E' possibile costruire curve di crescita per malattie rare? La sindrome di Shwachman come primo esempio italiano.
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yi 3 i = 1, . . . , n3 %:6%%# (6 4"6 ;6:&6=&)# >'" C4"5&'"# (& (#"$&%F f (yi ; θi )3 >'" θi = (θ1i , θ2i , θ3i , θ4i ) = (µi , σi , νi , τi ) ;#%%':# (& 746%%:' 86:6*#%:&3 '<"4"' (#& 746)& 849 #$$#:# C4"5&'"# (& ;6:&6=&)& #$8)&>6%&;#@ A 8:&*& (4# 86:6*#%:&3 µi # σi 3 $'"' 86:6*#%:& (& 8'$&5&'"# # $>6)63 :&$8#%G %&;6*#"%#3 *#"%:# <)& 6)%:& (4#3 $# 8:#$#"%&3 $'"' 86:6*#%:& (& C':*6 1*'(#)G )6"' :&$8#%%&;6*#"%# 6$&**#%:&6 # >4:%'$&2@ A) *'(#))' 6$$4"%' 849 >'*4"74# #$$#:# <#"#:6)&556%' >'" 8&H (& 746%%:' 86:6*#%:&@ E&<=I # -%6$&"'8'4)'$ 1JKKL2 (#M"&$>'"' )6 C':*4)65&'"# ':&<&"6)# (#) *'(#))' /0,+-- >'*# $#<4#@ N#: k = 1, . . . , 43 $&6 gk (.) 4"6 C4"5&'"# *'"'%'"6 "'%6 >?# *#%%# &" :#)65&'"# & 86:6*#%:& (#))6 (&$%:&=45&'"# >'" )# '88':%4"# ;6:&6=&)& #$8)&>6%&;#3 '$$&6. gk (θk ) = ηk = Xk βk +. Jk X. Zjk γjk. j=1. O4&"(&3 ;6)#. g1 (µ) = η1 = X1 β1 +. J1 X. Zj1 γj1. J2 X. Zj2 γj2. J3 X. Zj3 γj3. J4 X. Zj4 γj4. j=1. g2 (σ) = η2 = X2 β2 +. j=1. g3 (ν) = η3 = X3 β3 +. j=1. g4 (τ ) = η4 = X4 β4 +. j=1. (';# µ, σ, ν, τ # ηk $'"' ;#%%':& (& )4"<?#556 n3 βk = (β1k , β2k , . . . , βJk k ) . 4" ;#%%':# (& )4"<?#556 Jk 3 Xk . 4"6 *6%:&># "'%6 (& (&$#<"' (& (&*#"G $&'"& n × Jk 3 Zjk . 4"P6)%:6 *6%:&># "'%6 (& (&$#<"' (& (&*#"$&'"& n × qjk # γjk . 4"6 ;6:&6=&)# 6)#6%':&6 (& (&*#"$&'"# qjk 3 8#: )6 746)# $& 6$$4*# −1 γjk ∼ Nqjk ( , G−1 jk )3 (';# Gjk . )P&";#:$6 <#"#:6)&556%6 (#))6 *6%:&># Gjk =.
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(17) !"! #$%&'& ()#*++ !"#$ % βj &"'" ("$)(%$'*% +$,-% $. T (pj ) =. p T j,. &$ pj 6= 0. -".(T ),. ,-*+%/$'*%0. 1$ 2"*$'3$ p1 , . . . , pm &"'" '4/$+% +$,-% !%&*%'*%0 5' 2"-%'"/%" ("'#$'3%"',-$ !% .+,!" m 6, pj = j 2$+ ".'% j 7 /$'*+$ '$% 89 -$ 2"*$'3$ pj &"'" &($-*$ !, 4' &$* 2+$!$:'%*" P 0 ;"<&*"' $ =-*/,' > ??@A &4..$+%&("'" !% ("'&%!$+,+$ P = {−2, −1, −0.5, 0, 0.5, 1, 2, . . . , /,B(3, p)}0 1, !$:'%3%"'$ ("/2-$*, !% 4' 89 ("/2+$'!$ ,'(6$ -, 2"&&%C%-%*D !% +%2$*$+$ %- .+,!" !$--$ 2"*$'3$7 "&&%,. β0 +. m X. β j Hj ,. j=1. !"#$ H1 = T (p1 ) $7 2$+ j = 2, . . . , m7. Hj =. (p ) T j ,. &$ pj 6= pj−1. H j−1 -".(T ),. ,-*+%/$'*%0. =! $&$/2%" 4' 89 !% .+,!" E $ 2"*$'3$ > 7 E7 EA 2+$&$'*, -, F"+/,G. β0 + β1 T + β2 T 3 + β3 T 3 -".(T ).. !"!". #$%&$''()*$ '+,(*$ -./(-0. 1, +$.+$&&%"'$ &2-%'$ H 4' 2,+*%("-,+$ /$*"!" !% +$.+$&&%"'$ C,&,*" &4--$ F4'3%"'% &2-%'$0 = !%I$+$'3, !$--, +$.+$&&%"'$ 2"-%'"/%,-$7 (6$ 4*%-%33, 4' &"-" 2"-%'"/%" 2$+ ,22+"&&%/,+$ -, F4'3%"'$ '$--J%'*$+#,--" !% !$:'%3%"'$7 -J%'*$+2"-,3%"'$ &2-%'$ H "**$'4*, !%#%!$'!" -J%'*$+#,--" %' 2%K &"**"L%'*$+#,--% $ &($.-%$'!" 2$+ (%,&(4'" !% $&&% 4' 2"-%'"/%" !% .+,!" d7 &"-%*,/$'*$ 2%(("-"0 M$'."'" 2"% %/2"&*$ 4-*$+%"+% ("'!%3%"'% %' /"!" (6$ % 2"-%'"/% &% &,-!%'" *+, !% -"+" %' /"!" -%&(%"7 %/2"'$'!" -, ("'*%'4%*D !$--$ 2+%/$ d − 1 !$+%#,*$0 N%,'" !,*% 4' %'*$+#,--" [a, b] $ 4' %'*$+" 2"&%*%#" n0 8"&*". h = (b − a)/n,. &% !$:'%&("'". xi = a + ih. $. yi = f (xi ),. 72$+ i = 0, 1, . . . , n.. N% (6%,/, !"#$% &'(#&) #$*%+!,")$*% !$--, F4'3%"'$ f (x) '$% '"!% x0 , x1 , . . . , xn 4', F4'3%"'$ s(x) *,-$ (6$ &$ x0 ≤ x ≤ x1 s1 (x), &$ x1 ≤ x ≤ x2 s(x) = ... sn−1 (x), &$ xn−1 ≤ x ≤ xn s0 (x), .
(18) !"#$%&% '( )*$%+# * )%+*&&# "*, &! -$#)! +*&&* .,/* +# ,*- #$!. !. "#$% #&'( si (x) ) *' +#,('#-(# "( &./"# ≤ 30 % 1/,% ('#,1.% 23% 4 s(xi ) = yi. ∀i5. !4 s(x) ∈ C 2 [a, b] ⇔ s(x) "%.($/6(,% 2#' 2#'1('*(17 8'# /, 9%2#'"# #."('% (' [a, b]4 :#(23) 2(/92*'# "%( +#,('#-( si (x)0 +%. i = 0, 1, . . . , n0 ) ('"($("*/1# "/( 9*#( ; 2#%<2(%'1(0 ,/ =*'>(#'% s(x) 9/.7 ('"($("*/1/ "/ 4n 2#%<2(%'1( 2#-? +,%99($/-%'1%4 @*11/$(/0 A*%91( 2#%<2(%'1( 9#'# $('2#,/1( "/,,% 2#'"(>(#'( "( ('1%.+#,/>(#'% B C0 23% 9#'# (' 1*11# n + 10 % "/,,% 2#'"(>(#'( "( .%&#,/.(17 B!C 23% 9#'# (' 1*11# 3(n + 1)4 D#11./%'"# (, '*-%.# "%( $('2#,( "/, '*-%.# "%( 2#%<2(%'1( 23% ("%'1(82/'# ,/ 9+,('% s(x)0 9( #11(%'%. 4n − (n + 1) − 3(n + 1) = 2 % A*('"(0 +%. ("%'1(82/.% 2#-+,%1/-%'1% ,/ =*'>(#'% 9+,('%0 9%.$#'# "*% 2#'"(>(#'( /&&(*'1($%4 D( 92%,&#'# ,% "*% 2#'"(>(#'( /&&(*'1($% 2#-%. s′′ (a) = s′′ (b) = 0, 23% ("%'1(82/'# *'/ =*'>(#'% 9+,('% "%11/ !"#$% &'(#&) $)*'+)"%4 :%. *,1%.(#.( /++.#=#'"(-%'1( 9( .(-/'"/ / "% E##. B!FF C4. !. "#$%&% '"( )#* &+$, -%- ,-&,)#-&#-$,. G/"% % H"%9 B IIJC 3/''# +.#+#91# *'K%91%'9(#'% "%, -%1#"# LMD "( N#,% B II!C +%. "/1( '#' '%2%99/.(/-%'1% ('"(+%'"%'1(4 D( /99*-% "( /$%.% *' $%11#.% "( n #99%.$/>(#'( y = (y1 , . . . , yn ) B/'? 23% "(+%'"%'1( 1./ ,#.#C %9%&*(1% /,,% %17 x = (x1 , . . . , xn ) % 9( %9%&*% *'/ 1./9=#.-/>(#'% "( E#O?N#O +%. .%'"%.% '#.-/,( ( "/1(4 L/ =*'>(#'% "( ,#&? $%.#9(-(&,(/'>/ +%. µ0 σ % λ ) n X
(19) ∂ωi
(20) 1 1
(21) l(µ, σ, λ) = − log |Σ| − ω T Σ−1 ω + log
(22)
(23) 2 2 ∂yi
(24) i=1.
(25).
(26). BP4 C. "#$% ω ) (, $%11#.% "( "(-%'9(#'( n × 1 "%,,% #99%.$/>(#'( 1./9=#.-/1%. ωi =. τ (xi , yi ) − µ(xi ) , σ(xi ). 2#'. τ (x, Y ) =. Y λ(x) − 1 , λ(x). µ(x), σ(x), λ(x) =*'>(#'( ,(92(/1% "%,,K%17 "(+%'"%'1( "/( $%11#.( "( +/./-%1.( µ, σ, λ0 .(9+%11($/-%'1%4 Q'#,1.%0 µ(x) % σ(x) 9#'#0 .(9+%11($/-%'1%0 ,/ -%"(/ % ,/ "%$(/>(#'% 91/'"/." "%,,/ $/.(/6(,% 1./9=#.-/1/ τ (x, Y )4 L/ 2#-+#'%'1%.
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