--- module: 101-01 language: en chapter: 101 title: "Quantitative Biostatistics for Biomedical and Clinical Science" module_title: "Variables, distributions, sampling, and estimation" source_sha256: 713180e1dd9679c97c77f784cf127a02fb4aafec62b80cfebde703ae1094fa6f --- # Variables, distributions, sampling, and estimation ## Quantities before arithmetic ### Dilution conserves the solute amount #### 2 mmol/L x 5 mL / 20 mL = 0.5 mmol/L #### 20 mL is final volume, not solvent added ### Absolute versus relative change #### 20 to 30: ten units, a fifty percent rise #### 30 to 20: a one third fall, reference changes ### Percentages alone show neither reproducibility nor cause ## Probability-based reasoning and design ### From variable data to uncertain conclusions #### Does not turn imperfect data into certainty ### Define population, sampling, variables, timing, target ### Correct maths can answer the wrong question #### Eligibility, measurement, follow-up differ from target ## Variables and measurement ### Categorical: nominal or ordinal #### Ordinal spacing not necessarily equal ### Quantitative: discrete counts or continuous ### Pain scale is ordered and analysed numerically #### One unit may differ across range and people ### Accuracy, precision, reliability, validity ### Random error usually weakens associations ### Systematic error can create or conceal associations ### Calibration plus evidence for subjective constructs ## Distributions and summaries ### Centre: mean, median, mode #### Mean uses every value, sensitive to extremes ### Spread: range, interquartile range, variance, SD ### Centre without spread hides heterogeneity ### Normal: symmetric, set by mean and SD #### Mixtures, detection limits, subgroups distort shape #### An approximation justified by purpose ### Log transformation symmetrises multiplicative variation #### Back-transformed summaries read as ratios ## Z scores and reference intervals ### Standard deviations from the reference mean ### Not the probability of pathology #### Unusual yet healthy, common yet dangerous ### Reference intervals hold a central proportion #### Some healthy outside, some diseased inside ## Count distributions and probability ### Binomial: successes in fixed independent trials #### Mean is trials times event probability ### Poisson: counts per time, area, or exposure #### Mean equals variance in the simplest model #### Overdispersion and clustering need extensions ### Independence differs from mutual exclusivity ### Bayes' rule updates a prior by likelihood #### Post-test probability depends on pre-test probability ## Sampling and the standard error ### Random sampling supports generalisation #### Convenience samples may differ systematically ### Random allocation supports causal comparison #### Allocation does not make participants representative #### Sampling alone does not remove confounding ### Standard error is SD of the sampling distribution #### SD for individuals, SE for estimate uncertainty #### Halving SE needs about four times the sample ### Central limit theorem: standardised means approach normal #### Does not normalise data or remove bias #### Clustered data carry less independent information ## Interval estimation and Bayesian inference ### 95% CI: procedure covers truth in 95% of repeats #### Realised interval: parameter fixed, not moving ### Narrow interval around bias still misleads #### Nonresponse, loss, misspecification, clustering unseen ### Crossing the null does not prove no effect ### Bayesian: prior plus likelihood gives posterior #### Direct probability statements about parameters #### Sensitivity analysis for contested priors ## Estimators, resampling, and sample size ### Judge by bias, variance, consistency, robustness #### Modest bias can reduce prediction error #### No universally best estimator ### Bootstrap resamples with replacement #### Must preserve clustering and dependence ### Neither repairs selection bias ### Sample size links effect, variability, events, missingness #### Define the minimally important effect beforehand #### Inflating the assumed effect does not create power ## Descriptive analysis and the estimand ### Describe before complex modelling ### Investigate outliers, do not auto-delete #### Data error, unusual valid patient, or incomplete model #### Exclusion rules chosen after seeing impact invite bias ### Name the estimand #### Population mean, risk difference, effect of intervention #### Then ask if design and analysis identify it