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概率论与数理统计第 24 讲. 本讲义可在网址 http://math.shekou.com 或 ftp://math.shekou.com 下载. 第八章 方差分析与回归分析. § 8.1 单因素试验的方差分析. - PowerPoint PPT Presentation
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24http://math.shekou.comftp://math.shekou.com
8.1
, , . , , , , ; , , , , , , , , . , , .
, , . . , (, , , ); ().
. , , , ; , . , A,B,C,, , A1,A2,.
1 , . , . :
, , , . , , , ., . , .
, Ar, A1,A2,,Ar, Ai(i=1,2,,r), , r, :(1) ;(2) ;(3) .
, , , imi, H0: m1=m2==mr.H1: m1,m2,,mr.H1.
Ai(i=1,2,,r), ni,
,
.
:
(1) rN(i,2) (i=1,2,,r);
(2) 1,2,,r,2.
_1211872623.unknown
_1211872669.unknown
, Ai
ST
()SA()SEST=SA+SE(1.4)
, SESAST=SA+SEH0, XijN(m,s2), , :(1) ST/s2~c2(n-1);(2) SE/s2~c2(n-r);(3) SA/s2~c2(r-1).(4) SESA.
, , , r, H0. , , F, H0, F~(r-1,n-r).
F
A
SA
r1
E
SE
nr
T
ST
n1
_1211894816.unknown
_1211894882.unknown
_1211894783.unknown
8.2
, . , . . , , . ..
, A,B, ArA1,A2,,Ar, BsB1,B2,,Bs. A,B(Ai,Bj), (i=1,2,,r,j=1,2,,s), rsXij,
BA
B1B2BsA1X11X12X1sA2X21X22X2sArXr1Xr2Xrs
1. , :(1) Xij~N(mij,s2), mij, s2, i=1,,r;j=1,,s.(2) ;(3) Xij, i=1,,r;j=1,,s
, , H0A: m1j=m2j==mrjj=1,,s,H0B: mi1=mi2==misi=1,,r.H1A: m1j,m2j,,mrj.H1B: mi1,mi2,,mis.
2.
:A:B::ST=SA+SB+SE,
H0A,H0B, (1) ST/s2, SA/s2, SB/s2, SE/s2rs-1, r-1, s-1, (r-1)(s-1)c2.(2) SA,SB,SE.(3)
3. a, H0AFAFa(r-1,(r-1)(s-1));H0BFBFa(s-1,(r-1)(s-1));
F
A
SA
r1
B
SB
s1
SE
(r1)(s1)
ST
rs1
_1211920101.unknown
_1211920262.unknown
_1211920397.unknown
_1211920170.unknown
_1211919982.unknown
, A,B. ArA1,A2,,Ar, BsB1,B2,,Bs. A,B(Ai,Bj), (i=1,2,,r, j=1,2,,s)t(t2)(), rstXijk (i=1,,r; j=1,,s; k=1,,t).
Xijk, Xijk~N(mij,s2),, mijmij=m+ai+bj+gij(i=1,,r; j=1,,s),aiAi, bijBj, gijAiBj.
:H0A: a1=a2==ar=0.H1A: a1,a2,,ar.H0B: b1=b2==bs=0.H1B: b1,b2,,bs.H0AB: g11=g12==grs=0,H1AB: g11,g12,,grs.
2. :
ST::ST=SE+SA+SB+SAB
, H0A,H0B, H0AB, (1) ST/s2, SA/s2, SB/s2, SAB/s2,SE/s2rst-1, r-1, s-1, (r-1)(s-1), rs(t-1)c2(2) SA,SB,SAB,SE.
3. H0A, a, H0AFAFa(r-1,rs(t-1))
H0B, a, H0BFBFa(s-1,rs(t-1))
H0AB, a, H0ABFABFa((r-1)(s-1),rs(t-1))
F
A
SA
r-1
B
SB
s-1
SA(B
(r-1)(s-1)
SE
rs(t-1)
ST
rst-1
_1211986573.unknown
_1211986835.unknown
_1211987081.unknown
_1211987198.unknown
_1211986922.unknown
_1211986605.unknown
_1211986502.unknown
8.3
., , , , , . . .
. Y=f(x), , , Y=f(x)+eY, x, e().
, , , , ., .
, xY. :xi, yi, xOy
xi100110120130140150160170180190yi45515461667074788589
xi100110120130140150160170180190yi45515461667074788589
Chart1
45
51
54
61
66
70
74
78
85
89
Sheet1
100110120130140150160170180190
45515461667074788589
Sheet1
0
0
0
0
0
0
0
0
0
0
Sheet2
Sheet3
, , , , :yi=b0+b1xi+ei,i=1,2,,10ei, .
, , Yx, Y=b0+b1x+e,(3.3)e~N(0,s2), b0,b1.(x1,Y1),(x2,Y2),,(xn,Yn)(x,Y), (x1,y1),(x2,y2),,(xn,yn).
Yi=0+1xi+i,
i=1,2,,n
(3.2)
1,2,,n. ,
Y~N(0+1x,2), E(Y)=0+1x
(3.3)
0,1
. x,
(3.4)
E(Y)=0+1x, (3.4)Yx, ,
.
_1212092545.unknown
_1212092676.unknown
_1212092502.unknown
,
(x1,y1),(x2,y2),,(xn,yn), xi,
yi
yi
.
_1212133066.unknown
_1212133108.unknown
_1212133195.unknown
_1212133014.unknown
xi, yi
, .
01
,
. , Q0, 1, ,
_1212133425.unknown
_1212133555.unknown
_1212133591.unknown
_1212133382.unknown
,
(3.5)
(3.5)(3.8)b0,b1. Yx.
,
1
0,1,
0,1,
.
_1212136873.unknown
_1212136900.unknown
_1212136819.unknown
, Y=b0+b1x+e, e~N(0,s2), b1=0, Yx, :H0:b1=0,H1:b10.H0, y1,y2,,yn, ,
S=S+S(3.10). S, S.
2 , H0,
,
~2(n2),
~2(1).
H0:
T, F, .
_1212138165.unknown
_1212138184.unknown
_1212138258.unknown
_1212138007.unknown
1. T-H0, Tt, |t|>ta/2(n-2), H0, , H0, .
2. F-H0, FF0, F0>Fa(1,n-2), H0, , H0, .
3. |r|>ra(n-2), H0, , H0, .ra(n-2).
, , , , , .
b0,b1,s2x.x'=1/x, :Y'=b0+b1x'+e, e~N(0,s2),
2. Y=aebxe, lne~N(0,s2),(3.23)a,b,s2x.Y=aebxelnY=lna+bx+lneY'=lnY, b0=lna, b1=b, x'=x, e'=lne, (3.23):Y'=b0+b1x'+e', e'~N(0,s2),
3. Y=axbe, lne~N(0,s2),(3.24)a,b,s2x.Y=axbelnY=lna+blnx+lneY'=lnY, b0=lna, b1=b, x'=lnx, e'=lne, (3.24):Y'=b0+b1x'+e', e'~N(0,s2),
4. Y=a+bh(x)+e, e~N(0,s2),(3.25)a,b,s2x. h(x)x, Y'=Y, b0=a, b1=b, x'=h(x), (3.25)Y'=b0+b1x'+e, e~N(0,s2).