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The Behrans -Fisher problem without tears
Let be two INDEPENDENT samples
X~N(mX, varX): nX , Y~N(mY, varY): nY.
One intend to test
H0: mX= mY against Ha: mX =/ mY.
It´s well-know if that the variances are different it wasn´t be possible to derive the distribution of mX-mY. This is the Behrans-Fisher problem.
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Let us define (as usual) the test statistics:
____ T = (mXo - mYo) / [s * sqr (1/nX +1/nY)]
____ s ^ 2 = (ssdX + ssdY) / (nX + nY- 2)
____ ssd = sum of squared deviations from the observed sample means mXo, mYo..
It is well-known that if varX= varY then T follows a T Student Distribution with nX + nY - 2 degrees of freedom.
Based on that nX, nY are given and that then W=T depends only from the ratio var X/ var Y one can simulate (Monte Carlo) the .95 and .99 fractiles of the test statistics.
More precisely, a Table of Critical Values, 5% and 1% significance levels, of W was made based on
X~N(0, sqrt(varX)):nX , Y~N(0, sqrt(varY)):nY for different sample size pairs nX, nY.
___rs = sigmaX / sigmaY = 2
______nX, nY_________5%________1%______
______10,10_____+/- 2.151____+/- 2.991____
______11, 9________ 2.295_______3.194____
______12, 8________ 2.458_______3.412____
______13, 7________ 2.630_______3.644____
______14, 8________ 2.881_______3.918____
___rs = 3
______nX, nY_________5%________1%______
______10,10_____+/- 2.191____+/- 3.079____
______11, 9________ 2.410_______3.392____
______12, 8________ 2.648_______3.739____
___rs = 4
______nX, nY_________5%________1%______
______10,10_____+/- 2.212____ +/- 3.135____
______11, 9________ 2.467_______3.507____
______12, 8________ 2.745_______3.936____
Note: For rs=1 exactly one have
_ T(18, 5%) = +/- 2.101 , T(18, 1%) = +/- 2.878.
Modus Operandi
____From data get rs,
____to calculate the samples taken rs by
____ X~N(0, 1).nX, Y~N(0, rs): nY
____to obtain T, comparison with the critical values in order to get a sound decision, what is rather doubtful taking into account the available methods.
Example for RIGHT-TAIL TEST
_X___ 120,107,110,116,114,111,113,117,114,112
_Y___ 110,111,107,108,110,105,107,106,111,111
____W = 3.484
_____rs = 1.6373
___p(W > 3.484) ~~0.00145 (1´000´000 samples)
__T(5%) = +/- 2.126, T(1%) = +/- 2.934
Luis Amaral Afonso |
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