Biostatistics for health sciences

In short

Biostatistics is the application of statistical methods to health and biological data: how to collect it, summarise it, draw inferences about a population from a sample, and quantify uncertainty. Every epidemiological measure, clinical trial result, NFHS table and programme evaluation in India rests on it, and no public-health claim is credible without the sample size, the confidence interval and the test behind it. It is examined in BPH/B.Sc Public Health university papers, in MPH and ICMR-JRF/UGC-NET entrance tests, and tested directly in placement rounds for monitoring-and-evaluation, clinical research and data-analyst posts. Employers who test it include ICMR institutes, NHSRC and State Health Societies, IQVIA, Parexel, Syneos, Fortis/Apollo clinical research units, PHFI, IHAT, Jhpiego, CARE India, the Gates Foundation's Indian partners and CRO/CDM teams. The practical skills asked for are Epi Info, SPSS, Stata, R or Python plus clean Excel work.

Explained

Data, variables and presentation

• Biostatistics vs epidemiology: epidemiology asks the health question and designs the study; biostatistics supplies the measurement, estimation and testing tools.
• Population vs sample: a population parameter (μ, σ, P) is fixed and usually unknown; a sample statistic (x̄, s, p) estimates it and varies from sample to sample.
• Qualitative (categorical) data: *nominal* - no order (blood group, religion, district); *binary/dichotomous* (alive/dead); *ordinal* - ordered but unequal gaps (mild/moderate/severe, Likert).
• Quantitative data: *discrete* - counts (number of episodes of diarrhoea); *continuous* - measured (height in cm, Hb in g/dL, BP in mmHg).
• Scales of measurement: nominal, ordinal, interval (temperature in °C - no true zero), ratio (weight, true zero, ratios meaningful).
• Variable roles: independent (exposure), dependent (outcome), confounder, effect modifier.
• Presentation: frequency and relative-frequency tables, cross-tabulation; bar and multiple/component bar diagrams for categorical; pie chart for a single composition; histogram, frequency polygon and ogive for continuous; box-and-whisker for spread and outliers; scatter diagram for two continuous variables; line/epidemic curve for time; spot map for place.
• Rule: a histogram has no gaps between bars (continuous classes); a bar diagram has gaps (discrete categories). Never use a pie chart for more than about six slices or for non-exhaustive categories.

Descriptive statistics

• Measures of central tendency: *mean* - sum/n, uses all values, affected by outliers; *median* - middle value, robust, used for skewed data (income, hospital stay, parasite counts); *mode* - most frequent, the only average for nominal data.
• Geometric mean is used for antibody titres and serial dilutions; harmonic mean for rates over equal distances.
• Measures of dispersion: range (max − min, unstable), interquartile range (Q3 − Q1, used with median), mean deviation, variance and standard deviation (same units as the data), coefficient of variation (CV = SD/mean × 100, unit-free, compares variability of Hb with that of weight).
• Percentiles and z-scores: WHO growth standards classify children by z-score - stunting is height-for-age below −2 SD, wasting weight-for-height below −2 SD, severe below −3 SD; this is how NFHS and Poshan Tracker report undernutrition.
• Shape: *symmetrical* - mean = median = mode; *positively (right) skewed* - mean > median (serum triglycerides, length of stay); *negatively skewed* - mean < median. Kurtosis describes peakedness.
• Outliers: commonly values beyond Q1 − 1.5×IQR or Q3 + 1.5×IQR; investigate, do not delete silently.

Probability and theoretical distributions

• Probability P(A) lies between 0 and 1. *Addition rule* for mutually exclusive events: P(A or B) = P(A) + P(B); general: P(A) + P(B) − P(A and B). *Multiplication rule* for independent events: P(A and B) = P(A)·P(B). Conditional probability P(A|B) = P(A and B)/P(B) - the basis of sensitivity, specificity and predictive values, and of Bayes' theorem.
• Binomial distribution: fixed n independent trials, two outcomes, constant p; mean = np, variance = npq. Used for proportions - number of seroconversions out of 50 vaccinated.
• Poisson distribution: rare events in a defined time or space; mean = variance = λ. Used for new cases per week, deaths per 1,000 person-years, bacterial colony counts.
• Normal (Gaussian) distribution: bell-shaped, symmetrical about μ, defined by μ and σ, total area 1, asymptotic tails. Empirical rule: μ ± 1σ covers ~68%, μ ± 2σ ~95% (exactly 1.96σ), μ ± 3σ ~99.7%.
• Standard normal variate z = (x − μ)/σ converts any normal variable to mean 0, SD 1 so one table serves all; used to set biological reference ranges (mean ± 1.96 SD of healthy people).
• Central limit theorem: whatever the shape of the parent population, the distribution of sample means approaches normal as n increases (n ≥ 30 is the usual working rule). This is why normal-based tests work on skewed data in large samples.
• Standard deviation vs standard error: SD describes variability among individuals; SE = SD/√n describes the precision of the sample mean. SE always shrinks as n grows; SD does not.

Sampling and estimation

• Why sample: cost, time, feasibility, and often better quality of measurement than a census.
• Probability sampling (allows statistical inference): *simple random* (lottery/random numbers), *systematic* (every kth unit after a random start - house-to-house surveys), *stratified* (divide by strata such as rural/urban or caste, sample within each; improves precision), *cluster* (sample whole villages/wards - used because listing everyone is impossible; cheaper but less precise), *multistage* (NFHS: district → PSU/village or census enumeration block → household), *probability proportional to size*.
• Non-probability sampling (no valid inference): convenience, purposive, quota, snowball (useful for hidden populations - injecting drug users, sex workers).
• Design effect (DEFF) is the inflation in variance from clustering; cluster surveys multiply the required sample size by DEFF (often 1.5-2). The classic WHO immunisation coverage survey uses 30 clusters × 7 children.
• Errors: *sampling error* - chance difference between sample and population, reduced by larger n, quantified by SE; *non-sampling error* - selection, measurement, recall, interviewer, non-response bias; not fixed by a bigger sample.
• Point vs interval estimate: a 95% confidence interval means that 95% of such intervals, computed from repeated samples, would contain the true parameter. A wider interval = less precision; the width falls as 1/√n.
• Precision is governed by absolute width for proportions and by SD and n for means.

Hypothesis testing

1. State the null hypothesis (H0) - no difference/no association - and the alternative (H1), one- or two-tailed (two-tailed by default in health research).
2. Fix the significance level α (usually 0.05) and the test, before looking at the data.
3. Compute the test statistic and its degrees of freedom.
4. Obtain the p-value - the probability of a result as extreme as, or more extreme than, the observed one *if H0 were true*.
5. Decide: p < α → reject H0 ("statistically significant"); p ≥ α → fail to reject (not "H0 is proved").
6. Type I error (α): rejecting a true H0 - a false positive. Type II error (β): failing to reject a false H0. Power = 1 − β, conventionally 80% or 90%.
7. Statistical vs clinical/public-health significance: a 1 mmHg fall in BP can be p < 0.001 in 50,000 people and worthless; a 10% drop in neonatal mortality can be p = 0.08 in a small trial and very important. Always report the effect size with its CI, not p alone.
8. Parametric tests assume normality, independence and (for t/ANOVA) similar variances, and need interval/ratio data: one-sample t, independent (unpaired) t, paired t, one-way and two-way ANOVA (F test, with post-hoc Tukey/Bonferroni), Pearson correlation, linear regression, z test for large samples and proportions.
9. Non-parametric tests (distribution-free, for ordinal or skewed data, small samples): sign test, Wilcoxon signed-rank (paired), Mann-Whitney U (two independent groups), Kruskal-Wallis (≥3 independent groups), Friedman (repeated measures), Spearman's rank correlation, chi-square tests.
10. Chi-square (χ²): tests association between two categorical variables, goodness of fit, and homogeneity of proportions; df = (r − 1)(c − 1). Requires expected frequency ≥ 5 in at least 80% of cells and none below 1; use Fisher's exact test for small or sparse 2×2 tables, McNemar's test for paired/matched categorical data (before-after, matched case-control).
11. Multiple testing: 20 independent tests at α = 0.05 give roughly a 64% chance of at least one false positive; correct with Bonferroni (α/k) or report that analyses are exploratory.
12. Beyond basics (commonly asked): logistic regression gives adjusted odds ratios for binary outcomes; Poisson/negative-binomial regression for counts and rates; Cox proportional-hazards regression and Kaplan-Meier survival curves with the log-rank test for time-to-event data (censoring is the reason ordinary methods fail); Cohen's kappa for inter-observer agreement; Bland-Altman plots for method comparison.

Correlation, regression and sample size

• Correlation measures the strength and direction of a linear relationship; Pearson r lies between −1 and +1, is unit-free, and r² is the proportion of variance explained. Spearman ρ uses ranks.
• r near 0 means no *linear* relation - a strong curve (U-shaped dose-response) can still give r ≈ 0. Correlation is not causation, and an ecological correlation between district-level averages does not apply to individuals (ecological fallacy).
• Simple linear regression Y = a + bX predicts the outcome; b is the change in Y per unit change in X, with its own SE, CI and t-test. Multiple regression adjusts for several predictors simultaneously - the standard way to control confounding in analysis.
• Sample size depends on: expected prevalence or effect size, desired precision (d) or the difference to be detected, α, power, the design effect, expected non-response and loss to follow-up, and whether the outcome is a proportion or a mean. It must be computed and reported *a priori* - journals, ethics committees and the New Drugs and Clinical Trials Rules, 2019 all expect it.
• Indian data sources and rules: NFHS (IIPS, Mumbai; NFHS-5 for 2019-21 is the round still most quoted, with NFHS-6 fieldwork and releases in progress), SRS and the Census (ORGI), NSSO/MoSPI health rounds, HMIS and RCH portals, Poshan Tracker, NCDIR cancer registries, IDSP/IHIP surveillance, and the Civil Registration System under the Registration of Births and Deaths Act, 1969 as amended in 2023.
• Ethics and data governance: ICMR National Ethical Guidelines for Biomedical and Health Research Involving Human Participants (2017), Institutional Ethics Committee approval and registration, mandatory prospective registration of clinical trials with CTRI (ICMR-NIMS), CDSCO oversight, Good Clinical Practice; under the Digital Personal Data Protection Rules, 2025 the data-fiduciary duties (notice, consent, breach intimation, erasure) take effect only from 13 May 2027 - until then IT Act section 43A and the SPDI Rules, 2011 govern sensitive personal health data. Reporting guidelines: CONSORT (trials), STROBE (observational), PRISMA (systematic reviews).

Key points

• Biostatistics - statistics applied to health data: collection, summarisation, inference and quantification of uncertainty.
• Variable types - nominal, ordinal, discrete, continuous; the type decides the summary measure and the test.
• Mean, median, mode - mean for symmetric data, median for skewed data, mode for nominal data.
• SD vs SE - SD describes individual variation; SE = SD/√n describes precision of an estimate.
• CV - SD as a percentage of the mean; the only fair way to compare variability across different units.
• Normal distribution - symmetric, defined by μ and σ; mean ± 1.96 SD covers 95% of individuals.
• Central limit theorem - sample means are approximately normal for large n regardless of parent shape.
• 95% CI - range of plausible values for the parameter; if a CI for a difference includes 0 (or an RR/OR CI includes 1), the result is not significant at 5%.
• p-value - probability of the observed or more extreme data if H0 is true; not the probability that H0 is true.
• Type I (α) - false positive; Type II (β) - false negative; power = 1 − β, aim for 80-90%.
• Chi-square - association between categorical variables, df = (r−1)(c−1); use Fisher's exact when expected counts are small.
• t-test - compares one or two means from small samples; paired t for before-after in the same subject.
• ANOVA - compares three or more means with one F test, avoiding inflated type I error from multiple t-tests.
• Non-parametric tests - Mann-Whitney, Wilcoxon, Kruskal-Wallis, Spearman; used for ordinal or skewed data and small n.
• Correlation - strength of linear association; r² is variance explained; never proof of causation.
• Regression - predicts an outcome and adjusts for confounders; logistic for binary, Cox for time-to-event.
• Sample size - driven by effect size, precision, α, power, design effect and non-response; computed before the study.
• Probability sampling - the only sampling that permits valid inference; stratified gains precision, cluster saves cost but inflates variance.

Formulas and examples

Example 1 - descriptive statistics. Systolic BP (mmHg) of 10 patients: 118, 124, 130, 126, 140, 122, 134, 128, 136, 132.

• Σx = 1,290 → mean = 1,290/10 = 129.0 mmHg.
• Sorted: 118, 122, 124, 126, 128, 130, 132, 134, 136, 140 → median = (128 + 130)/2 = 129.0 mmHg (so the data are symmetric).
• Deviations: −11, −7, −5, −3, −1, +1, +3, +5, +7, +11; squares sum to 410.
• s² = 410/9 = 45.56 → s = 6.75 mmHg; CV = 6.75/129 × 100 = 5.2%.
• SE = 6.75/√10 = 2.13 mmHg; 95% CI (t = 2.262, df 9) = 129 ± 2.262 × 2.13 = 124.2 to 133.8 mmHg.

Example 2 - CI for a prevalence. 180 of 400 women surveyed in a block are anaemic.

• p = 180/400 = 0.45 → 45.0%.
• SE(p) = √(0.45 × 0.55/400) = √0.00061875 = 0.0249 (2.49 percentage points).
• 1.96 × SE = 0.0488 → 95% CI = 0.401 to 0.499 → 40.1% to 49.9%. The block estimate is not significantly different from a State figure of 43%, but is clearly below 55%.

Example 3 - chi-square and odds ratio for a 2×2 table. Typhoid vaccination and disease over one season:

• Risk: 30/200 = 15% vs 60/200 = 30% → RR = 0.50, so vaccine efficacy = (1 − 0.50) × 100 = 50%.
• χ² = 400 × (30×140 − 170×60)² / (200 × 200 × 90 × 310) = 400 × (4,200 − 10,200)² / 1.116 × 10⁹ = 400 × 36,000,000 / 1.116 × 10⁹ = 12.90, df = 1.
• Critical χ² at α = 0.05, df 1 = 3.84 (6.63 at 0.01) → p < 0.001, a significant association.
• OR = (30 × 140)/(170 × 60) = 4,200/10,200 = 0.412; ln OR = −0.887; √(1/30 + 1/170 + 1/60 + 1/140) = √0.0630 = 0.251; 95% CI = exp(−0.887 ± 0.492) = 0.25 to 0.67 - excludes 1, consistent with the χ².

Example 4 - unpaired t-test. Haemoglobin after 3 months of IFA tablets: group A (n = 25) mean 10.2 g/dL, SD 1.5; group B (n = 25) mean 11.0 g/dL, SD 1.3.

• s_p² = [24 × 2.25 + 24 × 1.69]/48 = (54 + 40.56)/48 = 1.970 → s_p = 1.404 g/dL.
• SE of difference = 1.404 × √(1/25 + 1/25) = 1.404 × 0.2828 = 0.397 g/dL.
• t = (11.0 − 10.2)/0.397 = 2.02, df = 48; critical t = 2.011 → p ≈ 0.049, just significant.
• 95% CI of the difference = 0.80 ± 2.011 × 0.397 = 0.002 to 1.60 g/dL - the lower limit is almost zero, so the evidence is weak and the study is underpowered; this is exactly why the CI is reported with the p-value.

Example 5 - sample size. (a) Prevalence survey, expected prevalence 30%, absolute precision ±5%, 95% confidence:

• n = 1.96² × 0.30 × 0.70 / 0.05² = 3.8416 × 0.21 / 0.0025 = 0.8067/0.0025 = 322.7 → 323.
• Cluster design with DEFF = 2 → 646; allowing 10% non-response (646/0.90) → 718 women.

(b) To detect a 0.8 g/dL difference in Hb with σ = 1.4 g/dL, α = 0.05, power 80%:

• n = 2 × (1.96 + 0.84)² × 1.4² / 0.8² = 2 × 7.84 × 1.96 / 0.64 = 30.73/0.64 = 48.0 → 49 per group (98 in all) - more than the 25 per group actually studied in Example 4.

Choosing the test

Where it is used

• National surveys and programme monitoring: NFHS, SRS, NSSO and HMIS estimates with design effects and confidence intervals; district league tables under NHM and Aspirational Districts; coverage evaluation surveys for immunisation (U-WIN/eVIN data).
• Clinical research and CROs: sample size and randomisation plans, statistical analysis plans, interim analyses and CONSORT reporting for CTRI-registered trials - roles such as biostatistician, statistical programmer (SAS/R), clinical data manager at IQVIA, Parexel, Syneos, Novotech and hospital CRUs.
• Surveillance and outbreak analytics: IDSP/IHIP weekly counts modelled as Poisson, threshold and aberration detection, test-positivity trends - State and district surveillance units, NCDC, ICMR institutes.
• Hospital quality and HMIS analytics: control charts for infection rates, length-of-stay medians, Bland-Altman and kappa for instrument and observer agreement, NABH and NQAS indicator dashboards.
• Health economics, insurance and actuarial work: claim rates and cost distributions under Ayushman Bharat PM-JAY, utilisation and premium modelling (amounts in rupees), SHA/NHA analytics.
• NGO and donor M&E posts: baseline-endline comparisons, cluster-randomised evaluations and difference-in-differences at PHFI, Jhpiego, CARE India, IHAT and State Health Resource Centres; everyday tools are Excel, Epi Info, OpenEpi, SPSS, Stata and R.

Interview questions

• Q: What is the difference between standard deviation and standard error? A: SD measures the spread of individual observations around the mean; SE measures how precisely the sample mean estimates the population mean, and equals SD/√n, so it falls as the sample grows.
• Q: Define the p-value exactly. A: The probability of obtaining a result as extreme as, or more extreme than, the one observed, assuming the null hypothesis is true. It is not the probability that the null hypothesis is true, nor a measure of effect size.
• Q: How do you interpret a 95% confidence interval? A: It is a range calculated so that 95% of such intervals from repeated samples would contain the true parameter; if the interval for a difference contains zero (or for an RR/OR contains 1), the result is not significant at the 5% level.
• Q: When would you choose the median over the mean? A: For skewed data or data with outliers - length of hospital stay, income, parasite counts - and for ordinal data, because the median is not pulled by extreme values.
• Q: Which test compares haemoglobin before and after iron supplementation in the same 30 women? A: A paired t-test on the within-person differences (Wilcoxon signed-rank if the differences are clearly non-normal).
• Q: Why not run several t-tests instead of ANOVA for four groups? A: Each test carries its own 5% type I error, so the overall chance of a false positive rises well above 5%; ANOVA tests all means with one F test, followed by a corrected post-hoc comparison.
• Q: What are the assumptions of the chi-square test and what do you do when they fail? A: Independent observations, categorical data from counts, and expected frequency of at least 5 in most cells; with sparse cells use Fisher's exact test, or combine categories. Use McNemar for paired data.
• Q: What inputs do you need to compute a sample size? A: The expected prevalence or effect size with its variability, the precision or difference to be detected, α, power, the design effect if clustered, and allowances for non-response and loss to follow-up.
• Q: A study reports p = 0.001 for a 1 mmHg fall in systolic BP. Is it important? A: Statistically significant but clinically trivial - with a very large sample tiny differences reach significance, so the effect size and its confidence interval decide importance.
• Q: What is the central limit theorem and why does it matter in public health? A: The distribution of sample means approaches normality as n increases, whatever the population shape; it lets us use z- and t-based intervals and tests on skewed health data in reasonably large samples.
• Q: Difference between type I and type II error, and how do you reduce each? A: Type I is rejecting a true null (reduce by lowering α and avoiding multiple uncorrected tests); type II is missing a real effect (reduce by increasing sample size, improving measurement precision or accepting a larger α).
• Q: Why is cluster sampling less precise than simple random sampling of the same size? A: People within a cluster resemble each other, so each additional person adds less new information; the variance is inflated by the design effect and the sample size must be multiplied by it.

Common mistakes

• Dividing by n instead of (n − 1) for a sample SD, and quoting SE when the question asks for SD (or plotting SE bars and calling them SD).
• Treating "not significant" as proof of no difference, when the study may simply be underpowered - always check the confidence interval and the planned power.
• Using a chi-square test on percentages or on mean values instead of on actual observed counts, or applying it to paired data where McNemar is required.
• Applying parametric tests to ordinal scores or badly skewed small samples without checking normality, and using a one-tailed test only after seeing the direction of the result.
• Reading causation into a correlation, reporting r for a clearly curved relationship, or inferring individual risk from district-level averages (ecological fallacy).
• Omitting the design effect and non-response allowance from a cluster-survey sample size, then reporting an interval that is falsely narrow.
• Reporting only p-values with no effect size, denominator or confidence interval - and rounding or recycling figures without recomputing them from the raw data.

Related topics

Epidemiology: principles, measures and study designs · Demography and vital statistics · Health management information systems and surveillance · Research methodology and questionnaire design · Health economics and programme evaluation · Statistical software: Epi Info, SPSS and R.