My notes taken from Kenny, D. A., Kashy, D. A., Bolger, N. (1998) Data analysis in social psychology. In D. Gilbert, S. Fiske, & G. Lindzey (eds.) Handbook of social psychology , vol. 1, pp. 233-251. Boston: McGraw-Hill.p233

majors strides have been made in the analysis of data in which persons interact with or rate multiple players
nonindependence of observations is a serious issue that is often just ignored then ANOVA is limited
replacing ANOVA by structural equation modeling OR multilevel modeling on certain kind of variables
which unit of analysis should be chosen: individual or group? If person is used as unit of analysis, the assumptions of independence is likely to be violated because persons within groups may influence one another (Kenny and Judd, 1986). Alternatively, if group (= couple, team, organization...) is used, the power of the statistical tests is likely to be reduced because there are fewer degress of freedom than there are in the analysis that uses person as the unit of analysis.
concerning the independent variable (IV) A, there is three cases: nested (when groups are assigned to levels of the IV such that every member of a given group has the same score on A with some groups at one level of A and other groups at other levels of A), crossed (when A varies within the group with some member in one level of A and other group members in another level of A) and mixed (both nested and crossed). [I often use nested variables like in my masters thesis then I will develop on this now]
for nested IV: there is a method to measure the nonindependence of the data using the intraclass correlation. Group effects occur if the scors of individual within a group are more similar to one another than are the scores of individuals who are in different groups. The intraclass correlation can be viewed as the amount of variance in the persons' scores that is due to the group, controlling for the effects of A. When the intraclass correlation is not large and total sample size and the group size are small, power is very low. Using the ANOVA: this correlation is equal to (n= number of persons per group): (mean square for groups within A - mean square for individual within groups within A)/(mean square for groups within A + (n-1)*mean square for individual within groups within A)
Summary: safer to make group as unit of analysis and so it is then necessary to collect data from a sufficient number of groups. general guideline: if there is nonindependence, then group must be used as the unit of analyis; if there is independence, the individual may be the unit of analysis. The usual standard for "sufficient power" is having and 80 percent chance of rejecting the null hypothesis.

Hopefully my data are nested then I can use ANOVA but with this discussion on which kind of unit of analysis I may choose. However, multilevel modeling might be useful and can be applied here. Here is an interesting resource to compute this index.