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