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and the Welsh approximation with regard to the two sample T testIntroductionCalculating a confidence interval or credible interval for the differences in means of two normally distributed samples is one of the more fundamental questions asked of statistics. However in text books it is nearly always assumed that these two sample problems have equal variance, so that C.I. can be calculated with ubiquitous Student’s T distribution. This is rarely the state of affairs in the real world. The common approach taught in numerous textbooks, is the use of the Welsh approximation, which tries to approximate the effects of unequal variance by reducing the degrees of freedom to some usually non integer value.The Behrens distribution, described in 1929, attempts to define the necessary two sample unequal variance distribution exactly, but even now only approximations exist for calculating densities, percentiles and quantiles. Another possible problem is that reference priors do not exist for the Behrens-Fisher problem that would lead to solutions that would agree exactly with a Classical statistician’s re-sampling perspective.Note About How Behrens Distribution Is Calculated.I have found the R code presented in Lee 2003 to be very slow and can give very unstable results. Instead an R package called “tdist” was used, which could efficiently calculate linear combinations of t random variables.The B-F distribution is distributed as...BF(df1,df2,θ) ~ T2cos(θ) – T1sin(θ)(1) [page 145 of Lee]whereθ = arctan( (s1/sqrt(n1)) * (s2/sqrt(n2)))T1 ~ T( df = n1-1 )T2 ~ T( df = n2-1 ) can loosely be interpreted as a relative coefficient of variability between the two samples.  = 45 degrees and n1 = n2 implies that that two samples have equal variance and equal sample sizes, and thus the T distribution is a special case of the B-F distribution.Thus the linear coefficients for the tdist package would be [cos(θ), -sin(θ)].Using The Behrens Distribution With Substantial Prior InformationPrior information for each sample can be specified in terms of v0, n0, θ0, and S0. Formulas from the top of Lee page 69 should be helpful.H4sIAAAAAAACC7t+9tgsBjBoYHNgYuAEsS5GMYIYzKKMDAxMDDJgWVYg5mSCsZgYGaEsRqb///+D WXqMElAxbrg6HiYHpgY2ISBLjY2fQYrhP0gxgwCQfwDIWgbEDmwMDM+AhnFD1fAw+CaWZIRUFqQy MCSA7f7N1PAP4sIJYFtZmBgEQjJzU4sV/FLLFYLycxPzGHYs+Vx+EIgVDL6W7xBMq2UBqtMFKuYA 0kZcDYxlrCCdeZUw/hkmEL8MzjdhgPDR7QN5FQgFgitzk/JzGBiCBbh+OnwUxmEfIwOI/oBkbwAL yAhtuD2LwD6whdqj8B/JHpL8BbGHEWrPA8ZCZpARBpUw/hGwfwwruaChygUOebA10DAVYGAH8/aA 44qRiUkpuLK4JDWXoRZkDwNQhyJDF1gDiIj5LMyA7kdmsH4Ag+zYh0ACAAA= H4sIAAAAAAACC7t+9tgsBjBYwObAxMgJYn2IYmQAMphFGRkYmBhkwLKsQMzJBGMxMTJCWYxM//// B7P0GCWgYtxwdTxMDkwL2ISALDU2fgYphv8gxQwCQP4BIGsZECewMTA8AxrGDVXDw+CbWJIRUlmQ ysCQALb7N1PDP4gLJ4BtZWFiEAjJzE0tVvBLLVcIys9NzGPYseRz+UEgVjD4Wr5DMK2UBahOF6iY A0gbcTUwLmMF6cyrhPFfMaHyrRggfHT7QF4FQoHgytyk/BwGhhcCXAwOH4Vx2MfIAKI/INnbwAIy Qhtuzy6wD2yh9ij8R7KHJH9B7GGE2vOAcSEzyAiDShj/Dtg/hpVc0FDlAoc82BpomAowsIN5e8Bx xcjEpBRcWVySmstQCrKHAahDkaELrAFExHwWZkD3IzNYPwCTBxXUQAIAAA== H4sIAAAAAAACC7t+9tgsBjBo4HZgYuAEsS6GM4IYzJsZgQSDDFiWFYg5mWAsJkZGKIuR6f///2CW HqMEVIwbro6HyYGpgVsIyFJj42eQYvgPUswgAOQfALKWAbEDNwPDM6Bh3FA1PAy+iSUZIZUFqQwM CQwcQNFfTBAdIMACxLqMELNFmGYxqHKAWMJA1lkOkDt/Myn8h6icwAhWz8QgEJKZm1qs4JdarhCU n5uYx7Bjyefyg0CsYPC1fIdQmizETEawXUZcDxhvcoG0GlbC+MGsIL4BnK/Igsq/wQBRD7G/4R95 9jOB/fYB7o4GxomcICP0K2H8+xwgviac/4AJxNfAsJeRRH9D7GWC22sO9n8e3B4JsL0VcH4EO6p8 FjOEj8X/TEAoEFyZm5Sfw8BwnZdrx4KPwkT734gNZIQ23J5tYJ/ZYvMvEwuSPYWcXNy47cH0bw04 Pgvh9ogxQPhc0BTJBU614OhkgNgtwMAO5u0Bp3NGJial4MriktRcBlmQPQxAHYoMXWANIAV7Pgsz oPuRGawfAMOSiyt8AwAA H4sIAAAAAAACC7t+9tgsBjBYIJLAxMAJYl30YAQxmM2ZgASDDFiWFYg5mWAsJkZGKIuR6f///2CW HqMEVIwbro6HKYFpgYgQkKXGxs8gxfAfpJhBAMg/AGRtB+IEEQYGcaB6bqgaHgbfxJKMkMqCVAaG BAYOoOgvJogOEGABYl1GiNkiTLsZlgiBWMJAlo8wyJ2/mRT+Q1ROYASrZ2IQCMnMTS1W8EstVwjK z03MY9ix5HP5QSBWMPhaXsOdxgIxkxFslxHXF4Y3wiCtRpUQPgNTmwCIb1AJkz/KB+IbwvkWvKj8 35zIfAamdE5k/QxMl5hR+V8ZIOoh7m/4R577mcBh8wHujwWMcWB/aFbC+F78IL4GnL+XF1VenQMi j+4ORhLDEeIOJrg7pguBTKiA2/OWG8TPg/M3cKDy77CC+MFwPgczKt+BAcLHEl5MQCgQXJmblJ/D wJDLw3XH4aMw0eGVKggyQhduzwkuEF8b4Q52VP5GFlT+JXBI2VZiSYco7vrAzTUFt7vQw+8Lgzcf sru+MOzjQeU3ckL4WOKNiQU1PO6SEh7rwemlsJILmjO5wLkXnCwZIHYJMLCDeXvA+Z2RiUkpuLK4 JDWXgQVkLgNQhyJDF1gDiIj5LMyA7kdmsH4A35iBKIQEAAA= H4sIAAAAAAACC5VSPUvDUBS976bpR1qwWBERkajgplSRDoLQWVCkLU4OVqgomLbaYgk4FCfp5OCU oQj1LygOChVBRJCOrh0EwUmtLgrG95EXSMXBFy6cc3Puved9PNzfWMBXW6sihBjaXiYMKKcEwAdD /K9KI4QSISEOImjbNkeTZMDJhV1dBKvY1nopGvf3wCDYTAxRypsUndE41gASVB92NBFYyJY3MmYx B7AKQZr9RFHBlo/GBBG9+7BO6oShGO6TFpFa3aMljrZGFUIbx3709onjMIp/KdhDkZOI5e4UkZuH otMjiS2n1oLZoMjpeBmQvnZU6auhdvtoqNLHs9rt402VM5f80odALHegsbv4Qt0WOzwkfJ8I0cym kSvpi7mKnioY2Tycn3QqVzT0+EflNrxuiTnIz3Na0/E9xEqnTMkfg14+FvDyhCK4mF/9lvPJv+Yr /E5eXB9NkuI+8qbkGe5j1+V5P+Nplx+hV58Exku/fLEnSr9o2jTWClsAT0HtOvka+/NchC/F9TXj Yy3mTM15lRp/ufy4QcyKQoCzC/7WCeJo2iyVcwZYrC/QihGo8QKFxkonBt1noPD6H22MxNaAAwAA H4sIAAAAAAACC6VTvy8EQRR+781wLGE5EhHFUuiOJf6Ai54CJckSJ0esX3dx2URErVIoRCEKhUKp UCgQodJoJGp/ABGRkFjzZvYWh8SPSWbzfW++9755MztXF2eboEc64UmsZOSOICggHhFAQoteLVOz koqIECOEFIahRh3YFMWqYl01eTKdqFeovbwWmiFkMdiKHym0zywB0CBVTqSphr7RfHYomMtw8Qr1 eSKTwUMJIYWmdgMtiRtklCRPNFJR63zQYqTNKoXR1shh+linRo5Hay6dkIkVEcdsaWJAaTAxIFfy uTyTExq3NdSeBPbQpJ/JOf2ZgjMw64/OwMHOfeFYTcd9KJzXTfjGm4D3220d0Xo5p3YFhr+IJfGe X+JUxI3fysvf/IQ+j9vY16NNySWWI58t4eo77QxK+8Jf9mV8RNxfj+5v8dP+8Z/7Py3jCvl4/9fE fCbmHjDPfeVL7DsY+GOz0wC9tpXavkv+sJ8F3NU+80GRZ8HwL+5H/SdvPhtV1t33PqX9LeCePqFU YEUvw9KvR187GC+bH48ah/q9IVHbYJDLZ3zwuS6ojFZY1QksOLxPQmlPQue/AlZXWPgEBAAA H4sIAAAAAAACC7t+9tgsBghga2Bi5AQxJkQxMgAZzFsZGRiYGGTAkqxAzMkEYzExMkJZjEz///8H s/QYJaBi3HB1PEwNTAxsQkCWGhs/gxTDf5BiBgEg/wCQtR2IDwCVmQPVc0PV8DD4JpZkhFQWpDIw FIDt/s2k8B/iwAkgSxlYmBgEQjJzU4sV/FLLFYLycxPzGHYs+Vx+EIgVDL6Wf+ZO+8sCVKcLVM0B pI24vjB4sYK0GlVC+AxM/swgvmElxPwEMs1nZADRH+D2WDDxsCKb2/CPXHdDzGWEmruAMQ4c3nWV WMKDCQgFgitzk/JzGBi2c3PNcvgoTKR7vzBcYAEZoYthLiNF7mVgWgc2twwjHBgpCt8FjLOYQCYE YzOXiQUpHDK5uVJxhwNm+J4Ep4fkShhfhgHET6vkgqZKLnDKBUcjNE0KMLCDeXvAaZ2RiUkpuLK4 JDWX4S/IHgagDkWGLrAGEBHzWZgB3U/MYP0AYqZb+YADAAA= Note that the reference prior implies v0 = -1, n0 = 0, S0 = 0 (θ0 need not be defined). Also be aware that θ represents two different things, it was used in section 2.13 for specifying prior information, and in section 5.3 as a parameter for the Behrens-Fisher distribution.To see how this prior information could be used in practice, observe that with a reference prior the statistic would beH4sIAAAAAAACC6WUPWhTURTHzz33vrZ5jTYmaZoGoVFEHPxKoUoFJTqELgqx2QQxaqKCSdq0EJ+U oqsUcXFxyiBaJwcdHDpEkZYOTh0cBCmCWjel1PoBjffzkRvaQfrCg/M7H/9z3sl97/27+Ucgr3qi 4UBAWOOniTBoPwVgsFtGHX4H0FhIiLYINptNaR0mce3r9vOC2HDqiTC39nf0QAKaIhlCnBvces7v SwmACk/t1jlBOJufvJ7zxgoAadLFvX9QVYiL8fsQUdpR7MULMWFF8DeZiZncpJVLdC7BGZ07i0sx W2cWP+jYAgz3KZ+xhO9lXPluM9PvIpuJ2fpFZvQHnHb9Acfoh6jRN5bwvdH6y5iIKt8yfoqLvf/F OxvqaR6IhSP/hUa90uXKTYBS0L1f/xGBV49Xa6/5nTy6VhsLFqdUbwSxj0G3n833i9J1T/EztDnr KP6l+QVR/NMz9adQ8Fe/3uaso3jFr1f8RfMwC1v9FtHmlBO2+i+RsNV/mOUirf0W0eaUo3jFr1ds +qfp1E7BJ30+7tqMjmK172TT7Ft4Gd937kapMJE8V6glz1dK+fIW+6byvH339/6RJeNCwfMMf462 8kOMyfgtn79FW7lOjwVb8+t0vUPF28/F/82Jek6q55xm2T6hUPYMn5BzTGgeQjs+hCNWPE2vdbfG 0zTQJV7/q0XDg3Luss8baMczIPhMZpPzvo39T7OnvULiSMbMbdjVfZ/0dHArX70ymS+bWd7KnR+s Gr7bqbj9bMC2dl6nIwEhMVg1vI8JTlU3+W+RtbzzC0F319bvfPsO0vTADqE7XnX119WVX2A5Pqhe IeiUNCe/2QRx76g3MVkowZTQBV6xB+7JApEwtxqB9meisv4fUpo2fUgGAAA= While when incorporating prior information (using the formulas above) the statistic would beH4sIAAAAAAACC6WYXUwcVRTH79z53AELAqawS2Vhl6EFlm7bJ0koqJE3jSnFRqspqNugsqAUiutH 5AGNkjQ2GmLqQ4tJRaTGkLTWpm5iTdUaEi0aE7RC11jFvtRUt5SGdlnnfsxkzoaFQXcz7P3de+45 /3Pnnpthpr/9+l1EPwPhphzkIa3vIwJpiCkNIRmV0lHZvDzYamFB4C0Bp9Np2qoTinlfjm2Xi5ty BsIFZstQ8pAXpYkxyjf5jNn62Lyawgj9pZtzuE0uur+9t2Nn7NkIQiOYxL6JD6UuUQ0HSVBsfvNb YtHHuzsR+iNHv/XglUJ06miy/wvz8ofn+z/J2TsmmYYh09pMAG3VL+Kr5WTqxthq/u6V9J+y+xMQ +b3q8NtVRVxsihFexCwz8rHikzUowovCcC1pFZqthhDTIC+tkFMquwbMNQhcwyye3OjMLbtfM7ef V88N2353h1huLvQuudeblEbKXOu94F5vUjoSdK03vRa9uNK13l/Wore4Gu4dDewdzPdOVPzAx/ZO VOyug/tqSjzmZWMnxBmv5ccP/IjcNm5aMNsOOeCDfjrkGh4jid7wsT6rRfrerHNRhyh77lLGmn6m 3nBfjzOrr6lk+x3k9chymFY+5XU3rezhdTe6Ug6C+xxOqr+BuhtdKYdZ9zmcVF8GdbeiXuxe74Ln vTLXei+617vg+SjoWq+4Fr1SpWu9ibXoNXjdwTrr097m+71PG8yos0vaO7x2zmnfeGFdnTd72Nhh PS+jrg7r67nPKbmP15XVIn1H61jfiNzNfYzIX/JaG1hC2ddRavrb7Tpe8IxtJi4WYoyf1iAvKpCH ZMheCfI4hjylM77B+YTA+HrMit+ACf9px4e8qEAekiF7JcjjGPKUzviyHZ/xHOfTnmt1ID8NsleF PC5DrpcgT2LIEzpjK/8fBcbX7fgfloD8NMheFfK4DLlegjyJIU/ojC/b8Rlb+ac8b9Ezscbm4gon x9XOGsIhm18NwvFpP+QrG5w8L0H/81IJ8J/ADwH/CdwdhOPn/ZCTwD9Sfigi3GDzoXw4/shtcHyL BseRzMZZXfnTKPs58mv2uso8R84qv1c588pXXzQgbw44eU74DthvwxEDchW3z9RJeiVT586nopF9 /gci/f4d3dH2Lpf1/5j+2ibiIRazuN6AnBuEfMbv5FKtqBpyzIAcCkC+WQZ5uNTJrfJLVM/zNm83 IItByF/5ndwo4mrIDxuQfQHIC2WQj5Q6uU05UODU16bcnQfZp0P7WYVx5jm9tvuUuZ9SnlP0eaLL rtPOSsjbA5BFui777bqcqHXqjqvHwPy4uqcCcgldlxabj/ucPC9BPfPSC5WQGwOQC4CeBH691rlu CTwM5ifwrgrIZUBPAp8DepDy+e1Oe6QMrnPGQ0qI/tv55F6Ld3movc3V9L7ttzmFoX0zInxP8zLn xP+qv+N0f4abLb6r3MnXlAR93tvabNXLgWrn+FllN70PW5qtc+UfA/JowMmlGuOwzes2OLlVhnpa 5XqgZz2eBHoaxX6gZ064D+jZhmcMyIeBnkbxINDTKBYBPW3K6TzIz+Q647cpO1Q4XiEx/8s8J/2n +tP5/b91h2K22nue6G3vsvaEr5C4re2x+H2d8TK1jyX4jKa6f0aLqwY9E5/rsfb+nYCR8kohY52/ JdHpmxSaMmJa8pFKKU7fvQgYV7TE9vVGomiMxEXmjHI0RCeQP48mC1HmOoh0/r8rIE7oEBIAAB== WinBugs ExampleI tried to extend the one sample test with mean and variance unknown for the two sample test with mean and variance unknown but was unsuccessful. Also it does not appear necessary for this problem. It is easy enough to generate random variables from the BF distribution in R using the fact that BF(df1,df2,θ) ~ T2cos(θ) – T1sin(θ).Sample DataData from Lee 5.7 problem 8, page 155, lengths of cuckoo bird’s eggs.>X = c(22.0, 23.9, 20.9, 23.8, 25.0, 24.0, 21.7, 23.8, 22.8, 23.1)>Y = c(23.2, 22.0, 22.2, 21.2, 21.6, 21.9, 22.0, 22.9, 22.8)Comparison With Welsh Approximation>t.test(X,Y)>bf.test(X,Y)>null_diff( mu.x=mean(X), mu.y=mean(Y), s.x=sd(X), s.y=sd(Y), n.x=length(X), n.y=length(Y) )Welsht = 1.9924, df = 13.796, p-value = 0.066595 percent confidence interval: [ -0.07019846 1.87019846 ]BFdf1 = 9, df2 = 8, theta = 61.33793 degrees, p-value = 0.0790650395 percent confidence interval: [ -0.1212955 1.9212955 ]H4sIAAAAAAACC+3dCZxN5R/H8fOcMbZBo5Uo+74LCRVlLQkplMqe3SD0L0pRURKRqJSiIgmFSGVJ su9apQkV2Wm1zf+cOedcz/l45v4maZ2Z1+ua+773OnPnuc885zm/s3yVZVmDnFsx5xbv3BrFWNZR 69TX1EaWVTiPZeWvc0Ndy1JWjTmW9UBWy3JeZjnfrG7+6+6JtayLM1hWTWVZBy18bc9gDZoaY9Wz 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The next two graphs show the same thing but compares CDFs instead. What is important to note, is that regardless of what significance level you choose, the Welsh T will give you larger C.I.s. This is easiest to see in the last figure because for quantiles less then zero, the Behrens distribution will give you a larger p value, while for quantiles larger then zero the opposite effect is observed.Substantial Prior Information ExampleUsing the same data as in the previous example, suppose some prior information was available, and we will use the same information for both samples.v0=20n0=5S0=40θ0=22Thus the sample means should be about 22mm and are worth perhaps 5 observations, and the variance is expected to be approximately 2.2 with standard deviation of 0.8.Df1=15,df2=14,> atan( sqrt(40+sd(X)*9 + 1/(1/5+1/10)*(22-mean(X))^2)+ /sqrt(40+sd(Y)*8 + 1/(1/5+1/9)*(22-mean(Y))^2)+ )*180/pi[1] 47.84711Thus the distribution would be BF( 15, 14, 47.84711 ). This is slightly narrower distribution due the increase in the degrees of freedom for the two samples. Using identical prior information has also biased theta toward 45 degrees; the value at which the BF distribution behaves like a pooled two sample t test.> pbf(.975, 9, 8,61.33793) # reference prior 1 0.8163274 > pbf(.975,15,14,47.84711) # substantial prior 1 0.8228224However at the tails it turns out the increase in the degrees of freedom did not counter act the effect of driving theta toward 45 degrees, and in fact the substantial prior information has made the C.I. wider.> y1=y2=x=seq(-5,5,0.01)> for( i in 1:length(x) ){+ y1[i]=dbf(x[i],15,14,47.84711)+ y2[i]=dbf(x[i], 9, 8,61.33793)}> plot(x,y2,main="POSTERIOR\nred = reference, blue = substantial prior",type="l",col="red")> lines(x,y1,type="l",col="blue")H4sIAAAAAAACC+3dCXwNV/sH8JlELEkQaomtYt+JfSc0tqKUUFW170vEvhOEWGOn1sYSexVVpVRt tVRtrVKlGkt5+UuEUktL/ufJzK3n/jKZJ7ztqyr5fEbmm8TNzdwzz5xzZu78dE3TgtXiqhYvtYS5 aNoStTg+QqpqWi+1+NRqWFvTdO3KDk2b5278vPqk9TR/brCbpmVNpml+uqbFaPBxKZkWvMZVq6P1 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mpEz5J/Ie6ClMd/rrjn68mr5fwYm2YfkuwAA ConclusionThe Welsh Approximation has always been a “fudge” that was needed when models had to be simplified such that tables of t distributions could be used instead of using a computer to calculate or precisely estimate these things. With the prevalence of computer packages like R for calculating pvalues instead of looking them up in a distribution table, the simplifications of the Welsh Approximation looks like a very dated technique. More stable answers are now obtainable with the use of the Behrens distribution.Source for the functions I created available at http://jbcolvin.fastmail.fm/behrens/,Namely project.R and bf.R.