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The extent to which a person follows an agreed treatment plan. Understanding, tolerability, access and expectations can all affect adherence, so nonadherence is not a diagnosis of irrationality.

A harmful or unintended response attributed to a medicine under specified conditions. Temporal association alone does not establish drug causation; disease, concomitant treatments and contextual influences must also be assessed.

An always-taker is a unit with potential treatment receipts $D(0)=D(1)=1$: it takes the treatment under either value of binary encouragement $Z$. This $(1,1)$ pair is a principal stratum defined by both potential receipts.

For example, a unit may participate whether or not it receives an invitation. Encouraged participants combine always-takers and compliers; unencouraged participants combine always-takers and defiers. Under instrumental-variable monotonicity, the latter group consists only of always-takers. Even then, comparing observed participants across assignment arms mixes different strata and is not automatically a within-stratum causal effect. The exclusion restriction, when imposed, says assignment cannot affect an always-taker's outcome through any channel other than the treatment whose receipt is unchanged.^[ref]

For one response per cell in an order-$q$ Latin square, fit an intercept and additive row, column and treatment effects with sum-to-zero constraints. Each factor contributes $q-1$ statistical degrees of freedom; the residual has $q^2-1-3(q-1)=(q-1)(q-2)$. The centered factor spaces are mutually orthogonal, so sequential sums of squares do not depend on fitting order. With an additional orthogonal symbol factor from a Graeco-Latin square, residual statistical degrees of freedom become $(q-1)(q-3)$. Exact F-tests require independent equal-variance normal errors; unmodeled interactions can invalidate the error term.

A binary block design has $t$ treatments, $b$ blocks of size $k<t$, every treatment in $r$ blocks, and every distinct pair in $\lambda$ blocks. Counting occurrences and pairs gives $tr=bk$ and $r(k-1)=\lambda (t-1)$. Its incidence matrix of a set system $N$ satisfies $N{N}^T=(r-\lambda )I+\lambda 11^T$. For statistical comparison of all treatment effects one also needs connectedness; $\lambda >0$ guarantees it, whereas singleton blocks with $\lambda =0$ do not.

Biostatistics develops and applies statistical inference to biological and health questions. Its problems include designing experiments, comparing treatments, measuring biological variation and analyzing survival or repeated observations. The experimental unit must match the independent unit of information: multiple measurements of one organism need not substitute for measurements of multiple organisms.

Suppose a randomized study observes an event in 10 of 100 treated participants and 20 of 100 controls. The estimated risk difference is $0.10-0.20=-0.10$, while the estimated relative risk is $0.10/0.20=0.5$. These express different scales of comparison, and neither alone expresses its uncertainty. Assignment, follow-up and the definition of the event determine what effect can be estimated. For incomplete follow-up, methods from survival analysis distinguish observing no event yet from establishing that an event will never occur.

A bivariate probit model for a binary treatment and outcome uses latent equationswhere $(U,V)$ has a bivariate normal distribution. Its correlation parameter represents dependence between the two latent disturbances; when nonzero, treatment is endogenous in the outcome equation.

An outcome assessor is kept unaware of randomized treatment assignment to limit differential questioning, interpretation and recording. This does not require the participant or intervention provider to be blinded. Participants can reveal their assignment, so standardized interviews and restricted records help preserve blinding. Blinding after assignment is different from allocation concealment before enrollment.

Withholding treatment or hypothesis information from participants, handlers or scorers where appropriate to reduce expectation-dependent differences. What was blinded and which cues remained available should be reported.

Partition a regular factorial design by fixed signs of independent factorial contrasts. Products of those block-generating columns span the block effects. A treatment contrast that is constant on each block is therefore confounded with block differences. Select generators so that only scientifically dispensable treatment effects enter this space; randomize the assignment of blocks and the run order within each block.

A block design assigns treatments to experimental units partitioned into blocks. Its incidence counts $n_{ij}$ record how often treatment $i$ occurs in block $j$. Balance and connectedness determine which treatment contrasts are estimable and how much adjustment for blocks costs.

A block groups experimental units expected to have similar background responses. Comparing treatments within blocks can reduce the variance of an estimator. Block allocation must be specified before responses are used to evaluate treatment effects.

A carryover effect is an effect of a previous treatment on a later response. Balanced transitions help prevent a treatment from being preceded disproportionately by one particular alternative, but do not ensure that carryover effects vanish or are separately identifiable in every model.

Causal inference uses data and explicit assumptions to learn effects of specified interventions. For binary treatment $A$, a target average treatment effect is $\mathbb{E}[Y(1)-Y(0)]$, comparing potential outcomes for the same population. Observing only $Y=Y(A)$ leaves one of each unit's potential outcomes unobserved.

An observed treated-minus-control difference can include confounding. For a concrete model, suppose half a population is in a low-baseline stratum with $(Y(0),Y(1))=(0,1)$ and half in a high-baseline stratum with $(2,3)$. Let treatment probabilities be $1/4$ and $3/4$ in the two strata. Every unit's causal effect is one. The treated mean is $\frac{1}{4}(1)+\frac{3}{4}(3)=2.5$, while the untreated mean is $\frac{3}{4}(0)+\frac{1}{4}(2)=0.5$. Their observed difference is two because their stratum compositions differ.

Under conditional exchangeability, positivity and consistency in causal inference, outcome standardization identifies the effect byIn the example, using the same half-and-half target weights for both strategies recovers one. The substantive assumptions, rather than the arithmetic alone, justify interpreting the standardized contrast as an intervention effect.^[ref]

A nineteenth-century explanation of changing therapeutic practice: inflammatory diseases were said to have become less vigorous or more debilitating, requiring less depletion. It competed with explanations based on improved pathology and better evaluation of treatment. This is a historical theory, not a general modern law. ^[ref]

Examining and revising interpretations or assumptions that influence emotion and action, typically with explicit alternatives and tests rather than unsupported reassurance. Its contribution to a treatment package requires component-level evidence.

A completely randomized design assigns treatments directly to available experimental units using complete randomization, without blocking. Fixed group sizes can be enforced. The residual then includes uncontrolled between-unit variation.

Complete randomization chooses uniformly among all treatment assignments having prespecified treatment-arm sizes.

A complier is a unit with potential treatment receipts $D(0)=0$ and $D(1)=1$: binary encouragement $Z$ changes its receipt from no treatment to treatment. This $(0,1)$ pair defines a principal stratum, rather than a behavior inferred from one observed assignment.

For example, a unit that participates exactly when invited is a complier. Under instrumental-variable monotonicity, which excludes defiers,Thus, with randomized assignment, the difference in treatment-receipt rates estimates the complier fraction. To interpret an outcome-rate ratio as the local average treatment effect, one additionally needs the relevant independence and exclusion assumptions and a nonzero complier fraction. An observed encouraged participant may instead be an always-taker.^[ref]

The use of mathematical models to investigate processes contributing to mental disorder and its treatment. Model fitting, parameter recovery and experimental intervention have different evidential roles; an estimated parameter is not automatically a diagnosis or a biological mechanism.