# Statistics

## Types of data:

 Categorical: Qualitative -        Nominal ·      Labelled ·      No quantity or order (e.g. black, white, Hispanic) -        Ordinal ·      Numbered ·      Variable increments (e.g. mild, moderate, severe) Numerical: Quantitative -        Discrete ·      Whole numbers only (e.g. number of admissions) -        Continuous ·      Any value ·      Constant increment ·      Interval data: false zero point (i.e. 20°C isn’t twice as hot as 10°C) ·      Ratio data: true zero point (e.g. 2/52 is twice as old as 1/52)

## Test selection:

 2 groups >2 groups Paired Unpaired Paired Unpaired Parametric Student T Student T ANOVA ANOVA Non-parametric -Nominal McNemar Chi Cochrane Q Cochrane Q Non-parametric -Ordinal / ratio Wilcoxon Rank Sum Mann Whitney U Friedman Kruskell Wallis

N.B. Parametric tests are used when data a) are numerical b) follow a distribution

## Measures of central tendency:

 Mean ·      Sum / number of observations ·      More affected by outliers Median ·      Half the observations are higher, other half are lower ·      Less affected by outliers Mode ·      Most frequently occurring value ·      Rarely relevant

## Measures of variability:

 Inter-quartile range ·      Middle 50% of observations ·      Often represented on box and whisker plot o  Box: 25, 50 and 75 o  Whiskers: 10 and 90 Standard deviation ·      SD = √variance ·      Variance = ε(x-ẋ)2/(n-1), where x: mean ẋ: each value n-1: degrees of freedom Standard error ·      SE = SD/√n, where n: number of observations ·      Indicates how far the sample mean is likely to be from the true mean ·      Used to derive confidence interval 95% confidence interval ·      CI = mean ± 0.95(SD/√n) ·      Indicates both magnitude and precision of difference (unlike P value) ·      If the confidence interval crosses 1.0, result is insignificant

## Comparisons:

Example:

 Pain No pain Fentanyl A B Nothing C D

 Relative risk ·      Risk of pain in fentanyl group = A/(A+B) = RF ·      Risk of pain in nothing group = C/(C+D) = RN ·      Relative risk = RF/RN ·      The risk of the event in the intervention group compared with the risk of the even in the control group” Odds ratio ·      Odds of pain in fentanyl group = A/B = OF ·      Odds of pain in nothing group = C/D = ON ·      Odds ratio = OF/ON ·      “A ratio of event to non-event in the intervention group compared with the control group” Hazard ratio ·      Hazard ratio is the relative risk of an event happening at time t ·      i.e. risk of pain now cf. risk of pain at some point Number needed to treat ·      Absolute risk reduction = RN - RF ·      NNT = 1 / (absolute risk reduction)

## Purpose of test:

 Screening Diagnostic Target Everyone Symptomatic; or Positive screening test Nature Non-invasive Invasive Thresholds High sensitivity (few false neg) High specificity (few false pos) Cost Cheap Expensive

## Sensitivity and specificity etc

 Difficult ETT Not difficult ETT Predictive values: High MP score a)True +ve b)False +ve PPV: a/(a+b) Low MP score c)False -ve d)True -ve NPV: d/(c+d) Sensitivity: a/(a+c) 30% Specificity: d/(b+d) 90%

 Likelihood ratios Positive LR = sensitivity / (1-specificity) = 3 Negative LR = (1-sensitivity) / specificity = 0.78 Receiver operator characteristic ·        Relationship between sensitivity, specificity, test quality ·        ↑AUC associated with high utility Youden’s J statistic ·        J = sensitivity + specificity - 1 ·        J = point of maximum divergence of the curve ·        Single representation of a test’s utility, between 0 to 1

## P values etc:

 P value ·      Probability of finding this result by chance if the null hypothesis were true ·      i.e. probability of this being a false positive result Problems: ·      Does not take into account prior probability ·      Does not quantify effect size (cf. confidence interval) ·      0.05 not low enough if the stakes are high ·      Inappropriate for multiple comparisons (see Bonferroni correction) Type 1 error ·      False positive ·      Threshold (α value) usually set 5% (since inappropriately finding a difference is terrible) Type 2 error ·      False negative ·      Threshold (β value) usually set at 20% (since inappropriately finding no difference isn’t so bad) Power ·      The ability to detect a difference if there is one ·      Power = 1 – β value = usually 80% Many determinants, e.g. ·      Sample size ·      Magnitude of difference ·      Threshold Fragility index ·      Number of patients whose change in status would turn a significant result into a non-significant result Bonferroni correction ·      Used if assessing for several outcomes simultaneously ·      Divide α (p = 0.05) by the number of tests being performed

## Types of Bias

 Type Explanation Mitigation Hawthorne effect Being studied = feel good Control group Selection Samples studied does not reflect the population Forced participation (i.e. no solution) Response Questions encourage a certain type of answer Control group Recall Patient reports symptoms differently depending upon allocation Patient blinding Detection Intervention and control group assessed differently Assessor blinding Observer Data collector able to be subjective about the outcome Assessor blinding Hard outcomes Publication Negative results don’t get published Clinical trial registries

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