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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