Sources and References

Textbook — Kestenbaum, B. (2019), Epidemiology and Biostatistics: An Introduction to Clinical Research Supplementary — Szklo and Nieto, Epidemiology: Beyond the Basics; Rothman, Epidemiology: An Introduction Online resources — Johns Hopkins Bloomberg School of Public Health OpenCourseWare; MIT OCW Epidemiology materials

Chapter 1: Introduction to Biostatistics

1.1 What Is Biostatistics?

Biostatistics is the application of statistical reasoning to problems in medicine, public health, and biology. It provides the quantitative backbone for evidence-based medicine: every claim about whether a drug works, whether a risk factor causes disease, or whether a screening test is accurate ultimately rests on biostatistical methods.

The field sits at the intersection of three disciplines. Epidemiology asks “who gets sick and why?” Statistics asks “how do we draw reliable conclusions from data?” Clinical medicine asks “what should we do for this patient?” Biostatistics supplies the formal tools that allow epidemiologists and clinicians to answer their questions with measured confidence.

1.2 Populations and Samples

A central distinction in all of statistics is between a population and a sample. In biostatistics, the population is the complete collection of individuals about whom we wish to draw conclusions. A sample is a subset of that population that we actually observe.

The validity of any biostatistical study depends on how well the sample represents the target population. Selection bias arises when the study population differs systematically from the target population in ways that affect the outcome of interest.

1.3 Types of Medical Data

Medical research generates several kinds of data, each requiring different statistical techniques.

Data Type	Description	Examples
Nominal (categorical)	Unordered categories	Blood type (A, B, AB, O); sex (male, female)
Ordinal	Ordered categories without equal spacing	Cancer stage (I, II, III, IV); pain scale (mild, moderate, severe)
Discrete numerical	Counts taking integer values	Number of hospitalizations; parity (number of live births)
Continuous numerical	Measurements on a continuous scale	Blood pressure (mmHg); serum cholesterol (mg/dL); BMI
Time-to-event (survival)	Time until an event occurs, possibly censored	Time from diagnosis to death; time from surgery to recurrence

Understanding the type of data is the first step in choosing an appropriate analysis. Nominal and ordinal data call for contingency-table methods and nonparametric tests; continuous data often permit parametric approaches such as t-tests and linear regression; time-to-event data require survival analysis techniques.

1.4 Descriptive Statistics

Before any formal inference, a researcher must describe the data. The two essential features of any distribution are its centre (location) and spread (variability).

Measures of Centre

The sample mean of \( n \) observations \( x_1, x_2, \ldots, x_n \) is

\[ \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i. \]

The sample median is the middle value when observations are ranked from smallest to largest. For an odd number of observations it is the \( \frac{n+1}{2} \)-th order statistic; for an even number it is the average of the two central values.

When the distribution is symmetric, the mean and median coincide. When the distribution is skewed (as is common with medical costs or hospital lengths of stay), the median is often more informative because it is resistant to extreme values.

Measures of Spread

The sample variance is

\[ s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2, \]

and the sample standard deviation is \( s = \sqrt{s^2} \). The interquartile range (IQR) is defined as \( Q_3 - Q_1 \), where \( Q_1 \) and \( Q_3 \) are the 25th and 75th percentiles respectively.

For skewed data, the IQR is preferred over the standard deviation for the same reason the median is preferred over the mean: it is robust to outliers.

1.5 Graphical Displays

Visual summaries are indispensable in biostatistics. The most commonly used displays include:

• Histograms and frequency polygons for continuous data, showing the shape of the distribution (symmetric, skewed, bimodal).
• Box plots, which display the median, IQR, and potential outliers, enabling rapid comparison across groups.
• Bar charts for categorical data, displaying the frequency or proportion in each category.
• Scatter plots for examining the relationship between two continuous variables (e.g., age versus systolic blood pressure).
• Kaplan-Meier curves for survival data, showing the estimated probability of surviving past each time point.

---

Chapter 2: Incidence, Prevalence, and Standardization of Rates

2.1 Measuring Disease Frequency

Quantifying how often disease occurs is the starting point of epidemiology. The two fundamental measures are prevalence and incidence.

Relationship Between Prevalence and Incidence

Under steady-state conditions (constant incidence rate \( I \) and constant average duration of disease \( D \)), prevalence and incidence are related by the approximation

\[ P \approx I \times D, \]

where \( P \) is the prevalence. This identity shows that prevalence can be high either because the disease is common (high incidence) or because it lasts a long time (high duration), or both.

2.2 Crude Rates

A crude rate is a summary measure computed for an entire population without adjusting for differences in the composition of the population (e.g., age, sex).

\[ \text{Crude death rate} = \frac{\text{Total number of deaths in a year}}{\text{Mid-year population}} \times 1{,}000 \]

Crude rates are simple to compute and directly reflect the actual experience of a population. However, they can be misleading when comparing populations with different age structures. A region with many elderly residents will have a higher crude death rate than a younger region, even if age-specific death rates are identical.

2.3 Direct Standardization

Direct standardization removes the confounding effect of age (or another variable) by computing the rate that each population would have experienced if it had the same age distribution as a chosen standard population.

The standard population may be a real population (e.g., the 2011 Canadian Census population, the WHO World Standard Population) or a combined total of the populations being compared.

Steps for Direct Standardization

1. Obtain age-specific rates \( r_i \) for each population being compared.
2. Choose a standard population and compute the proportion \( w_i \) in each age stratum.
3. For each population, multiply each age-specific rate by the corresponding weight and sum: \( \text{ASR} = \sum_i w_i \cdot r_i \).
4. Compare the adjusted rates across populations.

Advantages and Limitations

Direct standardization produces rates that are easy to interpret and compare. However, it requires reliable age-specific rates in each study population, which may be unstable when the study population is small or when events are rare in certain age groups.

2.4 Indirect Standardization and the SMR

When age-specific rates in the study population are unreliable (because the population is small), indirect standardization is used instead. Rather than applying the study population’s rates to a standard population, indirect standardization applies the standard population’s rates to the study population’s age structure.

When to Use Direct vs. Indirect Standardization

Feature	Direct	Indirect
Requires age-specific rates in study pop.	Yes	No
Produces an adjusted rate	Yes	No (produces a ratio, the SMR)
Suitable for small populations	No	Yes
Allows direct comparison between study populations	Yes	Not straightforward (different expected values)

---

Chapter 3: Medical Study Design

3.1 Overview of Study Designs

Epidemiological studies can be broadly classified as observational or experimental. In observational studies, the investigator merely observes exposures and outcomes without intervening. In experimental studies (clinical trials), the investigator assigns exposures (treatments) to participants.

The major observational designs, in order of increasing strength of evidence, are:

1. Cross-sectional studies
2. Case-control studies
3. Cohort studies (prospective and retrospective)

Each design involves trade-offs between cost, time, and the strength of causal inference it supports.

3.2 Cross-Sectional Studies

A cross-sectional study measures exposure and disease status simultaneously at a single point in time (or over a short period). It provides a snapshot of the population.

Strengths and Limitations

Cross-sectional studies are relatively quick and inexpensive. They are useful for estimating disease prevalence and for generating hypotheses. However, because exposure and disease are measured simultaneously, it is generally impossible to establish temporal sequence: we cannot determine whether the exposure preceded the disease or vice versa. This makes causal inference weak.

Measures of Association in Cross-Sectional Studies

The prevalence ratio (PR) and the prevalence odds ratio (POR) are used:

\[ \text{PR} = \frac{\text{Prevalence in exposed}}{\text{Prevalence in unexposed}}, \qquad \text{POR} = \frac{\text{Odds of disease in exposed}}{\text{Odds of disease in unexposed}}. \]

3.3 Cohort Studies

A cohort study follows a group of individuals over time to observe who develops the outcome of interest. Participants are classified by exposure status at baseline, and outcomes are ascertained during follow-up.

Measures of Association in Cohort Studies

The primary measures are the relative risk (risk ratio) and the risk difference.

3.4 Hypothesis Testing for 2x2 Tables

When comparing proportions across two groups in a 2x2 table, the chi-squared test is the standard approach.

For small samples (any expected cell count less than 5), Fisher’s exact test should be used instead.

Confidence Interval for the Relative Risk

A 95% confidence interval for \( \ln(\text{RR}) \) is

\[ \ln(\text{RR}) \pm 1.96 \sqrt{\frac{1}{a} - \frac{1}{a+b} + \frac{1}{c} - \frac{1}{c+d}}. \]

Exponentiating the endpoints gives the 95% CI for RR. If the interval does not contain 1, the association is statistically significant at the 5% level.

---

Chapter 4: Causation, Confounding, and Regression

4.1 Causation in Epidemiology

Establishing that an exposure causes a disease is one of the most important and difficult goals in epidemiology. Statistical association alone does not prove causation. In 1965, Sir Austin Bradford Hill proposed nine considerations (often called “criteria”) for evaluating whether an observed association is causal.

4.2 Confounding

Confounding occurs when an extraneous variable distorts the association between the exposure and the outcome.

Detecting Confounding

A practical rule of thumb: if adjusting for a potential confounder changes the crude measure of association by more than 10%, confounding is present. Formally, compare the crude (unadjusted) RR or OR to the stratum-specific or adjusted estimates. If they differ meaningfully, confounding was present.

4.3 Effect Modification (Interaction)

Effect modification occurs when the magnitude of the association between exposure and outcome differs across levels of a third variable. Unlike confounding, effect modification is a biological phenomenon of interest, not a bias to be eliminated.

Distinguishing Confounding from Effect Modification

Given stratum-specific estimates of the measure of association (e.g., \( \text{RR}_1 \) and \( \text{RR}_2 \) for two strata of a third variable):

• If \( \text{RR}_1 \approx \text{RR}_2 \) but both differ from the crude RR, then confounding is present (but not effect modification). Report the adjusted estimate.
• If \( \text{RR}_1 \neq \text{RR}_2 \), then effect modification is present. Report the stratum-specific estimates.

4.4 Stratified Analysis

Stratification is the most transparent method for dealing with confounding. The analysis is conducted separately within strata of the confounding variable, and then a summary adjusted estimate is computed.

4.5 Introduction to Multiple Linear Regression

Multiple linear regression provides a more flexible alternative to stratification for controlling confounding, especially when there are many confounders or when confounders are continuous.

In biostatistics, the key distinction is between crude and adjusted estimates:

• The crude estimate of the association between \( X_1 \) and \( Y \) comes from a simple regression of \( Y \) on \( X_1 \) alone.
• The adjusted estimate comes from a multiple regression that includes confounders \( X_2, \ldots, X_p \). The coefficient \( \beta_1 \) in the multiple regression represents the association between \( X_1 \) and \( Y \) after controlling for the other variables.

Model Assumptions and Diagnostics

The assumptions of multiple linear regression are:

1. Linearity: the relationship between each predictor and the outcome is linear.
2. Independence: observations are independent of one another.
3. Homoscedasticity: the variance of the residuals is constant across levels of the predictors.
4. Normality: the residuals are approximately normally distributed.

Residual plots (residuals vs. fitted values, Q-Q plots) are used to check these assumptions.

---

Chapter 5: Case-Control Studies

5.1 Design Principles

In a case-control study, the investigator starts by identifying individuals who have the disease (cases) and individuals who do not (controls), and then looks back in time to compare their past exposures.

Case-control studies are particularly efficient for studying rare diseases because they start by assembling a sufficient number of cases. A cohort study of a rare disease would require following an enormous population to observe enough cases.

Selection of Cases and Controls

• Cases should be representative of all cases in the target population (or a well-defined subset). Incident cases (newly diagnosed) are preferred over prevalent cases to avoid survival bias.
• Controls should come from the same source population that gave rise to the cases. If cases come from a hospital, controls should ideally come from the same catchment area (population-based controls) or from the same hospital (hospital-based controls, with appropriate exclusions).

5.2 The Odds Ratio

In a case-control study, we cannot directly compute the incidence of disease in exposed and unexposed groups (because we selected on disease status, not exposure). Instead, we compute the odds ratio.

5.3 Confidence Interval for the Odds Ratio

Since the sampling distribution of the OR is right-skewed, we work on the natural log scale where the distribution is approximately normal.

5.4 Mantel-Haenszel Adjusted Odds Ratio

When confounding is present, stratified analysis using the Mantel-Haenszel method provides an adjusted OR.

5.5 Matched Case-Control Studies

In a matched case-control study, each case is individually matched with one or more controls on potential confounders (e.g., age, sex). The most common design is 1:1 matching.

5.6 McNemar’s Test

McNemar’s test is used to test the null hypothesis that the matched OR equals 1 (i.e., no association between exposure and disease after accounting for matching).

5.7 Logistic Regression

When there are multiple confounders or when matching on many variables is impractical, logistic regression provides a flexible modelling approach for binary outcomes.

Advantages of Logistic Regression

1. It can adjust for many confounders simultaneously, including continuous variables.
2. It provides adjusted odds ratios with confidence intervals for each predictor.
3. It can test for interaction (effect modification) by including product terms.
4. It is the standard analytical method for case-control studies and for any study with a binary outcome.

---

Chapter 6: Randomized Controlled Trials

6.1 Design Principles

The randomized controlled trial (RCT) is considered the gold standard for evaluating the efficacy of medical interventions. By randomly assigning participants to treatment or control groups, the RCT eliminates confounding (both known and unknown) in expectation.

Key Design Elements

The fundamental elements of an RCT are:

1. A clearly defined study population with inclusion and exclusion criteria.
2. Random allocation of participants to intervention and control groups.
3. An intervention (new treatment, vaccine, behavioural program, etc.).
4. A comparison group (placebo, active comparator, or usual care).
5. Blinding (masking) of participants, investigators, and/or outcome assessors.
6. Objective outcome measurement.
7. Adequate follow-up.

6.2 Randomization

Randomization is the defining feature of the RCT. It serves two purposes: (1) it eliminates selection bias in the assignment of treatments, and (2) in large trials, it tends to balance both known and unknown confounders between groups.

Common randomization methods include:

• Simple randomization: each participant is assigned by an independent random mechanism (like a coin flip), giving equal probability to each group.
• Block randomization: participants are randomized within blocks of a fixed size to ensure that the groups remain balanced throughout the trial.
• Stratified randomization: randomization is performed within strata defined by important prognostic factors (e.g., age group, disease severity) to ensure balance on these factors.

6.3 Blinding

Blinding (masking) prevents knowledge of treatment assignment from influencing the assessment of outcomes.

Level	Who is blinded	Purpose
Single-blind	Participant only	Prevents placebo effect and reporting bias
Double-blind	Participant and investigator	Additionally prevents observer bias
Triple-blind	Participant, investigator, and data analyst	Additionally prevents analysis bias

Double-blinding is the standard for drug trials. It is not always possible (e.g., surgical vs. medical treatment), in which case blinded outcome assessment is used.

6.4 Intention-to-Treat vs. Per-Protocol Analysis

6.5 Crossover Designs

Advantages of crossover designs include:

• Elimination of between-subject variability, since each subject is compared to themselves.
• Smaller sample sizes are typically needed.

Limitations include:

• A sufficient washout period is required to eliminate carryover effects.
• The design is inappropriate for conditions that change over time or for treatments with permanent effects.
• Period effects (the natural course of disease changing over time) can bias results.

6.6 Measures of Treatment Effect in RCTs

Several measures quantify the treatment effect:

6.7 Vaccine Efficacy and Infectious Disease Epidemiology

RCTs are the standard method for evaluating vaccines. Vaccine efficacy (VE) measures the proportionate reduction in disease incidence among vaccinated individuals compared to unvaccinated individuals.

Basic Concepts in Infectious Disease Epidemiology

---

Chapter 7: Diagnostic Testing and Screening

7.1 Sensitivity and Specificity

Diagnostic tests are evaluated by their ability to correctly identify those with and without disease. The two fundamental properties are sensitivity and specificity.

Sensitivity and specificity are intrinsic properties of the test and do not depend on the prevalence of disease in the population being tested. A highly sensitive test is good for ruling out disease (mnemonic: SnNOut – Sensitivity, Negative result, rules Out). A highly specific test is good for ruling in disease (mnemonic: SpPIn – Specificity, Positive result, rules In).

7.2 Predictive Values

While sensitivity and specificity describe the test’s performance from the perspective of the disease status, clinicians care about the reverse question: given a test result, what is the probability the patient has the disease?

The Effect of Prevalence on Predictive Values

Unlike sensitivity and specificity, predictive values depend heavily on the prevalence of disease. Using Bayes’ theorem:

\[ \text{PPV} = \frac{\text{Sensitivity} \times \text{Prevalence}}{\text{Sensitivity} \times \text{Prevalence} + (1 - \text{Specificity}) \times (1 - \text{Prevalence})} \]\[ \text{NPV} = \frac{\text{Specificity} \times (1 - \text{Prevalence})}{\text{Specificity} \times (1 - \text{Prevalence}) + (1 - \text{Sensitivity}) \times \text{Prevalence}} \]

7.3 Likelihood Ratios

Likelihood ratios provide an alternative way to express test performance that is independent of prevalence and directly links pre-test to post-test probability.

Likelihood ratios are used with pre-test odds to compute post-test odds:

\[ \text{Post-test odds} = \text{Pre-test odds} \times \text{LR} \]

where Pre-test odds \( = \text{Prevalence} / (1 - \text{Prevalence}) \). A test with LR+ greater than 10 or LR– less than 0.1 is generally considered clinically useful.

7.4 ROC Curves

Many diagnostic tests produce a continuous measurement (e.g., blood glucose level, tumour marker concentration) rather than a simple positive/negative result. Different cutoff values produce different combinations of sensitivity and specificity.

Interpreting ROC Curves

• A test with no discriminatory power produces a diagonal line from (0, 0) to (1, 1), representing random guessing.
• A perfect test produces a curve that passes through the point (0, 1), indicating 100% sensitivity and 100% specificity.
• The closer the curve hugs the upper-left corner, the better the test.

Area Under the Curve (AUC)

Choosing the Optimal Cutoff

The choice of cutoff depends on the clinical context:

• When the cost of missing a disease is high (e.g., cancer screening), choose a cutoff that maximizes sensitivity, accepting lower specificity.
• When the cost of a false positive is high (e.g., confirmatory tests for HIV), choose a cutoff that maximizes specificity.
• A common approach is to choose the cutoff that maximizes the Youden index: \( J = \text{Sensitivity} + \text{Specificity} - 1 \).

---

Chapter 8: Survival Analysis

8.1 Introduction to Survival Data

Survival analysis deals with time-to-event data: the time from a defined starting point (e.g., diagnosis, enrollment in a study, start of treatment) until the occurrence of an event of interest (e.g., death, disease recurrence, hospital readmission).

The key feature that distinguishes survival data from ordinary continuous data is censoring.

Standard statistical methods (e.g., comparing mean times using a t-test) cannot be directly applied to survival data because censored observations are not simply “missing data” – they provide partial information (the individual survived at least this long).

Survival and Hazard Functions

8.2 Period Life Tables

Life tables are among the oldest statistical tools, used for centuries by actuaries and demographers. A period life table describes the mortality experience of a population during a fixed time period.

Constructing a Period Life Table

The standard columns of a period life table are:

Column	Symbol	Description
Age interval	\( [x, x+n) \)	Age range (e.g., 0–1, 1–5, 5–10, …)
Mortality rate	\( {}_nM_x \)	Observed death rate in the interval
Probability of dying	\( {}_nq_x \)	Probability of dying during the interval, given alive at start
Number surviving	\( l_x \)	Number alive at exact age \( x \) (starting from a radix, e.g., \( l_0 = 100{,}000 \))
Deaths	\( {}_nd_x \)	Number dying during the interval: \( {}_nd_x = l_x \cdot {}_nq_x \)
Person-years lived	\( {}_nL_x \)	Total person-years lived in the interval
Total person-years remaining	\( T_x \)	Total person-years lived from age \( x \) onward: \( T_x = \sum_{y \geq x} {}_nL_y \)
Life expectancy	\( e_x \)	Expected remaining years of life at age \( x \): \( e_x = T_x / l_x \)

The conversion from the observed mortality rate to the probability of dying within the interval is:

[

]

where \( {}_na_x \) is the average number of years lived in the interval by those who die during it (often approximated as \( n/2 \) for most age groups).

8.3 Clinical (Actuarial) Life Tables

Clinical life tables adapt the period life table methodology to clinical data, where follow-up times vary and censoring (withdrawal) occurs.

8.4 The Kaplan-Meier Estimator

The Kaplan-Meier (KM) estimator, introduced by Edward Kaplan and Paul Meier in 1958, is the most widely used method for estimating the survival function. Unlike the actuarial life table, which uses fixed intervals, the KM estimator recalculates the survival probability at each observed event time.

The resulting KM curve is a step function that decreases at each event time and remains flat between events. Censored observations are typically marked with tick marks on the curve.

Properties of the KM Estimator

• It is a nonparametric method: no assumptions about the underlying distribution of survival times.
• It provides a consistent estimate of the survival function.
• It handles censoring naturally, using the number at risk at each event time.
• Confidence intervals can be constructed using Greenwood’s formula for the variance of \( \hat{S}(t) \).

8.5 The Log-Rank Test

The log-rank test is the standard method for comparing survival curves between two or more groups.

8.6 Introduction to the Cox Proportional Hazards Model

While the log-rank test compares survival curves between groups, it cannot adjust for multiple covariates simultaneously. The Cox proportional hazards (PH) model, introduced by Sir David Cox in 1972, is the most widely used regression model for survival data.

Interpreting Hazard Ratios

The hazard ratio (HR) for a one-unit increase in \( X_j \), holding other covariates constant, is:

\[ \text{HR} = e^{\beta_j} \]

• HR = 1: no effect of the covariate on survival
• HR > 1: the covariate is associated with increased hazard (worse survival)
• HR < 1: the covariate is associated with decreased hazard (better survival)

The Proportional Hazards Assumption

The fundamental assumption is that the hazard ratio between any two individuals is constant over time. That is, the covariate effects multiply the baseline hazard by a fixed factor at all time points. This can be checked by:

1. Plotting \( \log[-\log \hat{S}(t)] \) vs. \( \log(t) \) for each group: the curves should be approximately parallel.
2. Including time-by-covariate interaction terms in the model: if these are statistically significant, the PH assumption is violated.
3. Examining Schoenfeld residuals: a systematic trend in these residuals over time indicates non-proportional hazards.

---

Summary of Key Formulas

Disease Frequency Measures

Measure	Formula
Prevalence	\( \frac{\text{Existing cases}}{\text{Total population}} \)
Incidence rate	\( \frac{\text{New cases}}{\text{Person-time at risk}} \)
Cumulative incidence	\( \frac{\text{New cases}}{\text{Population at risk at start}} \)

Measures of Association

Measure	Formula	Study design
Relative Risk (RR)	\( \frac{a/(a+b)}{c/(c+d)} \)	Cohort, RCT
Odds Ratio (OR)	\( \frac{ad}{bc} \)	Case-control, cross-sectional
Risk Difference (RD)	\( \frac{a}{a+b} - \frac{c}{c+d} \)	Cohort, RCT
NNT	\( \frac{1}{\text{ARR}} \)	RCT

Diagnostic Testing

Measure	Formula
Sensitivity	\( \frac{\text{TP}}{\text{TP} + \text{FN}} \)
Specificity	\( \frac{\text{TN}}{\text{FP} + \text{TN}} \)
PPV	\( \frac{\text{TP}}{\text{TP} + \text{FP}} \)
NPV	\( \frac{\text{TN}}{\text{FN} + \text{TN}} \)
LR+	\( \frac{\text{Sensitivity}}{1 - \text{Specificity}} \)
LR–	\( \frac{1 - \text{Sensitivity}}{\text{Specificity}} \)

Standardization

Method	Key quantity
Direct standardization	\( \text{ASR} = \sum_i w_i \cdot r_i \)
Indirect standardization	\( \text{SMR} = \frac{O}{E} \times 100 \)

Survival Analysis

Method	Formula
Kaplan-Meier	\( \hat{S}(t) = \prod_{j: t_j \leq t} \left(1 - \frac{d_j}{n_j}\right) \)
Cox PH model	\( h(t \mid \mathbf{X}) = h_0(t) \exp(\boldsymbol{\beta}^T \mathbf{X}) \)
Hazard ratio	\( \text{HR} = e^{\beta_j} \)