8 9 Latent Variablesjarrettbyrnes.info/ubc_sem/notes/8_9_Latent_Variables.pdfg&...
19
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
Transcript of 8 9 Latent Variablesjarrettbyrnes.info/ubc_sem/notes/8_9_Latent_Variables.pdfg&...
!"#$"%!&
%&
'()*+)&,(-.(/0*1&
23)4(0&5.67(11&
8197()*:&5.67(11&
;*(14-*7*+)&8--6-&
%&
'()*+)&,(-.(/0*1&
%< =>()&.1&(&0()*+)&?(-.(/0*@&
#< '()*+)&?(-.(/0*1&A.)>&7409B0*&.+:.3()6-1&
!< C.D+E&(&0()*+)&?(-.(/0*&
F< C(3)6-&2+(0G1.1&H< '()*+)&,(-.(/0*1&(1&(&I*1B6+1*&$< J6B.+E&A.)>&7*(14-*7*+)&*--6-&
=>()&.1&(&0()*+)&?(-.(/0*@&
!&K&"L&#L&
&!&M&2&0()*+)&?(-.(/0*&.1&(&?(-.(/0*&)>()&.1&&&&&4+7*(14-*:N&/4)&.1&>GB6)>*1.O*:&)6&*L.1)&&
=>()&.1&(&0()*+)&?(-.(/0*@&
!&K&"L&#L&
&#L&M&P>*&-*0(96+1>.B&/*)A**+&(&0()*+)&?(-.(/0*&(+:&.)1&6/1*-?*:&364+)*-B(-)&
!"#$"%!&
#&
=>()&.1&(&0()*+)&?(-.(/0*@&
!&K&"L&#L&
&"L&M&P>*&*--6-&.+&)>*&7*(14-*7*+)&6Q&L&/G&!&&
'()*+)&8L6E*+641&,(-.(/0*1&
!&K&"L&#L&
KR&#L!S"L&
'()*+)&8+:6E*+641&,(-.(/0*1&
$&T&%G&#G&
G&R&#G$&S&%G&
&&'()*+)&8+:6E*+641&,(-.(/0*1&
$&T&%G&#G&
&M&,(-.(+3*&.+&-*1B6+1*&)6&B-*:.3)6-1&
&&
!"#$"%!&
!&
'()*+)&,(-.(/0*1&
%< =>()&.1&(&0()*+)&?(-.(/0*@&
#< '()*+)&?(-.(/0*1&A.)>&7409B0*&.+:.3()6-1&
!< C.D+E&(&0()*+)&?(-.(/0*&
F< C(3)6-&2+(0G1.1&H< '()*+)&,(-.(/0*1&(1&(&I*1B6+1*&$< J6B.+E&A.)>&7*(14-*7*+)&*--6-&
=>()&.Q&U+*&;*(14-*7*+)&206+*&.1+V)&,*-G&W66:@&
X.)-6E*+&
X&Y)-.B&J606-&
Z%&
[+(334-()*&;(3>.+*&
Z#&
Y)4\1&P(1)*:&X&
Z!&
'()*+)&,(-.(/0*1&(1&P>*6-*93(0&J6+1)-43)1&
Y)6-7&].1)4-/(+3*&
=(?*&^*.E>)1&
"%&
_-*3.B.)(96+&[+)*+1.)G&
U+1>6-*&]*)-.)41&
"#&
"!&
'()*+)&,(-.(/0*1&(1&P>*6-*93(0&J6+1)-43)1&
Y)6-7&].1)4-/(+3*&
=(?*&^*.E>)1&
"%&
_-*3.B.)(96+&[+)*+1.)G&
U+1>6-*&]*)-.)41&
"#&
"!&
"’1&(-*&?(-.(96+&.+&6/1*-?(96+1&+6)&*LB0(.+*:&/G&1)6-7&:.1)4-/(+3*&
!"#$"%!&
F&
J6+3*B)&,(0.:.)GM&&P>*&C(00(3G&6Q&X(7.+E&
_-*:(96+&[+)*+1.)G&
Y4+&Y)(-&]*+1.)G&
"%&
'6/1)*-&]*+1.)G&
Y>**B>*(:&]*+1.)G&
"#&
"!&
"’1&(-*&?(-.(96+&.+&6/1*-?(96+1&+6)&*LB0(.+*:&/G&B-*:(96+&.+)*+1.)G@@@&
“The skepticism regarding 'latent variables' among many statisticians can probably be attributed to the metaphysical status of hypothetical constructs. On the other hand ... the concept of a 'good statistician' is not real, but nevertheless useful …”
- Skrondal and Rabe-Hesketh#
'()*+)&,(-.(/0*1&
%< =>()&.1&(&0()*+)&?(-.(/0*@&
#< '()*+)&?(-.(/0*1&A.)>&7409B0*&.+:.3()6-1&
!< C(3)6-&2+(0G1.1&F< '()*+)&,(-.(/0*1&(1&(&I*1B6+1*&H< J6B.+E&A.)>&7*(14-*7*+)&*--6-&
!"#$"%!&
H&
'()*+)&,(-.(/0*1&A.)>&;409B0*&[+:.3()6-1&
!&
K%&"L%&
K#&
K!&
"L#&
"L!&
#L%&#L#&
#L!&
'()*+)&?(-.(/0*1&-*B-*1*+)1&1>(-*:&.+Q6-7(96+&6Q&.+:.3()6-1&
J6776+&36+3*B)4(0&:.(E-(7&6Q&YB*(-7(+’1&(+(0G1.1&6Q&“W`P>*6-GN”&)>*&.:*(&6Q&(&E*+*-(0.O*:&.+)*00.E*+3*&Q(3)6-&4+:*-0G.+E&)*1)&B*-Q6-7(+3*<&&X6)*&1>(-*:&?(-.(+3*&6Q&)*1)1&.+:.3()*&“E<”&
'()*+)&?(-.(/0*1&-*B-*1*+)1&1>(-*:&.+Q6-7(96+&6Q&.+:.3()6-1&
!&K%&
"L%&
K#&K!&"L#&
"L!&
[+:.3()6-1&7(G&J6?(-G&Q6-&U)>*-&I*(16+1&
!&
K%&"L%&
K#&
K!&
"L#&
"L!&
#L%&#L#&
#L!&
!"#$"%!&
$&
[+:.3()6-1&7(G&J6?(-G&Q6-&J(41(0&I*(16+1&
!&
K%&"L%&
K#&
K!&
"L#&
"L!&
#L%&#L#&
#L!&
;(G/*&-*1B*3.QG&G64-&76:*0@&
].a*-*+)&;*(14-*7*+)1&
20E(0&J6?*-&
b&J6?*-&"L%&
c4(:-()&]*+1.9*1&
5(+:&P-(+1*3)1&
"L#&
"L!&
;409B0*&_-6B*-9*1&
W-(O.+E&].1)4-/(+3*&
20E(*&'61)&
"L%&
J>(+E*&.+&W-(O*-&]*+1.)G&
5.)*1&B*-&(0E(&
"L#&
"L!&
I*B*()*:&;*(14-*1&
C.1>&]*+1.)G&
C.1>&()&)%&"L%&
C.1>&()&)#&
C.1>&()&)!&
"L#&
"L!&
!"#$"%!&
d&
;409`Y(7B0*&
C.1>&]*+1.)G&
J64+)&/G&U/1*-?*-&%&
"L%&
J64+)&/G&U/1*-?*-&#&
J64+)&/G&U/1*-?*-&!&
"L#&
"L!&
8?(04(9+E&=>*)>*-&[+:.3()6-1&=.00&;(e*&(&W66:&'()*+)&,(-.(/0*&
!"#$%&$'()*%%$+,-*.#/(&&&&&&&&&&&&G%&&&&&&&&&&&&&&G#&&&&&&&&&&G!&&&&&&&& &&GF&&&&&&&&&&&GH&G%&&&&&&&&%<fff&G#&&&&&&&&f<g!!&&&&&&%<fff&G!&&&&&&&&f<h%!&&&&&&f<h!F&&&&&%<fff&GF&&&&&&&&f<dd!&&&&&&f<d#h&&&&&f<$g!&&&&%<fff&GH&&&&&&&&f<d!f&&&&&&f<$F$&&&&&f<$f!&&&&f<g$g&&&&&&%<fff&
GF&
GH&
$1'
G%&
G#&
G!&
i%j J6--*0(96+1&(76+E&3(+:.:()*&.+:.3()6-1&)*00&41&A>*)>*-&:()(&.1&36+1.1)*+)&A.)>&A>()&.1&.7B0.*:&/G&64-&76:*0<&&
i#j X6)*&36--*0(96+1&(-*&(00&1)-6+EN&/4)&+6)&(00&*k4(00G&1)-6+E<&P>.1&1>6A1&41&)>()&)>*1*&(-*&+6)&-*:4+:(+)&.+:.3()6-1&)>()&(-*&367B0*)*0G&.+)*-3>(+E*(/0*<&
i!j &[+&B(-9340(-N&?(-.(/0*1&GF&(+:&GH&(-*&76-*&1)-6+E0G&36--*0()*:&A.)>&*(3>&6)>*-&)>(+&A.)>&)>*&6)>*-&?(-1<&
C.L.+E&_(-(7*)*-1&Q6-&[:*+9l(/.0.)G&
GF&
GH&
$1'
G%&
G#&
G!&
i%j X6)*&A*&+**:&)6&mlLn&167*&B(-(7*)*-1&i1B*3.QG&)>*.-&?(04*1j&Q6-&.:*+9l(/.0.)G<&&
i#j [+&)>.1&3(1*N&[&3>61*&)6&1*)&?(-.(+3*&6Q&0()*+)&?(-.(/0*&R&%<f<&&
i!j P>*&6)>*-&3>6.3*&A640:&/*&)6&lL&6+*&6Q&)>*&B()>&36*o3.*+)1&)6&%<f<&&
iFj C.L.+E&(&06(:.+E&)6&%&B4)1&)>*&0()*+)&?(-.(/0*&6+&)>*&13(0*&6Q&)>()&.+:.3()6-<&&
iHj P*1)&76:*0&A.)>&:.a*-*+)&B()>1&lL*:&)6&%&)6&*+14-*&)>()&G64-&0()*+)&?(-.(/0*&.1&E66:&
%1'
%2'
%3'
%4'
%5'
&1'0(
'()*+)&,(-.(/0*&A.)>&PA6&[+:.3()6-1&
$1'G%&
G#&
%1'
%2'
%< _-6/0*7&`&A*&>(?*&6+0G&6+*&B.*3*&6Q&.+Q6-7(96+&(/64)&G%&(+:&G#&`&)>*.-&36--*0(96+&iR&f<g!!j<&&
#< ;6:*0&>(1&)A6&B()>&36*o3.*+)1N&B041&)>*&?(-.(+3*&6Q&64-&0()*+)&?(-.(/0*<&&
!< &=*&3(+&lL&)>*&?(04*&6Q&64-&',&)6&%N&/4)&)>()&1900&0*(?*1&41&A.)>&6+*&e+6A&(+:&)A6&4+e+6A+1<&
XUP&[]8XP[C[8]<&&
!.$(1*+2-*./(34$.(54$%$(,%$(*.+6(53*(7.'78,5*%#9(54$6(4,&$($:2,+(3$7;45(7.(54$($#-<,-*.(*=(54$(>?(@,"#$.5(*54$%(7.=*%<,-*.AB((
1*9(3$(8,.(#5,.',%'7C$(54$(53*(<$,#2%$#9(,.'(*.+6($#-<,5$(,(#7.;+$(D,%,<$5$%(=*%("*54(D,54#B(
(1'
(1'
!"#$"%!&
h&
=>G&p1*&'()*+)&,(-.(/0*1&A.)>&;409B0*&[+:.3()6-1@&
%< 5*q*-&(334-(3G&.+&7*(14-*7*+)&6Q&-*0(96+1>.B1&:4*&)6&1>(-*:&?(-.(96+&/*)A**+&.+:.3()6-1<&
#< T64&3(++6)&7*(14-*&(&)>*6-*93(0&36+1)-43)r&&&
'()*+)&,(-.(/0*1&
%< =>()&.1&(&0()*+)&?(-.(/0*@&
#< '()*+)&?(-.(/0*1&A.)>&7409B0*&.+:.3()6-1&
!< C.D+E&(&0()*+)&?(-.(/0*&
F< C(3)6-&2+(0G1.1&H< '()*+)&,(-.(/0*1&(1&(&I*1B6+1*&$< J6B.+E&A.)>&7*(14-*7*+)&*--6-&
EF,<D+$/(GD*#$<,-#<(7.(H*7#*.(I,%5(J%*;#(
Y(+)61N&s<J<&t&J(++()*00(N&]<J<&i#f%%j<&_>*+6)GB.3&.+)*E-(96+&*7*-E*1&Q-67&(B61*7(917&(+:&13(0*&.+&B6.16+&Q-6E1<&!"#$%&'()*%&+$(,%&-$.%&/%-%+%0&1230&415674132%&
=>()&:-.?*1&)>*&*?60496+&6Q&A(-+.+E&3606-(96+@&
K*'6(8*.'7-*.L(M*F78756L( I7$5L(
!"#$"%!&
g&
2&_>G06E*+*93&2BB-6(3>&)6&Y8;&41.+E&[+:*B*+:*+)&J6+)-(1)1&
Y(+)61&(+:&J(++()*00(&#f%%&_X2Y&
santosCov <-read.table("santosCov.txt", na.strings=".")santosCov <-as.matrix(santosCov)
santosCFA1<-' Aposematism =~ Alkaloid.quantity + Alkaloid.diversity +
Conspicuous.coloration'
santosFit1<-sem(santosCFA1, sample.cov=santoCov, sample.nobs=21)
2B61*7(917&(1&(&'()*+)&,(-.(/0*&
2B61*7(917&
20e(06.:&c4(+9)G&
"L%&
20e(06.:&&].?*-1.)G&
J6+1B.34641&J606-(96+&
"L#&
"L!&
#L%&
#L#&
#L!&
&&
Y(+)61&t&J(++()*00(&#f%%&_X2Y&&
2B61*7(917&(1&(&'()*+)&,(-.(/0*&
Estimate Std.err Z-value P(>|z|)Latent variables: Aposematism =~ Alkaloid.quan 1.000 Alkaloid.dive 2.358 0.558 4.223 0.000 Conspicuous.c 0.753 0.215 3.502 0.000
Variances: Alkaloid.quan -0.000 0.000 -0.190 0.849 Alkaloid.dive 0.004 0.002 2.376 0.018 Conspicuous.c 0.001 0.000 2.905 0.004 Aposematism 0.001 0.000 2.915 0.004
2B61*7(917&
20e(06.:&c4(+9)G&
`f<f!&
20e(06.:&&].?*-1.)G&
J6+1B.34641&J606-(96+&
f<!d&
f<H!&
%<f%&
f<dg&
f<df&
&&%&
Y(+)61&t&J(++()*00(&#f%%&_X2Y&&
^(GA66:&3(1*M&+*E(9?*&*197()*&6Q&?(-.(+3*<&&J(+&.+:.3()*&6?*-l)&.+:.3()6-<&&200&A.)>.+&-64+:.+E&*--6-&>*-*N&(1&.+:.3()6-&A(1&lL*:N&(+:&?(-.(+3*&+6)&:.a*-*+)&Q-67&f&
8L*-3.1*M&C.D+E&'()*+)&,(-.(/0*1&
!&K%&"L%&
K#&
K!&
"L#&
"L!&
#L%&#L#&
#L!&
• &P>*&Y(+)61&36?(-.(+3*&7()-.L&>(1&7(+G&6)>*-&?(-.(/0*1&-*0()*:&)6&Q-6E&:.*)&(+:&Q-6E&1.O*&u&)-G&64)&V/6:G&1.O*V&(1&(&0()*+)&?(-.(/0*&
!"#$"%!&
%f&
8L*-3.1*M&C.D+E&'()*+)&,(-.(/0*1&
Y.O*&
'6E<;(11&"L%&
'6E<I;I&
'6E<Y36B*&
"L#&
"L!&
#L%&#L#&
#L!&
santosSize<-'! Size =~ Log.Mass + Log.RMR + Log.Scope'!
santosSizeFit<-sem(santosSize, sample.co=santosCov, sample.nobs=21)!
8L*-3.1*M&C.D+E&'()*+)&,(-.(/0*1&
Y.O*&
'6E<;(11&f<fF&
'6E<I;I&
'6E<Y36B*&
f<%F&
f<%#&
f<gh&f<g!&
f<gF&
Estimate Std.err Z-value P(>|z|) Std.lv Std.all!Latent variables:! Size =~! Log.Mass 1.000 0.096 0.981! Log.RMR 0.815 0.083 9.771 0.000 0.078 0.930! Log.Scope 0.861 0.084 10.228 0.000 0.082 0.938!
c4*196+1@&
'()*+)&,(-.(/0*1&
%< =>()&.1&(&0()*+)&?(-.(/0*@&
#< '()*+)&?(-.(/0*1&A.)>&7409B0*&.+:.3()6-1&
!< C.D+E&(&0()*+)&?(-.(/0*&
F< C(3)6-&2+(0G1.1&H< '()*+)&,(-.(/0*1&(1&(&I*1B6+1*&$< J6B.+E&A.)>&7*(14-*7*+)&*--6-&
!"#$"%!&
%%&
8L(7B0*M&_>G06E*+*93&JC2r&
2B61*7(917&
20e(06.:&c4(+9)G&
`f<f!&
20e(06.:&&].?*-1.)G&
J6+1B.34641&J606-(96+&
f<!d&
f<H!&
%<f%&
f<dg&
f<df&
&&%&
Y(+)61&t&J(++()*00(&#f%%&_X2Y&&
iJ6+l-7()6-G&C(3)6-&2+(0G1.1j&
8LB06-()6-G&C(3)6-&2+(0G1.1&
_>61B>6-641&
X.)-6E*+&
Y6.0&C(3)6-&%&
J(-/6+&
B^&
W-(.+&Y.O*&Y6.0&C(3)6-&#&
^4741&
)%#&
v &U/1*-?*:&?(-.(/0*1&*LB0(.+*:&/G&36--*0()*:&Q(3)6-1&v &8--6-&.+&6/1*-?*:&?(-.(/0*1&v &54)w*LB06-()6-G&`&XUP&^T_UP^8Y[Y&P8YP[XW<&
"L%&
"L#&
"L!&
"LF&
"LH&
"L$&
J6+l-7()6-G&C(3)6-&2+(0G1.1&
v &P*1)1&A>*)>*-&?(-.(/0*1&1*B(-()*&.+)6&E-64B1&v &p1*Q40&)6&)*1)&(&“7*(14-*7*+)&76:*0”&Q6-&Y8;&v &)%#&-*B-*1*+)1&36776+&?(-.(96+&:4*&)6&6)>*-&Q(3)6-1&
_>61B>6-641&
X.)-6E*+&
Y6.0&&Y)6.3>.67*)-G&
J(-/6+&
B^&
W-(.+&Y.O*& Y6.0&J67B61.96+&
^4741&
)%#&
"L%&
"L#&
"L!&
"LF&
"LH&
"L$&
&%&
&#&
[:*+9l3(96+M&C.L.+E&Y3(0*&Q6-&(&&Y)(+:(-:.O*:&;6:*0&
v &[+&761)&3(1*1N&A*&+**:&)6&B-6?.:*&(&13(0*&Q6-&64-&0()*+)&?(-.(/0*1<&
v &P*1)&)>()&-*140)1&:6+V)&3>(+E*&.Q&G64&3>(+E*&13(0*<&
_>61B>6-641&
X.)-6E*+&
Y6.0&&Y)6.3>.67*)-G&
J(-/6+&
B^&
W-(.+&Y.O*& Y6.0&J67B61.96+&
^4741&
)%#&
"L%&
"L#&
"L!&
"LF&
"LH&
"L$&
%&
%&
&%&
&#&
!"#$"%!&
%#&
[:*+9l3(96+M&C.L.+E&,(-.(+3*&Q6-&(+&&p+1)(+:(-:.O*:&;6:*0&
v &J>*3e&)6&7(e*&14-*&7*)>6:&6Q&.:*+9l3(96+&:6*1+V)&(0)*-&-*140)1&6Q&JC2&v Y)(+:(-:.O*:&6-&p+1)(+:(-:.O*:&(BB-6(3>&X8J8YY2IT&Q6-&.:*+9l3(96+&
_>61B>6-641&
X.)-6E*+&
Y6.0&&Y)6.3>.67*)-G&
J(-/6+&
B^&
W-(.+&Y.O*& Y6.0&J67B61.96+&
^4741&
)%#&
"L%&
"L#&
"L!&
"LF&
"LH&
"L$&
%&
%&
&%&
&#&
[:*+9l3(96+M&200&'()*+)&,(-.(/0*1&>(?*&P>-**&[+:.3()6-1&A.)>64)&J6--*0()*:&8--6-&
!&
K%&"L%&
K#&
K!&
"L#&
"L!&
%&#L#&
#L!&
P>-**&[+:.3()6-&I40*&`&YpCC[J[8XP&
[:*+9l3(96+M&PA6&[+:.3()6-1&B*-&'()*+)N&;409B0*&'()*+)1N&p+36--*0()*:&[+:.3()6-&,(-.(+3*&
!%&K%&"L%&
K#&"L#&
%
#L#&
PA6&[+:.3()6-&I40*&`&YpCC[J[8XP&
!#&K!&"L!&
KF&"LF&
%
#LF&
[:*+9l3(96+M&[Q&Y67*&[+:.3()6-1&J6?(-GN&8(3>&;41)&^(?*&2)&'*(1)&U+*&p+36--*0()*:&[+:.3()6-&
!%&K%&"L%&
K#&"L#&
%
#L#&
J6--*0()*:&[+:.3()6-&I40*&&`&X8J8YY2IT&
!#&K!&"L!&
KF&"LF&
%
#LF&
!"#$"%!&
%!&
[:*+9l3(96+M&[Q&[+:.3()6-1&Y>(-*:N&8(3>&'()*+)&X**:1&U+*&p+.k4*&[+:.3()6-&
!%&K%&"L%&
K#&"L#&
%
#L#&
Y>(-*:&[+:.3()6-&I40*&`&X8J8YY2IT&
!#&K!&"L!&
KF&"LF&
#L!#&
%&
#L!%&
87B.-.3(0&p+:*-.:*+9l3(96+&Y900&_611./0*&
!&
K%&"L%&
K#&
K!&
"L#&
"L!&
%&
W*+*-(0&I40*1&Q6-&[:*+9l3(96+&%< P̀ -40*&1900&>60:1&u&+*3*11(-G&
#< Y)(+:(-:.O(96+&`&+*3*11(-G&
!< P>-**&.+:.3()6-&-40*&u&14o3.*+)&
F< PA6&[+:.3()6-&-40*&u&14o3.*+)&
H< J6--*0()*:&[+:.3()6-&-40*&u&+*3*11(-G&
$< Y>(-*:&[+:.3()6-&I40*&`&+*3*11(-G&
NBKB(N*.$(*=(54$#$(,%$("*54(.$8$##,%6(,.'(#2O87$.5P(
8L*-3.1*M&_>G06E*+*93&JC2r&
santosCFA2<-paste(santosCFA1, ' Aposematism =~ Ant.Mite.Specialization+log.Prey
Scale =~ Log.Mass+Log.RMR+Log.Scope+Conspicuous.coloration',
sep="\n")
!"#$"%!&
%F&
'()*+)&,(-.(/0*1&
%< =>()&.1&(&0()*+)&?(-.(/0*@&
#< '()*+)&?(-.(/0*1&A.)>&7409B0*&.+:.3()6-1&
!< C.D+E&(&0()*+)&?(-.(/0*&
F< C(3)6-&2+(0G1.1&H< '()*+)&,(-.(/0*1&(1&(&I*1B6+1*&$< J6B.+E&A.)>&7*(14-*7*+)&*--6-&
M4$(EF,<D+$/(M4$(;$.$%,+(D$%=*%<,.8$(*=(5%,.#D+,.5$'(D+,.5#(,#(,(=2.8-*.(*=(54$7%(;$.$-8(
'7##7<7+,%756(5*(+*8,+(D*D2+,-*.#B((
Q-67M&
P-(?.1N&Y<8<&(+:&W-(3*N&s<5<&#f%f<&_-*:.39+E&B*-Q6-7(+3*&Q6-&*3606E.3(0&-*1)6-(96+M&(&3(1*&1)4:G&41.+E&-8("9:(&(*);":.<#"(<&=$#*#>.$(*&+88*.$(9#:?&#fM%g#`#fF<&
P>*&P>*6-G&]-.?.+E&)>*&;6:*0.+E&
P>*6-G&14EE*1)1&Q6006A.+E&Q6-&)-(+1B0(+)*:&-8("9:(<&
Path Distance
Latitude Genetic Distance
Performance
Distance effects Gene Flow
/4)&A>()&:6&A*&7*(+&/G&B*-Q6-7(+3*@&
_*-Q6-7(+3*&(1&(&0()*+)&36+1)-43)&_*-Q6-7(+3*&.7B0.*1&367B0*LN&.+)*-36--*0()*:&-*1B6+1*&/G&7(+G&)-(.)1&-*x*39+E&167*&4+:*-0G.+EN&4+7*(14-*:&3(41*&6-&3(41*1<&&
5*&(A(-*&)>()&1.7B0G&0.+e.+E&(&/4+3>&6Q&7*(14-*1&)6&(&0()*+)&?(-.(/0*&:6*1&+6)&7*(+&G64&>(?*&36--*3)0G&1B*3.l*:&)>*&76:*0<&&
U+*&+**:1&)6&1)(G&Q6341*:&6+&)>*&k4*196+&6Q&mA>()&(3)4(00G&3(41*1&)>*&-*1B6+1*1&[&1**@n<&P>.1&.1&>6A&G64&(--.?*&()&B-6B*-&76:*0&1B*3.l3(96+N&+6)&1.7B0G&)>-64E>&(BB0G.+E&)>*&76:*0&1)-43)4-*1&)6&G64-&:()(<&&
GF&
GH&
H$%=*%<,.8$(
G%&
G#&
G!&
N*5$(547#(<*'$+(46D*54$#7C$#(3$(4,&$(Q&$(*"#$%&$'(%$#D*.#$#(34*#$(7.5$%8*%%$+,-*.#(,%$(8*.#7#5$.5(3754(,(#7.;+$(2.'$%+67.;(8,2#$B((
M4$%$(<,6("$(*54$%(547.;#(54,5(7.R2$.8$(60S6T(,.'(,U$85(54$7%(*"#$%&$'(7.5$%8*%%$+,-*.#B((
G+#*9(D$%=*%<,.8$(7#(%$D%$#$.5$'(,#(,(2.7'7<$.#7*.,+(8,2#$(@.*5(,(8*++$8-*.(*=('7#D,%,5$(8,2#$#AB(
!"#$"%!&
%H&
0(?((+N&'()*+)&[+:.3()6-1N&(+:&I*E-*11.6+&
B*-Q6-7(+3*& 306+*:.(7&
performance ~ clonediam!
B*-Q6-7(+3*&306+*:.(7&
performance =~ clonediam!
p1.+E&'()*+)&,(-.(/0*1&.+&Y8;&
spartinaModel<-'performance =~ clonediam +!!! ! ! ! ! !numbstems + numbinfl + meanht +!!! ! ! ! ! !meanwidth! ! !!!! ! ! ! ! !meanht ~~ meanwidth!
performance ~ geneticdist'!
B*-Q6-7(+3*&
+47/1)*71&
+47/.+x&
7*(+>)&
7*(+A.:)>&
306+*:.(7&
E*+*93:.1)&
%1'
%2'
%3'
%4'
%5'
&'%&
211*11.+E&C.)&
> spartinaFit<-sem(spartinaModel, data=spartina)!> summary(spartinaFit, standardized=T, rsquare=T)!lavaan (0.5-12) converged normally after 154 iterations!
Number of observations 23!
Estimator ML! Minimum Function Test Statistic 12.237! Degrees of freedom 8! P-value (Chi-square) 0.141!
B*-Q6-7(+3*&
+47/1)*71&
+47/.+x&
7*(+>)&
7*(+A.:)>&
306+*:.(7&
E*+*93:.1)&
*%&
*#&
*!&
*F&
*H&
&'%&
J6*o3.*+)1&
Estimate Std.err Z-value P(>|z|) Std.lv Std.all!Latent variables:! performance =~! clonediam 1.000 17.199 0.969! numbstems 0.904 0.079 11.508 0.000 15.555 0.962! numbinfl 0.106 0.015 7.030 0.000 1.822 0.853! meanht 0.643 0.114 5.654 0.000 11.066 0.785! meanwidth 0.076 0.016 4.680 0.000 1.308 0.718!
Regressions:! performance ~! geneticdist -57.134 12.465 -4.584 0.000 -3.322 -0.708!
B*-Q6-7(+3*&
+47/1)*71&
+47/.+x&
7*(+>)&
7*(+A.:)>&
306+*:.(7&
E*+*93:.1)&
*%&
*#&
*!&
*F&
*H&
&'%&
!"#$"%!&
%$&
C.)&6Q&,(-.(/0*1&
R-Square:!
clonediam 0.938! numbstems 0.925! numbinfl 0.727! meanht 0.617! meanwidth 0.515! performance 0.502!
B*-Q6-7(+3*&
+47/1)*71&
+47/.+x&
7*(+>)&
7*(+A.:)>&
306+*:.(7&
E*+*93:.1)&
*%&
*#&
*!&
*F&
*H&
&'%&
2&'()*+)&8L*-3.1*&
B*-Q6-7(+3*&
+47/1)*71&
+47/.+x&
7*(+>)&
7*(+A.:)>&
306+*:.(7&
E*+*93:.1)&
0(9)4:*&
*%&
*#&
*!&
*F&
*H&
2&'()*+)&8L*-3.1*&
B*-Q6-7(+3*&
+47/1)*71&
+47/.+x&
7*(+>)&
7*(+A.:)>&
306+*:.(7&
E*+*93:.1)&
0(9)4:*&
%1'
%2'
%3'
%4'
%5'
&'
spartinaModel2<-paste(spartinaModel, '! meanht ~ latitude! meanwidth ~ latitude', sep="\n")!
'()*+)&,(-.(/0*1&
%< =>()&.1&(&0()*+)&?(-.(/0*@&
#< '()*+)&?(-.(/0*1&A.)>&7409B0*&.+:.3()6-1&
!< C.D+E&(&0()*+)&?(-.(/0*&
F< C(3)6-&2+(0G1.1&H< '()*+)&,(-.(/0*1&(1&(&I*1B6+1*&$< J6B.+E&A.)>&7*(14-*7*+)&*--6-&
!"#$"%!&
%d&
'()*+)&,(-.(/0*1&(+:&;*(14-*7*+)&8--6-&
P-4*&J6?*-&UQ&20E(*&
8197()*:&J6?*-&&6Q&20E(*&
8--6-&.+&;*(14-*7*+)&6Q&J6?*-&
%&
'2X]Y2P&;*(14-*7*+)1&6Q&y*0B&J(+6BG&
• &.7(E*-G&(3k4.-*:&(BB-6L<&*?*-G&#&A**e1&Q-67&%ghF`B-*1*+)&&
• &!f&7&-*160496+&].?*-&)-(+1*3)1&
I2Y&_[JPpI8&
J(+6BG&/.67(11R&e*0B.+*11z%HF<hg`$h<$#&
I#&&R&f<$#&
'2X]Y2P&y*0B.+*11&?1<&J(+6BG&5.67(11&
!"#$"%!&
%h&
=*&3(+&)-(+1Q6-7&1()*00.)*&:()(&)6&3(+6BG&/.67(11N&(+:&lL&)>*&4+1)(+:(-:.O*:&06(:.+E&)6&%<&&&
'2X]Y2P&y*0B.+*11&?1<&J(+6BG&5.67(11&
P-4*&J(+6BG&5.67(11&
;*(14-*:&J(+6BG&5.67(11&
54)&A>()&(/64)&*--6-@&
=*&e+6A&)>()&I#R%`*197()*:&?(-"6/1*-?*:&?(-&
"LR&i%`I#j&
p+1)(+:(-:.O*:&;*(14-*7*+)&8--6-&R"Lz?(-i;*(14-*:&J(+6BG&5.67(11j&
"L& i%j&
'*)V1&'66e&()&)>*&'P8I&:()(M&]()(&_-*B&
library(lavaan) lter<-read.csv("./lter_kelp.csv")
#1) Calculate fitted values for spring biomass #landsat observations to biomass lter$landsat_spring_biomass<-154.89*lter$spring_canopy+68.62
#2) Calculate fitted values for summer biomass #summer kelp counts to biomass y=0.08x+0.01 r^2=0.79 lter$summer_kelp_biomass<-0.08*lter$kelp+0.01
#3) Transform fitted values for easier fitting #transformation for easier fitting lter$summer_kelp_biomass<-log(lter$summer_kelp_biomass+1) lter$landsat_spring_biomass <-log(lter$landsat_spring_biomass +1)
f<dH&
'2X]Y2P&y*0B.+*11&?1<&J(+6BG&5.67(11&C.)&)>.1&;6:*0r&
'2X]Y2P&YB-.+E&&J(+6BG&5.67(11&
;*(14-*:&Y477*-&y*0B&5.67(11&
i4+1)(+:(-:.O*:&36*o3.*+)1j&
f<Hf&
if<#$j&
noerror<-'summer_kelp_biomass ~ landsat_spring_biomass'
'2X]Y2P&y*0B.+*11&?1<&J(+6BG&5.67(11&
P-4*&YB-.+E&y*0B&&5.67(11&
;*(14-*:&YB-.+E&&J(+6BG&5.67(11&
f<!h&
i%j&
;*(14-*:&Y477*-&y*0B&5.67(11&
{
var(lter$landsat_spring_biomass, na.rm=T)*(0.38)
[1] 3.762301
errorCanopy<-'true_spring_biomass =~ 1*landsat_spring_biomasssummer_kelp_biomass ~ true_spring_biomass
#errorlandsat_spring_biomass ~~ 3.762301*landsat_spring_biomass
'
I#Rf<$#N&%`f<$#Rf<!h&
!"#$"%!&
%g&
[+36-B6-(9+E&7*(14-*7*+)&*--6-&.+&)>*&B-*:.3)6-&.+3-*(1*1&)>*&36*o3.*+)&1.O*&(+:&(764+)&6Q&
?(-.(+3*&*LB0(.+*:&
'2X]Y2P&y*0B.+*11&?1<&J(+6BG&5.67(11&
P-4*&YB-.+E&y*0B&&5.67(11&
'2X]Y2P&YB-.+E&&J(+6BG&5.67(11&
f<!h&
i%j&
;*(14-*:&Y477*-&y*0B&5.67(11&
f<$!&
if<F%j&
f<dg&
f<$f&
8L*-3.1*M&J6:*&)>.1&76:*0r&P-4*&YB-.+E&y*0B&&
5.67(11&
'2X]Y2P&YB-.+E&&J(+6BG&5.67(11&
f<!h&
i%j&
;*(14-*:&Y477*-&y*0B&5.67(11&
f<#%&
Y477*-&y*0B&5.67(11& &
i%j&
var(lter$summer_kelp_biomass, na.rm=T)*(0.21)[1] 0.5495345
error_both<-'true_spring_biomass =~ 1*landsat_spring_biomasstrue_summer_biomass =~ 1*summer_kelp_biomass
true_summer_biomass ~ true_spring_biomass
#errorlandsat_spring_biomass ~~ 3.762301*landsat_spring_biomasssummer_kelp_biomass ~~ 0.5495345* summer_kelp_biomass
'
'2X]Y2P&y*0B.+*11&?1<&J(+6BG&5.67(11&
P-4*&YB-.+E&y*0B&&J(+6BG&5.67(11&
;*(14-*:&YB-.+E&&J(+6BG&5.67(11&
f<!h&
i%j&
;*(14-*:&Y477*-&y*0B&5.67(11&
f<#%&
Y477*-&y*0B&5.67(11&
f<Hf&
f<d%&
if<F%j&
;*(14-*7*+)&*--6-&.+&)>*&-*1B6+1*&(a*3)1&1)(+:(-:.O*:&36*o3.*+)1&(+:&*197()*1&6Q&*--6-N&
/4)&+6)&)>*&4+1)(+:(-:.O*:&36*o3.*+)<&
i%j&f<dg& f<hg&
I*(16+1&)6&P>.+e&(/64)&;*(14-*7*+)&8--6-&
%< =*&e+6A&64-&7*(14-*7*+)1&(-*&+6)&B*-Q*3)r&
#< [+3-*(1*:&(334-(3G&.+&*197(9+E&-*0(96+1>.B1&/*)A**+&?(-.(/0*1<&
!< [+3-*(1.+E&*LB0(+()6-G&B6A*-&6Q&G64-&>(-:`*(-+*:&7*(14-*7*+)1<&
