When data come from several independent studies for the purpose of estimating treatment control differences, meta-analysis can be carried out either on the best linear unbiased estimators computed from each study or on the pooled individual patient data modelled as a two-way model without interaction, where the two factors represent the different studies and the different treatments. Assuming that observations within and between studies are independent having a common variance, Olkin and Sampson (1998) have obtained the surprising result that the two meta-analytic procedures are equivalent, i.e., they both produce the same estimator. In this article, the same equivalence is established for the two-way fixed-effects model without interaction with the only assumption that the observations across studies be independent. A consequence of the equivalence result is that, regardless of the covariance structure, it is possible to get an explicit representation for the best linear unbiased estimator of any vector of treatment contrasts in a two-way fixed-effects model without interaction as long as the studies are independent. Another interesting consequence is that, for the purpose of best linear unbiased estimation, an unbalanced two-way fixed-effects model without interaction can be treated as several independent unbalanced one-way models, regardless of the covariance structure, when the studies are independent.