Hkratni preizkusi neinferiornosti : delo diplomskega seminarja

Šraj, Eva (author), Smrekar, Jaka (mentor)

Diplomska naloga temelji na preizkušanju domnev glede učinkovitosti treh med seboj neodvisnih skupin zdravljenj -- eksperimentalno zdravljenje (E), referenčno zdravljenje (R) in zdravljenje s placebom (P). Pomembni predpostavki hkratnega preizkušanja domnev sta homoskedastičnost in normalnost. Opazovanje $k$-tega bolnika v skupini $i$ je označeno z $X_{ik}$ in porazdeljeno $X_{ik} \sim N(\mu_i, \sigma^2)$, za $i= E, P, R$; $k = 1, \ldots, n_i$. Oznaka $n_i$ predstavlja število bolnikov posamezne vrste zdravljenja $i$, $\delta_{ij} \geq 0$ dopustno odstopanje, za $i = E, P, R$; $i \neq j$, $z_\alpha$ pa zgornji $\alpha$-percentil standardne normalne porazdelitve. Z zavrnitvijo glavne ničelne domneve $H_0$ dokažemo superiornost eksperimentalnega zdravljenja (E) nad zdravljenjem s placebom (P) in hkrati neinferiornost eksperimentalnega (E) proti referenčnemu zdravljenju (R). Asimptotična moč za zavrnitev $H_0$ je enaka $$\begin{align*} \Phi^{\Sigma} \Big(\frac{\mu_E - \mu_P - \delta_{EP}}{\sigma \sqrt{\frac{1}{n_E} + \frac{1}{n_P}}} - z_{\alpha}, \frac{\mu_E - \mu_R + \delta_{ER}}{\sigma \sqrt{\frac{1}{n_E} + \frac{1}{n_R}}} - z_{\alpha} \Big). \end{align*}$$ Z zavrnitvijo razširjene ničelne domneve $\widetilde{H}_0$ dokažemo še superiornost referenčnega zdravljenja (R) nad zdravljenju s placebom (P). Asimptotična moč za zavrnitev $\widetilde{H}_0$ pri stopnji značilnosti $\alpha$ je enaka $$\begin{align*} \Phi^{\widetilde{\Sigma}} \Big( &\frac{\mu_E - \mu_P - \delta_{EP}}{\sigma \sqrt{\frac{1}{n_E} + \frac{1}{n_P}}} - z_{\alpha}, \frac{\mu_E - \mu_R + \delta_{ER}}{\sigma \sqrt{\frac{1}{n_E} + \frac{1}{n_R}}} - z_{\alpha}, \frac{\mu_R - \mu_P - \delta_{RP}}{\sigma \sqrt{\frac{1}{n_R} + \frac{1}{n_P}}} - z_{\alpha} \Big). \end{align*}$$

• Language: Slovenian
• Keywords: neinferiornost, načelna superiornost, asimptotična moč, združena cenilka, vzorčna varianca, zvezno porazdeljeni podatki
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2019
• PID: 20.500.12556/RUL-109975
• UDC: 519.2
• COBISS.SI-ID: 18740057
• Publication date in RUL: 11.09.2019

ŠRAJ, Eva, 2019, Hkratni preizkusi neinferiornosti : delo diplomskega seminarja [online]. Bachelor’s thesis. [Accessed 17 May 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=109975

Secondary language

• Language: English
• Title: Simultaneous non-inferiority trials

Abstract

This work is based on hypothesis testing regarding efficacy of 3 independent groups of treatment -- experimental treatment (E), active reference (R) and placebo (P). The important assumptions here are homoscedasticity and normality. The observation for the $k$th patient in group $i$ is denoted as $X_{ik}$ and is distributed as $X_{ik} \sim N(\mu, \sigma^2)$, for $i = E, P, R$; $k=1, \ldots, n_i$. The symbol $n_i$ represents the number of patients in group $i$, $\delta_{ij} \geq 0$ the inferiority/superiority margin, for $i = E, P, R$; $i \neq j$ and $z_\alpha$ upper $\alpha$-percentile of standard normal distribution. With the rejection of the principal null hypothesis $H_0$ we prove the superiority of experimental treatment (E) versus placebo (P) and, simultaneously, non-inferiority of experimental treatment (E) versus active reference (R). The asymptotic power for the rejection of $H_0$ equals $$\begin{align*} \Phi^{\Sigma} \Big(\frac{\mu_E - \mu_P - \delta_{EP}}{\sigma \sqrt{\frac{1}{n_E} + \frac{1}{n_P}}} - z_{\alpha}, \frac{\mu_E - \mu_R + \delta_{ER}}{\sigma \sqrt{\frac{1}{n_E} + \frac{1}{n_R}}} - z_{\alpha} \Big). \end{align*}$$ With the rejection of the extended null hypothesis $\widetilde{H}_0$ we prove, in addition, the superiority of active reference (R) versus placebo (P). The asymptotic power for the rejection of $\widetilde{H}_0$ equals $$\begin{align*} \Phi^{\widetilde{\Sigma}} \Big( \frac{\mu_E - \mu_P - \delta_{EP}}{\sigma \sqrt{\frac{1}{n_E} + \frac{1}{n_P}}} - z_{\alpha}, \frac{\mu_E - \mu_R + \delta_{ER}}{\sigma \sqrt{\frac{1}{n_E} + \frac{1}{n_R}}} - z_{\alpha}, \frac{\mu_R - \mu_P - \delta_{RP}}{\sigma \sqrt{\frac{1}{n_R} + \frac{1}{n_P}}} - z_{\alpha} \Big). \end{align*}$$

• Keywords: non-inferiority, superiority, asymptotic power, pooled estimator, sample variance, continuous data