ࡱ> uwt5@ Vbjbj22 .XXv98,4#"222222\"^"^"^"^"^"^"$$R&")22222"22#BDN  + W b   ' p | QRTWhoui &1ǿǷǿǯǷh xmH sH htEhtEH*mH sH htEhtEmH sH h mH sH htEmH sH h>7mH sH h mH sH hwmH sH h}`#h}`#mH sH h}`#mH sH hw>*H*mH sH h=JX>*mH sH hw>*mH sH 5CDEN  "!"""8#9#gdw$a$gdw"T <P^pqrBD\`ɽ{sks_shFh?vv6mH sH h xmH sH h?vvmH sH hFh 6mH sH h xh 6mH sH h4 mH sH hFhF6mH sH hFmH sH hFhF6H*mH sH hFh>76mH sH h>7mH sH hh6mH sH h6mH sH hmH sH h mH sH h H*mH sH $4Z\,.2"Vft f| "b÷莇 hh xh. Th xmHsHh xH*mH sH h?vvH*mH sH h?vvh?vv6mH sH h&Uh?vv6mH sH h&UmH sH heYH*mH sH hmH sH hFh?vv6mH sH h xmH sH h?vvmH sH heYmH sH 0 &hjlnpvxz|8 N p r N!~!ƾ~vj־־־a־h6mH sH h xh6mH sH h0DhH*h. Th6mHsHhFh6H*mH sH hFh6mH sH  h0Dhh. ThmHsHh&UmH sH hmH sH h?vvmH sH hh xH*h xmH sH  hh xhFh x6H*mH sH hFh x6mH sH $~!!!!!!!!" "P"`" ##$$$$$ $2$4$H$J$L$N$~$$$$$$&&&`&&&&'''''+'2':'Z'['z'{'|'~''''(((((ڭɭɤɘh?vvh x6mH sH h xH*mH sH hK_mH sH h?vvmH sH h xh6mH sH h xmH sH h6mH sH h&UmH sH hH*mH sH hmH sH h?vvh6mH sH <9#$%}'~'++U/V/T0V00.101N1P1v1x1338888T;V;>>gdcgdu*gdw(("(((,*z*++ +"+++.-------------'.S...(/1/8/T/U/V/a///R0T0V0潶涔vnh4 mH sH hK_h xmH sH hK_hK_mH sH h&UmH sH hhK_H*hFhK_6H*mH sH hFhK_6mH sH  hhK_h. ThK_mHsHhK_H*mH sH h?vvhK_6mH sH hK_6mH sH hK_mH sH h xmH sH h xH*mH sH )V00000001 1>1@1B1D1H1J1`1b1d1f1h1j1l1n1x1z1111111223333v4|4+50525t5|555666666666666666ƽƽƵҵҵh9.H*mH sH h9.H*mH sH h?vvh9.6mH sH h9.mH sH h&U6mH sH h&UmH sH hu*h?/mH sH h4 H*mH sH hu*H*mH sH h?vvhu*6mH sH hu*mH sH ;66667788888888'9/929A9:::J:L:X:Z:v:x:::: ; ;;;;`;d;f;z;|;;;;;;=<=~h9.H*mH sH h?_h9.H*mH sH h?/mH sH h?_h6yjH*mH sH h6yjH*mH sH h6yjmH sH hcmH sH hh6mH sH h6mH sH hmH sH hu*mH sH h9.6mH sH h9.mH sH h9.H*mH sH .<=>=@=B=T=V=Z=\=============>>^>`>h>j>v>x>~>>>>&?0??@[@@@@@@@@@@@@@@@@@@@ A=A>A?AAABAFA۳䫣hcmH sH hmH sH hqmH sH h?/mH sH h9.h6yj6mH sH h9.mH sH h?_h6yjH*mH sH h6yjH*mH sH h6yjmH sH h?_h?_H*mH sH h?_mH sH =>>A?A$^&^x^z^^^\___bb,c.ccc\d^dddddLfNffgd*gdaigd?_gdcgd?/FAGAHAIAJAB^^^ ^^^^"^$^&^^^`^x^~^^^^^^^^^^^^_D_T_V_X_Z_\_j`````.a0a2aabbbbb4cԟԟԎԆh~smmH sH h mH sH hqH*mH sH h+5hqH*mH sH hH*mH sH h?vvh6mH sH hmH sH h+5mH sH h?_hqmH sH Uhqh?_H*mH sH h?_mH sH h?_h?_H*mH sH 42(1/N1 + 1/N2). From which we can estimate 2 itself as (m1  m2)2/(1/N1 + 1/N2). This is the estimate from the actual sample mean difference, analogous to N*(m  )2 above in the single sample case. The analogue of the estimate from within the sample is more complicated, because we have two samples and not just one. We can estimate the variance within sample 1 as s12 and that from sample 2 as s22 , and it seems reasonable to combine these by weighting them by the size of the corresponding degrees of freedom: larger samples should give a more accurate estimate of 2 than smaller ones. The actual formula for this weighted variance estimate is {(N1  1)*s12 + (N2  1)*s22}/( N1 + N2  2)} If we call this  pooled variance estimate sp2, we get, taking the ratio of the two variance estimates, {(m1  m2)2/(1/N1 + 1/N2)}/sp2 = {(m1  m2)2/{sp2 *(1/N1 + 1/N2)}. Taking square roots of top and bottom, we have the usual formula for a t-statistic, now expressed, if you like to think of it that way, as the ratio of two standard deviations: t = (m1  m2)/"{sp2 *(1/N1 + 1/N2)}. The Behrens-Fisher case: population variances not necessarily equal Surprisingly, this approach even works with the difficult Behrens-Fisher problem, in which the samples are drawn from populations which may have different variances, but where the null hypothesis is still that they have the same mean. In this case, (m1  m2)2 is still an unbiassed estimator of the variance of a certain random variable. This variable is a little complicated to define but we can do so as follows: on the assumption that the null hypothesis is true, so 1 = 2, say, the statistic is the difference between the means of samples of size N1 and N2 drawn from the two populations respectively (note that we must now talk about two populations even if the null hypothesis is true, because the null hypothesis ensures that the population means are equal, but not necessarily the standard deviations). Even without assuming the null hypothesis, we can calculate the variance of the m1  m2 statistic using the data internal to the two samples to estimate 1 and 2. The obvious estimates to take for these are s1 and s2 respectively. In that case, the variance of m1  m2 will be the sum of the individual variances. The variance of m1 is then estimated as s12/N1, and that of m2 as s22/N2. So the variance of m1  m2 on this calculation is (s12/N1 + s22/N2). Taking the ratio gives us (m1  m2)2/(s12/N1 + s22/N2) and taking the square root in the usual way gives the standard expression for the Behrens-Fisher statistic t': t' = (m1  m2)/"(s12/N1 + s22/N2). Which is what we set out to prove, as it shows that the variance-ratio approach produces the same result as the approach in the textbooks. APPENDIX The total sum of squares in a t-test can be dissected just as it is in an ANOVA, but the algebra is simpler, considerably so for the one-sample case which is the only one dealt with in detail here. We can write in the usual way: (xi  )2 = (xi  m + m  )2 = (xi  m)2 + (m  )2 + 2(xi  m)(m  ). The third term on the right hand side is equal to zero, as (xi  m) = 0 by definition of m as the sample mean, so finally (xi  )2 = (xi  m)2 + N*(m  )2 & & & & ( ) Hidden in this formula, like the animals in a puzzle picture, there are in fact three estimates of the variance of the underlying population which are unbiassed provided that the null hypothesis is true. In this case, where we have no doubts that the sample did come from the population we can estimate the unknown variance as (xi  )2/N, with N and not N  1 in the denominator (the correction to N  1 is only necessary when we are estimating the population mean from the sample itself and therefore lose a degree of freedom doing so: it is not needed when we know the population mean already). Call this estimate, est1. There is another way in which we can estimate the variance. The formula (xi  m)2/(N  1) is known to provide an unbiassed estimate of variance. If we write (xi  m)2/(N  1) as est2, the second estimate, the formula (*) above becomes N*est1 = (N  1)* est2 + N.(m  )2 Suppose we now take the expectations of both sides. Because both est1 and est2 are unbiassed, their expectations are both equal to 2. Therefore we get N*2 = (N  1)*2 + E{N.(m  )2} And finally E{N.(m  )2} = 2, which for sceptics is the long-awaited proof that N.(m  )2 really does provide an unbiassed estimator for the population variance, on the assumption that the null hypothesis is true. PAGE  PAGE 1 4c6cDcFcHcRcTcbcdcfclcncpcrczc|c~cccccc dXdZd^d`dddfdndpdrdtdvd~ddddddd뺲zzqzzh+5H*mH sH h?_h+5H*mH sH h+5mH sH haihaimH sH h+5h H*mH sH h+5h H*mH sH h mH sH hqmH sH h~smH*mH sH h~smh~smmH sH h~smH*mH sH h+5h~smH*mH sH h~smmH sH h?_h~smH*mH sH (dddddddddddddddddddddeeLfNfVfZf\fdfffpfrftfffffϿϿ϶ϪϿϿז׎vvj^vvh+5h?H*mH sH h+5h?H*mH sH h?_h?H*mH sH h?mH sH h?h<mH sH hmH sH h+5h*H*mH sH h+5h*H*mH sH h*H*mH sH h?_h*H*mH sH h*h*mH sH h mH sH haimH sH h+5h+5H*mH sH h+5h+5H*mH sH $ffff g"g$gFhZhxhhhhii i(i*i,i.ijjjjjjjjjk|k~kkkk*l0lmm"m$mVm|mm&n(n*n˘~~~vhRtmH sH h?hu]S6mH sH hhu]S6H*mH sH hhu]S6mH sH h?H*mH sH hu]SH*mH sH h?_h?H*mH sH hu]S6mH sH hu]SmH sH hh6mH sH h6mH sH hmH sH h mH sH h?mH sH .ff"g$ghhmmp q"qXqZqqqtrvrrrsssssuugd$a$gdu*gd?gd gdai*n2n4nnnnnn*o,o8o:oporoooooop p"p$pNpPpRpTpXpZpvpxpzp|ppppppppppppppppppqqqqqqqqq"qVqXq\q^q`qhqjqlqhu]SmH sH h?H*mH sH h?h?H*mH sH hmH sH h?hRtH*mH sH h?H*mH sH hRtmH sH h?_h?H*mH sH h?mH sH Clqnqpqtqvqxq|q~qqqqqqqrrrrrrrrrrrrrrrrrrssssuuuuuuuthFhu*6H*mH sH hFhu*6mH sH hu*hu*mHsHh. Thu*mHsHhmH sH hu*mH sH h9 mH sH h?_h?H*mH sH h?H*mH sH h?H*mH sH h?h?H*mH sH h?mH sH hRtmH sH hRtH*mH sH *uuuuuuuuuuuuuuuuuvv v vvvvvvv(v*v0v2v4v6vFvHvJvPvRvVvXv^v`vfvhvvvvvvvvvbwfwhwjwpwrwtwvw|wŷŷѯѣѯѣŷѯѯќŷŷѣhu* h. Thu*h. Thu*H*mHsHhu*mH sH hFhu*6H*mH sH hFhu*6mH sH h. Thu*mHsHhu*hu*H*mHsHhu*hu*mHsHh. Thu*6mHsH|ȱꝒ|sllld[hu*6mH sH h0Dhu*H* h0Dhu*hj6mH sH hjmH sH  hjhu* hjh*AhjhjmHsHh*AmHsHh. Thu*6mHsHhjh*AmHsHh. Thu*H*mHsHhu*mH sH hFhu*6H*mH sH hFhu*6mH sH h. Thu*mHsHh +thu*mHsH#>|z||||B}D}F}H}J}P}R}T}V}}}}}}~~~~$~&~~~~~~~~~~~~~~~~~djɻճɻճ}ynh\JhjmH sH hjhu*huShu*H*h. Thu*6mHsH huShu*h\Jhu*mH sH h\Jhu*H*mH sH hhu*H*hFhu*6H*mH sH hFhu*6mH sH  hhu*h. Thu*mHsHhjH*mH sH hjmH sH hu*mH sH *jlnv|~468BDHJPRTVxz~ ¾¾˷˷˷˷¾˷˷h mH sH hu]ShRtmH sH hu*mH sH huShaH*h. Tha6mHsH huShahahaH*mH sH hamH sH hahamH sH hjmH sH hjhjH*mH sH h\JhjH*mH sH 2 "$0248:FHJLNRTVh}`#h}`#0JmHnHuh9 h9 0Jjh9 0JU h7kehtEhuSmH sH  huShc "468NPRTVh]hgdai &`#$gdXdgdw ,1h/ =!"#$% @@@ NormalCJ_HaJmH sH tH DA@D Default Paragraph FontRi@R  Table Normal4 l4a (k@(No List4 @4 aiFooter  !.)@. ai Page Number9CDEN 3 4 C D o p 45stXY-.tR"S"""##Y%Z%W'X'w'x'''''(5(6())))**+*******++++++,,'/(/00011/1011111P2Q2R2[2\2A3B33344J4K4667Q7R77777^8_8888n9o9p9q9r9s9t9u9v9999999900000p0p0p0000000p00000000p00p00p00p00p00p0000p0000000000000p0 0p0000p000000 0 0 00 0 0 0p0p0 0p00 0 00 0 000 00(0(000000(0000(0000000(00000(000000000 0000000000p000p0000000p0p0000000@0@0@0@0@0@0"LCD3 4 C D o p 45XY-.t""##x'''''(5(6(*****+++'/(/0110111P2R77777^8_888n9q99M90M90O90M90, ļ!O90O90O90O90O90O90O90O90O90O90O90 O90 O90O90O90O90O90O90O90O90O90O90O90O90O90O90O90O90, M90M90O90M90O90!M90!O90M90O90% , M90%M90%M90M90M90M90M90O90M90, T!O90O90hO90O90O90O90O90, !O90O90O90O90O90O90M90=, xM90=M90=M90,M90,M90,M90,M90,M90,O90,RD ~!(V06<=FA4cdf*nlqu|w>|jV!$%&')*+,.BCDFGHJKLM9#>fuV"(-EINT# !!9;;ABDtuw  ######&&))*1O2777p9u999;;ABDtuw  ######&&))*1O2777p9u99EN X'x''')+*1/111R2Z2"3B366v9999;;ABDtuw  ######&&))*1O2777p9u99ps401ra<;w 9 >7ai x 4 }`#9.;@tEuIOu]SuS. T&U=JXkLYXd7ke6yj~smyo +t?vv?caq?_\JjRt*AF<<0DeYBX}:*Nu*&$;0iL}& K_?/+5@XM$  !#$%&e'9P@PPPPPPP PD@P$PL@P(P*P,P\@P0Pd@P6Pp@P:P<P>P@P^UnknownGz Times New Roman5Symbol3& z Arial"1h:t&$<&0h0hq4dY9Y9 3H?w'Understanding the t-test: why is t2 = Fps401raps401raOh+'0x  4 @ LX`hp(Understanding the t-test: why is t2 = Fndeps401ras40 Normal.dotnps401ra840Microsoft Word 10.0@캃@\5@ʽ0՜.+,0 hp   GoldsmithshY9A (Understanding the t-test: why is t2 = F Title  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOQRSTUVWXYZ[\]^_`abcefghijkmnopqrsvRoot Entry Fpùx1TableP'WordDocument.SummaryInformation(dDocumentSummaryInformation8lCompObjj  FMicrosoft Word Document MSWordDocWord.Document.89q