Title every definition and result callout (base for #872 diagram)

_subfiles/Linear-models-overview/_sec_linreg_diag_residuals.qmd

@@ -123,6 +123,8 @@ $$\hat{\vY} = H\vY$$
 :::
 
 :::{#thm-resid-unbiased}
+#### Mean and variance of residuals
+
 For an ordinary least squares linear model
 with fitted values $\hat y_i = \dprodf{\vx_i}{\vb}$
 (and fitted-value vector $\hat{\vY}$),

_subfiles/intro-MLEs/_sec-loglik.qmd

@@ -8,6 +8,7 @@ It is typically easier to work with the log of the likelihood function:
 ---
 
 :::{#thm-mle-use-log}
+#### Maximize the log-likelihood instead of the likelihood
 
 The likelihood and log-likelihood have the same maximizer:
 

_subfiles/intro-MLEs/_sec_likelihood.qmd

@@ -103,6 +103,7 @@ $$\Lik_i(\theta) = \P(X_i=x_i)$$
 ---
 
 :::{#thm-ds-lik-obs-lik}
+#### Dataset likelihood as a product of observation likelihoods
 
 For $\iid$ data $\vx \eqdef \x1n$,
 the likelihood of the dataset is equal to the product of the observation-specific likelihood factors:

_subfiles/intro-to-survival-analysis/_sec-cuhaz.qmd

@@ -2,6 +2,8 @@
 Since $\haz(t) = \deriv{t}\cb{-\log{\surv(t)}}$ (see @thm-h-logS), we also have:
 
 :::{#cor-surv-int-haz}
+#### Survival function from the cumulative hazard
+
 $$\surv(t) = \exp{-\int_{u=0}^t \haz(u)du}$${#eq-surv-int-haz}
 :::
 

_subfiles/intro-to-survival-analysis/_sec-exp-dist.qmd

@@ -75,6 +75,8 @@ $$
 ---
 
 :::{#thm-mle-exp}
+#### MLE of the exponential rate parameter
+
 Let $T=\sum t_i$ and $U=\sum u_j$. Then:
 
 $$

_subfiles/intro-to-survival-analysis/_sec-inv-survf.qmd

@@ -167,6 +167,7 @@ qexp(p = 0.5, rate = 2)
 {{< slidebreak >}}
 
 :::{#thm-inv-surv-is-quantile}
+#### Inverse survival function is the quantile function
 
 The inverse survival function equals the $(1-p)$th
 [population quantile](probability.qmd#def-quantile-function)

_subfiles/intro-to-survival-analysis/_sec-survf.qmd

@@ -27,6 +27,7 @@ $$\surv(t) \eqdef \Pr(T > t)$$
 ---
 
 :::{#thm-survival-expressions-1}
+#### Equivalent expressions for the survival function
 
 $$
 \begin{aligned}
@@ -109,6 +110,7 @@ ggplot() +
 ---
 
 :::{#thm-surv-fn-as-mean-status}
+#### Survival function as expected survival status
 
 If $A_t$ represents survival status at time $t$, with $A_t = 1$ denoting alive at time $t$ and $A_t = 0$ denoting deceased at time $t$, then:
 
@@ -119,6 +121,7 @@ $$\surv(t) = \P(A_t=1) = \E{A_t}$$
 ---
 
 :::{#thm-surv-and-mean}
+#### Mean as the integral of the survival function
 
 If $T$ is a nonnegative random variable, then:
 

_subfiles/logistic-regression/_sec-d_odds-d_logodds.qmd

@@ -1,4 +1,6 @@
 :::{#lem-deriv-invodds}
+#### Derivative of odds w.r.t. log-odds
+
 $$\derivf{\odds}{\logodds} = \odds$$
 
 :::
@@ -26,6 +28,8 @@ $$
 
 :::{#thm-d_odds-d_logodds}
 
+#### Derivative of odds in terms of probability
+
 $$\derivf{\omega}{\eta} = \frac{\pi}{1-\pi}$${#eq-d_omega-d_eta}
 
 :::

_subfiles/logistic-regression/_sec_OR-ratio-ratio.qmd

@@ -5,6 +5,8 @@ so odds ratios are ratios of ratios:
 :::
 
 :::{#thm-or-ratio-ratio}
+#### Odds ratio as a ratio of ratios
+
 $$
 \ba
 \ratio(\odds_1, \odds_2)

_subfiles/logistic-regression/_sec_OR_logistic.qmd

@@ -197,6 +197,8 @@ $$
 
 :::{#thm-logistic-OR}
 
+#### Odds ratio from difference in covariate patterns
+
 The odds ratio comparing covariate patterns $\vx$ and $\vxs$ is:
 
 {{< include _subfiles/logistic-regression/_eq_OR_delta.qmd >}}
@@ -211,6 +213,8 @@ By @sol-simplify-logistic-OR.
 
 :::{#cor-log-or}
 
+#### Log odds ratio equals the difference in log-odds
+
 $$\logf {\ror(\vx,\vxs)} = \difflogodds$$
 
 :::

_subfiles/logistic-regression/_sec_d-pi_d-eta.qmd

@@ -1,5 +1,7 @@
 :::{#thm-d_prob-d_logodds}
 
+#### Derivative of probability w.r.t. log-odds
+
 $$\derivf{\prob}{\logodds} = \pi (1-\pi)$$
 :::
 
@@ -39,6 +41,8 @@ $$
 
 :::{#cor-d_pi-d_eta-var}
 
+#### Derivative of probability w.r.t. linear predictor as a variance
+
 If $\pi = \Pr(Y=1| \vX=\vx)$, then:
 
 $$\derivf{\pi}{\eta} = \Varf{Y|X=x}$$

_subfiles/logistic-regression/_sec_derive_logistic_loglik.qmd

@@ -41,6 +41,8 @@ $$
 
 :::{#lem-logistic-loglik-component}
 
+#### Per-observation log-likelihood component
+
 $$\ell_i(\pi_i) = y_i \eta_i - \logf{1+\odds_i}$$
 
 :::

_subfiles/logistic-regression/_sec_expit.qmd

@@ -4,6 +4,8 @@
 
 :::{#thm-prob-from-logodds}
 
+#### Probability as a function of log-odds
+
 ::: notes
 If $\prob$ is the probability of an event $A$,
 $\odds$ is the corresponding odds of $A$,
@@ -61,6 +63,7 @@ Details left to the reader.
 ---
 
 :::{#thm-expit-prob-logodds}
+#### Probability via the expit function
 If $\prob$ is the probability of an event $A$,
 $\odds$ is the corresponding odds of $A$,
 and $\logodds$ is the corresponding log-odds of $A$,

_subfiles/logistic-regression/_sec_invodds.qmd

@@ -52,6 +52,8 @@ $$
 
 :::{#thm-odds-to-prob}
 
+#### Probability as a function of odds
+
 If $\pi$ is the probability of an event
 and $\omega$ is the corresponding odds of that event,
 then:
@@ -86,6 +88,8 @@ can be called the **inverse-odds function**.
 
 :::{#cor-invodds-pi}
 
+#### Probability via the inverse-odds function
+
 $$\prob = \invoddsf{\odds}$$
 :::
 
@@ -100,6 +104,8 @@ By @def-inv-odds and @thm-odds-to-prob.
 
 :::{#cor-invodds-odds-inv}
 
+#### Inverse-odds function inverts the odds function
+
 $$\invoddsf{\odds} = \oddsinvf{\odds}$$
 
 :::
@@ -252,6 +258,7 @@ $$
 ---
 
 :::{#cor-inverse-odds-nonevent}
+#### One plus odds in terms of non-event probability
 $$1+\odds = \frac{1}{1-\prob}$$
 :::
 

_subfiles/logistic-regression/_sec_logistic_score_fn.qmd

@@ -4,6 +4,8 @@ As usual, by independence, we have:
 
 :::{#lem-score-logistic}
 
+#### Score function decomposes over observations
+
 $$
 \ba
 \brown{\vec{\llik'}(\vb)}
@@ -22,6 +24,8 @@ we can apply the [vector chain rule](math-prereqs.qmd#thm-chain-vec):
 
 :::{#lem-logistic-score-comp}
 
+#### Chain rule applied to the score component
+
 $$
 \ba
 \magenta{\vec{\llik_i'}(\vb)}
@@ -38,6 +42,8 @@ $$
 
 :::{#lem-d_logodds-d_vb}
 
+#### Derivative of log-odds with respect to coefficients
+
 By [the derivative of a linear combination](math-prereqs.qmd#thm-deriv-lincom):
 
 $$
@@ -90,6 +96,7 @@ $$
 
 
 :::{#thm-logistic-score-comp}
+#### Score component for one observation
 $$\magenta{\llik_i'(\vb)} = \magenta{\vx_i \err_i}$${#eq-score-comp}
 :::
 
@@ -106,6 +113,8 @@ we have:
 
 :::{#thm-logistic-score-fn}
 
+#### Logistic-model score function
+
 $$
 \ba
 \brown{\vec{\llik'}(\vb)} &= \sumin \magenta{\llik_i'(\vb)}\\

_subfiles/logistic-regression/_sec_logistic_slope_mean.qmd

@@ -2,6 +2,8 @@
 
 :::{#lem-d_logodds-d_x}
 
+#### Derivative of log-odds w.r.t. predictor
+
 By [the derivative of a linear combination](math-prereqs.qmd#thm-deriv-lincom):
 
 $$

_subfiles/logistic-regression/_sec_logit.qmd

@@ -15,6 +15,8 @@ $$\logodds \eqdef \logf{\omega}$${#eq-def-logodds}
 
 :::{#thm-logodds-pi}
 
+#### Log-odds as a function of probability
+
 If $\prob$ is the probability of an event $A$,
 $\odds$ is the corresponding odds of $A$,
 and $\eta$ is the corresponding log-odds of $A$,
@@ -81,6 +83,7 @@ Apply @def-logit-fn and then @def-odds (details left to the reader).
 ---
 
 :::{#cor-logodds-logit}
+#### Log-odds via the logit function
 If $\prob$ is the probability of an event $A$
 and $\logodds$ is the corresponding log-odds of $A$,
 then:

_subfiles/logistic-regression/_sec_odds_fn.qmd

@@ -22,6 +22,7 @@ $$
 ---
 
 :::{#thm-prob-to-odds}
+#### Odds as a function of probability
 If $\prob$ is the probability of an event $A$
 and $\odds$ is the corresponding odds of $A$,
 then:
@@ -64,6 +65,7 @@ which is easier to remember and manipulate:
 :::
 
 :::{#cor-oddsf-to-odds}
+#### Odds via the odds function
 If $\prob$ is the probability of an outcome $A$
 and $\odds$ is the corresponding odds of $A$,
 then:

_subfiles/logistic-regression/_sec_odds_of_rare_events.qmd

@@ -50,6 +50,8 @@ $$
 
 :::{#thm-odds-minus-probs}
 
+#### Difference between odds and probability
+
 Let $\odds = \frac{\pi}{1-\pi}$. Then:
 
 $$\odds - \pi = \frac{\pi^2}{1-\pi}$$

_subfiles/logistic-regression/_sec_overview_bernoulli_models.qmd

@@ -11,6 +11,8 @@ What is logistic regression?
 :::{#sol-def-logistic-regression}
 
 :::{#def-logistic-regression}
+#### Logistic regression model
+
 **Logistic regression** is a framework for modeling [binary](data.qmd#def-binary) outcomes, conditional on one or more *predictors* (a.k.a. *covariates*).
 :::
 

_subfiles/logistic-regression/_thm-d_odds_d_beta.qmd

@@ -1,4 +1,5 @@
 :::{#thm-d_odds_d_beta}
+#### Gradient of odds w.r.t. coefficients
 
 ::: notes
 To derive $\derivf{\odds}{\vb}$,
@@ -19,6 +20,8 @@ $$
 
 :::{#cor-d_odds_d_beta}
 
+#### Gradient of odds w.r.t. coefficients in terms of probability
+
 $$
 \ba
 \derivf{\odds}{\vb}

_subfiles/logistic-regression/_thm-d_pi_d_beta.qmd

@@ -2,7 +2,9 @@
 
 :::{#thm-d_pi_d_beta}
 
-Using 
+#### Gradient of fitted probability w.r.t. coefficients
+
+Using
 @lem-d_logodds-d_vb and 
 @thm-d_prob-d_logodds:
 

_subfiles/logistic-regression/_thm_odds-from-logodds.qmd

@@ -1,4 +1,5 @@
 :::{#lem-odds-from-logodds}
+#### Odds from log-odds
 
 ::: notes
 If $\odds$ is the odds of an event $A$

_subfiles/logistic-regression/_thms-deriv-odds.qmd

@@ -30,6 +30,8 @@ $$
 
 :::{#cor-deriv-odds}
 
+#### Derivative of odds function in terms of odds
+
 $$\derivf{\odds}{\prob} = \sqf{1+\odds}$$
 
 :::

_subfiles/misc/_cor-deriv-expit.qmd

@@ -1,3 +1,5 @@
 :::{#cor-deriv-expit}
+#### Derivative of expit
+
 $$\dexpitf{\logodds} = (\expitf{\logodds}) (1 - \expitf{\logodds})$$
 :::

_subfiles/misc/_cor-deriv-invodds.qmd

@@ -1,5 +1,7 @@
 
 :::{#cor-deriv-invodds}
 
+#### Derivative of inverse-odds function
+
 $$\doddsinvf{\odds} = \sqf{1 - \invoddsf{\odds}}$$
 :::

_subfiles/misc/_cor_prob-nonevent.qmd

@@ -1,5 +1,7 @@
 :::{#cor-inverse-odds-nonevent2}
 
+#### Probability of a non-event from the odds
+
 If $\prob$ is the probability of event $A$
 and $\odds$ is the corresponding odds of event $A$,
 then the probability that $A$ does not occur is:

_subfiles/misc/_lem-one-minus-expit.qmd

@@ -1,4 +1,6 @@
 :::{#lem-one-minus-expit}
+#### One minus expit
+
 $$1-\expitf{\logodds} = \inv{1+\exp{\logodds}}$$
 :::
 

_subfiles/proportional-hazards-models/_cor-hazard-ratio-vs-baseline.qmd

@@ -1,5 +1,7 @@
 :::{#cor-hazard-ratio-vs-baseline}
 
+#### Hazard factor from difference of log-hazard from baseline
+
 $$\hazfactor(t|\vx)= \expf{\diffloghaz(t|\vx)}$$
 
 :::

_subfiles/proportional-hazards-models/_def-ph-model.qmd

@@ -27,6 +27,8 @@ Equivalently:
 
 :::{#lem-ph-lincomp}
 
+#### Log-hazard as baseline plus a linear combination
+
 In a proportional hazards model (that is, if @eq-ph-diffloghaz holds):
 
 $$