.- help for ^nct^ .- Non-central t programs ---------------------- ^nctprob^ t' delta df /* yields p ^nctinv^ p delta df /* yields t' ^nctncp^ t' p df /* yields delta ^nctn^ t' delta p /* yields df + 1 ^nct2^ t' delta df /* yields two-sided p ^nct2inv^ p delta df /* yields two-sided t' where ^t'^ is the observed t-value ^delta^ is the noncentrality parameter ^df^ is the degrees of freedom (^df^ is a positive integer) ^p^ is the probability (0 < ^p^ < 1) For each program, entering the program name with no parameters displays the command syntax. Description ----------- Let Y and Z be independent random variables, where Z ~ N(delta,1) (Z is distributed Normal with mean delta and variance 1) Y ~ ChiSq(n) (Y is distributed chi-square with degrees of freedom n). Then, X = Z / sqrt(Y/n) is said to have a noncentral t distribution with noncentrality delta and degrees of freedom n. That is, X ~ t(delta, n). ^nct^xxxx is a family of immediate programs, all related to the noncentral t distribution. Each one-sided program computes a missing parameter, given the other parameters, such that P(t<=^t'^| ^delta^, ^df^) = ^p^. for | use ------+-------------------------------------------------------- ^p^ | ^nctprob^ -- Cumulative non-central t probabilities ^t'^ | ^nctinv^ -- Inverse cumulative non-central t values ^delta^ | ^nctncp^ -- Noncentrality parameter of the non-central t ^df^ | ^nctn^ -- Sample size for the cumulative non-central t There are also two programs that yield two-sided values from a noncentral t, defined such that P(|t|<=^t'^| ^delta^, ^df^) = 1 - ^p^. for | use ------+-------------------------------------------------------- ^p^ | ^nct2^ -- 2-sided non-central t probabilities ^t'^ | ^nct2inv^ -- Inverse 2-sided non-central t values The core program, ^nctprob^, computes probabilities from the cumulative non-central t distribution from negative infinity to ^t'^ for noncentrality parameter, ^delta^, and positive, integer degrees of freedom, ^df^. That is, ^nctprob^ computes ^p^ such that P(t<=^t'^| ^delta^, ^df^) = ^p^. Each program prints its computed value and returns it in global ^S_1^ and in a result ^r()^. The value computed and the name of the returned parameter for each program are: ^nctprob^ -- ^r(p)^ -- the probability p ^nctinv^ -- ^r(t)^ -- the critical t' ^nctncp^ -- ^r(delta)^ -- the noncentrality parameter delta ^nctn^ -- ^r(n)^ -- the minimum n ^nct2^ -- ^r(p)^ -- the two-sided tail probability ^nct2inv^ -- ^r(t)^ -- the two-sided critical t' ^nctn^ computes the minimum ^n^ such that, for ^df^ = ^n^ - 1, when ^p^ < 0.5, P(t<=^t'^|^delta^, ^df^) <= ^p^, and when ^p^ > 0.5, 1 - P(t<=^t'^|^delta^, ^df^) <= ^p^. Warning: Convergence time for ^nctn^ is a function of the computed minimum ^n^ and increases greatly when that ^n^ gets large. Because the limit of the noncentral t (as n gets large) is the noncentral z, a comparison of the desired probability ^p^ to P(z < ^t'^ - ^delta^) is informative. If these values are within .005 of each other, convergence time will be noticably non-negligible. Notes ----- These programs call ^integ^. The user is cautioned that the initial Stata 6.0 release of ^integ^ (version 3.0.4) has a bug. Please install version 3.0.5 or later. Some of these programs require installation of programs ^ridder^ (see STB-24, insert ssi5.4 for ^ridder^). Author ------ Thomas J. Steichen Examples -------- . ^nctprob 3.6 3.0 11^ > gives p for t' = 3.6, delta = 3.0, df = 11 . ^nctinv .95 2.0 17^ > gives t' for p = .95, delta = 2.0, df = 17 . ^nctncp 4.045 .95 17^ > gives delta for t' = 4.045, p = .95, df = 17 . ^nctinv 4 2 .95^ > gives n for t' = 4, delta = 2, p = .95 . ^nct2 3.1 2.0 17^ > gives p for |t'| = 3.1, delta = 2.0, df = 17 . ^nct2inv .05 2.0 17^ > gives |t'| for p = .05, delta = 2.0, df = 17 . ^nctprob^ > displays the command syntax