Rediscovering a Little Known Fact about the t-test and the F-test: Algebraic, Geometric, Distributional and Graphical Considerations
Keywords:Binomial proportion, F -test, Nested models, Null hypothesis, Orthogonal sum of squares decomposition, Test statistic
We discuss the role that the null hypothesis should play in the construction of a test statistic used to make a decision about that hypothesis. To construct the test statistic for a point null hypothesis about a binomial proportion, a common recommendation is to act as if the null hypothesis is true. We argue that, on the surface, the one-sample t -test of a point null hypothesis about a Gaussian population mean does not appear to follow the recommendation. We show how simple algebraic manipulations of the usual t-statistic lead to an equivalent test procedure consistent with the recommendation. We provide geometric intuition regarding this equivalence and we consider extensions to testing nested hypotheses in Gaussian linear models. We discuss an application to graphical residual diagnostics where the form of the test statistic makes a practical difference. By examining the formulation of the test statistic from multiple perspectives in this familiar example, we provide simple, concrete illustrations of some important issues that can guide the formulation of effective solutions to more complex statistical problems.
A. AGRESTI, B. A. COULL (1998). Approximate is better than “exact” for interval estimation of binomial proportions. The American Statistician, 52, no. 2, pp. 119–126.
G. CASELLA, R. BERGER (2002). Statistical Inference. Duxbury-Thomson Learning, Pacific Grove.
F. CHAO, P. GERLAND, A. R. COOK, L. ALKEMA (2019). Systematic assessment of the sex ratio at birth for all countries and estimation of national imbalances and regional reference levels. Proceedings of the National Academy of Sciences, 116, no. 19, pp. 9303–9311.
R. F. ENGLE (1984). Chapter 13 Wald, likelihood ratio, and Lagrange multiplier tests in econometrics. Elsevier, vol. 2 of Handbook of Econometrics, pp. 775–826.
I. GOOD (1986). Comments, conjectures, and conclusions: C258 editorial note on c257 regarding the t-test. Journal of Statistical Computation and Simulation, 25, no. 3-4, pp. 296–297.
L. R. LAMOTTE (1994). A note on the role of independence in t statistics constructed from linear statistics in regression models. The American Statistician, 48, no. 3, pp. 238–240.
J. J. LEFANTE JR, A. K. SHAH (1986). C257. a note on the one-sample t-test. Journal of Statistical Computation and Simulation, 25, no. 3-4, pp. 295–296.
E. L. LEHMANN (1986). Testing Statistical Hypotheses. JohnWiley & Sons, New York.
D. S. MOORE, G. P. MCCABE, B. A. CRAIG (2012). Introduction to the Practice of Statistics. WH Freeman, New York.
G. A. SACHER, E. F. STAFFELDT (1974). Relation of gestation time to brain weight for placental mammals: implications for the theory of vertebrate growth. The American Naturalist, 108, no. 963, pp. 593–615.
A. K. SHAH, K. KRISHNAMOORTHY (1993). Testing means using hypothesis-dependent variance estimates. The American Statistician, 47, no. 2, pp. 115–117.
A. K. SHAH, J. J. LEFANTE JR (1987). C293. a note on using a hypothesis-dependent variance estimate. Journal of Statistical Computation and Simulation, 28, no. 4, pp. 347–349.
S. WEISBERG (2014). Applied Linear Regression. Wiley Series in Probability and Statistics. Wiley, New York. URL https://books.google.com/books?id=FHt-AwAAQBAJ.
S. YANG, K. BLACK (2019). Using the standard wald confidence interval for a population proportion hypothesis test is a common mistake. Teaching Statistics, 41, no. 2, pp. 65–68.
How to Cite
Copyright (c) 2022 Statistica
This work is licensed under a Creative Commons Attribution 3.0 Unported License.