frank_loglik_copula_scale

<code>frank_loglik_copula_scale()</code> computes the loglikelihood on the copula
scale for the Frank copula which is parameterized by <code>theta</code> as follows:
$$ C(u, v) = - \frac{1}{\theta} \log \left[ 1 - \frac{(1 - e^{-\theta u})(1 - e^{-\theta v})}{1 - e^{-\theta}} \right]$$

In a clinical trial, it frequently occurs that the most credible
outcome to evaluate the effectiveness of a new therapy (the true endpoint) is
difficult to measure. In such a situation, it can be an effective strategy to
replace the true endpoint by a (bio)marker that is easier to measure and that
allows for a prediction of the treatment effect on the true endpoint (a
surrogate endpoint). The package 'Surrogate' allows for an evaluation of the
appropriateness of a candidate surrogate endpoint based on the meta-analytic,
information-theoretic, and causal-inference frameworks. Part of this software
has been developed using funding provided from the European Union's Seventh
Framework Programme for research, technological development and demonstration
(Grant Agreement no 602552), the Special Research Fund (BOF) of Hasselt
University (BOF-number: BOF2OCPO3), GlaxoSmithKline Biologicals, Baekeland
Mandaat (HBC.2022.0145), and Johnson & Johnson Innovative Medicine.

Wim der Elst

Surrogate

Evaluation of Surrogate Endpoints in Clinical Trials

Wim Van Der Elst

Florian Stijven

Fenny Ong

Dries De Witte

Paul Meyvisch

Alvaro Poveda

Ariel Alonso

Hannah Ensor

Christoper Weir

Geert Molenberghs

frank_loglik_copula_scale function

<dl><dt>theta</dt>
<dd>Copula parameter</dd>
<dt>u</dt>
<dd>A numeric vector. Corresponds to first variable on the copula scale.</dd>
<dt>v</dt>
<dd>A numeric vector. Corresponds to second variable on the copula
scale.</dd>
<dt>d1</dt>
<dd>An integer vector. Indicates whether first variable is observed or
right-censored,<ul>
<li><code>d1[i] = 1</code> if <code>u[i]</code> corresponds to non-censored value</li>
<li><code>d1[i] = 0</code> if <code>u[i]</code> corresponds to right-censored value</li>
<li><code>d1[i] = -1</code> if <code>u[i]</code> corresponds to left-censored value</li>
</ul></dd>
<dt>d2</dt>
<dd>An integer vector. Indicates whether first variable is observed or
right-censored,<ul>
<li><code>d2[i] = 1</code> if <code>v[i]</code> corresponds to non-censored value</li>
<li><code>d2[i] = 0</code> if <code>v[i]</code> corresponds to right-censored value</li>
<li><code>d2[i] = -1</code> if <code>v[i]</code> corresponds to left-censored value</li>
</ul></dd>
<dt>return_sum</dt>
<dd>Return the sum of the individual loglikelihoods? If <code>FALSE</code>,
a vector with the individual loglikelihood contributions is returned.</dd></dl>

Arguments

Loglikelihood on the Copula Scale for the Frank Copula — frank_loglik_copula_scale

<dl>

<dt>theta</dt>
<dd>Copula parameter</dd>


<dt>u</dt>
<dd>A numeric vector. Corresponds to first variable on the copula scale.</dd>


<dt>v</dt>
<dd>A numeric vector. Corresponds to second variable on the copula
scale.</dd>


<dt>d1</dt>
<dd>An integer vector. Indicates whether first variable is observed or
right-censored,<ul>
<li><code>d1[i] = 1</code> if <code>u[i]</code> corresponds to non-censored value</li>
<li><code>d1[i] = 0</code> if <code>u[i]</code> corresponds to right-censored value</li>
<li><code>d1[i] = -1</code> if <code>u[i]</code> corresponds to left-censored value</li>
</ul></dd>


<dt>d2</dt>
<dd>An integer vector. Indicates whether first variable is observed or
right-censored,<ul>
<li><code>d2[i] = 1</code> if <code>v[i]</code> corresponds to non-censored value</li>
<li><code>d2[i] = 0</code> if <code>v[i]</code> corresponds to right-censored value</li>
<li><code>d2[i] = -1</code> if <code>v[i]</code> corresponds to left-censored value</li>
</ul></dd>


<dt>return_sum</dt>
<dd>Return the sum of the individual loglikelihoods? If <code>FALSE</code>,
a vector with the individual loglikelihood contributions is returned.</dd>

</dl>

Loglikelihood on the Copula Scale for the Frank Copula

frank_loglik_copula_scale: Loglikelihood on the Copula Scale for the Frank Copula

Description

Usage

Value

Arguments