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Topic: Chi-squared to normal approximation
Replies: 1   Last Post: May 19, 2013 9:10 AM

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 Luis A. Afonso Posts: 4,743 From: LIsbon (Portugal) Registered: 2/16/05
Chi-squared to normal approximation
Posted: May 18, 2013 6:47 PM

Our remote colleagues Wilson, E. B. & Hilferty M. M., did provide a paper: The distribution of chi-square able to attain the transformation Chi-squared to Standard normal reported at Proceedings of the National Academy of Sciences USA, 17, 684-688, Dec. 1931. As cited at Matthew Shultz www.rasch.org/rmt/rmt162g.htm the transform is, Y Chi-squared r df, p=1/3, approximately W~N(0,1),

___W(Y) = [(Y/r)^p - (1- 2*p*p / r)] / sqrt( 2*p*p / r

For r= 100 (50) 600 we obtained correctly 1.245(.900), 1.645(.950), 1.960(.975), 2.326(.990) as the normal fractiles.
Note: the inputted Chi-squares fractiles are those Kalkulator provide. See routine below.
Well done, Wilson, Hilferty . . .

Luis A. Afonso

REM "VINCE"
CLS
DEFDBL A-Z
DIM CHI(9, 12)
REM "0.010 ,0.025 ,0.050 ,0.100 ,0.900 ,0.950 ,0.975 ,0.990 "
REM
DATA 70.06,74.22,77.93,82.36,118.50,124.34,129.56,135.81
FOR j = 1 TO 8: READ CHI(j, 1): NEXT j
DATA 112.67,117.98,122.69,128.28,172.58,179.58,185.80,193.21
FOR j = 1 TO 8: READ CHI(j, 2): NEXT j
DATA 156.43,162.73,168.28,174.84,226.02,233.99,241.06,249.45
FOR j = 1 TO 8: READ CHI(j, 3): NEXT j
DATA 200.94,208.10,214.39,221.81,279.05,287.88,295.69,304.94
FOR j = 1 TO 8: READ CHI(j, 4): NEXT j
DATA 245.97,253.91,260.88,269.07,331.79,341.40,349.87,359.91
FOR j = 1 TO 8: READ CHI(j, 5): NEXT j
DATA 291.41,300.06,307.65,316.55,384.31,394.63,403.72,414.47
FOR j = 1 TO 8: READ CHI(j, 6): NEXT j
DATA 337.16,346.48,354.64,364.21,436.65,447.63,457.31,468.72
FOR j = 1 TO 8: READ CHI(j, 7): NEXT j
DATA 383.16,393.12,401.82,412.01,488.85,500.46,510.67,522.72
FOR j = 1 TO 8: READ CHI(j, 8): NEXT j
DATA 429.39,439.94,449.15,459.93,540.93,553.13,563.85,576.49
FOR j = 1 TO 8: READ CHI(j, 9): NEXT j
DATA 475.80,486.91,496.61,507.95,592.91,605.67,616.88,630.08
FOR j = 1 TO 8: READ CHI(j, 10): NEXT j
DATA 522.37,534.02,544.18,556.06,644.80,658.09,669.77,683.52
FOR j = 1 TO 8: READ CHI(j, 11): NEXT j
REM
p = 1 / 3
FOR k = 1 TO 11: r = 100 + (k - 1) * 50
FOR j = 1 TO 8
y = CHI(j, k)
y1 = (y / r) ^ p: u = 2 * p * p / r
y2 = y1 - (1 - u)
y3 = y2 / SQR(u)
PRINT USING "###.### "; y3;
NEXT j
PRINT
NEXT k
END

Date Subject Author
5/18/13 Luis A. Afonso
5/19/13 Luis A. Afonso