log linear model - @@ home - kku web hosting file2 log linear model difference from logitmodel-do...

1

Log Linear Model

��. ��

�� !��"��#�Email: [email protected]

Web Page: http://home.kku.ac.th/nikom

�"��1. �:;<��;�=��>�! <� �� (fitting models) A��>��"�:"�� #��"��

2. �:;<��#�:�� (estimating parameter)��C>D��#�:��"�� "� (main effect) ��"�� interaction effect

2

Log Linear Model

difference from logit model

-do not distinguish between response variable

(dependent) and explanatory variable

(independent)-relation among independent variable

Ex. ��"�:"�� ECC� � < ��>;<�� A�D��:��>��"��#�

=��>��F��D� (Log Linear Model)

�� ( �� 2)

( �� 1) 1 2 ��

1 n11 n12 n1.

2 n21 n22 n2.

�� n.1 n.2 n

Log linear model for 2-way table

3

Independent modely

j

x λλλµ ++=i

)ij

ln(

��

�� Fatal

(1)

Non Fatal (0)

�� (1) 1601 162527

�� (0) 510 412368

�"��#�� >��F��">��"��C�>�GFC

inj

j

belt λλλµ ++=i

)ij

ln(

=��>��CC��CE�� (Saturated Model)-Log Linear Model G��C>D��#��F�� GH��D�E�A��#��I�� "�� "� (main effect) ��"��#�� (interaction effect)

xy

ij

y

j

x λλλλµ +++=i

)ij

ln(

4

uij = GH��D�E�A��#��I��= �#� intercept !"K� �>= �!��:��"�� "��"��! < 1= �!��:��"�� "��"��! < 2= �!��:��"��#��

xy

ij

y

j

x λλλλµ +++=i

)ij

ln(

x

iλ

y

jλ

λ

xy

ijλ

��D��=��>��H� �"C�� Log Linear model �H� �"C�I��! < 1 …(1)�H� �"C�I��! < 2 …(2)�H� �"C�I��! < 3 …(3)�H� �"C�I��! < 4 …(4)

xyyx

1111)11

ln( λλλλµ +++=

xyyx

1221)12

ln( λλλλµ +++=

xyyx

2112)21

ln( λλλλµ +++=

xyyx

2222)22

ln( λλλλµ +++=

5

��&� '(��

A��H��#� A>Q !H�R>D=>��D�� (1)-(4)=>��H� �>A D ��;<�!��#�A��!H�A DR>D��A �#>"�� K

�H� �"C�I��! < 1 T(5)�H� �"C�I��! < 2 T(6)�H� �"C�I��! < 3 T(7)�H� �"C�I��! < 4 T(8)

xy

ij

x

i λλλλ ,,, y

j

0,0,0,0,0 222112

y

22 ===== xyxyxyx λλλλλ

xyyx

1111)11

ln( λλλλµ +++=x

1)12

ln( λλµ +=

y

1)21

ln( λλµ +=

λµ =)22

ln(

G��D�� (5)-(8) ��H�� R>D>"�� K

=−=

22

12ln)22

ln()12

ln(x

1 µ

µµµλ

=−=

22

21ln)22

ln()21

ln(x

1 µ

µµµλ

=−−=

2112

2211ln)22

ln()21

(ln(-)12

ln()11

ln(11 µµ

µµµµµµλxy

xy

ij

x

i λλλλ ,,, y

j

)log( 22µλ =

6

��

�� Fatal

(1)

Non Fatal (0)

�� (1) 1601 162527

�� (0) 510 412368

�� !"#$%#�!&'�(��)*+��,�!"-$,

>"��"K�G��"��#��H��#� R>D>"�� K

12.9297ln(412368) ==λ

0.9311412368

162527lnln(412368)ln(162527)x

1 −=

=−=λ

6.6953412568

510lnln(412368)ln(510)y

1 −=

=−=λ

2.075010)(162527)(5

368)(1601)(412ln

)ln(412368)(ln(510)-ln(162527)ln(1601)xy

11

=

=

−−=λ

��

�� Fatal (1) Non Fatal (0)

�� (1) 1601 162527

�� (0) 510 412368

xy

ij

x

i λλλλ ,,, y

j

7

. gen bj=b*j

. poisson freq b j bj

Iteration 0: log likelihood = -23529.833







Poisson regression Number of obs = 4

LR chi2(3) = 884872.49

Prob > chi2 = 0.0000

Log likelihood = -22.946503 Pseudo R2 = 0.9999

------------------------------------------------------------------------------

freq | Coef. Std. Err. z P>|z| [95% Conf. Interval]

-------------+----------------------------------------------------------------

b | -.931072 .0029288 -317.90 0.000 -.9368123 -.9253317

j | -6.695261 .0443081 -151.11 0.000 -6.782103 -6.608419

bj | 2.075045 .0509311 40.74 0.000 1.975222 2.174868

_cons | 12.92967 .0015572 8302.90 0.000 12.92662 12.93272

------------------------------------------------------------------------------

. glm freq b j bj, f(po) l(log) ef








Generalized linear models No. of obs = 4

Optimization : ML Residual df = 0

Scale parameter = 1

Deviance = 1.44111e-11 (1/df) Deviance = .

Pearson = 5.49837e-24 (1/df) Pearson = .

Variance function: V(u) = u [Poisson]

Link function : g(u) = ln(u) [Log]

AIC = 13.47325

Log likelihood = -22.94650294 BIC = 1.44e-11

------------------------------------------------------------------------------

| OIM

freq | IRR Std. Err. z P>|z| [95% Conf. Interval]

-------------+----------------------------------------------------------------

b | .394131 .0011543 -317.90 0.000 .391875 .3963999

j | .0012368 .0000548 -151.11 0.000 .0011339 .001349

bj | 7.964905 .405661 40.74 0.000 7.20822 8.801024

------------------------------------------------------------------------------

8

. loglin count qbelt injury ,fit( qbelt, injury, qbelt injury)

Variable qbelt = A

Variable injury = B

Margins fit: qbelt, injury, qbelt injury

Note: Regression-like constraints are assumed. The first level of each

variable (and all iteractions with it) will be dropped from estimation.









LR chi2(3) = 884872.49

Prob > chi2 = 0.0000


----------------------------------------------------------------------------

count | Coef. Std. Err. z P>|z| [95% Conf. Interval]

-------------+----------------------------------------------------------------

A2 | -.931072 .0029288 -317.90 0.000 -.9368123 -.9253317

AB22 | 2.075045 .0509311 40.74 0.000 1.975222 2.174868

B2 | -6.695261 .0443081 -151.11 0.000 -6.782103 -6.608419

_cons | 12.92967 .0015572 8302.90 0.000 12.92662 12.93272

------------------------------------------------------------------------------

��)*+'��,��

� ��"�:"�� #�� A�D selt belt �"C��C�>�GFC. poisson freq b j bj,irr









LR chi2(3) = 884872.49

Prob > chi2 = 0.0000


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

b | .394131 .0011543 -317.90 0.000 .391875 .3963999

j | .0012368 .0000548 -151.11 0.000 .0011339 .001349

bj | 7.964907 .4056611 40.74 0.000 7.208222 8.801026

------------------------------------------------------------------------------

xy

11λ

. list

+---------------------+

| b j freq bj |

|---------------------|

1. | 1 1 1601 1 |

2. | 1 0 162527 0 |

3. | 0 1 510 0 |

4. | 0 0 412368 0 |

+---------------------+

. di (1601*412368)/(162527*510)

7.9649049

9

��)*+-��.)*+/ +��0(1 odds ratio

. poisson freq b j bj,irr









LR chi2(3) = 884872.49

Prob > chi2 = 0.0000


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

b | .394131 .0011543 -317.90 0.000 .391875 .3963999

j | .0012368 .0000548 -151.11 0.000 .0011339 .001349

bj | 7.964907 .4056611 40.74 0.000 7.208222 8.801026

------------------------------------------------------------------------------

��>��C�>�GFC��EDR�#��> Selt Belt � �#��!#��"C 7.96�#��;<�"<�� #��"K��# 7.21 - 8.80

------------------------------------------

| m

a and c | 0 1

----------+-------------------------------

0 |

0 | 279(279.6168) 2(1.38317)

1 | 43(42.38317) 3(3.61683)

----------+-------------------------------

1 |

0 | 456(455.3832) 44(44.61683)

1 | 538(538.6168) 911(310.3832)

------------------------------------------

Log linear model for 3-way tables�� !�"!#$% &�' ())*+',- (cigarette) &�'01-� *'� (alcohol)345 &�'6789�: ";<0 (marijuana) >?@�!9'*A#

acmcmam

ik

acmca

ijkjkijkji)

ij( λλλλλλλλµ +++++++=ln

10

Log linear model for 3-way tables

-Homogeneous association modelAll Pairwise Association Present-No independent

( )

-Conditional independence model

yzxz

ik

xyzyx

jkijkji)

ij( λλλλλλλµ ++++++=ln

0=xyz

ijkλ

0== xyzxy

ijkijλλyzxz

ik

zyx

jkkji)

ij( λλλλλλµ +++++=ln

yzxyzyx

jkijkji)


xz

ik

xyzyx

ijkji)


0== xyzxz

ijkikλλ

0== xyzyz

ijkjkλλ

-joint independent (Partial Independence Model )

-Complete Independence Model

0=== xyzxzxy

ijkikijλλλ

xyzyx

ijkji)

ij( λλλλλµ ++++=ln

xz

ik

zyx

kji)


yzzyx

jkkji)


0, === xyzxzxz

k

xy

ijkikijλλλλ

zyx

kji)

ij( λλλλµ +++=ln

11

Model df. symbol

Homogeneous association model (All Pairwise Asoociation)

1. l XY,XZ YZ

Conditional Independence

2. (r-1) XZ, YZ

3. (c-1) XY, YZ

4. (l-1) XY, XZ

joint independent (Partial Independence)

5. (r-1)(c-1) Z , XY

6. (r-1)(l-1) Y, XZ

7. (c-1)(l-1) X, YZ

Complete Independence (mutual independent model )

8. (r-1)(c-1)(l-1) X,Y,Z

Three-factor interaction model XYZ

0=xyz

ijkλ

0== xyz

ijk

xy

ij λλ0== xyz

ijk

xz

ik λλ0== xyz

ijk

yz

jk λλ

0=== xyzyzxz

ijkjkijλλλ

0=== xyzyzxy

ijkjkijλλλ

0=== xyzxzxy

ijkikijλλλ

0, === xyzxzxz

k

xy

ijkikijλλλλ

12

Inference for Loglinear Model �� fit Model [3 Way] :�G��G�� 1. goodness of fit

2. residuals3. tests about partial association4. Odds ratio & CI

goodness of fit

-Likelihood ratio Chi-square-Pearson Statistics

13

��"�:"�� ECC� � < ��>;<�� A�D��:��>��"��#� a = alcohol, c=cigarette,m=marijuama

+------------------+

| a c m freq |

|------------------|

1. | 1 1 1 911 |

2. | 1 1 0 538 |

3. | 1 0 1 44 |

4. | 1 0 0 456 |

5. | 0 1 1 3 |

|------------------|

6. | 0 1 0 43 |

7. | 0 0 1 2 |

8. | 0 0 0 279 |

+------------------+

�� fit Model & Selected Model. ipf [fw=freq] ,fit(a+c+m+a*c+a*m+c*m+a*c*m) exp

...

N.B. structural/sampling zeroes may lead to an incorrect df

Residual degrees of freedom = 0

Number of parameters = 8

Number of cells = 8

Loglikelihood = 12010.79937313193

Loglikelihood = 12010.79937313193

Goodness of Fit Tests

---------------------

df = 0

Likelihood Ratio Statistic G² = 0.0000 p-value = .

Pearson Statistic X² = 0.0000 p-value = .

+---------------------------------------+

| a c m Efreq Ofreq prob |

|---------------------------------------|

| 0 0 0 279 279 .12258348 |

| 0 0 1 2 2 .00087873 |

| 0 1 0 43 43 .01889279 |

| 0 1 1 3 3 .0013181 |

| 1 0 0 456 456 .20035149 |

|---------------------------------------|

| 1 0 1 44 44 .01933216 |

| 1 1 0 538 538 .23637961 |

| 1 1 1 911 911 .40026362 |

+---------------------------------------+

14

. poisson freq a c m ac am cm acm








LR chi2(7) = 2851.46

Prob > chi2 = 0.0000


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

a | .491281 .076008 6.46 0.000 .3423081 .6402539

c | -1.870012 .1638293 -11.41 0.000 -2.191111 -1.548912

m | -4.938065 .7096367 -6.96 0.000 -6.328927 -3.547202

ac | 2.035377 .1757605 11.58 0.000 1.690893 2.379862

am | 2.599761 .7269831 3.58 0.000 1.174901 4.024622

cm | 2.275477 .9274553 2.45 0.014 .4576977 4.093256

acm | .5895107 .9423641 0.63 0.532 -1.257489 2.43651

_cons | 5.631212 .0598684 94.06 0.000 5.513872 5.748552

------------------------------------------------------------------------------

. poisgof

Goodness-of-fit chi2 = -.0014935

Prob > chi2(0) = .

. poisgof,pearson

Goodness-of-fit chi2 = 0

Prob > chi2(0) = .

. predict u,n

. list freq u

+------------+

| freq u |

|------------|

1. | 911 911 |

2. | 538 538 |

3. | 44 44 |

4. | 456 456 |

5. | 3 3 |

6. | 43 43 |

7. | 2 2 |

8. | 279 279 |

+------------+

15

. glm freq a c m ac am cm acm, f(po) l(log)








Generalized linear models No. of obs = 8

Optimization : ML: Newton-Raphson Residual df = 0

Scale parameter = 1

Deviance = 2.23377e-13 (1/df) Deviance = .

Pearson = 5.63860e-21 (1/df) Pearson = .

Variance function: V(u) = u [Poisson]

Link function : g(u) = ln(u) [Log]

Standard errors : OIM

Log likelihood = -24.52171418 AIC = 8.130429

BIC = 2.23377e-13

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

a | .491281 .076008 6.46 0.000 .3423081 .6402539

c | -1.870012 .1638293 -11.41 0.000 -2.191111 -1.548912

m | -4.938065 .7096367 -6.96 0.000 -6.328927 -3.547202

ac | 2.035377 .1757605 11.58 0.000 1.690893 2.379862

am | 2.599761 .7269831 3.58 0.000 1.174901 4.024622

cm | 2.275477 .9274553 2.45 0.014 .4576977 4.093256

acm | .5895107 .9423641 0.63 0.532 -1.257489 2.43651

_cons | 5.631212 .0598684 94.06 0.000 5.513872 5.748552

------------------------------------------------------------------------------

. ipf [fw=freq] ,fit(a+c+m+a*c+a*m+c*m) exp




Number of cells = 8

Loglikelihood = 11977.97067255003

…

Loglikelihood = 12010.6123801968


---------------------

df = 1

Likelihood Ratio Statistic G² = 0.3740 p-value = 0.541

Pearson Statistic X² = 0.4011 p-value = 0.527

+-------------------------------------------+


|-------------------------------------------|

| 0 0 0 279.61669 279 .12285443 |

| 0 0 1 1.38317 2 .00060772 |

| 0 1 0 42.383186 43 .01862179 |

| 0 1 1 3.6168357 3 .00158912 |

| 1 0 0 455.38331 456 .20008054 |

|-------------------------------------------|

| 1 0 1 44.61683 44 .01960318 |

| 1 1 0 538.61681 538 .23665062 |

| 1 1 1 910.38316 911 .3999926 |

+-------------------------------------------+

16

. ipf [fw=freq] ,fit(a+c+m+a*c+a*m) exp

…

Loglikelihood = 11762.11473553461

Loglikelihood = 11762.11473553461


---------------------

df = 2



+-------------------------------------------+


|-------------------------------------------|

| 0 0 0 276.70336 279 .12157441 |

| 0 0 1 4.2966361 2 .0018878 |

| 0 1 0 45.296636 43 .01990186 |

| 0 1 1 .70336391 3 .00030904 |

| 1 0 0 255.00257 456 .11203979 |

|-------------------------------------------|

| 1 0 1 244.99743 44 .10764386 |

| 1 1 0 738.99743 538 .32469132 |

| 1 1 1 710.00257 911 .31195192 |

+-------------------------------------------+

.

. ipf [fw=freq] ,fit(a+c+m+a*c+c*m) exp

…

Loglikelihood = 11964.79019282218

Loglikelihood = 11964.79019282218


---------------------

df = 2



+-------------------------------------------+


|-------------------------------------------|

| 0 0 0 264.44942 279 .11619043 |

| 0 0 1 16.550576 2 .00727178 |

| 0 1 0 17.876923 43 .00785454 |

| 0 1 1 28.123077 3 .01235636 |

| 1 0 0 470.55058 456 .20674454 |

|-------------------------------------------|

| 1 0 1 29.449424 44 .01293911 |

| 1 1 0 563.12308 538 .24741787 |

| 1 1 1 885.87692 911 .38922536 |

+-------------------------------------------+

17

. ipf [fw=freq] ,fit(a+c+m+a*m+c*m) exp

Deleting all matrices......




Number of cells = 8

Loglikelihood = 11916.92222167878

Loglikelihood = 11916.92222167878


---------------------

df = 2



+-------------------------------------------+


|-------------------------------------------|

| 0 0 0 179.84043 279 .079016 |

| 0 0 1 .23958333 2 .00010527 |

| 0 1 0 142.15957 43 .06246027 |

| 0 1 1 4.7604167 3 .00209157 |

| 1 0 0 555.15957 456 .24391897 |

|-------------------------------------------|

| 1 0 1 45.760417 44 .02010563 |

| 1 1 0 438.84043 538 .19281214 |

| 1 1 1 909.23958 911 .39949015 |

+-------------------------------------------+

. ipf [fw=freq] ,fit(a+c+m+a*c) exp

Loglikelihood = 11588.88605129906

Loglikelihood = 11588.88605129906


---------------------

df = 3



+-------------------------------------------+


|-------------------------------------------|

| 0 0 0 162.47627 279 .07138676 |

| 0 0 1 118.52373 2 .05207545 |

| 0 1 0 26.59754 43 .01168609 |

| 0 1 1 19.40246 3 .00852481 |

| 1 0 0 289.10369 456 .12702271 |

|-------------------------------------------|

| 1 0 1 210.89631 44 .09266094 |

| 1 1 0 837.8225 538 .36811182 |

| 1 1 1 611.1775 911 .26853142 |

+-------------------------------------------+

18

. ipf [fw=freq] ,fit(a+c+m+a*m) exp

…

Loglikelihood = 11541.01808015565

Loglikelihood = 11541.01808015565


---------------------

df = 3



+-------------------------------------------+


|-------------------------------------------|

| 0 0 0 110.49297 279 .048547 |

| 0 0 1 1.7157293 2 .00075384 |

| 0 1 0 211.50703 43 .09292927 |

| 0 1 1 3.2842707 3 .001443 |

| 1 0 0 341.08699 456 .14986248 |

|-------------------------------------------|

| 1 0 1 327.70431 44 .14398256 |

| 1 1 0 652.91301 538 .28686863 |

| 1 1 1 627.29569 911 .27561322 |

+-------------------------------------------+

. ipf [fw=freq] ,fit(a+c+m+c*m) exp

…

Loglikelihood = 11743.69353744323

Loglikelihood = 11743.69353744323


---------------------

df = 3



+-------------------------------------------+


|-------------------------------------------|

| 0 0 0 105.59974 279 .04639707 |

| 0 0 1 6.6089631 2 .00290376 |

| 0 1 0 83.474077 43 .03667578 |

| 0 1 1 131.31722 3 .0576965 |

| 1 0 0 629.40026 456 .2765379 |

|-------------------------------------------|

| 1 0 1 39.391037 44 .01730713 |

| 1 1 0 497.52592 538 .21859663 |

| 1 1 1 782.68278 911 .34388523 |

+-------------------------------------------+

19

. ipf [fw=freq] ,fit(a+c+m) exp



Number of cells = 8




---------------------

df = 4

Likelihood Ratio Statistic G² = 1.3e+03 p-value = 0.000

Pearson Statistic X² = 1.4e+03 p-value = 0.000

+-------------------------------------------+


|-------------------------------------------|

| 0 0 0 64.879898 279 .02850611 |

| 0 0 1 47.328801 2 .02079473 |

| 0 1 0 124.19392 43 .05456675 |

| 0 1 1 90.597385 3 .03980553 |

| 1 0 0 386.70007 456 .16990337 |

|-------------------------------------------|

| 1 0 1 282.09123 44 .12394167 |

| 1 1 0 740.22612 538 .32523116 |

| 1 1 1 539.98258 911 .23725069 |

+-------------------------------------------+

Model G2 X2 df. P-value

Homogeneous association model (All Pairwise Asoociation)

1. .37 .40 .54,.53

Conditional Independence

2. 187.75 177.61 .00,.00

3. 92.02 80.81 .00,.00

4. 497.37 443.76 .00,.00

joint independent (Partial Independence)

5. 843.83 704.91 .00, .00

6. 893.52 824.16 .00, .00

7. 534.21 503.60 .00, .00

Complete Independence (Mutual independent model)

8. 1286.02 1411.39 .00, .00

Three-factor interaction model 0 0

0=xyz

ijkλ

0== xyz

ijk

xy

ij λλ0== xyz

ijk

xz

ik λλ0== xyz

ijk

yz

jk λλ

0=== xyzyzxz

ijkjkijλλλ

0=== xyzyzxy

ijkjkijλλλ

0=== xyzxzxy

ijkikijλλλ

0, === xyzxzxz

k

xy

ijkikijλλλλ

20

��U V'��-W�+XWY (Dissimilarity Index)

>"�� R�#�>��D� (Gini,1914 D��g�A� Agresti,2002, Kula & Firth,2005) ��h�>"�� ! <��>�g��A��D�� "��GH��D�E�A��I��#��"�� #��> �"�! <R>DG�� fitted �� H��>"�� K

-� �#��E#�� #�� 0 g� 1-�#��D�A��D 0 GH��D�E�A��I��#��"�� #�A��D�� "C�#��> �"� ! <R>DG�� fitted �� ->"�� R�#�>��D�= 0 ��>��#�� fitted R>D��CE�� -!��i�C"��#�>"�� R�#�>��D�� #��D��#� .02 �; .03

2/ˆ2/ˆˆ ∑∑ −=−=∆i

ii

i

ii pnn πµ

Z�'W��Y G�� =��>��F��D� �� #�>"�� R�#�>��D�>"�� K

yz

jk

xz

ik

xy

ij

wz

hk

wy

hj

wx

hi

z

k

y

j

x

i

w

hhijk λλλλλλλλλλλµ ++++++++++=)log(

008219.)68694(2/205.11292/ˆˆ ==−=∆ ∑ nni

ii µ

21

1129.20568694��

5.242859518.24295131100

118.37166811.37266930100

45.920411038.0810841000

77.693856045.30661230000

7.558838387.55883801110

131.172910837.83109690110

33.11871845.11878121010

90.4951210471.5103810010

24.89276781.89287571101

148.50685985.49361340101

15.78479988.78489731001

107.82933353.82932460001

37.69446721.30557591111

161.308611748.31115870111

2.983093993.01699961011

120.63137166.36972870011

di = ni - uiuinizyxw

. gen wx =w*x

. gen wy =w*y

. gen wz =w*z

. gen xy =x*y

. gen xz =x*z

. gen yz =y*z

. qui poisson ni w x y z wx wy wz xy xz yz

. gini ni

Gini Dissimilarity Index = 0.00822

22

Tests about Partial Association

-�� C�! �C��#�� #��=��>��#�� C�! �C=��>� AC, AM, CM�"C AM, CM��h��!>�CA�D Likelihood ratio Statistic -2(L0-L1)G2(AM, CM)-G2(AC, AM, CM)

0:0 =acijH λ

. ipf [fw=freq] , fit(a*c+a*m+c*m) exp

Deleting all matrices…

...


---------------------

df = 1

Likelihood Ratio Statistic G^2 = 0.3740 p-value = 0.541

Pearson Statistic X^2 = 0.4011 p-value = 0.527

+-------------------------------------------+


|-------------------------------------------|

| 0 0 0 279.61673 279 .12285445 |

| 0 0 1 1.3831706 2 .00060772 |

| 0 1 0 42.383171 43 .01862178 |

| 0 1 1 3.6168352 3 .00158912 |

| 1 0 0 455.38327 456 .20008052 |

|-------------------------------------------|

| 1 0 1 44.616829 44 .01960318 |

| 1 1 0 538.61683 538 .23665063 |

| 1 1 1 910.38316 911 .3999926 |

+-------------------------------------------+

23

. ipf [fw=freq] , fit(a*m+c*m) exp

Deleting all matrices…

...


---------------------

df = 2

Likelihood Ratio Statistic G^2 = 187.7543 p-value = 0.000

Pearson Statistic X^2 = 177.6149 p-value = 0.000

+-------------------------------------------+

| a m c Efreq Ofreq prob |

|-------------------------------------------|

| 0 0 0 179.84043 279 .079016 |

| 0 0 1 142.15957 43 .06246027 |

| 0 1 0 .23958333 2 .00010527 |

| 0 1 1 4.7604167 3 .00209157 |

| 1 0 0 555.15957 456 .24391897 |

|-------------------------------------------|

| 1 0 1 438.84043 538 .19281214 |

| 1 1 0 45.760417 44 .02010563 |

| 1 1 1 909.23958 911 .39949015 |

+-------------------------------------------+

G2(AM, CM)-G2(AC, AM, CM)187.7543-0.3740 = 187.3803(df=2) (df=1) = df=2-1 = 1

. disp chiprob(1,187.3803)

1.186e-42

)-�Y'�� Strong Evidence Ho -> Strong evidenceA-C Partial association >"��"K� AC �D��E#A� Model

24

Odds ratio & Confidence Interval

)()(

)()(

/ˆ

kjki

kjjkii

kijnn

nnm

++

=

------------------------------------------

| m

a and c | 0 1

----------+-------------------------------

0 |

0 | 279(279.6168) 2(1.38317)

1 | 43(42.38317) 3(3.61683)

----------+-------------------------------

1 |

0 | 456(455.3832) 44(44.61683)

1 | 538(538.6168) 911(310.3832)

------------------------------------------

. list a c m freq u

+-----------------------------+

| a c m freq u |

|-----------------------------|

1. | 1 1 1 911 910.3832 |

2. | 1 1 0 538 538.6168 |

3. | 1 0 1 44 44.61683 |

4. | 1 0 0 456 455.3832 |

5. | 0 1 1 3 3.61683 |

|-----------------------------|

6. | 0 1 0 43 42.38317 |

7. | 0 0 1 2 1.38317 |

8. | 0 0 0 279 279.6168 |

+-----------------------------+

0110

0011ˆnn

nnmac =

778.7

)62.3(62.44

)38.1(38.910ˆ

=

=acm

25

�� #� odds ratio ��>��#�� "�:"��"��E�� #��>;<��ECC� � < !"K�A��#��EC�"p��R�#�EC�"p�� odds ratio = 7.8; 95%CI �!#��"C (7.80± 1.96*1.36) �!#��"C (5.55-10.98) Ig<��>��D��"C�� Wald (Z) �!#��"C 11.80 �� p-value < .001

778.7

)62.3(62.44

)38.1(38.910ˆ

=

=acm

0110

0011ˆnn

nnmac =

+-----------------------------+

| a c m freq u |

|-----------------------------|

1. | 1 1 1 911 910.3832 |

2. | 1 1 0 538 538.6168 |

3. | 1 0 1 44 44.61683 |

4. | 1 0 0 456 455.3832 |

5. | 0 1 1 3 3.61683 |

|-----------------------------|

6. | 0 1 0 43 42.38317 |

7. | 0 0 1 2 1.38317 |

8. | 0 0 0 279 279.6168 |

+-----------------------------+

. poisson freq a c m ac am cm, irr








LR chi2(6) = 2851.09

Prob > chi2 = 0.0000


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

a | 1.628597 .1233943 6.44 0.000 1.403849 1.889326

c | .1515759 .0246609 -11.60 0.000 .11019 .2085058

m | .0049467 .0023506 -11.17 0.000 .0019491 .0125545

ac | 7.803201 1.358259 11.80 0.000 5.547649 10.97581

am | 19.80658 9.203684 6.43 0.000 7.966636 49.24297

cm | 17.25133 2.826447 17.38 0.000 12.51302 23.78389

------------------------------------------------------------------------------

26

Strategies in model selection

��">��;��#�� "K��. sw poisson freq a c m ac am cm acm,pr(.15)

begin with full model

p = 0.5316 >= 0.1500 removing acm


LR chi2(6) = 2851.09

Prob > chi2 = 0.0000


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

a | .487719 .0757672 6.44 0.000 .339218 .63622

c | -1.886669 .162697 -11.60 0.000 -2.205549 -1.567789

m | -5.309042 .475197 -11.17 0.000 -6.240411 -4.377673

ac | 2.054534 .1740643 11.80 0.000 1.713374 2.395694

am | 2.986014 .464678 6.43 0.000 2.075262 3.896767

cm | 2.847889 .1638394 17.38 0.000 2.52677 3.169009

_cons | 5.63342 .0597008 94.36 0.000 5.516409 5.750432

------------------------------------------------------------------------------

. poisson freq f v





LR chi2(2) = 72.24

Prob > chi2 = 0.0000


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

f | -.0223473 .1057099 -0.21 0.833 -.2295349 .1848403

v | .9477894 .1177963 8.05 0.000 .7169129 1.178666

_cons | 3.923134 .1128342 34.77 0.000 3.701983 4.144285

------------------------------------------------------------------------------

. poisgof

Goodness-of-fit chi2 = 37.35134

Prob > chi2(1) = 0.0000

27

. loglink count f v,fit(f,v)

Variable f = A

Variable v = B

Margins fit: f,v



Iteration 0: Log Likelihood = -32.40625



Poissok regression Number of obs = 4

Goodness-of-fit chi2(1) = 37.351 Model chi2(2) = 72.238

Prob > chi2 = 0.0000 Prob > chi2 = 0.0000

Log Likelihood = -30.954 Pseudo R2 = 0.5385

------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

A2 | .0223473 .1057099 0.21 0.833 -.1848403 .2295349

B2 | -.9477896 .1177963 -8.05 0.000 -1.178666 -.7169131

_cons | 4.848577 .0820511 59.09 0.000 4.687759 5.009394

------------------------------------------------------------------------------

. poisson freq




LR chi2(0) = 0.00

Prob > chi2 = .


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

_cons | 4.494239 .0528516 85.03 0.000 4.390651 4.597826

------------------------------------------------------------------------------

. poisgof


Prob > chi2(3) = 0.0000

28

. loglink count f v,fit()

Note: Only the grand mean will be fit (one poissok parameter

for all cells).

Variable f = A

Variable v = B

Margins fit: Grand mean only









Prob > chi2 = 0.0000 Prob > chi2 = .


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

_cons | 4.494239 .0528516 85.03 0.000 4.390651 4.597826

------------------------------------------------------------------------------

. poisson freq v





LR chi2(1) = 72.19

Prob > chi2 = 0.0000


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

v | .9477894 .1177963 8.05 0.000 .7169129 1.178666

_cons | 3.912023 .1 39.12 0.000 3.716027 4.108019

------------------------------------------------------------------------------

. poisgof


Prob > chi2(2) = 0.0000

29

. loglink count v,fit(v)

Variable v = A

Margins fit: v








Prob > chi2 = 0.0000 Prob > chi2 = 0.0000


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

A2 | -.9477894 .1177963 -8.05 0.000 -1.178666 -.7169129

_cons | 4.859812 .0622573 78.06 0.000 4.73779 4.981835

------------------------------------------------------------------------------

. poisson freq f




LR chi2(1) = 0.04

Prob > chi2 = 0.8326


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

f | -.0223473 .1057099 -0.21 0.833 -.2295349 .1848403

_cons | 4.50535 .0743294 60.61 0.000 4.359667 4.651033

------------------------------------------------------------------------------

. poisgof


Prob > chi2(2) = 0.0000

30

loglink count f,fit(f)

Variable f = A

Margins fit: f









Prob > chi2 = 0.0000 Prob > chi2 = 0.8321


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

A2 | .0223472 .1057099 0.21 0.833 -.1848403 .2295348

_cons | 4.483003 .0751646 59.64 0.000 4.335683 4.630323

------------------------------------------------------------------------------

��"�:"�� #��=��>��CC�F��D��>��G��

α==== )],|1([log kzixyPit

xz

ik

z

k

y

j

x

iijk λλλλλµ ++++=log

x

ikzixyPit βα +==== )],|1([logyz

jk

xz

ik

xy

ij

z

k

y

j

x

iijk λλλλλλλµ ++++++=log

31

z

kkzixyPit βα +==== )],|1([log

yz

jk

xz

ik

z

k

y

j

x

iijk λλλλλλµ +++++=log

z

k

x

ikzixyPit ββα ++==== )],|1([log

xz

ik

xy

ij

z

k

y

j

x

iijk λλλλλλµ +++++=log

. list x y z count

+-------------------+

| x y z count |

|-------------------|

1. | 1 1 1 19 |

2. | 1 0 1 132 |

3. | 0 1 1 0 |

4. | 0 0 1 9 |

5. | 1 1 0 11 |

|-------------------|

6. | 1 0 0 52 |

7. | 0 1 0 6 |

8. | 0 0 0 97 |

+-------------------+

�"��#�� "�:"�� #��=��>��CC�F��D��>��G��=!r�ED��!H��> (death penalty: y) �ED�� D�� (victim’s race: z) ��GH�� (dependant’s race :x)

32

. logit y [fw=count]


Logit estimates Number of obs = 326

LR chi2(0) = -0.00

Prob > chi2 = .

Log likelihood = -113.2564 Pseudo R2 = -0.0000

------------------------------------------------------------------------------

y | Coef. Std. Err. z P>|z| [95% Conf. Interval]

-------------+----------------------------------------------------------------

_cons | -2.086362 .176709 -11.81 0.000 -2.432705 -1.740019

------------------------------------------------------------------------------

. poisson count x y z xz







LR chi2(4) = 387.78

Prob > chi2 = 0.0000


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

x | -.4915943 .159943 -3.07 0.002 -.8050767 -.1781118

y | -2.086362 .176709 -11.81 0.000 -2.432705 -1.740019

z | -2.437504 .3475915 -7.01 0.000 -3.118771 -1.756238

xz | 3.31165 .3785702 8.75 0.000 2.569666 4.053633

_cons | 4.517713 .1004466 44.98 0.000 4.320841 4.714584

------------------------------------------------------------------------------

. logit y x [fw=count], nolog


LR chi2(1) = 6.25

Prob > chi2 = 0.0124


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

x | 1.057941 .4635394 2.28 0.022 .1494208 1.966462

_cons | -2.87168 .4196435 -6.84 0.000 -3.694166 -2.049194

------------------------------------------------------------------------------

. poisson count x y z xy xz


…


LR chi2(5) = 394.03

Prob > chi2 = 0.0000


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

x | -.5875747 .1638567 -3.59 0.000 -.908728 -.2664214

y | -2.87168 .4196435 -6.84 0.000 -3.694166 -2.049194

z | -2.437504 .3475915 -7.01 0.000 -3.118771 -1.756238

xy | 1.057941 .4635394 2.28 0.022 .1494208 1.966462

xz | 3.31165 .3785702 8.75 0.000 2.569666 4.053633

_cons | 4.579669 .101065 45.31 0.000 4.381586 4.777753

------------------------------------------------------------------------------

33

. logit y z [fw=count], nolog


LR chi2(1) = 0.22

Prob > chi2 = 0.6379


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

z | .1664121 .3539187 0.47 0.638 -.5272558 .8600799

_cons | -2.170733 .2559976 -8.48 0.000 -2.672479 -1.668987

------------------------------------------------------------------------------

. poisson count x y z yz xz


…


LR chi2(5) = 388.01

Prob > chi2 = 0.0000


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

x | -.4915943 .159943 -3.07 0.002 -.8050767 -.1781118

y | -2.170733 .2559979 -8.48 0.000 -2.67248 -1.668986

z | -2.455877 .3497847 -7.02 0.000 -3.141442 -1.770311

yz | .1664121 .3539191 0.47 0.638 -.5272566 .8600807

xz | 3.31165 .3785702 8.75 0.000 2.569666 4.053633

_cons | 4.526688 .101961 44.40 0.000 4.326848 4.726527

------------------------------------------------------------------------------

. logit y x z [fw=count]


…


LR chi2(2) = 7.43

Prob > chi2 = 0.0243


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

x | 1.324213 .5193463 2.55 0.011 .3063128 2.342113

z | -.4402222 .4008893 -1.10 0.272 -1.225951 .3455064

_cons | -2.842105 .4203379 -6.76 0.000 -3.665952 -2.018258

------------------------------------------------------------------------------

. poisson count x y z xz xy yz



LR chi2(6) = 395.21

Prob > chi2 = 0.0000


------------------------------------------------------------------------------


-------------+----------------------------------------------------------------

x | -.6331008 .1714027 -3.69 0.000 -.969044 -.2971577

y | -2.842105 .4203379 -6.76 0.000 -3.665952 -2.018258

z | -2.417691 .3480023 -6.95 0.000 -3.099763 -1.735619

xz | 3.357995 .3819713 8.79 0.000 2.609345 4.106645

xy | 1.324213 .5193463 2.55 0.011 .3063128 2.342113

yz | -.4402222 .4008893 -1.10 0.272 -1.225951 .3455064

_cons | 4.578062 .1012175 45.23 0.000 4.37968 4.776445

------------------------------------------------------------------------------

34

��(V&� '( odds ratio. logit y x z [fw=count],or


LR chi2(2) = 7.43

Prob > chi2 = 0.0243


------------------------------------------------------------------------------

y | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]

-------------+----------------------------------------------------------------

x | 3.759225 1.95234 2.55 0.011 1.358407 10.40319

z | .6438933 .25813 -1.10 0.272 .2934785 1.412705

------------------------------------------------------------------------------

. poisson count x y z xz xy yz,irr


LR chi2(6) = 395.21

Prob > chi2 = 0.0000


------------------------------------------------------------------------------

count | IRR Std. Err. z P>|z| [95% Conf. Interval]

-------------+----------------------------------------------------------------

x | .5309429 .0910051 -3.69 0.000 .3794456 .7429269

y | .0583028 .0245069 -6.76 0.000 .0255798 .1328867

z | .0891272 .0310165 -6.95 0.000 .0450599 .176291

xz | 28.73154 10.97462 8.79 0.000 13.59015 60.74261

xy | 3.759225 1.95234 2.55 0.011 1.358407 10.40319

yz | .6438933 .25813 -1.10 0.272 .2934785 1.412705

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