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  <head>
    <title>101-01 Variables, distributions, sampling, and estimation</title>
    <ownerName>Integrated Medical Foundations</ownerName>
  </head>
  <body>
    <outline text="Variables, distributions, sampling, and estimation">
      <outline text="Quantities before arithmetic">
        <outline text="Dilution conserves the solute amount">
          <outline text="2 mmol/L x 5 mL / 20 mL = 0.5 mmol/L"/>
          <outline text="20 mL is final volume, not solvent added"/>
        </outline>
        <outline text="Absolute versus relative change">
          <outline text="20 to 30: ten units, a fifty percent rise"/>
          <outline text="30 to 20: a one third fall, reference changes"/>
        </outline>
        <outline text="Percentages alone show neither reproducibility nor cause"/>
      </outline>
      <outline text="Probability-based reasoning and design">
        <outline text="From variable data to uncertain conclusions">
          <outline text="Does not turn imperfect data into certainty"/>
        </outline>
        <outline text="Define population, sampling, variables, timing, target"/>
        <outline text="Correct maths can answer the wrong question">
          <outline text="Eligibility, measurement, follow-up differ from target"/>
        </outline>
      </outline>
      <outline text="Variables and measurement">
        <outline text="Categorical: nominal or ordinal">
          <outline text="Ordinal spacing not necessarily equal"/>
        </outline>
        <outline text="Quantitative: discrete counts or continuous"/>
        <outline text="Pain scale is ordered and analysed numerically">
          <outline text="One unit may differ across range and people"/>
        </outline>
        <outline text="Accuracy, precision, reliability, validity"/>
        <outline text="Random error usually weakens associations"/>
        <outline text="Systematic error can create or conceal associations"/>
        <outline text="Calibration plus evidence for subjective constructs"/>
      </outline>
      <outline text="Distributions and summaries">
        <outline text="Centre: mean, median, mode">
          <outline text="Mean uses every value, sensitive to extremes"/>
        </outline>
        <outline text="Spread: range, interquartile range, variance, SD"/>
        <outline text="Centre without spread hides heterogeneity"/>
        <outline text="Normal: symmetric, set by mean and SD">
          <outline text="Mixtures, detection limits, subgroups distort shape"/>
          <outline text="An approximation justified by purpose"/>
        </outline>
        <outline text="Log transformation symmetrises multiplicative variation">
          <outline text="Back-transformed summaries read as ratios"/>
        </outline>
      </outline>
      <outline text="Z scores and reference intervals">
        <outline text="Standard deviations from the reference mean"/>
        <outline text="Not the probability of pathology">
          <outline text="Unusual yet healthy, common yet dangerous"/>
        </outline>
        <outline text="Reference intervals hold a central proportion">
          <outline text="Some healthy outside, some diseased inside"/>
        </outline>
      </outline>
      <outline text="Count distributions and probability">
        <outline text="Binomial: successes in fixed independent trials">
          <outline text="Mean is trials times event probability"/>
        </outline>
        <outline text="Poisson: counts per time, area, or exposure">
          <outline text="Mean equals variance in the simplest model"/>
          <outline text="Overdispersion and clustering need extensions"/>
        </outline>
        <outline text="Independence differs from mutual exclusivity"/>
        <outline text="Bayes&#x27; rule updates a prior by likelihood">
          <outline text="Post-test probability depends on pre-test probability"/>
        </outline>
      </outline>
      <outline text="Sampling and the standard error">
        <outline text="Random sampling supports generalisation">
          <outline text="Convenience samples may differ systematically"/>
        </outline>
        <outline text="Random allocation supports causal comparison">
          <outline text="Allocation does not make participants representative"/>
          <outline text="Sampling alone does not remove confounding"/>
        </outline>
        <outline text="Standard error is SD of the sampling distribution">
          <outline text="SD for individuals, SE for estimate uncertainty"/>
          <outline text="Halving SE needs about four times the sample"/>
        </outline>
        <outline text="Central limit theorem: standardised means approach normal">
          <outline text="Does not normalise data or remove bias"/>
          <outline text="Clustered data carry less independent information"/>
        </outline>
      </outline>
      <outline text="Interval estimation and Bayesian inference">
        <outline text="95% CI: procedure covers truth in 95% of repeats">
          <outline text="Realised interval: parameter fixed, not moving"/>
        </outline>
        <outline text="Narrow interval around bias still misleads">
          <outline text="Nonresponse, loss, misspecification, clustering unseen"/>
        </outline>
        <outline text="Crossing the null does not prove no effect"/>
        <outline text="Bayesian: prior plus likelihood gives posterior">
          <outline text="Direct probability statements about parameters"/>
          <outline text="Sensitivity analysis for contested priors"/>
        </outline>
      </outline>
      <outline text="Estimators, resampling, and sample size">
        <outline text="Judge by bias, variance, consistency, robustness">
          <outline text="Modest bias can reduce prediction error"/>
          <outline text="No universally best estimator"/>
        </outline>
        <outline text="Bootstrap resamples with replacement">
          <outline text="Must preserve clustering and dependence"/>
        </outline>
        <outline text="Neither repairs selection bias"/>
        <outline text="Sample size links effect, variability, events, missingness">
          <outline text="Define the minimally important effect beforehand"/>
          <outline text="Inflating the assumed effect does not create power"/>
        </outline>
      </outline>
      <outline text="Descriptive analysis and the estimand">
        <outline text="Describe before complex modelling"/>
        <outline text="Investigate outliers, do not auto-delete">
          <outline text="Data error, unusual valid patient, or incomplete model"/>
          <outline text="Exclusion rules chosen after seeing impact invite bias"/>
        </outline>
        <outline text="Name the estimand">
          <outline text="Population mean, risk difference, effect of intervention"/>
          <outline text="Then ask if design and analysis identify it"/>
        </outline>
      </outline>
    </outline>
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