The big ideas of medicine, with the fine print

Statement

With distinct event times $t_1<t_2<\cdots$, $d_j$ events at $t_j$ and $n_j$ patients still at risk just before $t_j$:

This is consistent when censoring is non-informative, meaning independent of the event time given covariates. Cox's proportional hazards model, $h(t\mid x)=h_0(t)e^{\beta^\top x}$, estimates $\beta$ from the partial likelihood without specifying $h_0$.

Why it's true

$S(t)=\prod\Pr(\text{survive interval}\mid\text{alive at its start})$. Each conditional probability is estimated only from patients actually observed during that interval. A patient censored at time $c$ contributes to every risk set up to $c$ and to none after. Dropping censored patients removes exactly the people who survived long enough to be censored, which biases survival down. Counting them as survivors to the end biases it up.

Statement

If the marker is $N(0,1)$ in non-diseased and $N(d',1)$ in diseased patients (the binormal model), then at threshold $c$, $\mathrm{Se}=\Phi(d'-c)$ and $1-\mathrm{Sp}=\Phi(-c)$. The area under the curve is

The threshold that minimizes expected cost is where the ROC slope equals $\frac{1-\pi}{\pi}\cdot\frac{C_{FP}}{C_{FN}}$, with $\pi$ the prevalence.

Why it's true

AUC is the Mann–Whitney probability: integrate $\mathrm{Se}(c)$ against the density of the non-diseased marker. For two normals, $X_{\text{dis}}-X_{\text{non}}\sim N(d',2)$. The optimal threshold follows from setting the derivative of the expected cost to zero: the slope of the ROC curve is the likelihood ratio at $c$.

Statement

Let true values be $T\sim N(\mu,\sigma_T^2)$ and measurements $X=T+e$ with $e\sim N(0,\sigma_e^2)$. With reliability $\rho=\sigma_T^2/(\sigma_T^2+\sigma_e^2)$,

so patients selected for $X_1>k$ show an expected fall of $(1-\rho)\big(\mathbb E[X_1\mid X_1>k]-\mu\big)$ at remeasurement, with no intervention at all.

Why it's true

$X_1$ and $X_2$ share $T$ but have independent errors. Selecting on a high $X_1$ selects partly for high $T$, which persists, and partly for positive $e_1$, which doesn't. The regression slope of $X_2$ on $X_1$ is $\operatorname{Cov}(X_1,X_2)/\operatorname{Var}X_1=\rho$.

Statement

Classify patients by ever receiving a treatment that starts at time $T_s$ after cohort entry, and compare outcomes from entry. The period $[0,T_s)$ is "immortal" for the treated: they can't die in it and still count as treated. Even with a hazard ratio of exactly 1, the naive comparison gives an apparent protective effect. Correct analyses treat exposure as time-varying: person-time before $T_s$ counts as unexposed. Alternatives are a landmark analysis or target-trial emulation with aligned time zero.

Why it's true

The naive approach moves the unexposed person-time $[0,T_s)$ of the eventually-treated into the treated group's denominator and removes it from the untreated group's. Meanwhile anyone who died before they could be treated is counted as untreated, adding their deaths to the comparison group. Both shifts favour treatment.

Statement

With $K$ equally spaced looks and standardized statistics $Z_k$, a naive procedure rejects if any $|Z_k|>1.96$. Its type I error is about 14% for $K=5$ (Armitage, McPherson & Rowe, 1969). The O'Brien–Fleming boundary rejects at look $k$ if

Pocock uses a constant boundary instead, 2.413 for $K=5$. The Lan–DeMets spending-function approach allows unplanned look times.

Why it's true

The sequence $Z_k\sqrt{k/K}$ is a discretely sampled Brownian motion, so the $Z_k$ are correlated with $\operatorname{Corr}(Z_j,Z_k)=\sqrt{j/k}$. Each additional look adds a new chance to cross 1.96. The constants are found by numerical integration (Armitage's recursion), so that the probability of crossing any boundary under $H_0$ equals $\alpha$.

Statement

With study estimates $y_i$ and variances $v_i$, the random-effects model $y_i\sim N(\theta,v_i+\tau^2)$ gives (DerSimonian–Laird)

Why it's true

Inverse-variance weighting is the minimum-variance unbiased combination of independent estimates of a common $\theta$. Cochran's $Q=\sum w_i(y_i-\bar y_w)^2$ has expectation $k-1$ under homogeneity, so the excess estimates $\tau^2$ by the method of moments. The funnel plot shows $y_i$ against precision. Without bias it's symmetric around $\theta$ and narrows as precision rises. Selective publication removes one lower corner, which Egger's regression tests for.

Statement

With elimination rate $\frac{V_{\max}C}{K_m+C}$ and dosing rate $R$, the steady state satisfies $R=\frac{V_{\max}C_{ss}}{K_m+C_{ss}}$, so

For phenytoin, typical adult values are around $V_{\max}\approx7$ mg/kg/day and $K_m\approx4$ mg/L, with a target range of 10–20 mg/L total concentration.

Why it's true

When $C\ll K_m$ elimination is first order, about $(V_{\max}/K_m)C$, and $C_{ss}$ is proportional to $R$. When $C\gg K_m$ it approaches the zero-order limit $V_{\max}$, and no steady state exists for $R\ge V_{\max}$. Phenytoin's therapeutic range lies above its $K_m$, right where the curve bends.

Statement

Why it's true

At equilibrium, agonist and antagonist compete for the same sites. The agonist's fractional occupancy becomes $\frac{C}{C+K_A(1+[B]/K_B)}$ (Gaddum's equation), which is the same curve with the apparent $K_A$ scaled by $1+[B]/K_B$. Enough agonist still occupies every receptor, so $E_{\max}$ is preserved. A non-competitive (allosteric or irreversible) antagonist removes receptors from play, which scales $E_{\max}$ down. With spare receptors, low doses of an irreversible antagonist first shift the curve and only later depress the maximum (Furchgott).

Statement

Model each case's offspring as negative binomial with mean $R$ and dispersion $k$ (variance $R(1+R/k)$). The probability $q$ that a single introduction goes extinct solves

For $R=3$: $k=\infty$ (Poisson) gives $q\approx0.06$, while $k=0.1$ gives $q\approx0.84$. For SARS in 2003, Lloyd-Smith et al. estimated $k\approx0.16$.

Why it's true

In a Galton–Watson branching process, the extinction probability is the smallest root of $q=G(q)$, where $G$ is the offspring probability generating function. For the negative binomial, $G(s)=(1+\frac Rk(1-s))^{-k}$. Small $k$ concentrates transmission: with $k=0.1$, roughly the top 10% of cases cause about 80% of infections. That's why contact tracing that looks backward to find the source cluster is efficient.

Statement

Venous return falls linearly as right atrial pressure rises toward the mean systemic filling pressure $P_{msf}$. Cardiac output rises with right atrial pressure (Frank–Starling):

Why it's true

$P_{msf}$, about 7 mmHg, is the pressure throughout the circulation when the heart stops, set by stressed volume and vascular compliance. Flow back to the heart is driven by the gradient from $P_{msf}$ to $P_{ra}$ across the venous resistance. Volume loading or venoconstriction raises $P_{msf}$ and shifts the VR curve right. Positive inotropes steepen the cardiac curve. The intersection gives the operating point. This framework explains why fluids help a preload-dependent heart and why inotropes alone do little when VR is limiting.

Statement

Surviving fraction after dose $d$ is $S=e^{-\alpha d-\beta d^2}$. For $n$ fractions of $d$ (total $D=nd$), the biologically effective dose is

Typical values are $\alpha/\beta\approx10$ Gy for many tumours and acutely responding tissue, about 3 Gy for late-responding normal tissue, and about 1.5 Gy for prostate cancer.

Why it's true

The linear term reflects lethal lesions from a single track. The quadratic term reflects pairs of sublethal lesions that interact before they are repaired. Between fractions, sublethal damage is repaired, so each fraction restarts the curve and the quadratic term counts only within a fraction. Tissues with low $\alpha/\beta$ have survival curves that bend more, so they are more sensitive to fraction size. Small fractions therefore protect late normal tissue relative to high-$\alpha/\beta$ tumours. But when a tumour's $\alpha/\beta$ is lower than that of the surrounding tissue, as in prostate cancer, larger fractions (hypofractionation) are favoured.

Statement

Skipper's log-kill hypothesis: a given dose kills the same fraction of sensitive cells regardless of tumour size. Tumour growth often follows the Gompertz law,

so the relative growth rate is highest when $N\ll K$. The Norton–Simon hypothesis: the regression produced by therapy is proportional to the growth rate of an untreated tumour of that size. Both point to dose density, shorter intervals between cycles, to give less time for fast regrowth.

Why it's true

In murine leukaemia (L1210), Skipper and Schabel found first-order kinetics of cell kill. A tumour detectable at about $10^9$ cells, roughly 1 cm³, needs about nine logs of kill to reach one cell, which is many cycles. Between cycles, Gompertzian regrowth from a reduced population is rapid, so the net reduction per cycle shrinks as intervals lengthen. CALGB 9741 found that two-weekly dose-dense adjuvant chemotherapy improved disease-free and overall survival in node-positive breast cancer compared with three-weekly dosing.