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199 lines
5.5 KiB
Plaintext
199 lines
5.5 KiB
Plaintext
*! version 1.0.1 TJS 9jun2000
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program define nctprob, rclass
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version 6.0
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args t delta df extra
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if "`df'" == "" | "`extra'" != "" {
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di in gr "Syntax for " in wh "nctprob" _c
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di in gr ", the cumulative non-central t"
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di in gr "distribution from negative infinity to t', is: " _n
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di in wh " nctprob " in gr "t' delta df" _n
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di in gr " where " in wh "t' " in gr "is the observed t"
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di in wh " delta " in gr "is the noncentrality parameter"
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di in wh " df " in gr "is the degrees of freedom" _n
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di in wh " nctprob " in gr "computes " in wh "p" _c
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di in gr " such that P(t<=" in wh "t'" in gr "| " in wh "delta" _c
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di in gr ", " in wh "df" in gr ") = " in wh "p"
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di in gr " and returns the value in result " _c
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di in wh "r(p) " in gr "and global " in wh "S_1" in gr "."
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global S_1 = .
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return scalar p = .
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exit 9
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}
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if `df' != int(`df') | `df' < 1 {
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di in re "degrees of freedom must be a positive integer"
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global S_1 = .
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return scalar p = .
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exit 498
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}
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local even = mod(`df',2) == 0
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/* numerical calculation of C(h,a) requires -preserve-
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but C(h,a) is only needed for odd df's */
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if !`even' { preserve }
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tempname A B h a p C M0 M1 M2 ak Mo Me k Mk
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tempvar x y
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scalar `A' = `t' / sqrt(`df')
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scalar `B' = `df' / (`df' + (`t')^2)
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scalar `h' = `delta' * sqrt(`B')
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if `even' { scalar `p' = normprob(-(`delta')) }
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else {
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scalar `p' = normprob(-(`delta' * sqrt(`B')))
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scalar `a' = abs(`A')
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qui range `x' 0 `a' 1001
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gen `y' = exp(-((`h')^2 / 2) * (1 + (`x')^2)) / (1 + (`x')^2)
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qui integ `y' `x'
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scalar `C' = r(integral)/ (2 * _pi)
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scalar `p' = `p' + 2 * `C' * sign(`A')
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}
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if `df' == 1 {
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di _n in gr " p =" in ye %10.6f `p'
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di _n in gr " P(t <= " `t' " | delta = " _c
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di in gr `delta' ", df = " `df' _c
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di in gr ") = " in ye `p'
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global S_1 = `p'
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return scalar p = `p'
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exit
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}
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scalar `M0' = `A' * sqrt(`B') * normd(`delta' * sqrt(`B'))
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scalar `M0' = `M0' * normprob(`delta' * `A' * sqrt(`B'))
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if `df' == 2 {
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scalar `p' = `p' + sqrt(2 * _pi) * `M0'
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di _n in gr " p =" in ye %10.6f `p'
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di _n in gr " P(t <= " `t' " | delta = " _c
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di in gr `delta' ", df = " `df' _c
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di in gr ") = " in ye `p'
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global S_1 = `p'
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return scalar p = `p'
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exit
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}
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scalar `M1' = `A' * normd(`delta') / sqrt(2 * _pi)
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scalar `M1' = `B' * (`delta' * `A' * `M0' + `M1')
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if `df' == 3 {
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scalar `p' = `p' + 2 * `M1'
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di _n in gr " p =" in ye %10.6f `p'
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di _n in gr " P(t <= " `t' " | delta = " _c
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di in gr `delta' ", df = " `df' _c
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di in gr ") = " in ye `p'
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global S_1 = `p'
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return scalar p = `p'
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exit
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}
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scalar `M2' = `B' * ( `delta' * `A' * `M1' + `M0') / 2
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if `df' == 4 {
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scalar `p' = `p' + sqrt(2 * _pi) * (`M0' + `M2')
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di _n in gr " p =" in ye %10.6f `p'
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di _n in gr " P(t <= " `t' " | delta = " _c
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di in gr `delta' ", df = " `df' _c
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di in gr ") = " in ye `p'
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global S_1 = `p'
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return scalar p = `p'
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exit
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}
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* calculate Mk's for k = 3 to `df'-2 and sum odds and evens
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scalar `ak' = 1
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scalar `Mo' = `M1'
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scalar `Me' = `M0' + `M2'
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scalar `k' = 3
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while `k' <= `df' - 2 {
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scalar `ak' = 1 / ((`k' - 2) * `ak')
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scalar `Mk' = `ak' * `delta' * `A' * `M2' + `M1'
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scalar `Mk' = (`k'-1) * `B' * `Mk' / `k'
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if mod(`k',2) == 0 { scalar `Me' = `Me' + `Mk' }
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else { scalar `Mo' = `Mo' + `Mk' }
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scalar `M1' = `M2'
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scalar `M2' = `Mk'
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scalar `k' = `k' + 1
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}
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if `even' { scalar `p' = `p' + sqrt(2 * _pi) * `Me' }
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else { scalar `p' = `p' + 2 * `Mo' }
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di _n in gr " p =" in ye %10.6f `p'
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di _n in gr " P(t <= " `t' " | delta = " _c
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di in gr `delta' ", df = " `df' _c
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di in gr ") = " in ye `p'
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global S_1 = `p'
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return scalar p = `p'
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exit
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end
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/* ------------------------------------------------------------
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Note: formula implemented above is from
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D. B. Owen (Technometrics 10(3):445-478, 1968)
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Let
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t be the observed t-value
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d be the non-centrality parameter
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v be the degrees of freedom
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G(z) be the cumulative standard Normal distribution
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G'(z) be the standard Normal density function
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Define
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A = t / sqrt(v)
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B = v / (v + t^2)
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then
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M = 0
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-1
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M = A * sqrt(B) * G'(d * sqrt(B)) * G(d * A * sqrt(B))
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0
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M = B * [ d * A * M + A * G'(d) / sqrt(2 * pi)]
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1 0
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M = B * [ d * A * M + M ]
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2 0 1
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and, for k>= 3,
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M = (k - 1) * B * [ a * d * A * M + M ] / k
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k k k-1 k-2
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where a = 1 / [(k - 2) * a ] and a = 1
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k k-1 2
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Finally, for even df's,
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P{T <= t | d, v} =
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G(-d) + sqrt(2 * pi) * [M + M + ... + M ]
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0 2 v-2
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and for odd df's,
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P{T <= t | d, v} =
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G(-d * sqrt(B)) + 2 * C(d * sqrt(B), A) + 2 * [M + M + ... + M ]
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1 3 v-2
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where
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abs(a)
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1 / exp[-h^2 / 2 * (1 + x^2)]
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C(h,a) = sign(a) -------- | --------------------------- dx
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2 * pi / 1 + x^2
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x=0
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Note: integral C(h,a) is computed numerically in this program.
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------------------------------------------------------------ */
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