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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*} 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*} |
