This paper seeks to distinguish between the principles upon which testing of statistical hypotheses may be based and the practical methods which these principles generate. 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 the principles. Simple relations between various test statistics and their distributions are outlined. These are then applied to two well-known methods of testing for structural change. It is found that the cusum of squares test using recursive residuals is based upon the Lagrange Multiplier Principle while the prediction interval (or Chow-) test is based upon the Wald Principle. Relations between these tests and some straightforward extensions are outlined.