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The Holly Seven Steps of Developing a Statistical Test

This guide outlines the seven crucial steps for developing a statistical test, covering topics such as data generation, model specification, hypotheses formulation, test statistic derivation, critical region construction, decision rule application, and the use of bootstrap for detecting violations of distributional assumptions.

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The Holly Seven Steps of Developing a Statistical Test

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  1. The Holly Seven Steps of Developing a Statistical Test Tugrul Temel Center for World Food Studies Free University, Amsterdam February 2000

  2. Step 1. A Data-Generating Process • Given a random sample of n i.i.d. random variables (r.v.) • Independence of obs. on a r.v • Identical (or constant) prob. dist. for r.v. • Stationary dist. for r.v. • The Borel field • The history does not forget • sample space, events, prob. of a set of events (or test size) • A Borel measurable function • Integration over events

  3. The Holy Seven Stepsfor a Statistical Test • A data-generating process • A model • The hypotheses • Asymptotic distributions of distance functions implied by the hypotheses • A test statistic • A critical region • A decision rule

  4. Step 2. A Model • Model specification • Parametric, semi-parametric, and non-parametric • Misspecification in • degree of polynomial, choice of variables, or both • Measurement error in • dependent, independent, or both • Nonlinearity in • policy variables, coefficients, or both

  5. Step 3. The Hypotheses • The null and alternative hypotheses • The underlying distance function (d) • The underlying loss and risk functions • Subjectivity of risk functions • Form of objective functions, i.e., Mean Squared Error • A transformation of d, T=T(d)

  6. Step 4. The Asymptotic Distribution of Tne • Tn is an estimator of T • Tne is the estimated Tn • A value of Tne far from zero is evidence against the null. To tell how far Tne must be from zero to reject the null, we find its asymptotic distribution. • Apply a CLT and a LLN to obtain a normal dist. for Tne • Estimate the variance of Tne

  7. Step 5. A Test Statistic (TS) • Define TS=f(Tne) • Using different random samples (i), find a distribution of TS(i) • Is the TS sufficient? • In reality, we have only one random sample, i.e., i=1. Obtain TS(i) by applying • The classical approach • The Bayesian approach • The bootstrap

  8. Step 6. A Critical Region • Construct a critical region given the distribution of TS(i) and the test size (alpha) • The link between the critical region and the Borel field

  9. Step 7. A Decision Rule • Reject the null if TS falls in the critical region or accept the null if it falls in the confidence region • Asymptotic versus the bootstraped values of critical region

  10. The Bootstrap • The bootstrap to detect • violations of parametric distributional assumptions, like a non-normal error structure assumed in OLS regression • characteristics of TS

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