Journal of Modern Applied Statistical Methods Volume 6 | Issue 1 Article 11 5-1-2007 Another Look at the Confidence Intervals for the Noncentral T Distribution Bruno Lecoutre Centre National de la Recherche Scientifique and Université de Rouen, France Follow this and additional works at: http://digitalcommons.wayne.edu/jmasm Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Lecoutre, Bruno (2007) "Another Look at the Confidence Intervals for the Noncentral T Distribution," Journal of Modern Applied Statistical Methods: Vol. 6 : Iss. 1 , Article 11. DOI: 10.22237/jmasm/1177992600 Available at: http://digitalcommons.wayne.edu/jmasm/vol6/iss1/11 This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState. Journal of Modern Applied Statistical Methods Copyright © 2007 JMASM, Inc. May, 2007, Vol. 6, No. 1, 107-116 1538 – 9472/07/$95.00 Another Look at Confidence Intervals for the Noncentral T Distribution Bruno Lecoutre Centre National de la Recherche Scientifique and Université de Rouen, France An alternative approach to the computation of confidence intervals for the noncentrality parameter of the Noncentral t distribution is proposed. It involves the percent points of a statistical distribution. This conceptual improvement renders the technical process for deriving the limits more comprehensible. Accurate approximations can be derived and easily used. Key words: Confidence intervals, noncentral t distribution, lambda-prime distribution, Bayesian inference. Introduction computed as the percent points of a statistical distribution. Unfortunately, this is not the case In spite of several recent presentations (see with the usual presentations. especially, Fidler & Thompson, 2001; Bird, Moreover, warnings about the accuracy 2002), many potential users, as well as statistical of some computer programs of the Noncentral t instructors, consider computing or teaching distribution (typically, the Noncentral t confidence intervals for the noncentrality algorithm fails for large sample size or effect parameter of the Noncentral t distribution to be size) cast doubt on some numerical results. very complex tasks. One of the conceptual Consequently, there remains the need for difficulties is the lack of explicit formula. accurate approximations that are not currently Although the considerable advances in easily available. Even when an exact computing techniques are supposed to render the computation is wanted, it needs an iterative task easy, they do not solve the conceptual algorithm, for which an accurate approximation difficulties. constitutes a good starting point. The latter state is all the more deceptive An alternative approach is proposed in in that when the number of degrees of freedom this article that results in computing the is large enough so that the Normal confidence limits as the percent points of a approximation holds the solution is very simple: statistical distribution as in the most familiar the confidence limits are given by the percent situations. An interesting consequence of this points of a Normal distribution, as for the conceptual improvement is that standard familiar case of an unstandardized difference techniques to approximate statistical between means. Thus, it can be expected that in distributions can be used in order to find easy to the general case the limits would also be use very accurate approximations. In conclusion, the question of the justification and interpretation of confidence intervals will be Bruno Lecoutre is Research Director, CNRS. briefly examined. His research interests are in experimental data Considerations and discussions analysis, Bayesian methods, foundations of regarding how and when to use confidence statistics, and the null hypothesis significance intervals for the Noncentral t distribution, may tests controversy. Contact him at ERIS, be found elsewhere. Therefore, this article is not Laboratoire de Mathématiques Raphaël Salem, methodological. In this perspective, it will be UMR 6085, C.N.R.S. et Université de Rouen sufficient, with no loss of generality, to consider Avenue de l’Université, BP 12, 76801 Saint- the elementary case of the inference about a Etienne-du-Rouvray, France. Email address: standardized difference between two means. [email protected] 107 108 CONFIDENCE INTERVALS FOR THE NONCENTRAL T DISTRIBUTION Computing confidence intervals from the pλ = Pr(t'df (λ) > tCALC) = α/2. (1) Noncentral t distribution When comparing two means, the t test The conceptual difficulties come from the fact λ statistic is the ratio (Y 1 −Y 2 ) / E of the two that finding the limit L involves as many λ − different distributions as considered values. A statistics, Y 1 Y 2 that is an estimate of the practical consequence is that it is a highly μ −μ population difference 1 2 and the standard difficult task to derive accurate approximations. error E of that estimate (see e.g., Fidler & Thomson, 2001, p. 587). In other words, E is an An alternative approach: computing confidence estimate of the standard deviation ε of the intervals as percent points of the Lambda-prime distribution sampling distribution for Y 1 −Y 2 . For instance, in the particular case of two independent groups, An alternative solution consists in computing the confidence limits for the assuming a common variance σ2, one has noncentrality parameter as percent points of a ε = σ + 1/ n1 1/ n2 . statistical distribution. When df is large enough The sampling distribution of the ratio so that the normal approximation holds, λL is α (Y 1 −Y 2 ) / E is a Noncentral t distribution with simply the 100 /2 percent point of the df degrees of freedom and a noncentrality standardized Normal distribution with mean t . This can be generalized by introducing an parameter λ, equal to (μ −μ ) /ε . This CALC 1 2 appropriate statistical distribution. Even if it has λ distribution is usually written t'df ( ). The not been made explicit in the usual noncentrality parameter is termed λ, as in Algina presentations, this distribution is in fact not and Keselman (2003), in order to avoid unfamiliar (without mentioning the fiducial and confusion with the population effect size. Bayesian presentations discussed in the Formally, the Noncentral t distribution is the conclusion). noncentrality parameter λ plus the standard Indeed, it is usual to plot pλ (or its Normal z distribution, all divided by the square complement 1–pλ) as a function of λ. An root of the usual Chi-square distribution divided illustration is given in Figure 1 for tCALC = by the degrees of freedom (see e.g., Fidler & +1.0076 with df = 22 (hence a p-value Thomson, 2001, p. 589): p = 0.3246, two-sided), which corresponds to the two-group A way data example given by Fidler λ λ + χ 2 & Thomson (2001, p. 586). The pλ value t'df ( ) = ( z) / df / df . increasingly varies from zero (when λ tends to - ∞) to one (when λ tends to +∞), so that the The traditional approach for finding the λ corresponding curve is nothing else than the lower (for instance) limit L of the noncentrality cumulative distribution function of a probability λ λ parameter uses the probability pλ that t'df ( ) distribution. Such a graphical representation is exceeds the value tCALC observed in the data in commonly proposed to get a graphical solution hand: for the confidence limits (see, for instance, Steiger & Fouladi, 1997, pp. 240), but the λ pλ = Pr(t'df ( ) > tCALC). proponents fail to recognize that, in doing this, they implicitly define the confidence limits as Then, one must vary the λ value in order to find, the percent points of this probability distribution. by successive approximations, the particular λ α value L such that pλL= /2: BRUNO LECOUTRE 109 pλ 1 0.975 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 P0=0.1623 0.10 0.025 0 -2 -1.5 -0.5 0 +0.5 +1 +1.5 +2 +2.5 +3.5 +4 λ =-0.986 λ =+2.979 L noncentrality parameter λ U Figure 1 - Plot of pλ as a function of λ for tCALC = +1.0076 and df = 22 and graphical solution for the 95% confidence interval for λ. The curve is the cumulative distribution function of the Λ'22(+1.0076) distribution. As for the Noncentral t, this distribution Thus, pλ can be computed from the distribution can be easily defined from the Normal and Chi- characterized by z + t χ 2 / df . This square distributions, but the result has not been CALC df distribution, which was considered (with no so popularized. (Y 1 −Y 2 ) / E > tCALC can be name) by Fisher (1990/1973, pp. 126-127) in the − − equivalently written as Y 1 Y 2 tCALC E > 0. fiducial framework, was called Lambda-prime in Consequently, pλ is the probability that Lecoutre (1999). It is also a noncentral distribution, again with df degrees of freedom, Y 1 −Y 2 − tCALC E exceeds zero. The sampling distribution of but with noncentrality tCALC. Formally: Y 1 −Y 2 − tCALC E can be formally defined from Λ + χ 2 independent standard Normal and Chi-square 'df (tCALC) = z tCALC df / df . distributions as: Consequently, it is possible to inverse in some 2 ε λ + − χ sense the problem in (1) and compute pλ as the ( z tCALC df / df ) . probability that the Lambda-prime distribution with noncentrality t is smaller than λ: so that CALC pλ = Pr(Λ'df (tCALC) < λ). pλ − − =Pr(Y 1 Y 2 tCALC E>0) Thus, the curve in Figure 1 is the cumulative =Pr( − z + t χ 2 / df < λ ) distribution function of the Lambda-prime CALC df distribution with 22 degrees of freedom and + χ 2 < λ noncentrality +1.0076.