17.1 Statistics & Critical Appraisal
Key Takeaways
Descriptive statistics summarise data (mean, median, mode, standard deviation, range, interquartile range); the median and IQR are preferred for skewed data, the mean and SD for symmetrical (normal) distributions.
Inferential statistics generalise from a sample to a population using hypothesis tests (t-test, chi-square, ANOVA) and estimation (confidence intervals); the choice of test depends on data type and distribution.
Type I (alpha) error is a false positive — rejecting a true null; Type II (beta) error is a false negative — failing to reject a false null; power = 1 − beta, typically targeted at 0.8.
Diagnostic test performance is described by sensitivity (true positive rate), specificity (true negative rate), positive and negative predictive value, and likelihood ratios; sensitivity and specificity are properties of the test, while predictive values depend on prevalence.
Correlation (r) measures association; regression models the relationship and predicts outcomes — correlation does not imply causation.
Descriptive Statistics
Descriptive statistics summarise a dataset without inferring beyond it. The choice depends on data type and distribution.
| Measure | Use |
|---|---|
| Mean | Average — for symmetrical (normal) continuous data |
| Median | Middle value — for skewed data (resistant to outliers) |
| Mode | Most common value — for categorical data |
| Standard deviation (SD) | Spread around the mean — for normal data |
| Interquartile range (IQR) | Spread around the median (25th–75th centile) — for skewed data |
| Range | Max to min — simplest spread, sensitive to outliers |
Data Types
- Categorical / qualitative — nominal (no order, e.g. blood group) or ordinal (ordered, e.g. pain score 0–10).
- Numerical / quantitative — discrete (counts) or continuous (measurements).
Skewness
- Normal (symmetrical) distribution — mean ≈ median ≈ mode; use mean and SD.
- Right (positive) skew — long tail to the right; mean > median; use median and IQR (e.g. income, length of stay).
- Left (negative) skew — long tail to the left; mean < median; use median and IQR.
The standard deviation measures how far values typically lie from the mean; in a normal distribution about 95% of values fall within mean ± 2 SD (the empirical rule).
Sampling and the Normal Distribution
The standard error (SE) measures the precision of a sample statistic (e.g. the mean) as an estimate of the population value; SE = SD / √n. As sample size grows, the SE shrinks — estimates become more precise.
The central limit theorem states that, with a sufficiently large sample, the sampling distribution of the mean is approximately normal regardless of the underlying distribution — the basis for many inferential tests.
Inferential Statistics
Inference generalises from a sample to a population. Two complementary approaches:
- Hypothesis testing — compare a null hypothesis (no effect) against an alternative; produce a p-value.
- Estimation — produce a confidence interval for the effect.
Choosing a Test
| Comparison | Data type | Test |
|---|---|---|
| Two independent groups | Continuous, normal | Independent (unpaired) t-test |
| Two related groups (paired) | Continuous, normal | Paired t-test |
| Two groups | Continuous, non-normal | Mann-Whitney U (independent) / Wilcoxon signed-rank (paired) |
| Three or more groups | Continuous, normal | ANOVA |
| Association between two categorical variables | Categorical | Chi-square test |
| Correlation | Continuous | Pearson (normal) / Spearman (non-normal) |
| Predicting an outcome from one or more predictors | Continuous outcome | Linear regression |
| Binary outcome from predictors | Binary outcome | Logistic regression |
The Null Hypothesis and p-values
- Null hypothesis (H₀): no difference / no association.
- Alternative hypothesis (H₁): there is a difference.
- p-value: the probability of the observed result (or more extreme) if H₀ were true.
- Significance level (α): conventionally 0.05. If p < α, reject H₀.
A p-value is not the probability that the null is true, nor the probability the result is due to chance; it is conditional on the null being true.
Type I and Type II Errors
| Error | Definition | Probability |
|---|---|---|
| Type I (alpha) | False positive — rejecting a true null | α (typically 0.05) |
| Type II (beta) | False negative — failing to reject a false null | β |
| Power | 1 − β — the probability of correctly rejecting a false null | Typically targeted at ≥ 0.8 (80%) |
Power is increased by a larger sample size, a larger true effect, and a higher significance level. Underpowered studies may miss a real effect (Type II error). Calculating sample size before a study (power calculation) prevents underpowering.
Confidence Intervals
A 95% confidence interval is the range that, over many samples, would contain the true population value 95% of the time. It conveys both the estimate and its precision:
- A narrow CI indicates a precise estimate (large sample, low variability).
- A wide CI indicates uncertainty.
- A CI that excludes the null value (0 for differences, 1 for ratios) implies statistical significance.
Confidence intervals are increasingly preferred to p-values because they convey effect size and precision, not just significance.
Diagnostic Tests
Evaluating a diagnostic test compares its result with a reference (gold) standard.
| Measure | Definition | Question answered |
|---|---|---|
| Sensitivity | True positive rate: TP / (TP + FN) | How good is the test at identifying disease in those who have it? |
| Specificity | True negative rate: TN / (TN + FP) | How good is the test at ruling disease out in those who don't have it? |
| Positive predictive value (PPV) | TP / (TP + FP) | If the test is positive, how likely is disease? |
| Negative predictive value (NPV) | TN / (TN + FN) | If the test is negative, how likely is the patient disease-free? |
Sensitivity and Specificity vs Predictive Values
- Sensitivity and specificity are properties of the test and do not change with prevalence.
- PPV and NPV depend on the prevalence of disease in the tested population: a positive test is more likely to be a true positive when disease is common; in a low-prevalence population even a specific test yields many false positives.
Likelihood Ratios
The likelihood ratio (LR) combines sensitivity and specificity and is independent of prevalence. LR+ = sensitivity / (1 − specificity); LR− = (1 − sensitivity) / specificity. A high LR+ strongly increases the probability of disease; a low LR− strongly decreases it. LRs are useful because they combine with pre-test probability to give a post-test probability.
SnNout and SpPin
- SnNout — when a test with high Sensitivity is Negative, disease is ruled out.
- SpPin — when a test with high Specificity is Positive, disease is ruled in.
Correlation and Regression
Correlation quantifies the strength and direction of a linear association between two variables, with r from −1 to +1. Pearson's r assumes normality; Spearman's rank is non-parametric. Correlation does not imply causation — a confounder may explain the association.
Regression models the relationship and predicts an outcome. Linear regression predicts a continuous outcome; logistic regression predicts a binary outcome (e.g. disease yes/no). Regression can adjust for confounders and quantify independent effects.
Critical Appraisal — Putting It Together
When appraising a paper:
- Is the question focused (PICO)?
- Is the design appropriate to the question?
- Were the groups comparable at baseline (randomised; matched)?
- Was there blinding and allocation concealment?
- Was the analysis intention-to-treat?
- Is the effect size and its confidence interval clinically meaningful?
- Are the results applicable to your patient?
CASP checklists structure this appraisal (see 14.4).
A study reports the lengths of hospital stay after a new extraction technique, and the data are markedly right-skewed with a few very long stays. Which pair of summary measures is most appropriate?
Mean and standard deviation
Median and interquartile range
Mode and range
Mean and p-value
A research team sets their significance level at 0.05 and their power at 0.80 for a trial of a new bonding agent. What does a power of 0.80 represent?
The probability of correctly detecting a true effect of the specified size (80% chance of avoiding a Type II error)
The probability of a false positive result
The probability that the null hypothesis is true
The width of the confidence interval
A new salivary test for oral cancer has high sensitivity. Using the 'SnNout' principle, what does this imply for clinical practice?
A positive test rules cancer in
The test is unaffected by prevalence
A negative test rules cancer out
The test has no false positives
A diagnostic test with high specificity is introduced into a low-prevalence screening population. What is the likely effect on the positive predictive value (PPV)?
The PPV will be high because specificity is high
The PPV is unaffected by prevalence
The PPV equals the sensitivity
The PPV will be low because the low prevalence means many positive results are false positives despite high specificity
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