This paper seeks to distinguish the principles upon which testing of statistical hypotheses may be based and the practical methods which these principles generate. Six examples are given for the case of nested hypotheses as illustrations. In particular, Seber's (1964) conclusion that the Wald, Lagrange Multiplier and Likelihood Ratio Principles all lead to exactly the same test statistic in the case of a linear hypothesis, is re-examined in the light of a strict interpretation of these principles. Simple relations between various test statistics and their distributions are outlined. The concept of an artificial model is analyzed. A distinction is made between an artificial model that is in some sense an 'unrestricted' specification and one that is simply an algorithm. For non-nested hypotheses, an artificial model with prior information on the parameters is regarded as conforming to the Wald Principle. When arbitrary numerical methods are used as 'identifying' restrictions, the artificial model reduces to an algorithm since it cannot reasonably be 'accepted'.