3.1.2. Independence of observation and pseudo-replication
A second factor in deciding on sample size, and a fundamental aspect of good experimental design, is independence of observations; what happens to one experimental unit should be independent of what happens to other experimental units before results of statistical analyses can be trusted. The experimental unit is the unit (subject, plant, pot, animal) that is randomly assigned to a treatment. Replication is the repetition of the experimental situation by replicating the experimental unit (Casella, 2008). Where observations are not independent, i.e. there are no true replicates within an experiment, we call this pseudo-replication or technical replication. Pseudo-replication can either be: i) temporal, involving repeated measures over time from the same bee, cage, hive or apiary; or ii) spatial, involving several measurements from the same vicinity. Pseudo-replication is a problem because one of the most important assumptions of standard statistical analysis is independence. Repeated measures through time on the same experimental unit will have non-independent (= dependent) errors because peculiarities of individuals will be reflected in all measurements made on them. Similarly, samples taken from the same vicinity may not have non-independent errors because peculiarities of locations will be common to all samples. For example, honey bees within the same cage might not be independent because measurements taken from one individual can be dependent on the state (behaviour, infection status, etc.) of another bee within the same cage (= spatial pseudo-replication), so each cage becomes the minimum unit to analyse statistically (i.e. the experimental unit). An alternative solution is to try estimating the covariance structure of the bees within a cage, i.e. allow for correlation within a cage in the statistical modelling.
But, are honey bees in different cages independent? This and similar issues have to be considered and were too often neglected in the past. Potential non-independence can be addressed by including cage, colony, and any other potentially confounding factors as random effects (or fixed effects in certain cases) in a more complex model (i.e. model the covariance structure imposed by cages, colonies, etc.). If pseudo-replication is not desired and is an unavoidable component of the experimental design, then it should be accounted for using the appropriate statistical tools, such as (generalised) linear mixed models ((G)LMM; see section 5.2.).
Some examples may clarify issues about independence of
Example 1: A researcher observes that the average number of Nosema spores per bee in a treated cage is significantly higher than in a control cage; one cannot rule out whether the observed effect was caused by the treatment or the cage.
Possible solution 1: Take cage as the experimental
unit and pool the observations per cage; including more cages is statistically
preferred (yields more power) to including more bees per cage.
Example 2: Relative to cages containing control bees, experimental cages were housed closer to a fan in the lab, resulting in higher levels of desiccation for the experimental cages and, in turn, higher mortality under constant airflow. In this case, the statistical difference between treatments would be confounded by the experimental design.
Possible solution 2: A rotation system could be
included to ensure all cages are exposed to the same environmental conditions
i.e. placed at identical distances from the fan and for the same periods of
Example 3: Honey bees from treated colonies had high levels of a virus and were A. mellifera mellifera, whereas control honey bees from untreated colonies that had low levels of a virus were A. mellifera ligustica. In such a case the statistical differences could be due to colony differences and/or to subspecies differences and/or due to the treatment and/or due to interactions.
Possible solution 3: Design the experiment using a factorial design with colony as the experimental unit. For half of the colonies in a treatment, use A. mellifera mellifera bees and for the other half use A. mellifera ligustica bees. Equal numbers of colonies of both subspecies should then be present in the treatment and control groups. Although equal numbers is not a requirement, it is nevertheless preferable to have a completely balanced design (equal numbers in each group or cell) for several reasons (e.g. highest power, efficiency, ease of parameter interpretation, especially interactions). It is, however, also possible to estimate and test with unbalanced designs. In a balanced design the differences between colonies, subspecies, and treatments (and their interactions!) can be properly quantified.
In essence, there are both environmental and genetic factors (which can also interact) that can profoundly affect independence and hence reliability of statistical inference. The preceding examples illustrate, among other things, the importance of randomising experimental units among different treatments. The final solutions of the experimental design are of course highly dependent on the research question and the variables measured.
In summary, randomisation and replication have two separate functions in an experiment. Variables that influence experimental units may be known or unknown, and random assignment of treatments to cages of honey bees is the safest way to avoid pitfalls of extraneous variables biasing results. Larger sample sizes (i.e. replication: number of colonies, cages, or bees per cage) improve the precision of an estimate (e.g. infection rate, mortality, etc.) and reduce the probability of uncontrolled factors producing spurious statistical insignificance or significance. Researchers should use as many honey bee colony sources from unrelated stock as possible if they want their results to be representative, and hence generalisable. One should also not be too cavalier about randomising honey bees to experimental treatments, or about arranging experimental treatments in any setting, including honey bee cage experiments; sound experimental design at this stage is critical to good science; more details are provided below.