假設二元反應曲線為簡單線性羅吉斯迴歸模型(Simple Linear Logistic Regression Model)，在樣本數為偶數的前題下，所謂的最佳D型設計(D-Optimal Design)是直接將半數的樣本點配置在第17.6個百分位數，而另一半則配置在第82.4個百分位數。很遺憾的是，這兩個位置在參數未知的情況下是無法決定的，因此逐次實驗設計法(Sequential Experimental Designs)在應用上就有其必要性。在大樣本的情況下，本文所探討的逐次實驗設計法在理論上具有良好的漸近最佳D型性質(Asymptotic D-Optimality)。尤其重要的是，這些特性並不會因為起始階段的配置不盡理想而消失，影響的只是收斂的快慢而已。但是在實際應用上，這些大樣本的理想性質卻不是我們關注的焦點。實驗步驟收斂速度的快慢，在小樣本的考慮下有決定性的重要性。基於這樣的考量，本文將提出三種起始階段設計的方法並透過模擬比較它們之間的優劣性。 / The D-optimal design is well known to be a two-point design for the simple linear logistic regression function model. Specif-ically , one half of the design points are allocated at the 17.6- th percentile, and the other half at the 82.4-th percentile. Since the locations of the two design points depend on the unknown parameters, the actual 2-locations can not be obtained. In order to dilemma, a sequential design is somehow necessary in practice. Sequential designs disscused in this context have some good properties that would not disappear even the initial stgae is not good enough under large sample size. The speed of converges of the sequential designs is influenced by the initial stage imposed under small sample size. Based on this, three initial stages will be provided in this study and will be compared through simulation conducted by C++ language.

A/B testing refers to the statistical procedure of experimental design and analysis to compare two treatments, A and B, applied to different testing subjects. It is widely used by technology companies such as Facebook, LinkedIn, and Netflix, to compare different algorithms, web-designs, and other online products and services. The subjects participating in these online A/B testing experiments are users who are connected in different scales of social networks. Two connected subjects are similar in terms of their social behaviors, education and financial background, and other demographic aspects. Hence, it is only natural to assume that their reactions to online products and services are related to their network adjacency. In this research, we propose to use the conditional autoregressive model (CAR) to present the network structure and include the network effects in the estimation and inference of the treatment effect. The following statistical designs are presented: D-optimal design for network A/B testing, a re-randomization experimental design approach for network A/B testing and covariate-assisted Bayesian sequential design for network A/B testing. The effectiveness of the proposed methods are shown through numerical results with synthetic networks and real social networks.

Problem statement: The D-optimal design is often used in clinical research. In multi-factor clinical experiments it is natural to restrict the experiment's design space so as not to give a patient the combination of several high dose treatments simultaneously. Under such design space restrictions it is unknown what designs are D-optimal. The goal of the thesis has been to find D-optimal designs for these design spaces. Approach: Two new algorithms for finding D-optimal designs with one, two or three factors with linear models has been developed and implemented in MATLAB. Two restricted design spaces were explored. In cases when the program could not find the D-optimal design an analytic approach was used. Results: Special attention was given to the two factor model with interaction. All of the D-optimal designs for this model, N less or equal to 30, and their permutations have been listed as well as their continous designs. Conclusion: In one of the restricted design regions a simple design pattern appeared for N greater than or equal to 7. In the other restricted design region no obvious pattern was found but its continuous design could be calculated through analysis. It turned out that the number of trials at the lowest dose combination did not change when moving from the full space design to the restricted design regions. / Frågeställning: D-optimala designer är vanliga i kliniska studier. När flera faktorer (läkemedel) prövas samtidigt kan det vara nödvändigt att begränsa försöksrummet så att patienterna undviker att få en hög dos av flera faktorer samtidigt. I sådana begränsade försöksrum är det okänt vilka designer som är D-optimala. Uppsatsens mål har varit att hitta D-optimala designer i begränsade försöksrum. Metod: Två nya algoritmer för att hitta D-optimala designer med en, två eller tre dimensioner och linjära modeller har utvecklats och implementerats i MATLAB. Två begränsade försöksrum har utforskats. I de fall då MATLAB-programmet inte kunde hitta de D-optimala designerna användes analytiska metoder. Resultat: Analys av en tvåfaktormodell med interaktion utforskades särskilt noggrant. Alla D-optimala designer och permutationer av dessa i de båda begränsade försöksrummen har listats för alla N mindre än eller lika med 30, samt även deras kontinuerliga designer. Slutsats: För det ena försöksrummet upptäcktes ett mönster i designen då N är större än eller lika med 7. I det andra försöksrummet upptäcktes inget mönster och det krävdes således analytiska metoder för att finna dess kontinuerliga design. Det visade sig att antalet försök i den lägsta doskombinationen förblev oförändrat då man bytte från det fulla designrummet till de båda begränsade designrummen.