Psychology 402 – p.4/43 New Zealand reoffender data...
11
Psychology 401 - Week 9 Binary response models Initially by Dr. Chris Sherlock; revised by Dr. David Lucy. [email protected] Lancaster University Psychology 402 – p.1/43 Binary response models In the name - the response is binary - that is yes/no The data under consideration for this lecture are: • New Zealand reoffender data (2005) • 1930 convicted criminals who were given sentences other than prison as disposals. • +579 in a scheme in which they met the victim (restorative justice). Psychology 402 – p.2/43 New Zealand reoffender data The variables are: 1. number of previous convictions (0/1-2/3+) 2. sex (female=0/male=1) 3. age 4. on scheme? (Y/N) 5. response: convicted of a new crime within a year of conviction? (Y/N) Psychology 402 – p.3/43 Binary outcome • To date all modelswe have seen have been of a continuous response • What to do about binary response? The solution is to use some “transformation” of the response. • Do not think in terms of yes/no, but: • the log(Odds) of getting a “yes” (or no). Psychology 402 – p.4/43
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Transcript of Psychology 402 – p.4/43 New Zealand reoffender data...
Psy
chol
ogy
401
-W
eek
9B
inar
yre
spon
sem
odel
sIn
itial
lyby
Dr.
Chr
isS
herlo
ck;r
evis
edby
Dr.
Dav
idLu
cy.
Lanc
aste
rU
nive
rsity
Psy
chol
ogy
402
–p.
1/43
Bin
ary
resp
onse
mod
els
Inth
ena
me
-th
ere
spon
seis
bina
ry-
that
isye
s/no
The
data
unde
rco
nsid
erat
ion
for
this
lect
ure
are:
•N
ewZ
eala
ndre
offe
nder
data
(200
5)
•19
30co
nvic
ted
crim
inal
sw
how
ere
give
nse
nten
ces
othe
rth
anpr
ison
asdi
spos
als.
•+
579
ina
sche
me
inw
hich
they
met
the
vict
im(r
esto
rativ
eju
stic
e).
Psy
chol
ogy
402
–p.
2/43
New
Zea
land
reof
fend
erda
ta
The
varia
bles
are:
1.nu
mbe
rof
prev
ious
conv
ictio
ns(0
/1-2
/3+
)
2.se
x(f
emal
e=0/
mal
e=1)
3.ag
e
4.on
sche
me?
(Y/N
)
5.re
spon
se:
conv
icte
dof
ane
wcr
ime
with
ina
year
ofco
nvic
tion?
(Y/N
)
Psy
chol
ogy
402
–p.
3/43
Bin
ary
outc
ome
•To
date
allm
odel
sw
eha
vese
enha
vebe
enof
aco
ntin
uous
resp
onse
•W
hatt
odo
abou
tbin
ary
resp
onse
?
The
solu
tion
isto
use
som
e“t
rans
form
atio
n”of
the
resp
onse
.
•D
ono
tthi
nkin
term
sof
yes/
no,b
ut:
•th
elog(O
dds)
ofge
tting
a“y
es”
(or
no).
Psy
chol
ogy
402
–p.
4/43
Odd
s
The
odds
ofan
even
t(E
)ar
e:
Odds=
theprobab
ilityof
theevent:Pr(E)
theprobab
ilityof
not
theevent:Pr(E)
Ina
bina
ryca
se,w
here
only
one
oftw
oou
tcom
esar
epo
ssib
le:
Oddsof
event:O(E
)=
Pr(E)
1−
Pr(E)
Psy
chol
ogy
402
–p.
5/43
Odd
s&
prob
abili
ty
Toex
chan
gebe
twee
nod
dsan
dpr
obab
ility
:
O(E
)=
Pr(E)
1−
Pr(E)
and:
Pr(E)=
O(E
)
1+O(E
)
Tous
eod
dsas
are
spon
sein
alin
ear
mod
elw
eus
eth
elog(O
),or
thelog
toba
see
ofth
eod
ds; u
ses
theln
butto
non
your
calc
ulat
or.
Psy
chol
ogy
402
–p.
6/43
Odd
s&
prob
abili
ty
verb
aleq
uiva
lenc
epr
obab
ility
(%)
odds
log(odds)
impo
ssib
le0.
00(0
)0
−∞
very
unlik
ely
0.01
(1%
)1/
99-4
.6un
likel
y0.
20(2
0%)
1/4
-1.4
even
s0.
50(5
0%)
10
likel
y0.
80(8
0%)
41.
4al
mos
tcer
tain
0.99
(99%
)99
4.6
cert
ain
1.00
(100
%)
∞∞
The
criti
calr
ealis
atio
nis
that
aqu
antit
yw
hich
isbo
unde
dx∈{0,1},
isno
win
the
dom
ainx∈{−
∞,∞
},co
ntin
uous
.
Psy
chol
ogy
402
–p.
7/43
The
logi
stic
tran
sfor
m
Taki
ngth
elog(O
dds)
ofa
prob
abili
tyis
know
nas
:
•a
“logi
stic
tran
sfor
m”
•th
e“lo
git”
ofth
eev
ent,
henc
eth
eco
mm
onna
me
for
abi
nary
resp
onse
mod
elas
“logi
stic
regr
essi
on”,
inth
esa
me
way
that
alin
ear
mod
elw
aste
rmed
“line
arre
gres
sion
”.
Inth
em
ore
tech
inca
llite
ratu
reyo
um
ayal
sose
eth
e“lo
git”
bein
gre
fere
dto
asa
“link
”
func
tion.
For
abi
nary
resp
onse
ther
ear
ein
fact
ava
riety
ofth
ese
link
func
tions
,ofw
hich
the
logi
tis
the
mos
tcom
mon
lyus
ed,a
ndea
sily
inte
rpre
ted.
Psy
chol
ogy
402
–p.
8/43
Bin
ary
mod
ellin
g
Whe
nw
eco
me
tobi
nary
mod
ellin
g:
•do
notw
orry
abou
tact
ually
tran
sfor
min
gth
eou
tcom
eyo
urse
lves
,
•co
mpu
ters
doit
all,
•yo
usi
mpl
yha
veto
spec
ifyth
eco
rrec
tres
pons
e.
The
proc
ess
from
aus
ers
poin
tofv
iew
isve
rysi
mila
rto
alin
ear
mod
elfo
ra
cont
inuo
usre
spon
se.
Mor
eto
the
poin
tdon
ein
the
sam
ew
ay.
Psy
chol
ogy
402
–p.
9/43
New
Zea
land
reof
fend
erda
ta
Bac
kto
the
data
.T
heso
rts
ofqu
estio
nsm
aybe
:
1.H
asth
esc
hem
ere
duce
dth
ere
conv
ictio
nra
tesi
gnifi
cant
ly?
2.F
orw
hich
grou
psm
ight
itbe
mos
teffe
ctiv
e?
3.Ih
ave
just
conv
icte
da
42ye
arol
dm
ale
ofan
offe
nce.
He
has
nopr
evio
usco
nvic
tions
.S
houl
dIp
uthi
mon
the
sche
me?
•In
othe
rw
ords
:ho
wlik
ely
ishe
tore
offe
ndif
Iput
him
onth
esc
hem
ean
dho
wlik
ely
ishe
tore
offe
ndif
Ido
not?
Psy
chol
ogy
402
–p.
10/4
3
Age
sof
sche
me
atte
ndee
s
020
4060
8010
0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
age
(yea
rs)
probability density
not o
n sc
hem
eon
sch
eme
Psy
chol
ogy
402
–p.
11/4
3
Reo
ffend
ing
and
sche
me
Sim
ple
tabl
esu
gges
tsth
aton
the
face
ofit
the
sche
me
has
little
effe
ctup
onre
offe
ndin
g.
%s
reof
fend
edno
yes
sche
me
no0.
650.
35ye
s0.
690.
31
prev
ious
01-
23+
fem
ale
0.50
0.18
0.32
mal
e0.
470.
210.
32
The
num
ber
ofpr
evio
usco
nvic
tions
seem
sno
tto
vary
muc
hby
sex.
Psy
chol
ogy
402
–p.
12/4
3
Spe
cify
ing
abi
nary
resp
onse P
sych
olog
y40
2–
p.13
/43
Spe
cify
ing
abi
nary
resp
onse P
sych
olog
y40
2–
p.14
/43
Spe
cify
ing
the
expl
anat
orie
s
Psy
chol
ogy
402
–p.
15/4
3
The
mod
el
Psy
chol
ogy
402
–p.
16/4
3
Sta
tistic
sdi
alog
ue Psy
chol
ogy
402
–p.
17/4
3
Nul
lmod
el
Fitt
ing
the
null
mod
el: m
odel
info
rmat
ion
depe
nden
tvat
riabl
ere
offe
nded
3
prob
abili
tydi
strib
utio
nbi
nom
ial
link
func
tion
logi
ta
the
proc
edur
em
odel
sye
sas
the
resp
onse
,tre
atin
gno
asth
ere
fere
nce
cate
gory
Her
eIh
ave
used
the
resp
onse
dial
ogue
toch
ange
the
mod
eled
cate
gory
from
“no”
to“y
es”.
Thi
sis
ofno
cons
eque
nce,
butm
akes
SP
SS
beha
vem
ore
like
othe
rm
odel
ling
func
tions
.P
sych
olog
y40
2–
p.18
/43
Goo
dnes
sof
fit
Goo
dnes
sof
fita
valu
edf
valu
e/df
devi
ance
563.
8839
11.
442
scal
edde
vian
ce56
3.88
391
Pea
rson
chi-s
quar
e45
9.90
391
1.17
6
scal
edP
ears
onch
i-squ
are
459.
9039
1
log
likel
ihoo
d-5
16.3
8
Aka
ike’
sin
form
atio
n
crite
rion
(AIC
)10
34.7
6
finite
sam
ple
corr
ecte
dA
IC10
34.7
8
Bay
esia
nin
form
atio
n
crite
rion
(BIC
)10
40.5
9
cons
iste
ntA
IC(C
AIC
)10
41.5
9
Psy
chol
ogy
402
–p.
19/4
3
Par
amet
eres
timat
es
The
para
met
eres
timat
esar
e:
Par
amet
eres
timat
es
95%
Wal
dC
Ihy
poth
esis
test
para
met
erβ
st.
dev
low
erup
per
Wal
dch
-sq
dfsi
g
inte
rcep
t-0
.647
0.04
20-0
.73
-0.0
523
2.20
10.
00
scal
e1
depe
nden
tvar
iabl
e:re
offe
nded
mod
el:
inte
rcep
t
The
sear
eve
rysi
mila
rto
the
para
met
ers
from
alin
ear
mod
elw
itha
cont
unuo
usre
spon
se, a
ndca
nbe
adde
dto
geth
erin
exac
tlyth
esa
me
way
.
Psy
chol
ogy
402
–p.
20/4
3
Par
amet
eres
timat
es
The
valu
eof
the
inte
rcep
tis
-0.6
47.
•T
his
isth
elo
g(O
dds)
ofge
tting
a“y
es”
from
the
sam
ple:
•as
“yes
”is
the
stat
ebe
ing
mod
elle
d.
•T
heod
dsof
getti
nga
“yes
”ar
eexp(−
0.647)
=0.524
Pr(yes)
=0.524
1+0.524=
0.3438
Psy
chol
ogy
402
–p.
21/4
3
Pro
babi
lity
of“y
es”
Pr(yes)
=0.524
1+0.524=
0.3438
reof
fend
ed
noye
s
1647
862
Pr(yes)
=862
1647
+862=
0.3435
Alit
tledi
ffere
nce,
butt
he0.
524
isro
unde
d.
Psy
chol
ogy
402
–p.
22/4
3
Fix
edef
fect
sm
odel
s
resp
onse
term
sB
IC
reof
fend
ing
∼1
1040
∼1
+sc
hem
e10
44∼
1+
prev
ious
991
∼1
+se
x10
16∼
1+
age
1000
∼1
+sc
hem
e+
prev
ious
991
∼1
+sc
hem
e+
sex
1023
∼1
+sc
hem
e+
age
1007
∼1
+pr
evio
us+
sex
965
Psy
chol
ogy
402
–p.
23/4
3
Fix
edef
fect
sm
odel
s
resp
onse
term
sB
IC
reof
fend
ing
∼1
+pr
evio
us+
age
945
∼1
+se
x+
age
975
∼1
+sc
hem
e+
prev
ious
+se
x96
9∼
1+
sche
me
+pr
evio
us+
age
949
∼1
+sc
hem
e+
sex
+ag
e97
9∼
1+
prev
ious
+se
x+
age
921
∼1
+sc
hem
e+
prev
ious
+se
x+
age
932
The
sche
me
appe
ars
tobe
doin
glit
tlefo
rth
epr
obab
ility
ofre
offe
ndin
g.
Psy
chol
ogy
402
–p.
24/4
3
Pro
babi
lity
of“y
es”
Fitt
ed:
reof
fend
ed∼
1+
prev
ious
(BIC
=99
1)P
aram
eter
estim
ates
95%
Wal
dC
Ihy
poth
esis
test
para
met
erβ
st.
dev
low
erup
per
Wal
dch
-sq
dfsi
g
inte
rcep
t-0
.625
0.07
4-0
.77
-0.4
770
.87
10.
00
prev
ious
=0
-0.3
250.
098
-0.5
2-0
.13
10.9
61
0.00
prev
ious
=1
0.57
70.
116
0.33
0.80
24.9
01
0.00
prev
ious
=2
0.
..
..
.
scal
e1
The
log(
odds
)an
indi
vidu
alw
ithze
ropr
ievi
ous
conv
ictio
nsw
illre
offe
ndis
-0.6
25+
-0.3
25=
-0.9
5.F
oran
indi
vidu
alw
ithon
epr
evio
usof
fens
e-0
.625
+0.
577
=-0
.048
,and
for
thre
eor
mor
e-0
.625
.P
sych
olog
y40
2–
p.25
/43
Pro
babi
lity
of“y
es”
•lo
g(od
ds)
=-0
.95,
sood
ds=exp(−
0.95)=
0.386
•lo
g(od
ds)
=-0
.048
,so
odds
=exp(−
0.048)
=0.953
•lo
g(od
ds)
=-0
.625
,so
odds
=exp(−
0.625)
=0.535
Con
vert
ing
topr
obab
ilitie
s:
•0.
386
/(1
+0.
386)
=0.
278
•0.
953
/(1
+0.
953)
=0.
488
•0.
535
/(1
+0.
535)
=0.
348
Psy
chol
ogy
402
–p.
26/4
3
Pro
babi
lity
of“y
es”
From
the
cros
sta
ble
ofpr
evio
usan
dre
offe
nded
:
prev
ious
noye
s
086
633
51-
226
024
83+
521
279
Em
pric
ally
:
•33
5/(
335
+86
6)=
0.27
4
•24
8/(
248
+26
0)=
0.48
8
•27
9/(
279
+52
1)=
0.34
8
Psy
chol
ogy
402
–p.
27/4
3
Bes
tfixe
def
fect
sm
odel
Bes
t:log(O
dds)
reof
fend
ed∼
1+
prev
ious
+se
x+
age;
try
som
efir
stor
der
inte
ract
ions
:
resp
onse
term
sB
IC
reof
fend
ing
∼1
+pr
evio
us+
sex
+ag
e92
1∼
1+
prev
ious
+se
x+
age
+se
x:ag
e92
9∼
1+
prev
ious
+se
x+
age
+pr
evio
us:a
ge93
6∼
1+
prev
ious
+se
x+
age
+pr
evio
us:s
ex93
9
Not
look
ing
hope
ful-
wou
ldtr
yth
irdor
der
inre
alin
vest
igat
ion.
Psy
chol
ogy
402
–p.
28/4
3
The
effe
ctof
the
sche
me
•T
hesc
hem
eef
fect
has
notb
een
reta
ined
inth
em
odel
.
•su
gges
tsit
isdo
ing
little
.
•H
owdo
we
quan
tify
the
exte
ntto
whi
chsc
hem
eis
not
doin
gan
ythi
ng?
•C
anw
e“t
est”
tose
ew
heth
erit
isha
ving
asi
gnifi
cant
effe
ct?
Per
sona
llyIa
mno
tin
favo
urof
sign
ifica
nce
test
ing,
butf
ora
jour
nal,
orfo
ra
repo
rt,a
p-v
alue
can
beof
imm
ense
help
Psy
chol
ogy
402
–p.
29/4
3
Like
ihoo
dra
tio
Goo
dnes
sof
fita
valu
edf
valu
e/df
devi
ance
413.
8538
71.
068
scal
edde
vian
ce41
3.85
387
Pea
rson
chi-s
quar
e38
3.48
387
0.99
1
scal
edP
ears
onch
i-squ
are
383.
4838
7
log
likel
ihoo
d-4
41.3
7
Aka
ike’
sin
form
atio
n
crite
rion
(AIC
)89
2.74
finite
sam
ple
corr
ecte
dA
IC89
2.77
Bay
esia
nin
form
atio
n
crite
rion
(BIC
)92
1.88
cons
iste
ntA
IC(C
AIC
)92
1.88
Psy
chol
ogy
402
–p.
30/4
3
Like
ihoo
dra
tio
An
easy
way
ofse
eing
how
muc
hm
ore
likel
yon
em
odel
isth
anth
eot
her:
1.Lo
okat
the
log
likel
ihoo
dfo
rth
em
odel
with
outs
chem
e(-
441.
37),
2.th
enlo
okat
the
log
likel
ihoo
dfo
rth
em
odel
(s)
with
sche
me
(-43
9.59
):
The
nta
keon
efr
omth
eot
her.
As
subt
ract
ion
oflo
gsis
equi
vale
ntto
the
divi
sion
ofth
eor
igin
alqu
antit
ies
you
will
have
the
log
ofth
elik
elih
ood
ratio
:-1
.78.
Exp
onen
tiate
toge
tthe
likel
ihoo
dra
tioexp(−
1.78)≈
0.168.
Psy
chol
ogy
402
–p.
31/4
3
Like
ihoo
dra
tio
The
likel
ihoo
dra
tioexp(−
1.78)≈
0.168.
Can
bein
terp
rete
das
:
•“t
hem
odel
with
sche
me
is0.
168
times
aslik
ely
asth
atw
ithou
tsch
eme”
.
•“t
hese
data
are
0.16
8tim
esm
ore
prob
able
wer
eth
em
odel
with
sche
me
true
asth
em
odel
with
outs
chem
e”
Psy
chol
ogy
402
–p.
32/4
3
Like
ihoo
dra
tio
The
likel
ihoo
dra
tioexp(−
1.78)≈
0.168.
Rea
llybe
tter
toph
rase
the
likel
ihoo
dra
tioth
eot
her
way
roun
d.R
ever
seth
elik
elih
ood
ratio
bych
angi
ngsi
gn:
exp(1.78)
≈5.93
Thi
sm
aybe
inte
rpre
ted
as:
•“t
hem
odel
with
outs
chem
eis
5.93
times
aslik
ely
asth
atw
ithsc
hem
e”.
•“t
hese
data
are
5.93
times
mor
epr
obab
lew
ere
the
mod
elw
ithou
tsch
eme
true
,as
the
mod
elw
ithsc
hem
e”
Psy
chol
ogy
402
–p.
33/4
3
Test
oflik
eiho
odra
tio
2×
[log(L
m1)−
log(L
m2)]∼
χ2
df=
m
whe
re:
1.Lmx
isth
elo
gof
the
likel
ihoo
dof
mod
elx
,
2.m
are
the
degr
ees
offr
eedo
mas
soci
ated
with
the
cova
riate
bein
gin
trod
uced
-for
cont
inuo
usdf
=1,
for
fact
ornu
mbe
rof
leve
ls-
1.
Her
eth
ere
are
two
leve
lsfo
r“s
chem
e”so
df=
1,an
dχ2=
2×
1.78
=3.56
Psy
chol
ogy
402
–p.
34/4
3
Test
oflik
eiho
odra
tio
χ2=
3.56
on1
degr
eeof
free
dom
:
Has
ap-v
alue
≈5.
9%,w
hich
isno
tsig
nific
anta
ta5%
leve
lof
sign
ifica
nce.
Toca
lcul
ate
this
you
can
use
any
one
ofa
num
ber
ofto
ols.
SP
SS
has
it’s
Cdf
.Chi
sq()
func
tion
from
the
“com
pute
varia
ble”
item
off
the
“tra
nsfo
rm”
men
u.T
hear
guem
ents
toth
isfu
nctio
nar
eth
eva
lue
ofth
eχ2
stat
istic
,3.5
6in
this
case
,and
the
degr
ees
offr
eedo
m,
1he
re.
You
have
toas
sign
the
valu
eit
toa
varia
ble,
and
that
you
take
the
valu
eaw
ayfr
om1
toge
tap-v
alue
for
the
test
.In
the
“num
eric
expr
essi
on”
box
from
the
“com
pute
varia
ble”
dial
ogue
itsh
ould
look
like:
1-
CD
F.C
HIS
Q(3
.56,
1).
Psy
chol
ogy
402
–p.
35/4
3
χ2
likel
ihoo
dra
tiote
st
χ2
probability density
01
23
45
0.0
0.5
1.0
1.5
3.55
Psy
chol
ogy
402
–p.
36/4
3
Res
idua
lplo
t
Are
sidu
alpl
otas
such
mak
eslit
tlese
nse:
•do
have
log
odds
of“r
eoffe
nded
”an
dw
heth
erth
eyre
offe
nded
•ca
nca
lcul
ate
prob
abili
ties
from
log
odds
Ifth
em
odel
isre
ason
able
then
:
•ab
out2
0%of
thos
ew
hose
pred
icte
dpr
obab
ility
ofre
offe
ndin
gis
0.20
will
have
reof
fend
ed,
•ab
out5
0%of
thos
ew
hose
pred
icte
dpr
obab
ility
ofre
offe
ndin
gis
0.50
will
have
reof
fend
ed,
•an
dso
on...
Psy
chol
ogy
402
–p.
37/4
3
Mod
eldi
agno
stic
s
Psy
chol
ogy
402
–p.
38/4
3
Mod
eldi
agno
stic
s
Psy
chol
ogy
402
–p.
39/4
3
Mod
eldi
agno
stic
s
Psy
chol
ogy
402
–p.
40/4
3
Mod
eldi
agno
stic
s
From
figur
e:
•ab
out2
0%of
thos
ew
how
ere
pred
icte
dw
itha
prob
abili
tyof
0.2
reof
fend
ed,
•ab
out5
0%of
thos
ew
how
ere
pred
icte
dw
itha
prob
abili
tyof
0.5
reof
fend
ed,
Thi
sis
nott
errib
lycl
ear
from
the
figur
e,bu
tdoe
sre
assu
reus
that
the
mod
elis
nott
otal
lyw
rong
.
Psy
chol
ogy
402
–p.
41/4
3
Mod
eldi
agno
stic
s
Inth
efu
ture
:
•m
aybe
able
toru
nso
meχ2
good
ness
offit
for
the
pred
icte
dpr
obab
ilitie
s,
•ot
her
mea
sure
,suc
has
empi
rical
cros
sen
trop
y(E
CE
)m
aybe
com
eav
aila
ble.
For
the
mom
entc
lass
ifica
tion
plot
sar
esi
mpl
eto
do,s
ow
illre
mai
nw
ithus
for
som
etim
e.
Psy
chol
ogy
402
–p.
42/4
3
Afte
rth
ebr
eak
We
shal
lbe:
•us
ing
abi
nary
resp
onse
mod
elto
exam
ine
your
num
ber
choi
ces
from
last
wee
k,
•an
dus
ing
abi
nary
resp
onse
mod
elto
see
whi
chfa
ctor
saf
fect
whe
ther
mar
ried
wom
enar
ein
the
wor
kpla
ceor
not.
Psy
chol
ogy
402
–p.
43/4
3
