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.