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Topic: Three ways to get the Chi-square critical values
Replies: 1   Last Post: Sep 25, 2013 5:12 PM

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Luis A. Afonso

Posts: 4,617
From: LIsbon (Portugal)
Registered: 2/16/05
Three ways to get the Chi-square critical values
Posted: Sep 25, 2013 5:57 AM
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Three ways to get the Chi-square critical values

The main goal is obviously to evaluate how exact is the Wilson-Hilferty (WH) approximation,Chi-square, 20 d.f. Distribution 2.5% , 97.5% , Critical Values. A second way is to use Monte Carlo, MC simulations: normal X ~ N(0,1):20 by an exact method (Box-Muller algorithm, BM) are simulated the Chi-square obtained by squaring & sum the values. Finally the third method consists in to use the W-H formula
__r=20, w~ N(0, 1), k=2/(9*r), and
Chi2= r*( w*sqrt(k) + (1-k) )^ 3
It is evident, for practical uses, the latter way interest: we discard the, say, 20-sized normal samples synthesis: one operation is enough to get the Chi2 value. Therefore we ask: how is it the amount of error associated to the approximation?
____

Results: < N222N > Routine
(2´000´000 M.C. samples)

__n=20___tables: 9.59, 34.17
___________9.59______34.12_______MC
___________9.57______34.18_______WH
__n=25_________13.12, 40.65
__________13.12______40.64_______
__________13.10______40.64_______
__n=30_________16.79, 46.98
__________16.79______46.96_______
__________16.78______46.98_______

This results are persuasive how accurate is the WH approximation. However the final objective of this confirmation is surely not the futile one to find out Critical Values. To obtain swiftly the ssd (sum of squared deviations) distribution of normal samples should have a rather different worth.
Luis A. Afonso
REM "N222N"
CLS : PRINT
DEFDBL A-Z
COLOR 4
LOCATE 4, 26: PRINT " < N222N > "
LOCATE 5, 20
INPUT " degrees of freedom "; n
LOCATE 6, 20
INPUT " how many samples "; all
REM
REM
COLOR 7
RANDOMIZE TIMER
DIM aw(8001), bw(8001)
pi = 4 * ATN(1)
REM
REM
FOR rpt = 1 TO all
14 rpt = rpt + 1
LOCATE 10, 50
PRINT USING "#########"; all - rpt
chi1 = 0
FOR t = 1 TO n
aa = SQR(-2 * LOG(RND))
x = 0 + 1 * aa * COS(2 * pi * RND)
chi1 = chi1 + x * x
NEXT t
REM
REM w
r = n: k = 2 / (9 * r)
aa = SQR(-2 * LOG(RND))
w = 0 + 1 * aa * COS(2 * pi * RND)
chi2 = r * (w * SQR(k) + (1 - k)) ^ 3
REM
REM
REM
IF chi1 > 80 THEN chi1 = 80
IF chi2 > 80 THEN chi2 = 80
ph1 = INT(100 * chi1 + .5)
ph2 = INT(100 * chi2 + .5)
aw(ph1) = aw(ph1) + 1
bw(ph2) = bw(ph2) + 1
bg = rpt / (all / 10)
IF bg <> INT(bg) THEN GOTO 14
REM
REM
c(1) = .025: c(2) = 1 - c(1)
FOR ci = 1 TO 2
cc = c(ci): s = 0
FOR t = 0 TO 8000
s = s + aw(t) / rpt
IF s > cc THEN GOTO 17
NEXT t
17 PRINT USING " ##.## #.#### "; t / 100; s;
NEXT ci: PRINT : PRINT
c(1) = .025: c(2) = 1 - c(1)
FOR ci = 1 TO 2
cc = c(ci): s = 0
FOR t = 0 TO 8000
s = s + bw(t) / rpt
IF s > cc THEN GOTO 18
NEXT t
18 PRINT USING " ##.## #.#### "; t / 100; s;
NEXT ci: PRINT : PRINT
NEXT rpt
REM
END



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