on what is known, so the reader can judge how far the findings reach.

Probability sampling — simple random, stratified, systematic and cluster

Probability sampling is any method in which every element of the population has a known, non-zero chance of being selected and the selection is made by chance rather than by the researcher's choice; it is the only family of methods that lets sampling error be estimated and findings be generalised with statistical confidence. Its four methods — simple random, stratified random, systematic and cluster — differ in how the chance is arranged, and examinations describe the procedure and ask for the name.

Simple random sampling gives every member of the population an equal and independent chance of selection — names drawn by lottery, by a random number table or by computer from a complete sampling frame; it is the standard against which the other methods are judged, but it needs a full list and may by chance miss a small subgroup.
Stratified random sampling divides the population into homogeneous strata on a variable that matters — age group, sex, ward, grade — and draws a random sample from each, proportionately (the same fraction from every stratum, such as 10 per cent of each of four age groups) or disproportionately to oversample a small stratum; it guarantees that every stratum is represented and gives more precision than a simple random sample of the same size.
Systematic sampling takes every kth element from the frame after a random start, where k is the population size divided by the sample size — 2,000 nurses and a sample of 200 give k = 10, so a random number between 1 and 10 and then every tenth name; it is quick and even, and it goes wrong only when the list has a periodic pattern that matches k.
Cluster (multistage) sampling divides the population into natural groups — villages, primary health centres, wards, schools — draws a random sample of the groups and studies every unit in the chosen groups, or draws a further random sample within them; it is the method for a scattered population with no individual list, it is cheaper, and its sampling error is larger than a simple random sample's of the same size because the units within a cluster resemble one another — the WHO thirty-cluster survey of immunisation coverage is its classic use.
Random sampling is not random assignment: sampling is how people get into the study from the population and serves generalisation, assignment is which group they land in within an experiment and serves the claim of cause, and a study may have either without the other.
The choice is made by the situation — a full list and a fairly homogeneous population call for simple random sampling, subgroups that must be represented for stratified, a long ordered list for systematic, and a scattered population with no list for cluster.

Non-probability sampling — convenience, purposive, quota and snowball

Non-probability sampling selects elements by non-random means — availability, the researcher's judgement, a quota or a referral — so that the chance of any one element being chosen is unknown and some elements may have no chance at all. It is quicker and cheaper than probability sampling and is the usual method in qualitative research and in much nursing research, at the price that sampling error cannot be estimated and generalisation is limited.

Convenience (accidental) sampling takes whoever is available and willing — the patients who happen to be in the outpatient department on a given day, the nurses on duty that shift; it is the weakest method and the most prone to bias, and it is also the commonest in nursing studies.
Quota sampling fixes in advance the proportions of subgroups the sample must contain — so many men and women, so many of each age group — to mirror the population and then fills each quota by convenience; it is stratified sampling without the randomness.
Purposive (judgemental) sampling has the researcher choose on purpose the people who can best inform the question — experts, typical cases, extreme cases, the widest variation — and it is the standard method of qualitative research; theoretical sampling in grounded theory chooses each next participant as the emerging theory demands.
Snowball (network, chain-referral) sampling asks each recruited participant to refer other eligible people she knows; it is the method for hidden or hard-to-reach populations — people who inject drugs, sex workers, people with a rare condition, women who have left abusive marriages — for whom no frame exists.
Consecutive sampling recruits every eligible person over a set period and is better than convenience because it takes everyone rather than only those at hand; whatever the method, the report must describe the sample in enough detail for a reader to judge to whom the findings might transfer.
The trade-off is fixed: non-probability methods are fast and cheap, but the sample may be unrepresentative, sampling error cannot be calculated and the findings apply strictly to the people studied — which is why a quantitative study that wants to generalise uses a probability method wherever a frame can be built.

Sampling error and sampling bias — representativeness and selection bias

Sampling error is the difference between a value calculated from a sample and the true value in the population that arises purely from chance in which units happened to be selected; sampling bias is a systematic difference that arises from how the units were selected, so that some kinds of people were more likely to be chosen than others. The first shrinks as the sample grows and can be estimated; the second does not shrink with size and can only be prevented by the method of selection.

Sampling error is random and unavoidable — two random samples from the same population give two different means — its size is the standard error, which falls with the square root of the sample size, and it is what a confidence interval expresses; a larger sample reduces it and a random method allows it to be calculated.
Sampling bias is systematic: recruiting only the patients who attend the morning clinic misses the working patients who come in the evening (selection bias), volunteers differ from those who decline (volunteer bias), those who answer differ from those who do not (non-response bias) and survivors differ from those who died (survivor bias); it is reduced by a probability method, a complete frame, a high response rate and consecutive recruitment — never by a larger sample.
A sample is representative when its key characteristics match the population's, which is checked by comparing the sample with known population figures for age, sex, diagnosis or place, by reporting the response rate and, where possible, by comparing responders with non-responders.
In longitudinal studies and trials attrition bias arises because those who drop out differ from those who stay; a full account of the losses and intention-to-treat analysis limit it, and generalisability (external validity) depends on the sample, the setting and the time of the study.
Other biases named in stems are recall bias in case-control studies, interviewer bias, measurement bias, the Hawthorne effect of being observed and social desirability in self-report; confounding is not a sampling bias but a third variable mixed with the exposure, and it is handled in the design or the analysis.
The practical rule is that the method is chosen to prevent bias and the size to limit error — a sample can be large and biased, or small and unbiased, and only the second can be corrected by adding participants.

Variables — independent, dependent, confounding and extraneous

A variable is any characteristic or quality that takes different values across the people or units studied — age, blood pressure, anxiety, the dressing used — as opposed to a constant, which is the same for all. Quantitative research is the study of relationships among variables, and the examinable content is the roles variables play in a design — independent, dependent, confounding, extraneous — and the way each is defined so that it can be measured.

The independent variable is the presumed cause — the one the researcher manipulates in an experiment or classifies people by in a non-experimental study: the drug, the teaching programme, the exposure; the dependent variable is the presumed effect, the outcome that is measured: the blood pressure, the anxiety score, the infection rate — and the dependent variable depends on the independent one.
Extraneous variables are all the other variables that could affect the dependent variable and blur the relationship being studied — age, severity of disease, which nurse gave the care; a confounding variable is an extraneous variable related to both the independent and the dependent variable, so that it offers a rival explanation for the result — coffee drinking, smoking and heart disease; an intervening (mediating) variable lies on the path between cause and effect and a moderating variable changes the strength of the relationship.
Extraneous variables are controlled by randomisation, by homogeneity (restricting the sample), by matching, by blocking or stratification, by statistical adjustment such as analysis of covariance, by repeated measures on the same people, by blinding and by a fixed protocol that treats every participant alike.
By nature a variable is continuous (any value on a scale — weight, temperature), discrete (whole numbers — the number of children), categorical (categories — blood group) or dichotomous (two categories — alive or dead at thirty days, male or female); an active variable is one the researcher manipulates and an attribute variable one the participant brings, such as age or sex.
A conceptual definition gives the abstract meaning of a concept — anxiety as a state of apprehension — and an operational definition says how it will be measured in this study — the score on the State-Trait Anxiety Inventory; operational definitions are what make a study replicable, and a conceptual or theoretical framework is the model that links the variables.
The proposal also names its assumptions (principles accepted as true without proof), its delimitations (the boundaries the researcher chooses) and its limitations (the weaknesses she cannot control), so that the reader can judge what the variables were allowed to show.

Hypothesis — null, alternative, directional and non-directional

A hypothesis is a formal, testable statement of the expected relationship between two or more variables in a population, written before the data are collected and either supported or not supported by them; it is the researcher's prediction of the answer to the research question. Hypotheses belong to quantitative studies that test relationships — a descriptive or a qualitative study has research questions instead — and the examinable content is the kinds of hypothesis and how each is worded.

A hypothesis states a relationship between an independent and a dependent variable in a named population, is testable because the variables can be measured, is derived from theory or the literature and is written as a declarative sentence in the present tense; a research question asks, a hypothesis predicts.
The null hypothesis (H0, the statistical hypothesis) states that there is no difference or no relationship between the variables in the population and that any difference seen in the sample is due to chance; it is the hypothesis the statistical test actually tests and tries to reject, while the alternative or research hypothesis (H1 or HA, the scientific hypothesis) states that a difference or relationship exists and is what the researcher believes.
A directional hypothesis predicts the direction of the relationship — 'patients who receive pre-operative teaching will report lower anxiety than those who do not' — and calls for a one-tailed test; a non-directional hypothesis predicts a relationship without saying which way — 'there is a difference in anxiety between patients who do and do not receive teaching' — and calls for a two-tailed test.
A simple hypothesis has one independent and one dependent variable and a complex (multivariate) hypothesis two or more of either; an associative hypothesis says the variables occur together and a causal hypothesis that one produces the other; a statistical hypothesis is written in terms of population parameters.
A hypothesis is never proved: the data support or fail to support the research hypothesis, and the null hypothesis is rejected or not rejected — 'accepted' is the word examiners mark wrong — and the level of significance and the Type I and Type II errors belong to the testing.
A good hypothesis is clear, specific, testable, consistent with what is known, states the relationship in plain language and names the population; it is written before the study begins, so that the data test it rather than suggest it.

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