The t-distribution (Student’s t-distribution) is a continuous probability distribution that arises in estimating the mean of a normally distributed population when the sample size is small and the population standard deviation is unknown. The larger the sample, the more the t-distribution resembles a normal distribution.
The t-distribution plays a role in many statistical analyses including Student’s t-test for assessing the statistical significance of the difference between two sample means and in the construction of confidence intervals for the difference between two population means. As the sample size increases, the t-distribution approaches the normal distribution.
Like the F-distribution the t-distribution can be obtained using the regularized incomplete beta function with the following inputs: X = d.f. * t / (d.f. * t * t), a = d.f. / 2 and b = 0.5, where t is the value obtained in the t-test, and d.f. is the degrees of freedom. The source code below applies that method to estimate the probability of the null hypothesis for a given t-value. The code includes the gamma function and the incomplete beta function, which are both used.
#COMPILE EXE #REGISTER NONE #DIM ALL ' FUNCTION PfromT(BYVAL T AS DOUBLE,BYVAL df AS DOUBLE) AS DOUBLE LOCAL Beta, betain AS EXT IF ABS(T) > 0.0 AND df > 0.0 THEN CALL IncomplBeta(df / (df + T * T), df * 0.5#, 0.5#, Beta, betain) FUNCTION = betain ELSE FUNCTION = 1 END IF END FUNCTION ' FUNCTION PBMAIN LOCAL Result AS STRING, P, t, df AS EXT LOCAL i1 AS LONG Result = INPUTBOX$("Enter t-value, d.f.", "Get p-value") i1 = INSTR(Result, ",") t = VAL(LEFT$(Result, i1 - 1)) df = VAL(MID$(Result,i1 + 1)) P = PfromT(t, df) MSGBOX "P=" + FORMAT$(P," ##.##################"), ,"Result:" END FUNCTION ' SUB IncomplBeta(BYVAL X AS EXT, BYVAL P AS EXT, BYVAL Q AS EXT, _ BYREF beta AS EXT, BYREF betain AS EXT) ' ' Derived from FORTRAN code based on: ' algorithm AS 63 Appl. Statist. (1973), vol.22, no.3 ' Computes incomplete beta function ratio for arguments ' X between zero and one, p (=a) and q (=b) positive. ' Returns log beta and Regularized Incomplete Beta ' LOCAL ns, indx AS LONG LOCAL zero, one, acu AS EXT LOCAL psq, cx, xx, pp, qq, term, ai, rx, temp AS EXT ' define accuracy and initialise zero = 0.0 : one = 1.0 : acu = 1.0E-18 betain = 0.0 : beta = 0.0 ' test for admissibility of arguments IF(p <= zero OR q <= zero) THEN EXIT SUB IF(x < zero OR x > one) THEN EXIT SUB IF(x = zero OR x = one) THEN EXIT SUB ' calculate log of beta by using function GammLn beta = gammln(p) + gammln(q) - gammln(p + q) betain = x ' change tail if necessary psq = p + q cx = one - x IF (p < psq * x) THEN xx = cx cx = x pp = q qq = p indx = 1 ELSE xx = x pp = p qq = q indx = 0 END IF term = one ai = one betain = one ns = qq + cx * psq ' use Soper's reduction formulae. rx = xx / cx temp = qq - ai IF (ns = 0) THEN rx = xx DO term = term * temp * rx / (pp + ai) betain = betain + term temp = ABS(term) IF(temp <= acu AND temp <= acu * betain) THEN EXIT DO ai = ai + one ns = ns - 1 IF (ns >= 0) THEN temp = qq - ai IF (ns = 0) THEN rx = xx ELSE temp = psq psq = psq + one END IF LOOP ' calculate Regularized Incomplete Beta betain = betain * EXP(pp * LOG(xx) + (qq - one) * LOG(cx) - beta) / pp IF indx = 1 THEN betain = one - betain END SUB ' FUNCTION GammLn(BYVAL x AS EXT) AS EXT ' Returns Ln(Gamma()) or 0 on error ' Based on Numerical Recipes gamma.h ' Lanczos, C. 1964, "A Precision Approximation ' of the Gamma Function," SIAM Journal on Numerical ' Analysis, ser. B, vol. 1, pp. 86-96. LOCAL j AS LONG, tmp, y, ser AS EXT DIM cof(0 TO 13) AS LOCAL EXT cof(0) = 57.1562356658629235 cof(1) = -59.5979603554754912 cof(2) = 14.1360979747417471 cof(3) = -0.491913816097620199 cof(4) = 0.339946499848118887e-4 cof(5) = 0.465236289270485756e-4 cof(6) = -0.983744753048795646e-4 cof(7) = 0.158088703224912494e-3 cof(8) = -0.210264441724104883e-3 cof(9) = 0.217439618115212643e-3 cof(10) = -0.164318106536763890e-3 cof(11) = 0.844182239838527433e-4 cof(12) = -0.261908384015814087e-4 cof(13) = 0.368991826595316234e-5 IF x <= 0.0 THEN FUNCTION = 0.0 : EXIT FUNCTION ' Bad argument y = x tmp = x + 5.2421875 tmp = (x + 0.5) * LOG(tmp) - tmp ser = 0.999999999999997092 FOR j = 0 TO 13 y = y + 1 ser = ser + cof(j)/y NEXT j FUNCTION = tmp + LOG(2.5066282746310005 * ser / x) END FUNCTION [/code]