Exam Formula Sheet
1 Epi 202: Probability
\[ \begin{aligned} \operatorname{Var}\mathopen{}\left(\tilde{a}\cdot \tilde{X}\right)\mathclose{} &= \operatorname{Var}\mathopen{}\left(\sum_{i=1}^na_i X_i\right)\mathclose{} \\ &= \tilde{a}^{\top} \operatorname{Var}\mathopen{}\left(\tilde{X}\right)\mathclose{} \tilde{a} \\ &= \sum_{i=1}^n\sum_{j=1}^n a_i a_j \operatorname{Cov}\mathopen{}\left(X_i,X_j\right)\mathclose{} \end{aligned} \]
\[\operatorname{E}\mathopen{}\left[Y\right]\mathclose{} = \operatorname{E}\mathopen{}\left[\operatorname{E}\mathopen{}\left[Y \mid X\right]\mathclose{}\right]\mathclose{}\]
\[\operatorname{E}\mathopen{}\left[Y \mid Z\right]\mathclose{} = \operatorname{E}\mathopen{}\left[\operatorname{E}\mathopen{}\left[Y \mid X,Z\right]\mathclose{} \mid Z\right]\mathclose{}\]
\[\operatorname{Var}\mathopen{}\left(Y\right)\mathclose{} = \operatorname{E}\mathopen{}\left[\operatorname{Var}\mathopen{}\left(Y \mid X\right)\mathclose{}\right]\mathclose{} + \operatorname{Var}\mathopen{}\left(\operatorname{E}\mathopen{}\left[Y \mid X\right]\mathclose{}\right)\mathclose{}\]
\[\operatorname{Cov}\mathopen{}\left(Y,Z\right)\mathclose{} = \operatorname{E}\mathopen{}\left[\operatorname{Cov}\mathopen{}\left(Y,Z \mid X\right)\mathclose{}\right]\mathclose{} + \operatorname{Cov}\mathopen{}\left(\operatorname{E}\mathopen{}\left[Y \mid X\right]\mathclose{}, \operatorname{E}\mathopen{}\left[Z \mid X\right]\mathclose{}\right)\mathclose{}\]
2 Epi 203: Statistical inference
\[\mathscr{L}(\theta) \stackrel{\text{def}}{=}\operatorname{p}(\tilde{X}= \tilde{x}| \Theta = \theta)\]
\[\ell\stackrel{\text{def}}{=}\operatorname{log}\mathopen{}\left\{\mathscr{L}(\tilde{x}|\theta)\right\}\mathclose{}\]
\[\ell'\stackrel{\text{def}}{=}\frac{\partial}{\partial \theta} \ell(\tilde{x}|\theta)\]
\[\ell''\stackrel{\text{def}}{=}\frac{\partial}{\partial \tilde{\theta}}\frac{\partial}{\partial \tilde{\theta}^{\top}} \ell(\tilde{x}| \tilde{\theta})\]
\[\ell_{ij}''= \frac{\partial}{\partial \theta_i}\frac{\partial}{\partial \theta_j} \ell(\tilde{X}= \tilde{x}| \tilde{\theta})\]
\[I\stackrel{\text{def}}{=}-\ell''(\tilde{x}|\tilde{\theta})\] \[\mathcal{I}\stackrel{\text{def}}{=}\operatorname{E}\mathopen{}\left[I(\tilde{x}|\theta)\right]\mathclose{}\]
\[\hat\theta_{ML}\ \dot \sim\ \operatorname{N}\mathopen{}\left(\theta,\mathopen{}\left[\mathcal{I}(\tilde{\theta})\right]\mathclose{}^{-1}\right)\mathclose{}\]
For one parameter \(\theta_k\):
\[ \text{CI}_{1-\alpha}(\theta_k) = \left[ \hat\theta_k \pm z_{1 - \frac{\alpha}{2}} \mathop{\widehat{\operatorname{SE}}}\nolimits\mathopen{}\left(\hat\theta_k\right)\mathclose{} \right] \]
\[ Z_k \stackrel{\text{def}}{=} \frac{\hat\theta_k-\theta_{k,0}}{\mathop{\widehat{\operatorname{SE}}}\nolimits\mathopen{}\left(\hat\theta_k\right)\mathclose{}} \ \dot{\sim} \ \operatorname{N}\mathopen{}\left(0,1\right)\mathclose{} \qquad \text{under }H_0:\theta_k=\theta_{k,0} \qquad z_k\text{ = observed }Z_k \]
\[ \begin{aligned} p\text{-value} &= 2\Pr \mathopen{}\left(\mathopen{}\left|Z\right|\mathclose{}\ge \mathopen{}\left|z_k\right|\mathclose{}\right)\mathclose{},\quad Z\sim \operatorname{N}\mathopen{}\left(0,1\right)\mathclose{} \\ &= 2\mathopen{}\left(1-\Phi\mathopen{}\left(\mathopen{}\left|z_k\right|\mathclose{}\right)\mathclose{}\right)\mathclose{} \end{aligned} \]
3 Sta 108: Linear regression
\[ t_k \stackrel{\text{def}}{=} \frac{\hat \beta_k-\beta_{k,0}}{\mathop{\widehat{\operatorname{SE}}}\nolimits\mathopen{}\left(\hat \beta_k\right)\mathclose{}} \ \dot{\sim} \ t_{n-p} \qquad \text{under }H_0:\beta_k=\beta_{k,0} \qquad t_k^{\text{obs}}\text{ = observed }t_k \]
\[ \text{CI}_{1-\alpha}(\beta_k) = \left[ \hat \beta_k \pm t_{n-p}\mathopen{}\left(1-\frac{\alpha}{2}\right)\mathclose{} \mathop{\widehat{\operatorname{SE}}}\nolimits\mathopen{}\left(\hat \beta_k\right)\mathclose{} \right] \]
Let \(T_{n-p}\sim t_{n-p}\).
\[ \begin{aligned} p\text{-value} &= 2\Pr \mathopen{}\left(\mathopen{}\left|T_{n-p}\right|\mathclose{}\ge \mathopen{}\left|t_k^{\text{obs}}\right|\mathclose{}\right)\mathclose{} \\ &= 2\mathopen{}\left(1-F_{t_{n-p}}\mathopen{}\left(\mathopen{}\left|t_k^{\text{obs}}\right|\mathclose{}\right)\mathclose{}\right)\mathclose{} \end{aligned} \]
\[ \text{CI}_{1-\alpha}\mathopen{}\left(\mu\mathopen{}\left({\tilde{x}^*}\right)\mathclose{}\right)\mathclose{} = \left[ \hat{\mu}\mathopen{}\left({\tilde{x}^*}\right)\mathclose{} \pm t_{n-p}\mathopen{}\left(1-\frac{\alpha}{2}\right)\mathclose{} \mathop{\widehat{\operatorname{SE}}}\nolimits\mathopen{}\left(\hat{\mu}\mathopen{}\left({\tilde{x}^*}\right)\mathclose{}\right)\mathclose{} \right] \]
\(Y^*\) denotes a new observation (not in the training data), with corresponding covariate pattern \({\tilde{x}^*}\). Let \(\hat{Y}^* \stackrel{\text{def}}{=}\hat{\mu}\mathopen{}\left({\tilde{x}^*}\right)\mathclose{}\).
\[ \text{PI}_{1-\alpha}\mathopen{}\left(Y^*|{\tilde{x}^*}\right)\mathclose{} = \left[ \hat{Y}^* \pm t_{n-p}\mathopen{}\left(1-\frac{\alpha}{2}\right)\mathclose{} \mathop{\widehat{\operatorname{SE}}}\nolimits\mathopen{}\left(Y^*-\hat{Y}^*\right)\mathclose{} \right] \]
\[ \mathop{\widehat{\operatorname{SE}}}\nolimits\mathopen{}\left(Y^*-\hat{Y}^*\right)\mathclose{} = \hat \sigma\sqrt{1 + {({\tilde{x}^*})}^{\top}(\mathbf{X}'\mathbf{X})^{-1}{\tilde{x}^*}} \]
\[ \operatorname{Var}\mathopen{}\left(Y^* - \hat{Y}^*\right)\mathclose{} = \sigma^2\mathopen{}\left(1 + {({\tilde{x}^*})}^{\top}(\mathbf{X}'\mathbf{X})^{-1}{\tilde{x}^*}\right)\mathclose{} \]
Let \(\hat{\Sigma} \stackrel{\text{def}}{=}\mathop{\widehat{\operatorname{Var}}}\nolimits\mathopen{}\left(\hat \beta\right)\mathclose{} = \hat \sigma^2(\mathbf{X}'\mathbf{X})^{-1}\).
\[ \mathop{\widehat{\operatorname{Var}}}\nolimits\mathopen{}\left(\hat{\mu}\mathopen{}\left(\tilde{x}\right)\mathclose{}\right)\mathclose{} = {\tilde{x}}^{\top}\hat{\Sigma}\tilde{x} \]
Let \(\Delta\mu\mathopen{}\left(\tilde{x}, {\tilde{x}^*}\right)\mathclose{} = \mu\mathopen{}\left(\tilde{x}\right)\mathclose{} - \mu\mathopen{}\left({\tilde{x}^*}\right)\mathclose{}\), and let \(\Delta\tilde{x}= \tilde{x}- {\tilde{x}^*}\); then:
\[ \mathop{\widehat{\operatorname{Var}}}\nolimits\mathopen{}\left(\widehat{\Delta\mu}\mathopen{}\left(\tilde{x}, {\tilde{x}^*}\right)\mathclose{}\right)\mathclose{} = {\Delta\tilde{x}}^{\top}\hat{\Sigma}\Delta\tilde{x} \]
4 Epi 204: Generalized linear models
Generalized linear models have three components:
The outcome distribution family: \(\operatorname{p}(Y|\mu(\tilde{x}))\)
The link function: \(g(\mu(\tilde{x})) = \eta(\tilde{x})\)
The linear component: \(\eta(\tilde{x}) = \tilde{x}\cdot \beta\)
\[\theta_{\omega}(\tilde{x},{\tilde{x}^*}) = \operatorname{exp}\mathopen{}\left\{(\Delta\tilde{x}) \cdot \tilde{\beta}\right\}\mathclose{}\]
4.1 Estimates of odds ratios from 2x2 contingency tables
\[\hat\theta=\frac{ad}{bc}\]
Summary of logistic regression definitions and results
Odds and log-odds
Odds \[\omega\stackrel{\text{def}}{=}\frac{\Pr(A)}{\Pr(\neg A)}\]
Conditional odds \[\omega(A|B) \stackrel{\text{def}}{=}\frac{\Pr(A|B)}{\Pr(\neg A|B)}\]
Odds function \[\operatorname{odds}\mathopen{}\left\{\pi\right\}\mathclose{} \stackrel{\text{def}}{=}\frac{\pi}{1-\pi}\]
Probability to odds \[\omega= \frac{\pi}{1-\pi}\]
Odds function equals odds \[\omega= \operatorname{odds}\mathopen{}\left\{\pi\right\}\mathclose{}\]
Simplified odds expressions \[\operatorname{odds}\mathopen{}\left\{\pi\right\}\mathclose{} = \frac{1}{\pi^{-1}-1} = \mathopen{}\left(\pi^{-1}-1\right)^{-1}\mathclose{}\]
Odds of a non-event \[\omega(\neg A) = \frac{1-\pi}{\pi} = \pi^{-1}-1\]
Odds ratio \[\theta(\omega_1, \omega_2) \stackrel{\text{def}}{=}\frac{\omega_1}{\omega_2}\]
OR as ratio of probability ratios \[\begin{aligned} \theta(\omega_1, \omega_2) &= \frac{\omega_1}{\omega_2} \\ &= \frac{\mathopen{}\left(\frac{\pi_1}{1-\pi_1}\right)\mathclose{}}{\mathopen{}\left(\frac{\pi_2}{1-\pi_2}\right)\mathclose{}} \end{aligned}\]
Odds ratios are reversible \[\theta_{\omega}(A|B) = \theta_{\omega}(B|A)\]
Conditional ORs are reversible \[\theta_{\omega}(A|B,C) = \theta_{\omega}(B|A,C)\]
Inverse-odds and probability recovery
Odds to probability \[\pi = \frac{\omega}{1+\omega}\]
Inverse-odds function \[\operatorname{invodds}\mathopen{}\left\{\omega\right\}\mathclose{} \stackrel{\text{def}}{=}\frac{\omega}{1 + \omega}\]
Probability as inverse-odds \[\pi= \operatorname{invodds}\mathopen{}\left\{\omega\right\}\mathclose{}\]
Simplified inverse-odds \[\operatorname{invodds}\mathopen{}\left\{\omega\right\}\mathclose{} = \frac{1}{1+\omega^{-1}} = (1+\omega^{-1})^{-1}\]
One minus inverse-odds \[1 - \pi= \frac{1}{1+\omega}\]
Complement of inverse-odds \[1+\omega= \frac{1}{1-\pi}\]
Log-odds (logit) and expit
Log-odds \[\eta\stackrel{\text{def}}{=}\operatorname{log}\mathopen{}\left\{\omega\right\}\mathclose{}\]
Log-odds from probability \[\eta= \operatorname{log}\mathopen{}\left\{\frac{\pi}{1-\pi}\right\}\mathclose{}\]
Logit function \[\operatorname{logit}(\pi) \stackrel{\text{def}}{=}\operatorname{log}\mathopen{}\left\{\operatorname{odds}\mathopen{}\left\{\pi\right\}\mathclose{}\right\}\mathclose{}\]
Logit expanded \[\operatorname{logit}(\pi) = \operatorname{log}\mathopen{}\left\{\frac{\pi}{1-\pi}\right\}\mathclose{}\]
Log-odds equals logit \[\eta= \operatorname{logit}\mathopen{}\left\{\pi\right\}\mathclose{}\]
Odds from log-odds \[\omega= \operatorname{exp}\mathopen{}\left\{\eta\right\}\mathclose{}\]
Probability from log-odds \[\pi= \frac{\operatorname{exp}\mathopen{}\left\{\eta\right\}\mathclose{}}{1+\operatorname{exp}\mathopen{}\left\{\eta\right\}\mathclose{}}\]
Expit / inverse-logit function \[\operatorname{expit}(\eta) \stackrel{\text{def}}{=}\operatorname{invodds}\mathopen{}\left\{\operatorname{exp}\mathopen{}\left\{\eta\right\}\mathclose{}\right\}\mathclose{}\]
Expit expressions \[\operatorname{expit}(\eta) = \frac{\operatorname{exp}\mathopen{}\left\{\eta\right\}\mathclose{}}{1+\operatorname{exp}\mathopen{}\left\{\eta\right\}\mathclose{}} = (1 + \operatorname{exp}\mathopen{}\left\{-\eta\right\}\mathclose{})^{-1}\]
Probability as expit \[\pi= \operatorname{expit}\mathopen{}\left\{\eta\right\}\mathclose{} \tag{1}\]
Logit and expit are inverses \[\operatorname{logit}\mathopen{}\left\{\operatorname{expit}\mathopen{}\left\{\eta\right\}\mathclose{}\right\}\mathclose{} = \eta\qquad \operatorname{expit}\mathopen{}\left\{\operatorname{logit}\mathopen{}\left\{\pi\right\}\mathclose{}\right\}\mathclose{} = \pi\]
\[ \underbrace{\pi}_{\Pr\mathopen{}\left(Y=1\right)\mathclose{}} \overbrace{ \underbrace{ \underset{ \xleftarrow[\frac{\omega}{1+\omega}]{} } { \xrightarrow{\frac{\pi}{1-\pi}} } \underbrace{\omega}_{\operatorname{odds}\mathopen{}\left\{Y=1\right\}\mathclose{}} \underset{ \xleftarrow[\operatorname{exp}\mathopen{}\left\{\eta\right\}\mathclose{}]{} } { \xrightarrow{\operatorname{log}\mathopen{}\left\{\omega\right\}\mathclose{}} } }_{\operatorname{expit}\mathopen{}\left\{\eta\right\}\mathclose{}} }^{\operatorname{logit}\mathopen{}\left\{\pi\right\}\mathclose{}} \underbrace{\eta}_{\text{log-odds}(Y=1)} \]
Rare events
Odds minus probability \[\omega- \pi = \frac{\pi^2}{1-\pi}, \quad \text{where } \omega= \frac{\pi}{1-\pi}\]
Derivatives
\[ \begin{array}{r|ccccc} & \pi& \omega& \eta& \tilde{x}& \tilde{\beta}\\ \hline \pi & 1 & \mathopen{}\left(1+\omega\right)^2\mathclose{} & \dfrac{\mathopen{}\left(1+\omega\right)^2\mathclose{}}{\omega} & \text{undef} & \text{undef} \\[10pt] \omega & \mathopen{}\left(1-\pi\right)^2\mathclose{} & 1 & \dfrac{1}{\omega} & \text{undef} & \text{undef} \\[10pt] \eta & \pi(1-\pi) & \omega & 1 & \text{undef} & \text{undef} \\[10pt] \tilde{x} & \tilde{\beta}\pi(1-\pi) & \tilde{\beta}\omega & \tilde{\beta} & \mathbb{I} & \mathbf{0} \\[10pt] \tilde{\beta} & \tilde{x}\pi(1-\pi) & \tilde{x}\omega & \tilde{x} & \mathbf{0} & \mathbb{I}\\ \end{array} \]
Column labels indicate the numerators of the derivatives; row labels indicate the denominators.
Log-likelihood and score function
Log-likelihood component \[\ell_i(\pi_i) = y_i \eta_i - \operatorname{log}\mathopen{}\left\{1+\omega_i\right\}\mathclose{}\]
Score function as sum \[\tilde{\ell'}(\tilde{\beta}) = \sum_{i=1}^n\tilde{\ell'_i}(\tilde{\beta})\]
Score component \[\ell_i'(\tilde{\beta}) = \tilde{x}_i e_i\]
Score function \[\tilde{\ell'}(\tilde{\beta}) = \sum_{i=1}^n\tilde{x}_ie_i = {\mathbf{X}}^{\top}\tilde{e}\]
MLE
One-sample MLE for odds \[\hat{\omega}= \frac{x}{n-x}\]
Odds ratios in logistic regression
General OR formula \[\theta_{\omega}(\tilde{x},{\tilde{x}^*}) = \operatorname{exp}\mathopen{}\left\{\eta(\tilde{x}) - \eta({\tilde{x}^*})\right\}\mathclose{}\]
Difference in log-odds \[\Delta\eta\stackrel{\text{def}}{=}\eta(\tilde{x}) - \eta({\tilde{x}^*})\]
OR in terms of \(\Delta\eta\) \[\theta_{\omega}(\tilde{x},{\tilde{x}^*}) = \operatorname{exp}\mathopen{}\left\{\Delta\eta\right\}\mathclose{}\]
\(\Delta\eta\) from covariates \[\Delta\eta= (\tilde{x}- {\tilde{x}^*})\cdot\tilde{\beta}\]
Difference in covariate patterns \[\Delta\tilde{x}\stackrel{\text{def}}{=}\tilde{x}- {\tilde{x}^*}\]
\(\Delta\eta\) from \(\Delta\tilde{x}\) \[\Delta\eta= \Delta\tilde{x}\cdot \tilde{\beta}\]
OR in terms of \(\Delta\tilde{x}\) \[\theta_{\omega}(\tilde{x},{\tilde{x}^*}) = \operatorname{exp}\mathopen{}\left\{(\Delta\tilde{x}) \cdot \tilde{\beta}\right\}\mathclose{}\]
Log OR equals \(\Delta\eta\) \[\operatorname{log}\mathopen{}\left\{\theta_{\omega}(\tilde{x},{\tilde{x}^*})\right\}\mathclose{} = \Delta\eta\]
Inference for log-odds and odds ratios
Estimated SE of log-odds \[\begin{aligned} \mathop{\widehat{\operatorname{Var}}}\nolimits\mathopen{}\left(\hat\eta(\tilde{x})\right)\mathclose{} &= {\tilde{x}}^{\top}\hat{\Sigma}\tilde{x}\\ \mathop{\widehat{\operatorname{SE}}}\nolimits\mathopen{}\left(\hat\eta(\tilde{x})\right)\mathclose{} &= \sqrt{{\tilde{x}}^{\top}\hat{\Sigma}\tilde{x}} \end{aligned}\]
Estimated SE of \(\Delta\widehat{\eta}\) \[\begin{aligned} \mathop{\widehat{\operatorname{Var}}}\nolimits\mathopen{}\left(\Delta{\hat\eta}\right)\mathclose{} &= {\Delta\tilde{x}}^{\top}\hat{\Sigma}(\Delta\tilde{x}) \\ \mathop{\widehat{\operatorname{SE}}}\nolimits\mathopen{}\left(\Delta{\hat\eta}\right)\mathclose{} &= \sqrt{{\Delta\tilde{x}}^{\top}\hat{\Sigma}(\Delta\tilde{x})} \end{aligned}\]
Comparing probabilities
Risk difference \[\delta(\pi_1,\pi_2) \stackrel{\text{def}}{=}\pi_1 - \pi_2\]
Risk ratio \[\rho(\pi_1,\pi_2) \stackrel{\text{def}}{=}\frac{\pi_1}{\pi_2}\]
Relative risk difference \[\xi(\pi_1,\pi_2) \stackrel{\text{def}}{=}\frac{\delta(\pi_1,\pi_2)}{\pi_2}\]
RRD equals RR minus 1 \[\xi(\pi_1,\pi_2) = \rho(\pi_1,\pi_2) - 1\]
Logistic regression model
Logistic regression \[Y_i|\tilde{X}_i \ \sim_{\perp\!\!\!\perp}\ \operatorname{Ber}(\pi(\tilde{X}_i)), \qquad \operatorname{logit}\mathopen{}\left\{\pi(\tilde{x})\right\}\mathclose{} = {\tilde{x}}^{\top}\tilde{\beta}\]
4.2 Survival analysis
4.2.1 Probability distribution functions
| Name | Symbols | Definition |
|---|---|---|
| Probability density function (PDF) | \(\operatorname{f}(t), \operatorname{p}(t)\) | \(\operatorname{p}(T=t)\) |
| Cumulative distribution function (CDF) | \(\operatorname{F}(t), \operatorname{P}(t)\) | \(\operatorname{P}(T\leq t)\) |
| Survival function | \(\operatorname{S}(t), \bar{\operatorname{F}}(t)\) | \(\operatorname{P}(T > t)\) |
| Hazard function | \(\lambda(t), \operatorname{h}(t)\) | \(\operatorname{p}(T=t|T\ge t)\) |
| Cumulative hazard function | \(\Lambda(t), \operatorname{H}(t)\) | \(\int_{u=-\infty}^t {\lambda}(u)du\) |
| Log-hazard function | \(\eta(t)\) | \(\operatorname{log}\mathopen{}\left\{{\lambda}(t)\right\}\mathclose{}\) |
4.2.2 Diagram of survival distribution function relationships
\[ \operatorname{f}(t) \xleftarrow[\operatorname{S}(t){\lambda}(t)]{-S'(t)} \operatorname{S}(t) \xleftarrow[]{\operatorname{exp}\mathopen{}\left\{-{\Lambda}(t)\right\}\mathclose{}} {\Lambda}(t) \xleftarrow[]{\int_{u=0}^t {\lambda}(u)du} {\lambda}(t) \xleftarrow[]{\operatorname{exp}\mathopen{}\left\{\eta(t)\right\}\mathclose{}} \eta(t) \]
\[ \operatorname{f}(t) \xrightarrow[\int_{u=t}^\infty \operatorname{f}(u)du]{\operatorname{f}(t)/{\lambda}(t)} \operatorname{S}(t) \xrightarrow[-\log{\operatorname{S}(t)}]{} {\Lambda}(t) \xrightarrow[{\Lambda}'(t)]{} {\lambda}(t) \xrightarrow[\operatorname{log}\mathopen{}\left\{{\lambda}(t)\right\}\mathclose{}]{} \eta(t) \]
4.2.3 Survival likelihood contributions, assuming non-informative censoring
\[ \begin{aligned} \operatorname{p}(Y=y,D=d) &= [\operatorname{f}_T(y)]^{d} [\operatorname{S}_T(y)]^{1-d} \\ &= [{\lambda}_T(y)]^{d} [\operatorname{S}_T(y)] \end{aligned} \]
4.2.4 Nonparametric time-to-event distribution estimators
\[\hat{{\lambda}}_i = \frac{d_i}{r_i}\]
\[\hat{\kappa}_i = 1 - \hat{{\lambda}}_i = \frac{r_i - d_i}{r_i}\]
\[\mathop{\hat{\operatorname{S}}}\nolimits_{KM}(t) \stackrel{\text{def}}{=}\prod_{\mathopen{}\left\{i:\ t_i \le t\right\}\mathclose{}} \hat{\kappa}_i\]
\[\hat{{\Lambda}}_{NA}(t) \stackrel{\text{def}}{=}\sum_{\mathopen{}\left\{i:\ t_i \le t\right\}\mathclose{}}\hat{{\lambda}}_i\]
4.2.5 Proportional hazards model structure
| Formula | Description |
|---|---|
| \({\lambda}(t\mid\tilde{x}) = {\lambda}_0(t)\cdot\theta_{{\lambda}}(\tilde{x})\) | Proportional hazards assumption |
| \({\Lambda}(t\mid\tilde{x}) = {\Lambda}_0(t)\cdot\theta_{{\lambda}}(\tilde{x})\) | Cumulative hazard factorization |
| \(\eta(t\mid\tilde{x}) = \eta_0(t) + \Delta\eta(\tilde{x})\) | Log-hazard decomposition |
| \(\Delta\eta(\tilde{x}) = \tilde{x} \cdot \tilde{\beta} = \beta_1 x_1 + \cdots + \beta_p x_p\) | Linear predictor |
| \(\theta_{{\lambda}}(\tilde{x}) = \operatorname{exp}\mathopen{}\left\{\Delta\eta(\tilde{x})\right\}\mathclose{}\) | Hazard multiplier |
| \(\theta_{{\lambda}}(t| \tilde{x}: {\tilde{x}^*}) \stackrel{\text{def}}{=}\frac{{\lambda}(t|\tilde{x})}{{\lambda}(t|{\tilde{x}^*})}\) | Hazard ratio definition |
| \(\theta_{{\lambda}}(t\mid\tilde{x}:{\tilde{x}^*}) = \operatorname{exp}\mathopen{}\left\{\Delta\eta(\tilde{x}) - \Delta\eta({\tilde{x}^*})\right\}\mathclose{} = \operatorname{exp}\mathopen{}\left\{(\tilde{x}-{\tilde{x}^*}) \cdot \tilde{\beta}\right\}\mathclose{}\) | Hazard ratio formula |
4.2.6 Proportional hazards model partial likelihood formula:
\[ \begin{aligned} \mathscr{L}^*_i &= \frac{\theta_{{\lambda}}(\tilde{x}_i)}{\sum_{k \in R(t_i)} \theta_{{\lambda}}(\tilde{x}_k)} \\ \mathscr{L}^* &= \prod_{\mathopen{}\left\{i:\ d_i = 1\right\}\mathclose{}} \mathscr{L}^*_i \end{aligned} \]
4.2.7 Proportional hazards model baseline cumulative hazard estimator:
\[\hat {\Lambda}_0(t) = \sum_{t_i < t} \frac{d_i}{\sum_{k\in R(t_i)} \theta_{{\lambda}}(x_k)}\]