Merge pull request #437 from Merck/katex

Use KaTeX for pkgdown math rendering

_pkgdown.yml

@@ -14,6 +14,7 @@ template:
     navbar-light-active-color: "#fff"
     dropdown-link-hover-color: "#fff"
     dropdown-link-hover-bg: "#00857c"
+  math-rendering: "katex"
 
 footer:
   structure:

pkgdown/extra.css

@@ -1,5 +1,6 @@
 /* navbar background */
-.bg-light, .navbar-light {
+.bg-light,
+.navbar-light {
     background-color: #00857c !important;
 }
 
@@ -21,4 +22,148 @@
 footer {
     padding-top: 1rem;
     padding-bottom: 1rem;
+}
+
+/* Redefine font locations for KaTeX so that they can be loaded */
+/* This is because pkgdown 2.1.0 doesn't bundle fonts when math-rendering: katex */
+/* CSS font definitions copied from katex/katex.css */
+/* Font files copied from katex/fonts/ */
+@font-face {
+    font-family: 'KaTeX_AMS';
+    src: url(fonts/KaTeX_AMS-Regular.woff2) format('woff2'), url(fonts/KaTeX_AMS-Regular.woff) format('woff'), url(fonts/KaTeX_AMS-Regular.ttf) format('truetype');
+    font-weight: normal;
+    font-style: normal;
+}
+
+@font-face {
+    font-family: 'KaTeX_Caligraphic';
+    src: url(fonts/KaTeX_Caligraphic-Bold.woff2) format('woff2'), url(fonts/KaTeX_Caligraphic-Bold.woff) format('woff'), url(fonts/KaTeX_Caligraphic-Bold.ttf) format('truetype');
+    font-weight: bold;
+    font-style: normal;
+}
+
+@font-face {
+    font-family: 'KaTeX_Caligraphic';
+    src: url(fonts/KaTeX_Caligraphic-Regular.woff2) format('woff2'), url(fonts/KaTeX_Caligraphic-Regular.woff) format('woff'), url(fonts/KaTeX_Caligraphic-Regular.ttf) format('truetype');
+    font-weight: normal;
+    font-style: normal;
+}
+
+@font-face {
+    font-family: 'KaTeX_Fraktur';
+    src: url(fonts/KaTeX_Fraktur-Bold.woff2) format('woff2'), url(fonts/KaTeX_Fraktur-Bold.woff) format('woff'), url(fonts/KaTeX_Fraktur-Bold.ttf) format('truetype');
+    font-weight: bold;
+    font-style: normal;
+}
+
+@font-face {
+    font-family: 'KaTeX_Fraktur';
+    src: url(fonts/KaTeX_Fraktur-Regular.woff2) format('woff2'), url(fonts/KaTeX_Fraktur-Regular.woff) format('woff'), url(fonts/KaTeX_Fraktur-Regular.ttf) format('truetype');
+    font-weight: normal;
+    font-style: normal;
+}
+
+@font-face {
+    font-family: 'KaTeX_Main';
+    src: url(fonts/KaTeX_Main-Bold.woff2) format('woff2'), url(fonts/KaTeX_Main-Bold.woff) format('woff'), url(fonts/KaTeX_Main-Bold.ttf) format('truetype');
+    font-weight: bold;
+    font-style: normal;
+}
+
+@font-face {
+    font-family: 'KaTeX_Main';
+    src: url(fonts/KaTeX_Main-BoldItalic.woff2) format('woff2'), url(fonts/KaTeX_Main-BoldItalic.woff) format('woff'), url(fonts/KaTeX_Main-BoldItalic.ttf) format('truetype');
+    font-weight: bold;
+    font-style: italic;
+}
+
+@font-face {
+    font-family: 'KaTeX_Main';
+    src: url(fonts/KaTeX_Main-Italic.woff2) format('woff2'), url(fonts/KaTeX_Main-Italic.woff) format('woff'), url(fonts/KaTeX_Main-Italic.ttf) format('truetype');
+    font-weight: normal;
+    font-style: italic;
+}
+
+@font-face {
+    font-family: 'KaTeX_Main';
+    src: url(fonts/KaTeX_Main-Regular.woff2) format('woff2'), url(fonts/KaTeX_Main-Regular.woff) format('woff'), url(fonts/KaTeX_Main-Regular.ttf) format('truetype');
+    font-weight: normal;
+    font-style: normal;
+}
+
+@font-face {
+    font-family: 'KaTeX_Math';
+    src: url(fonts/KaTeX_Math-BoldItalic.woff2) format('woff2'), url(fonts/KaTeX_Math-BoldItalic.woff) format('woff'), url(fonts/KaTeX_Math-BoldItalic.ttf) format('truetype');
+    font-weight: bold;
+    font-style: italic;
+}
+
+@font-face {
+    font-family: 'KaTeX_Math';
+    src: url(fonts/KaTeX_Math-Italic.woff2) format('woff2'), url(fonts/KaTeX_Math-Italic.woff) format('woff'), url(fonts/KaTeX_Math-Italic.ttf) format('truetype');
+    font-weight: normal;
+    font-style: italic;
+}
+
+@font-face {
+    font-family: 'KaTeX_SansSerif';
+    src: url(fonts/KaTeX_SansSerif-Bold.woff2) format('woff2'), url(fonts/KaTeX_SansSerif-Bold.woff) format('woff'), url(fonts/KaTeX_SansSerif-Bold.ttf) format('truetype');
+    font-weight: bold;
+    font-style: normal;
+}
+
+@font-face {
+    font-family: 'KaTeX_SansSerif';
+    src: url(fonts/KaTeX_SansSerif-Italic.woff2) format('woff2'), url(fonts/KaTeX_SansSerif-Italic.woff) format('woff'), url(fonts/KaTeX_SansSerif-Italic.ttf) format('truetype');
+    font-weight: normal;
+    font-style: italic;
+}
+
+@font-face {
+    font-family: 'KaTeX_SansSerif';
+    src: url(fonts/KaTeX_SansSerif-Regular.woff2) format('woff2'), url(fonts/KaTeX_SansSerif-Regular.woff) format('woff'), url(fonts/KaTeX_SansSerif-Regular.ttf) format('truetype');
+    font-weight: normal;
+    font-style: normal;
+}
+
+@font-face {
+    font-family: 'KaTeX_Script';
+    src: url(fonts/KaTeX_Script-Regular.woff2) format('woff2'), url(fonts/KaTeX_Script-Regular.woff) format('woff'), url(fonts/KaTeX_Script-Regular.ttf) format('truetype');
+    font-weight: normal;
+    font-style: normal;
+}
+
+@font-face {
+    font-family: 'KaTeX_Size1';
+    src: url(fonts/KaTeX_Size1-Regular.woff2) format('woff2'), url(fonts/KaTeX_Size1-Regular.woff) format('woff'), url(fonts/KaTeX_Size1-Regular.ttf) format('truetype');
+    font-weight: normal;
+    font-style: normal;
+}
+
+@font-face {
+    font-family: 'KaTeX_Size2';
+    src: url(fonts/KaTeX_Size2-Regular.woff2) format('woff2'), url(fonts/KaTeX_Size2-Regular.woff) format('woff'), url(fonts/KaTeX_Size2-Regular.ttf) format('truetype');
+    font-weight: normal;
+    font-style: normal;
+}
+
+@font-face {
+    font-family: 'KaTeX_Size3';
+    src: url(fonts/KaTeX_Size3-Regular.woff2) format('woff2'), url(fonts/KaTeX_Size3-Regular.woff) format('woff'), url(fonts/KaTeX_Size3-Regular.ttf) format('truetype');
+    font-weight: normal;
+    font-style: normal;
+}
+
+@font-face {
+    font-family: 'KaTeX_Size4';
+    src: url(fonts/KaTeX_Size4-Regular.woff2) format('woff2'), url(fonts/KaTeX_Size4-Regular.woff) format('woff'), url(fonts/KaTeX_Size4-Regular.ttf) format('truetype');
+    font-weight: normal;
+    font-style: normal;
+}
+
+@font-face {
+    font-family: 'KaTeX_Typewriter';
+    src: url(fonts/KaTeX_Typewriter-Regular.woff2) format('woff2'), url(fonts/KaTeX_Typewriter-Regular.woff) format('woff'), url(fonts/KaTeX_Typewriter-Regular.ttf) format('truetype');
+    font-weight: normal;
+    font-style: normal;
 }

vignettes/articles/story-npe-background.Rmd

@@ -65,23 +65,23 @@ Denoting $t_k=n_k/N$, we assume that for some real-valued function $\theta(t)$ f
 $$E(\hat{\theta}_k) =\theta(t_k) =E(\bar X_k).$$
 In the models of @proschan2006statistical and @jennison1999group we would have $\theta(t)$ equal to some constant $\theta$.
 We assume further that for $i=1,2,\ldots$
-$$\hbox{Var}(X_{i})=1.$$
+$$\text{Var}(X_{i})=1.$$
 The sample average variance under this assumption is for $1\leq k\leq K$
 
-$$\hbox{Var}(\hat\theta(t_k))=\hbox{Var}(\bar X_k) =  1/ n_k.$$
+$$\text{Var}(\hat\theta(t_k))=\text{Var}(\bar X_k) =  1/ n_k.$$
 The statistical information for the estimate $\hat\theta(t_k)$ for $1\leq k\leq K$ for this case is
-$$ \mathcal{I}_k \equiv \frac{1}{\hbox{Var}(\hat\theta(t_k))} = n_k.$$
+$$ \mathcal{I}_k \equiv \frac{1}{\text{Var}(\hat\theta(t_k))} = n_k.$$
 We now see that $t_k$, $1\leq k\leq K$ is the so-called information fraction at analysis $k$ in that
 $t_k=\mathcal{I}_k/\mathcal{I}_K.$
 
 ## Z-process
 
 The Z-process is commonly used (e.g., @jennison1999group) and will be used below to extend the computational algorithm in Chapter 19 of @jennison1999group by defining equivalently in the first and second lines below for $k=1,\ldots,K$
 
-$$Z_{k} = \frac{\hat\theta_k}{\sqrt{\hbox{Var}(\hat\theta_k)}}= \sqrt{\mathcal{I}_k}\hat\theta_k= \sqrt{n_k}\bar X_k.$$
+$$Z_{k} = \frac{\hat\theta_k}{\sqrt{\text{Var}(\hat\theta_k)}}= \sqrt{\mathcal{I}_k}\hat\theta_k= \sqrt{n_k}\bar X_k.$$
 
 The variance for $1\leq k\leq K$ is
-$$\hbox{Var}(Z_k) = 1$$
+$$\text{Var}(Z_k) = 1$$
 and the expected value is
 
 $$E(Z_{k})= \sqrt{\mathcal{I}_k}\theta(t_{k})= \sqrt{n_k}E(\bar X_k) .$$
@@ -94,7 +94,7 @@ $$B_{k}=\sqrt{t_k}Z_k$$
 which implies
 $$ E(B_{k}) = \sqrt{t_{k}\mathcal{I}_k}\theta(t_k) = t_k \sqrt{\mathcal{I}_K} \theta(t_k) = \mathcal{I}_k\theta(t_k)/\sqrt{\mathcal{I}_K}$$
 and
-$$\hbox{Var}(B_k) = t_k.$$
+$$\text{Var}(B_k) = t_k.$$
 
 For our example, we have
 
@@ -123,14 +123,14 @@ mt %>% kable(escape = FALSE)
 
 We assume independent increments in the B-process.
 That is, for $1\leq j < k\leq K$
-$$\tag{1} B_k - B_j \sim \hbox{Normal} (\sqrt{\mathcal{I}_K}(t_k\theta(t_k)- t_j\theta(t_j)), t_k-t_j)$$
+$$\tag{1} B_k - B_j \sim \text{Normal} (\sqrt{\mathcal{I}_K}(t_k\theta(t_k)- t_j\theta(t_j)), t_k-t_j)$$
 independent of $B_1,\ldots,B_j$.
 As noted above, for a given $1\leq k\leq K$ we have for our example
 $$B_j=\sum_{i=1}^{n_j}X_i / \sqrt N.$$
 Because of independence of the sequence $X_i$, $i=1,2,\ldots$, we  immediately have for $1\leq j\leq k\leq K$
-$$\hbox{Cov}(B_j,B_k) = \hbox{Var}(B_j) = t_j.$$
+$$\text{Cov}(B_j,B_k) = \text{Var}(B_j) = t_j.$$
 This leads further to
-$$\hbox{Corr}(B_j,B_k)=\frac{t_j}{\sqrt{t_jt_k}}=\sqrt{t_j/t_k}=\hbox{Corr}(Z_j,Z_k)=\hbox{Cov}(Z_j,Z_k)$$
+$$\text{Corr}(B_j,B_k)=\frac{t_j}{\sqrt{t_jt_k}}=\sqrt{t_j/t_k}=\text{Corr}(Z_j,Z_k)=\text{Cov}(Z_j,Z_k)$$
 which is the covariance structure in the so-called *canonical form* of @jennison1999group.
 For our example, we have
 $$B_k=\frac{1}{\sqrt N}\sum_{i=1}^{n_k}X_i$$
@@ -139,14 +139,14 @@ $$B_k-B_j=\frac{1}{\sqrt N}\sum_{i=n_j + 1}^{n_k}X_i$$
 and the covariance is obvious.
 We assume independent increments in the B-process that will be demonstrated for the simple example above.
 That is, for $1\leq j < k\leq K$
-$$\tag{1} B_k - B_j \sim \hbox{Normal} (\mathcal{I}_k\theta(t_k)- \mathcal{I}_j\theta(t_j), t_k-t_j)$$
+$$\tag{1} B_k - B_j \sim \text{Normal} (\mathcal{I}_k\theta(t_k)- \mathcal{I}_j\theta(t_j), t_k-t_j)$$
 independent of $B_1,\ldots,B_j$.
 For a given $1\leq j\leq k\leq K$ we have for our example
 $$B_j=\sum_{i=1}^{n_j}X_i / (\sqrt N\sigma).$$
 Because of independence of the sequence $X_i$, $i=1,2,\ldots$, we  immediately have for $1\leq j\leq k\leq K$
-$$\hbox{Cov}(B_j,B_k) = \hbox{Var}(B_j) = t_j/t_k =\mathcal{I}_j/\mathcal{I}_k.$$
+$$\text{Cov}(B_j,B_k) = \text{Var}(B_j) = t_j/t_k =\mathcal{I}_j/\mathcal{I}_k.$$
 This leads to
-$$\mathcal{I}_j/\mathcal{I}_k=\sqrt{t_j/t_k}=\hbox{Corr}(B_j,B_k)=\hbox{Corr}(Z_j,Z_k)=\hbox{Cov}(Z_j,Z_k)$$
+$$\mathcal{I}_j/\mathcal{I}_k=\sqrt{t_j/t_k}=\text{Corr}(B_j,B_k)=\text{Corr}(Z_j,Z_k)=\text{Cov}(Z_j,Z_k)$$
 which is the covariance structure in the so-called *canonical form* of @jennison1999group.
 The independence of $B_j$ and
 $$B_k-B_j=\sum_{i=n_j + 1}^{n_k}X_i/(\sqrt N\sigma)$$

vignettes/articles/story-npe-integration.Rmd

@@ -40,7 +40,7 @@ This vignettes generalize computational algorithms provided in Chapter 19 of @je
 ## Asymptotic normal and boundary crossing probabilities
 
 We assume $Z_1,\cdots,Z_K$ has a multivariate normal distribution with variance for $1\leq k\leq K$ of
-$$\hbox{Var}(Z_k) = 1$$
+$$\text{Var}(Z_k) = 1$$
 and the expected value is
 
 $$E(Z_{k})= \sqrt{\mathcal{I}_k}\theta(t_{k})= \sqrt{n_k}E(\bar X_k) .$$

vignettes/articles/story-power-evaluation-with-spending-bound.Rmd

@@ -74,7 +74,7 @@ We assume
 -   $n_{ik}$: the number of subjects in group $i$ and analysis $k$;
 -   $n_k$: the number of subjects at analysis $k$, i.e., $n_k = n_{1k} + n_{2k}$;
 -   $X_{ij}$: the independent random variable whether the $j$-th subject in group $i$ has response, i.e, $$
-    X_{ij} \sim \hbox{Bernoulli}(p_i);
+    X_{ij} \sim \text{Bernoulli}(p_i);
     $$
 -   $Y_{ik}$: the number of subject having response in group $i$ and analysis $k$, i.e., $$
     Y_{ik} = \sum_{j = 1}^{n_{ik}} X_{ij};
@@ -89,13 +89,13 @@ In this section, we will discuss the estimation of statistical information and v
 Under the alternative hypothesis, one can estimate the proportion of failures in group $i$ at analysis $k$ as $$
   \hat{p}_{ik} = Y_{ik}/n_{ik}.
 $$ We note its variance is $$
- \hbox{Var}(\hat p_{ik})=\frac{p_{i}(1-p_i)}{n_{ik}},
+ \text{Var}(\hat p_{ik})=\frac{p_{i}(1-p_i)}{n_{ik}},
 $$ and its consistent estimator $$
-  \widehat{\hbox{Var}}(\hat p_{ik})=\frac{\hat p_{ik}(1-\hat p_{ik})}{n_{ik}},
+  \widehat{\text{Var}}(\hat p_{ik})=\frac{\hat p_{ik}(1-\hat p_{ik})}{n_{ik}},
 $$ for any $i = 1, 2$ and $k = 1, 2, \ldots, K$. Letting $\hat\theta_k = \hat p_{1k} - \hat p_{2k},$ we also have $$
   \sigma^2_k
   \equiv
-  \hbox{Var}(\hat\theta_k)
+  \text{Var}(\hat\theta_k)
   =
   \frac{p_1(1-p_1)}{n_{1k}}+\frac{p_2(1-p_2)}{n_{2k}},
 $$ its consistent estimator $$
@@ -162,7 +162,7 @@ Then, we have the asymptotic distribution
 $$
   Z_k
   \sim
-  \hbox{Normal}
+  \text{Normal}
   \left(
     \sqrt{n_k}\frac{p_1 - p_2}{\sqrt{(1/\xi_1+ 1/\xi_2) p_0(1- p_0)} },
     \sigma^2_{0k}/\sigma^2_{1k}

vignettes/articles/story-risk-difference.Rmd

@@ -125,8 +125,8 @@ Simulation is used throughout to check the examples presented.
 
 -   $X_{C,k,s}, X_{E,k,s}$: random variables indicating the number of
     subjects failed in control/treatment arm, i.e.,
-    $X_{C,k,s} \sim \hbox{Binomial}(N_{C,k,s}, p_{C,k,s})$,
-    $X_{E,k,s} \sim \hbox{Binomial}(N_{E,k,s}, p_{E,k,s})$ at the $k$-th
+    $X_{C,k,s} \sim \text{Binomial}(N_{C,k,s}, p_{C,k,s})$,
+    $X_{E,k,s} \sim \text{Binomial}(N_{E,k,s}, p_{E,k,s})$ at the $k$-th
     analysis of the $s$-th strata.
 
 -   $x_{C,k,s}, x_{E,k,s}$: the observed outcome of
@@ -164,7 +164,7 @@ The test statistics at the $k$-th analysis is $$
     \sqrt{\sum_{s=1}^S \widehat w_{s,k}^2 \widehat\sigma_{H_0,k,s}^2}
   }
 $$ where
-$\widehat\sigma^2_{k,s} = \widehat{\hbox{Var}}(\widehat p_C -\widehat p_E)$.
+$\widehat\sigma^2_{k,s} = \widehat{\text{Var}}(\widehat p_C -\widehat p_E)$.
 And the value of $\widehat\sigma^2_{k,s}$ depends on the hypothesis and
 design, i.e., whether it is a superiority design, or non-inferiority
 design, or super-superiority design. We will discuss