In fashioning an appropriate experimental design, the structure of the problem at hand guides us to the appropriate statistical test to use. The structure of the problem at hand has two factors. One factor incorporates related samples, and the other factor comprises independent samples. This combination of structures requires a specific type of statistical test.
Let's go through the given options step by step:
Option A, the two-way within-subjects ANOVA, is not suitable in this situation. This would be suitable if both factors were based on related samples, i.e., if each subject was undergoing multiple forms of testing. However, this is not the case in our situation since we have one factor related to independent samples. Therefore, we discard option A.
Option B is the two-way between-subjects ANOVA, which is ideal if both factors involve independent samples, meaning that each group of subjects or objects is evaluated separately, and no subject is subject to multiple tests or forms of treatment. However, this is not the case in our problem since we have one factor relating to related samples. Hence, we discard option B.
Option D is the two-way independent-related samples ANOVA. This option does not exist and the term "independent-related" is contradictory in nature and hence we can rule out this option.
Finally, we have Option C, the two-way mixed-design ANOVA. In a mixed-design ANOVA, you treat one factor as an independent-samples factor and the other as a related-samples factor. This design is most beneficial in investigating the influence of an independent variable on a dependent variable at different levels of another independent variable. Therefore, a two-way mixed-design ANOVA is appropriate in situations where an experiment design has two factors but one factor involves related samples while the other factor involves independent samples.
Therefore, the best option for an experiment design that has two factors but one factor involves related samples while the other factor involves independent samples is a two-way mixed-design ANOVA.