Log Linear Model. . Web Page: 2. (estimating parameter) ก (main effect) interaction effect

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1 Log Lnear Model. ก ก Emal: Web Page: ก (fttng models) ก 2. (estmatng parameter) ก ก ก (man effect) nteracton effect

2 Log Lnear Model dfference from logt model -do not dstngush between response varable (dependent) and explanatory varable (ndependent) -relaton among ndependent varable Ex. ก ก ก (Log Lnear Model) Log lnear model for 2-way table ( 2) ( ) 2 n n2 n. 2 n2 n22 n2. n. n.2 n 2

3 Independent model x y ln( µ ) = λ+ λ + λ j ก ก ก ก Fatal Non Fatal () () () () ln( belt µ ) λ+ λ + = λ nj j (Saturated Model) -Log Lnear Model ก ก (man effect) (nteracton effect) ln( x y µ ) λ+ λ + λ j + = λ xy 3

4 ln( x y µ ) λ+ λ + λ j + = λ xy u = = ntercept = ก λ j = ก 2 = λ x λ y xy λ ก ก Log Lnear model x y xy ln( µ ) = λ+ λ + λ + λ () x y xy 2 ln( µ ) = λ+ λ + λ2 + λ2 (2) 2 x y xy 3 ln( µ ) = λ+ λ2+ λ + λ2 2 (3) x y xy 4 ln( µ ) = λ+ λ2+ λ2 + λ22 (4) 22 4

5 ก x y xy ก λ, λ, λj, λ ก ก ก ()-(4) ก x y xy xy xy λ2 =, λ2 =, λ2 =, λ2=, λ22 = ก ก x y xy ln( µ ) = λ+ λ (5) + λ + λ x 2 ln( µ ) = λ+ λ 2 (6) y 3 ln( µ ) = λ+ λ (7) 2 4 ln( µ ) =λ (8) 22 x y xy กก ก ก (5)-(8) λ, λ, λ, λ j λ= log( µ 22 ) µ x λ = = 2 ln( µ ) ln( µ ) ln 2 22 µ 22 µ x λ = = 2 ln( µ ) ln( µ ) ln 2 22 µ 22 λ xy µ µ = ln( µ ) ln( µ ) - (ln( µ ) ln( µ ) = ln µ µ 2 2 5

6 ก ก ก ก Fatal Non Fatal () () () () x y xy ก λ, λ, λj, λ λ = ln(42368) = x λ = ln(62527) ln(42368) = ln = y 5 λ = ln(5) ln(42368) = ln = λ xy = ln(6) ln(62527)- (ln(5) ln(42368)) (6)(42368) = ln = 2.75 (62527)(5) ก ก Fatal () Non Fatal () () ()

7 . gen bj=b*j. posson freq b j bj Iteraton : log lkelhood = Iteraton : log lkelhood = Iteraton 2: log lkelhood = Iteraton 3: log lkelhood = Iteraton 4: log lkelhood = Iteraton 5: log lkelhood = Iteraton 6: log lkelhood = Posson regresson Number of obs = 4 LR ch2(3) = Prob > ch2 =. Log lkelhood = Pseudo R2 =.9999 freq Coef. Std. Err. z P> z [95% Conf. Interval] b j bj _cons glm freq b j bj, f(po) l(log) ef Iteraton : log lkelhood = Iteraton : log lkelhood = Iteraton 2: log lkelhood = Iteraton 3: log lkelhood = Iteraton 4: log lkelhood = Iteraton 5: log lkelhood = Iteraton 6: log lkelhood = Generalzed lnear models No. of obs = 4 Optmzaton : ML Resdual df = Scale parameter = Devance =.44e- (/df) Devance =. Pearson = e-24 (/df) Pearson =. Varance functon: V(u) = u Lnk functon : g(u) = ln(u) [Posson] [Log] AIC = Log lkelhood = BIC =.44e- OIM freq IRR Std. Err. z P> z [95% Conf. Interval] b j bj

8 . logln count qbelt njury,ft( qbelt, njury, qbelt njury) Varable qbelt = A Varable njury = B Margns ft: qbelt, njury, qbelt njury Note: Regresson-lke constrants are assumed. The frst level of each varable (and all teractons wth t) wll be dropped from estmaton. Iteraton : log lkelhood = Iteraton : log lkelhood = Iteraton 2: log lkelhood = Iteraton 3: log lkelhood = Iteraton 4: log lkelhood = Iteraton 5: log lkelhood = Iteraton 6: log lkelhood = Posson regresson Number of obs = 4 LR ch2(3) = Prob > ch2 =. Log lkelhood = Pseudo R2 = count Coef. Std. Err. z P> z [95% Conf. Interval] A AB B _cons ก ก selt belt ก ก. posson freq b j bj,rr Iteraton : log lkelhood = Iteraton : log lkelhood = Iteraton 2: log lkelhood = Iteraton 3: log lkelhood = Iteraton 4: log lkelhood = Iteraton 5: log lkelhood = Iteraton 6: log lkelhood = Posson regresson Number of obs = 4 LR ch2(3) = Prob > ch2 =. Log lkelhood = Pseudo R2 =.9999 freq IRR Std. Err. z P> z [95% Conf. Interval] b j bj xy λ. lst b j freq bj d (6*42368)/(62527*5)

9 ก ก odds rato. posson freq b j bj,rr Iteraton : log lkelhood = Iteraton : log lkelhood = Iteraton 2: log lkelhood = Iteraton 3: log lkelhood = Iteraton 4: log lkelhood = Iteraton 5: log lkelhood = Iteraton 6: log lkelhood = Posson regresson Number of obs = 4 LR ch2(3) = Prob > ch2 =. Log lkelhood = Pseudo R2 =.9999 freq IRR Std. Err. z P> z [95% Conf. Interval] b j bj ก ก ก Selt Belt ก Log lnear model for 3-way tables ก (cgarette) ก (alcohol) ก (maruana) m a and c ( ) 2(.3837) 43( ) 3(3.6683) ( ) 44( ) 538( ) 9(3.3832) ln a c m ac am cm acm ( µ ) = λ+ λ + λ + λ + λ + λk + λ + λ j k jk k 9

10 Log lnear model for 3-way tables -Homogeneous assocaton model All Parwse Assocaton Present-No ndependent xyz ( = ) λ k ln x y z xy xz yz ( µ ) = λ+ λ + λ + λ + λ + λk + λ -Condtonal ndependence model x y z xz yz ( µ ) = λ+ λ + λ + λ + λk + λ j k jk λ xy = xyz λ ln j k jk k x y z xy yz ln( µ ) = λ+ λ + λ + λ + λ + λ λ xz j k jk k λk x y z xy xz ln( µ ) = λ+ λ + λ + λ + λ + λ j k k λ yz jk λk = xyz = xyz = = = -jont ndependent (Partal Independence Model ) xy λ xz = λ = λ k k x y z xy ln( µ ) = λ+ λ + λ + λ + λ j k x y z xz ln( µ ) = λ+ λ + λ + λ + λ j k k x y z yz ln( µ ) = λ+ λ + λ + λ + λ j k jk xy xz xz xyz λ, λ = λ = λ = -Complete Independence Model k k k xyz = ln x y z ( µ ) = λ+ λ + λ + λ j k

11 Model df. symbol Homogeneous assocaton model (All Parwse Asoocaton) xyz. λ k = l XY,XZ YZ Condtonal Independence xy 2. λ = xyz λk = (r-) XZ, YZ 3. xz λ = xyz k λ (c-) XY, YZ k = 4. yz λ = xyz jk λ (l-) XY, XZ k = jont ndependent(partal Independence) xz yz xyz 5. λ = λ = λ = (r-)(c-) Z, XY jk k xy yz xyz 6. λ = λ = λ = (r-)(l-) Y, XZ jk k 7. xy xz xyz λ = λ = λ = (c-)(l-) X, YZ k k Complete Independence (mutual ndependent model ) xy xz xz xyz 8. λ, λ = = = (r-)(c-)(l-) X,Y,Z k λ λ k k Three-factor nteracton model XYZ

12 Inference for Loglnear Model ก ft Model [3 Way] ก. goodness of ft 2. resduals 3. tests about partal assocaton 4. Odds rato & CI goodness of ft -Lkelhood rato Ch-square -Pearson Statstcs 2

13 ก ก ก a = alcohol, c=cgarette, m=maruama a c m freq ก ft Model & Selected Model. pf [fw=freq],ft(a+c+m+a*c+a*m+c*m+a*c*m) exp... N.B. structural/samplng zeroes may lead to an ncorrect df Resdual degrees of freedom = Number of parameters = 8 Number of cells = 8 Loglkelhood = Loglkelhood = Goodness of Ft Tests df = Lkelhood Rato Statstc G² =. p-value =. Pearson Statstc X² =. p-value = a c m Efreq Ofreq prob

14 . posson freq a c m ac am cm acm Iteraton : log lkelhood = Iteraton : log lkelhood = Iteraton 2: log lkelhood = Iteraton 3: log lkelhood = Iteraton 4: log lkelhood = Iteraton 5: log lkelhood = Posson regresson Number of obs = 8 LR ch2(7) = Prob > ch2 =. Log lkelhood = Pseudo R2 =.983 freq Coef. Std. Err. z P> z [95% Conf. Interval] a c m ac am cm acm _cons posgof Goodness-of-ft ch2 = Prob > ch2() =.. posgof,pearson Goodness-of-ft ch2 = Prob > ch2() =.. predct u,n. lst freq u freq u

15 . glm freq a c m ac am cm acm, f(po) l(log) Iteraton : log lkelhood = Iteraton : log lkelhood = Iteraton 2: log lkelhood = Iteraton 3: log lkelhood = Iteraton 4: log lkelhood = Iteraton 5: log lkelhood = Iteraton 6: log lkelhood = Generalzed lnear models No. of obs = 8 Optmzaton : ML: Newton-Raphson Resdual df = Scale parameter = Devance = e-3 (/df) Devance =. Pearson = e-2 (/df) Pearson =. Varance functon: V(u) = u [Posson] Lnk functon : g(u) = ln(u) [Log] Standard errors : OIM Log lkelhood = AIC = BIC = e-3 freq Coef. Std. Err. z P> z [95% Conf. Interval] a c m ac am cm acm _cons pf [fw=freq],ft(a+c+m+a*c+a*m+c*m) exp N.B. structural/samplng zeroes may lead to an ncorrect df Resdual degrees of freedom = Number of parameters = 7 Number of cells = 8 Loglkelhood = Loglkelhood = Goodness of Ft Tests df = Lkelhood Rato Statstc G² =.374 p-value =.54 Pearson Statstc X² =.4 p-value =.527 a c m Efreq Ofreq prob

16 . pf [fw=freq],ft(a+c+m+a*c+a*m) exp Loglkelhood = Loglkelhood = Goodness of Ft Tests df = 2 Lkelhood Rato Statstc G² = p-value =. Pearson Statstc X² = p-value =.. a c m Efreq Ofreq prob pf [fw=freq],ft(a+c+m+a*c+c*m) exp Loglkelhood = Loglkelhood = Goodness of Ft Tests df = 2 Lkelhood Rato Statstc G² = p-value =. Pearson Statstc X² = p-value =. a c m Efreq Ofreq prob

17 . pf [fw=freq],ft(a+c+m+a*m+c*m) exp Deletng all matrces... N.B. structural/samplng zeroes may lead to an ncorrect df Resdual degrees of freedom = 2 Number of parameters = 6 Number of cells = 8 Loglkelhood = Loglkelhood = Goodness of Ft Tests df = 2 Lkelhood Rato Statstc G² = p-value =. Pearson Statstc X² = p-value =. a c m Efreq Ofreq prob pf [fw=freq],ft(a+c+m+a*c) exp Loglkelhood = Loglkelhood = Goodness of Ft Tests df = 3 Lkelhood Rato Statstc G² = p-value =. Pearson Statstc X² = p-value =. a c m Efreq Ofreq prob

18 . pf [fw=freq],ft(a+c+m+a*m) exp Loglkelhood = Loglkelhood = Goodness of Ft Tests df = 3 Lkelhood Rato Statstc G² = p-value =. Pearson Statstc X² = p-value =. a c m Efreq Ofreq prob pf [fw=freq],ft(a+c+m+c*m) exp Loglkelhood = Loglkelhood = Goodness of Ft Tests df = 3 Lkelhood Rato Statstc G² = p-value =. Pearson Statstc X² = p-value =. a c m Efreq Ofreq prob

19 . pf [fw=freq],ft(a+c+m) exp Resdual degrees of freedom = 4 Number of parameters = 4 Number of cells = 8 Loglkelhood = Loglkelhood = Goodness of Ft Tests df = 4 Lkelhood Rato Statstc G² =.3e+3 p-value =. Pearson Statstc X² =.4e+3 p-value =. a c m Efreq Ofreq prob Model G 2 X 2 df. P-value Homogeneous assocaton model (All Parwse Asoocaton) xyz. λ k = ,.53 Condtonal Independence 2. xy λ = xyz λ ,. k = xz 3. λ = xyz k λ ,. k = 4. yz λ = xyz jk λ ,. k = jont ndependent (Partal Independence) xz yz xyz 5. λ = λ = λ = ,. jk k xy yz xyz 6. λ = λ = λ = ,. jk k 7. xy xz xyz λ = λ = λ = ,. k k Complete Independence (Mutual ndependent model) xy xz xz xyz 8. λ, λ = = = ,. k λ λ k k Three-factor nteracton model 9

20 (Dssmlarty Index) (Gn,94 Agrest,22, Kula & Frth,25) ก ก ก กก ftted ก = ˆ n ˆ µ / 2n= p ˆ π / ก ก ก ก กก ftted ก - = ก ftted - ก.2.3 กก log( µ ) = λ+ λ + λ + λ + λ + λ + λ + λ + λ + λ + λ hk w h x y j z k wx h wy hj wz hk xy xz k yz jk ˆ = n ˆ / 2n= / 2(68694) = µ.829 2

21 w x y z n u d = n - u gen wx =w*x. gen wy =w*y. gen wz =w*z. gen xy =x*y. gen xz =x*z. gen yz =y*z. qu posson n w x y z wx wy wz xy xz yz. gn n Gn Dssmlarty Index =.822 2

22 Tests about Partal Assocaton - ก AC, AM, CM ก AM, CM ก H : λ = Lkelhood rato Statstc -2(L -L ) G 2 (AM, CM)-G 2 (AC, AM, CM) ac. pf [fw=freq], ft(a*c+a*m+c*m) exp Deletng all matrces... Goodness of Ft Tests df = Lkelhood Rato Statstc G^2 =.374 p-value =.54 Pearson Statstc X^2 =.4 p-value =.527 a c m Efreq Ofreq prob

23 . pf [fw=freq], ft(a*m+c*m) exp Deletng all matrces... Goodness of Ft Tests df = 2 Lkelhood Rato Statstc G^2 = p-value =. Pearson Statstc X^2 = p-value =. a m c Efreq Ofreq prob G 2 (AM, CM)-G 2 (AC, AM, CM) = (df=2) (df=) = df=2- =. dsp chprob(,87.383).86e-42 Strong Evdence Ho -> Strong evdence A-C Partal assocaton AC Model 23

24 Odds rato & Confdence Interval mˆ / k = n n ( k ) + ( k ) n n jj ( k ) j+ ( k ) m a and c ( ) 2(.3837) 43( ) 3(3.6683) ( ) 44( ) 538( ) 9(3.3832) mˆ ac = n n n n. lst a c m freq u a c m freq u ˆ m ac = 9.38(.38) 44.62(3.62) =

25 a c m freq u n mˆ ac = n ˆ m ac = n n 9.38(.38) 44.62(3.62) = ก odds rato ก ก ก ก ก ก odds rato = 7.8; 95%CI ก (7.8±.96*.36) ก ( ) ก Wald (Z) ก.8 p-value <.. posson freq a c m ac am cm, rr Iteraton : log lkelhood = Iteraton : log lkelhood = Iteraton 2: log lkelhood = Iteraton 3: log lkelhood = Iteraton 4: log lkelhood = Iteraton 5: log lkelhood = Posson regresson Number of obs = 8 LR ch2(6) = Prob > ch2 =. Log lkelhood = Pseudo R2 =.983 freq IRR Std. Err. z P> z [95% Conf. Interval] a c m ac am cm

26 Strateges n model selecton ก ก. sw posson freq a c m ac am cm acm,pr(.5) begn wth full model p =.536 >=.5 removng acm Posson regresson Number of obs = 8 LR ch2(6) = Prob > ch2 =. Log lkelhood = Pseudo R2 =.983 freq Coef. Std. Err. z P> z [95% Conf. Interval] a c m ac am cm _cons posson freq f v Iteraton : log lkelhood = Iteraton : log lkelhood = Iteraton 2: log lkelhood = Posson regresson Number of obs = 4 LR ch2(2) = Prob > ch2 =. Log lkelhood = Pseudo R2 =.5385 freq Coef. Std. Err. z P> z [95% Conf. Interval] f v _cons posgof Goodness-of-ft ch2 = Prob > ch2() =. 26

27 . loglnk count f v,ft(f,v) Varable f = A Varable v = B Margns ft: f,v Note: Regresson-lke constrants are assumed. The frst level of each varable (and all teractons wth t) wll be dropped from estmaton. Iteraton : Log Lkelhood = Iteraton : Log Lkelhood = Iteraton 2: Log Lkelhood = Possok regresson Number of obs = 4 Goodness-of-ft ch2() = Model ch2(2) = Prob > ch2 =. Prob > ch2 =. Log Lkelhood = Pseudo R2 =.5385 count Coef. Std. Err. z P> z [95% Conf. Interval] A B _cons posson freq Iteraton : log lkelhood = Iteraton : log lkelhood = Posson regresson Number of obs = 4 LR ch2() =. Prob > ch2 =. Log lkelhood = Pseudo R2 =. freq Coef. Std. Err. z P> z [95% Conf. Interval] _cons posgof Goodness-of-ft ch2 = Prob > ch2(3) =. 27

28 . loglnk count f v,ft() Note: Only the grand mean wll be ft (one possok parameter for all cells). Varable f = A Varable v = B Margns ft: Grand mean only Note: Regresson-lke constrants are assumed. The frst level of each varable (and all teractons wth t) wll be dropped from estmaton. Iteraton : Log Lkelhood = Iteraton : Log Lkelhood = Iteraton 2: Log Lkelhood = Iteraton 3: Log Lkelhood = Possok regresson Number of obs = 4 Goodness-of-ft ch2(3) = Model ch2() =. Prob > ch2 =. Prob > ch2 =. Log Lkelhood = Pseudo R2 =. count Coef. Std. Err. z P> z [95% Conf. Interval] _cons posson freq v Iteraton : log lkelhood = Iteraton : log lkelhood = Iteraton 2: log lkelhood = Posson regresson Number of obs = 4 LR ch2() = 72.9 Prob > ch2 =. Log lkelhood = Pseudo R2 =.5382 freq Coef. Std. Err. z P> z [95% Conf. Interval] v _cons posgof Goodness-of-ft ch2 = Prob > ch2(2) =. 28

29 . loglnk count v,ft(v) Varable v = A Margns ft: v Note: Regresson-lke constrants are assumed. The frst level of each varable (and all teractons wth t) wll be dropped from estmaton. Iteraton : Log Lkelhood = Iteraton : Log Lkelhood = Iteraton 2: Log Lkelhood = Possok regresson Number of obs = 4 Goodness-of-ft ch2(2) = Model ch2() = Prob > ch2 =. Prob > ch2 =. Log Lkelhood = Pseudo R2 =.5382 count Coef. Std. Err. z P> z [95% Conf. Interval] A _cons posson freq f Iteraton : log lkelhood = Iteraton : log lkelhood = Posson regresson Number of obs = 4 LR ch2() =.4 Prob > ch2 =.8326 Log lkelhood = Pseudo R2 =.3 freq Coef. Std. Err. z P> z [95% Conf. Interval] f _cons posgof Goodness-of-ft ch2 = Prob > ch2(2) =. 29

30 loglnk count f,ft(f) Varable f = A Margns ft: f Note: Regresson-lke constrants are assumed. The frst level of each varable (and all teractons wth t) wll be dropped from estmaton. Iteraton : Log Lkelhood = Iteraton : Log Lkelhood = Iteraton 2: Log Lkelhood = Iteraton 3: Log Lkelhood = Possok regresson Number of obs = 4 Goodness-of-ft ch2(2) = Model ch2() =.45 Prob > ch2 =. Prob > ch2 =.832 Log Lkelhood = Pseudo R2 =.3 count Coef. Std. Err. z P> z [95% Conf. Interval] A _cons ก ก log t[ P( y = x=, z = k)] =α logµ = λ+ λ + λ + λ + λ k x y j z k xz k log t[ P( y = x=, z = k)] =α + β logµ = λ+ λ + λ + λ + λ + λ + λ k x y j z k xy xz k x yz jk 3

31 log t[ P( y= x=, z = k)] =α + β logµ = λ+ λ + λ + λ + λ + λ k x y j z k xz k z k yz jk log t[ P( y= x=, z = k)] = α + β + β logµ = λ+ λ + λ + λ + λ + λ k x y j z k xy x xz k z k ก ก ก ก (death penalty: y) (vctm s race: z) (dependant s race :x). lst x y z count x y z count

32 . logt y [fw=count] Iteraton : log lkelhood = Logt estmates Number of obs = 326 LR ch2() = -. Prob > ch2 =. Log lkelhood = Pseudo R2 = -. y Coef. Std. Err. z P> z [95% Conf. Interval] _cons posson count x y z xz Iteraton : log lkelhood = Iteraton : log lkelhood = Iteraton 2: log lkelhood = Iteraton 3: log lkelhood = Iteraton 4: log lkelhood = Posson regresson Number of obs = 8 LR ch2(4) = Prob > ch2 =. Log lkelhood = Pseudo R2 =.8985 count Coef. Std. Err. z P> z [95% Conf. Interval] x y z xz _cons logt y x [fw=count], nolog Logt estmates Number of obs = 326 LR ch2() = 6.25 Prob > ch2 =.24 Log lkelhood = Pseudo R2 =.276 y Coef. Std. Err. z P> z [95% Conf. Interval] x _cons posson count x y z xy xz Iteraton : log lkelhood = Posson regresson Number of obs = 8 LR ch2(5) = Prob > ch2 =. Log lkelhood = Pseudo R2 =.93 count Coef. Std. Err. z P> z [95% Conf. Interval] x y z xy xz _cons

33 . logt y z [fw=count], nolog Logt estmates Number of obs = 326 LR ch2() =.22 Prob > ch2 =.6379 Log lkelhood = Pseudo R2 =. y Coef. Std. Err. z P> z [95% Conf. Interval] z _cons posson count x y z yz xz Iteraton : log lkelhood = Posson regresson Number of obs = 8 LR ch2(5) = 388. Prob > ch2 =. Log lkelhood = Pseudo R2 =.899 count Coef. Std. Err. z P> z [95% Conf. Interval] x y z yz xz _cons logt y x z [fw=count] Iteraton : log lkelhood = Logt estmates Number of obs = 326 LR ch2(2) = 7.43 Prob > ch2 =.243 Log lkelhood = Pseudo R2 =.328 y Coef. Std. Err. z P> z [95% Conf. Interval] x z _cons posson count x y z xz xy yz Iteraton : log lkelhood = Posson regresson Number of obs = 8 LR ch2(6) = Prob > ch2 =. Log lkelhood = Pseudo R2 =.957 count Coef. Std. Err. z P> z [95% Conf. Interval] x y z xz xy yz _cons

34 ก odds rato. logt y x z [fw=count],or Logt estmates Number of obs = 326 LR ch2(2) = 7.43 Prob > ch2 =.243 Log lkelhood = Pseudo R2 =.328 y Odds Rato Std. Err. z P> z [95% Conf. Interval] x z posson count x y z xz xy yz,rr Posson regresson Number of obs = 8 LR ch2(6) = Prob > ch2 =. Log lkelhood = Pseudo R2 =.957 count IRR Std. Err. z P> z [95% Conf. Interval] x y z xz xy yz