The well known pure error-lack of fit test, which can be used to assess the adequacy of a linear regression model, is generalized to accommodate the case of nonreplication. The asymptotic null distribution of the proposed test statistic is derived. Also, the proposed test statistic is shown to be asymptotically comparable under general alternatives to the test statistic obtained in the case of replication. Consistency properties associated with pseudo lack of fit and pure error mean squares are given which parallel those obtained in the case of replication. In addition, the test statistic is invariant with respect to location and scale changes made to the regression variables.
"Testing Linear Regression Function Adequacy without Replication." Ann. Statist. 13 (4) 1482 - 1489, December, 1985. https://doi.org/10.1214/aos/1176349749