c. logit model, logistic regression, and log-linear model a comparison
TRANSCRIPT
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C. Logit model, logistic regression, and log-linear model
A comparison
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R o w i , C o l u m n j S e x : A , B
u u u u ln ABij
Bj
Aiij
o r
o r
w i t h A T I M E [ e a r l y = 0 ; l a t e = 1 ] a n d B S E X [ f e m a l e = 0 ; m a l e = 1 ]
E A R L Y i s r e f e r e n c e c a t e g o r y
... ln xxx 3322110
ijjiij ln
Leaving home
Models of counts: log-linear model
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Model 1: null model
= 4.887 ij = 133.5 for all i and j (=530/4)
Model 2: + TIME
= 4.649
i = 0.4291
ln = exp[4.649 + 0.4291 t] 104.5 for ‘early’ (t=0) and 160.5 for ‘late’ (t=1)
or
ln = exp[4.649] = 104.5 for early
ln = exp[4.649 + 0.4291] = 160.5 for late
Leaving home
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M o d e l 3 : T I M E A N D S E X
= 4 . 6 9 7 ; 2 = 0 . 4 2 9 1 ; 2 = - 0 . 0 9 8 2
R e f e r e n c e c a t e g o r i e s : ‘ e a r l y ’ [ 1 = 0 ] a n d ‘ F e m a l e s ’ [ 1 = 0 ]
jiij ln
TablePredicted number of young adults leaving home by age and sex
(unsaturated log-linear model)Females Males Total
< 20 109.6 99.4 209
20 168.4 152.6 321
Total 278 252 530
Leaving home
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11 = exp[4.697] = 109.6
21 = exp[4.697 + 0.4291] = 168.4
12 = exp[4.697 - 0.0982] = 99.4
22 = exp[4.697 + 0.4291 - 0.0982] = 152.8
Model 3: Time and Sex (unsaturated log-linear model)
jiij ln
jiij exp
Leaving home
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M o d e l 4 : T I M E A N D S E X A N D T I M E * S E X i n t e r a c t i o n( S a t u r a t e d l o g - l i n e a r m o d e l
= 4 . 9 0 5 o v e r a l l e f f e c t 2 = 0 . 0 5 7 6 T I M E 2 = - 0 . 6 0 1 2 G E N D E R 2 2 = 0 . 8 2 0 1 T I M E * G E N D E R
o r
1 i = 0 f o r < 2 0x 1 i = 1 f o r 2 0
x 2 i = 0 f e m a l e sx 2 i = 1 m a l e s
x 3 i = 0 < 2 0 a n d f e m a l e sx 3 i = 0 < 2 0 a n d m a l e sx 3 i = 0 2 0 a n d f e m a l e sx 3 i = 1 2 0 a n d m a l e s
S a t u r a t e d m o d e l p r e d i c t s p e r f e c t l y
i jjii j ln
x i332 i21 i10ij ln xx
Leaving home
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M o d e l 4 : T I M E A N D S E X A N D T I M E * S E X i n t e r a c t i o n
= 4 . 9 0 5 o v e r a l l e f f e c t 2 = 0 . 0 5 7 5 7 T I M E ( 2 ) 2 = - 0 . 6 0 1 2 S E X ( 2 ) 2 2 = 0 . 8 2 0 1 T I M E ( 2 ) * S E X ( 2 )
ijjiij ln
TablePredicted number of young adults leaving home by age and sex
(saturated log-linear model)Females Males Total
< 20 135 74 209
20 143 178 321
Total 278 252 530
Leaving home
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Model 4: TIME AND SEX AND TIME*SEX interaction
11 = exp[4.905
= 135
21 = exp[4.905 + 0.0576]
= 143
12 = exp[4.905 - 0.6012]
= 74
22 = exp[4.905 + 0.0576 - 0.6012 + 0.8201]
= 178
ijjiij ln
ijjiij exp
Leaving home
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Log-linear and logit model
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Log-linear model: μ ln μμμλAB
ij
B
j
A
iij
Select one variable as a dependent variable: response variable, e.g. does voting behaviour differ by sex
Are females more likely to vote conservative than males?
Logit model: γ ln B
j
2j
1j
λλ γ
Political attitudes
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μμμμμμλλ AB
21
B
1
A
2
AB
11
B
1
A
1
21
11 μ μ ln
Males voting conservative rather than labour:
Females voting conservative rather than labour:
μμμμμμλλ AB
22
B
2
A
2
AB
12
B
2
A
1
22
12 μ μ ln
Are females more likely to vote conservative than males?
Log-odds = logit
2 - - ln μ2μμμμμλλ AB
21
A
1
AB
21
AB
11
A
2
A
1
21
11
2 - - ln μ2μμμμμλλ AB
22
A
1
AB
22
AB
12
A
2
A
1
22
12
Effect coding (1)
θγγ B
1
B
1ln
θγγ B
2
B
2ln
A = Party; B = Sex
Political attitudes
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Are women more conservative than men? Do women vote more conservative than men? The odds ratio.
γγγγθθ B
1
B
2
B
1
B
2B
1
B
2 - γ γ ln
If the odds ratio is positive, then the odds of voting conservative rather than labour is larger for women than men. In that case, women vote more conservative than men.
0* - γ ln γγγθB
1
B
2
B
1
B
1
1* - γ ln γγγθB
1
B
2
B
1
B
2
bx a p-1
pln ln logit(p) η
pp
2
1 Logit model:
with a = γB
1 γ
and b = γγB
1
B
2
Log odds of reference category (males)
Log odds ratio (odds females / odds males)
with x = 0, 1
Political attitudes
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The logit model as a regression model
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• Select a response variable proportion
• Dependent variable of logit model is the log of (odds of) being in one category rather than in another.
• Number of observations in each subpopulation (males, females) is assumed to be fixed.
• Intercept (a) = log odds of reference category
• Slope (b) = log odds ratio
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DATA SexParty Male Female TotalConservative 279 352 631Labour 335 291 626Total 614 643 1257
Logit model: descriptive statisticsCounts in terms of odds and odds ratio
Male Female TotalOdds 0.8328 1.2096 1.0080Odds ratio (ref.cat: males): 1.4524
Sex
Reference categories: Labour; Males
Party Odds Odds ratioConservative 1.2616Labour 0.8687Total 1.0472 1.4524
F11 = 279
F21 = 335 = 279 * 335/279 = 279 / 0.8328
F12 = 352 = 279 * 352/279 = 279 1.2616
F22 = 291 = 279 * 352/279 * 291/352 = 279 * 1.2616 * [1/1.2096]
Political attitudes
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DATA SexParty Male Female TotalConservative 279 352 631Labour 335 291 626Total 614 643 1257
Proportion voting conservative: SexParty Male Female Males Females Conservative 0.454 0.547 0.8328 1.2096
Are females more likely to vote conservative than males?Logit model: logit(p) = a + bX (males reference category)
v exp(v) pln(odds) (odds)
a = -0.18292 0.8328 0.454 Males = 0.833/(1+0.833)b = 0.37323 1.4524 Odds ratioa+b = 0.19031 1.2096 0.547 Females = 1.2096/(1+1.2096)
logit(p) = -0.18292 + 0.37323X (with X = 0 for males and X = 1 for females)
If number of males and number of females are known, the counts can be calculated.
Odds of voting cons. rather than labour
LOGIT MODEL
Political attitudes
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Logistic regression SPSS
Variable Param S.E. Exp(param) SEX(1) .3732 .1133 1.4524Constant -.1903 .0792
Females voting labour: 1/[1+exp[-(-0.1903)]] = 45% 291/626 (females ref.cat)Males voting labour: 1/[1+exp[-(-0.1903+0.3732)]] = 55% 335/626
Reference category: females (X = 1 for males and X = 0 for females)
Different parameter coding: X = -0.5 for males and X = 0.5 for females
Variable Param S.E. Exp(param)SEX(1) -.3732 .1133 0.6885 Constant -.0037 .0567
Females voting labour: 1/[1+exp[-(-0.0037 + 0.5*(-0.3732))]] = 45% 291/626Males voting labour: 1/[1+exp[-(-0.0037 - 0.5 * (-0.3732))]] = 55% 335/626
Political attitudes
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Observation from a binomial distribution with parameter p and index m
The logit model andthe logistic regression
Leaving parental home
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L o g i t m o d e l a n d l o g i s t i c r e g r e s s i o n
N u m b e r o f y o u n g a d u l t s l e a v i n g h o m e e a r l y : 2 0 9T o t a l n u m b e r o f y o u n g a d u l t s l e a v i n g h o m e : 5 3 0P r o b a b i l i t y o f l e a v i n g h o m e e a r l y : 2 0 9 / 5 3 0 = 0 . 3 9 4
R E F E R E N C E C A T E G O R Y : l e a v i n g h o m e l a t e ( l a t e = 0 ; e a r l y = 1 )
O D D S o f l e a v i n g h o m e e a r l y v e r s u s l a t e : 2 0 9 / ( 5 3 0 - 2 0 9 ) = 0 . 6 5 1 1L o g i t o f l e a v i n g h o m e e a r l y : l n 0 . 6 5 1 1 = - 0 . 4 2 9 1
S p e c i f y a m o d e l :
L o g i t m o d e l
0.4291- 0 .394-1
0 .394ln
p-1
pln pLogit
Leaving Home
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L o g i s t i c r e g r e s s i o n
0.394 (-0.4291)-exp1
1 p
S t a n d a r d e r r o r :
0.0889 321
1
209
1
C o n fi d e n c e i n t e r v a l : - 0 . 4 2 9 1 1 . 9 6 * 0 . 0 8 8 9 = ( - 0 . 6 0 3 , - 0 . 2 5 5 ) O N L O G I T S C A L E
a n d
0.4366) (0.3546, 549)]exp[-(-0.21
1 ,
)][-(-0.6033exp1
1
O N P R O B A B I L I T Y S C A L E
Leaving home
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Relation logit and log-linear modelThe unsaturated model
Log-linear model:
With i effect of timing and j effect of sex
Odds of leaving parental home late rather than early: females:
ln jiij
1.536 109.6
168.4
11
21
21ODDS
1.536 0-0.4291exp -exp
exp
exp 2112
11
12
11
21
21ODDS
Leaving home
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Relation logit and log-linear modelThe unsaturated model
Odds of leaving parental home late rather than early: males:
1.536 99.4
152.6
12
22
21ODDS
1.536 0-0.4291exp -exp
exp
exp 2112
21
22
12
22
21ODDS
0.0889) (s.e.result same gives modellogit ofOutput
males. and femalesfor 0.4291 Logit pp
early
late
Leaving home
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Relation logit and log-linear modelThe saturated model
Log-linear model:
With i effect of timing and j effect of sex and ij the effect of interaction between timing and sex
Odds of leaving parental home late rather than early: females (ref):
ijjiij ln
1.059 135
143
11
21
21ODDS
1.059 0) - (0 0)-(0.0576exp
) - ( ) -exp exp
exp 21112112
1111
2112
11
21
21 (ODDS
Leaving home
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Relation logit and log-linear modelThe saturated model
Odds of leaving parental home late rather than early: males:
2.405 74
178
12
22
22ODDS
males)for 1 and femalesfor 0 X(with X 0.8201 0.0573 logit(p) :modellogit
[ref]) females odds / males (odds RATIO ODDS log is 0.8201 0.0573 - 0.8775
malesfor odds log is 0.8775 2.405ln
cat) ref. (females modellogit ofeffect overall is 0.0573 1.059ln
2.405 0) -(0.8201 0)-(0.0576exp
) - ( ) -exp exp
exp 22122212
1221
2222
12
22
22 (ODDS
Leaving home
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females 278
143 0.514
0.8201)]-77exp[-(0.871
1 p
males 252
178 0.706
77)]exp[-(0.871
1 p
0.8201X - 0.8777 p-1
pln Logit(p)
Logit model:
Logistic regression: probability of leaving home late
X=0 for males
X=1 for females
Leaving home
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T a b l eN u m b e r o f y o u n g a d u l t s l e a v i n g h o m e b y a g e a n d s e x
F e m a l e s M a l e s T o t a l
< 2 0 1 3 5 7 4 2 0 9
2 0 1 4 3 1 7 8 3 2 1
T o t a l 2 7 8 2 5 2 5 3 0
D u m m y c o d i n g : r e f e r e n c e c a t e g o r y : ( i ) f e m a l e s ; ( i i ) l e a v i n g h o m e l a t e
L o g i t m o d e l : xx ii10i
i 0.8201 - 0.05757- p-1
pln pLogit
x i i s 0 f o r f e m a l e s a n d 1 f o r m a l e s
L O G I T p i s – 0 . 0 5 7 5 7 f o r f e m a l e s a n d – 0 . 0 5 7 5 7 – 0 . 8 2 0 1 = - 0 . 8 7 7 7 f o r m a l e s
O D D SF e m a l e s ( r e f e r e n c e ) : e x p [ - 0 . 0 5 7 5 7 ] = 0 . 9 4 4 0 = 1 3 5 / 1 4 3M a l e s : e x p [ - 0 . 8 7 7 7 ] = 0 . 4 1 5 7 = 7 4 / 1 7 8
O D D S R A T I OO D D S m a l e s / O D D S f e m a l e s = e x p [ - 0 . 8 2 0 1 ] = 0 . 4 4 0 4 = 0 . 4 1 5 7 / 0 . 9 4 4 0
A r e m a l e s m o r e l i k e l y t o l e a v e h o m e e a r l y t h a n f e m a l e s ?
Leaving home
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L o g i s t i c r e g r e s s i o n
0.486 (-0.05757)-exp1
1 p f
0.294 0.8201) - (-0.05757-exp1
1 p m
xx ii10i
i 0.4101 0.4676- p-1
pln pLogit
x i i s 1 f o r f e m a l e s a n d - 1 f o r m a l e s
L o g i t p i s – 0 . 4 6 7 6 + 0 . 4 1 0 1 = - 0 . 0 5 7 6 f o r f e m a l e s a n d - 0 . 4 6 7 6 + 0 . 4 1 0 1 * ( - 1 ) = - 0 . 8 7 7 7 f o r m a l e s
xx ii10
i
i 0.8201 - 0.05757- p-1
pln pLogit
Dummy coding: ref.cat: females, late
Effect coding or marginal coding: females +1; males –1
Leaving home
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The logistic regression in SPSS
Micro data and tabulated data
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SPSS: Micro-data
• Micro-data: age at leaving home in months
• Crosstabs: Number leaving home by reason (row) and sex (column)
• Create variable: Age in years• Age = TRUNC[(month-1)/12]
• Create variable: TIMING2 based on MONTH: • TIMING2 =1 (early) if month 240 & reason < 4
• TIMING2 =2 (late) if month > 240 & reason < 4
• For analysis: select cases that are NOT censored: SELECT CASES with reason < 4
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SPSS: tabulated data
• Number of observations: WEIGHT cases (in data)
• No difference between model for tabulated data and
micro-data
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The logistic regression in SPSS
SPSS: regression/logisticNote: Dependent variable: TIMING2 (p = probability of leaving home LATE)
Covariate: sex (CATEGORICAL)
Logit[p/(1-p)] = 0.8777 – 0.8201 X with males reference categoryMales coded 0; hence X is 1 for females
OUTPUT SPSS:
---------------------- Variables in the Equation -----------
Variable B S.E. Wald df Sig R Exp(B)
SEX(1) -.8201 .1831 20.0598 1 .0000 -.1594 .4404Constant .8777 .1383 40.2681 1 .0000
Leaving home
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Related models
• Poisson distribution: counts have Poisson distribution (total number not fixed)
• Poisson regression
• Log-linear model: model of count data (log of counts)
• Binomial and multinomial distributions: counts follow multinomial distribution (total number is fixed)
• Logit model: model of proportions [and odds (log of odds)]
• Logistic regression
• Log-rate model: log-linear model with OFFSET (constant term)
Parameters of these models are related