Generalized Linear Model [GLM]

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1 Generalzed Lnear Model [GLM]. ก. ก Emal:

2 A Lttle Hstory Multple lnear regresson normal dstrbuton & dentty lnk (Legendre, Guass: early 19th century). ANOVA normal dstrbuton & dentty lnk (Fsher: 1920 s 1935). Lkelhood functon a general approach to nference about any statstcal model (Fsher, 1922). Dluton assays a bnomal dstrbuton wth complementary log-log lnk (Fsher, 1922). Exponental famly class of dstrbutons wth suffcent statstcs for parameters (Fsher, 1934). Probt analyss bnomal dstrbuton & probt lnk (Blss, 1935). Logt for proportons bnomal dstrbuton & logt lnk (Berkson, 1944; Dyke & Patterson, 1952)

3 A Lttle Hstory (contnued) Item analyss Bernoull dstrbuton & logt lnk (Rasch, 1960). Log lnear models for counts Posson dstrbuton & log lnk (Brch, 1963). Regressons for survval data exponental dstrbuton & recprocal or log lnk (Fegl & Zelen, 1965; Zppn & Armtage, 1966; Glasser, 1967). Inverse polynomals Gamma dstrbuton & recprocal lnk (Nelder, 1966). Nelder & Wedderburn (1972): provded unfcaton. They showed - All the prevously mentoned models are specal cases of general model, Generalzed Lnear Models - The MLE for all these models could be obtaned usng same algorthm. - All of the models lsted have dstrbutons n the Exponental Dsperson Famly

4 Generalzed Lnear Model (Generalzed Lnear Model: GLM) ก Nelder & Wedderburn (1972) Contnuous data-contnuous data Regresson Contnuous data-categorcal data Anova

5 Generalzed Lnear Model [GLM] ก 3 ก ก - ก (random component) - ก (systematc component) - ก ก (lnk functon) E(Y)=α + β 1 x β k x k

6 - ก (random component) ก ก ก ก ก ก (response varable) ก ก ก (type of exponental famly) E(Y)=α + β 1 x β k x k

7 - ก (systematc component) ก ก ก ก ก E(Y) = α + β x β x k k ก ก (lnear combnaton) ก ก ก (lnear predctor)

8 X ก X 3 = X I X 2 (X 3 ก nteracton X I X 2 ) X 3 = X 2 1

9 ก ก (lnk functon) ก ก ก ก ก ก µ =E(Y) ก ก (lnear predctor)

10 µ g(µ) = α + β 1 x β k x k ก g(.) ก ก (lnk functon) ก ก ก g(µ) = µ ก ก ก ก (dentty lnk) µ = α + β 1 x β k x k

11 ก ก -loglnear model ก g(µ) = log(µ) log(µ) = α + β 1 x β k x k

12 ก ก -logt model ก µ g(µ) = log 1µ µ log = α+ β x + 1 µ +.. β x 1 1 k k

13 1 ก ก ก ก Normal Identty Regresson Normal Identty ก Analyss of varance Normal Identty Analyss of covarance Bernoull Logt Logstc regresson Posson Log Log lnear Multnomal Gernalzed logt Multnomal response

14 STATA lnk functons are Lnk functon glm opton dentty lnk(dentty) log lnk(log) logt lnk(logt) probt lnk(probt) complementary log-log lnk(cloglog) odds power lnk(opower #) power lnk(power #) negatve bnomal lnk(nbnomal) log-log lnk(loglog) log-complment lnk(logc) STATA dstrbuton famles are Famly glm opton Gaussan(normal) famly(gaussan) Inverse Gaussan famly(gaussan) Bernoull/bnomal famly(bnomal) Posson famly(posson) Negatve bnomal famly(nbnomal) Gamma famly(gamma)

15 ก ก ก ก ก ก HD NHD µ log 1µ = α + β 1x 1 glm hd1 snore, famly(bnomal n) lnk(logt)

16 GLM. nput snore hd1 hd0 snore hd end hd0. generate n=hd0+hd1

17 . glm hd1 snore, famly(bnomal n) lnk(logt) Iteraton 0: log lkelhood = Iteraton 1: log lkelhood = Iteraton 2: log lkelhood = Generalzed lnear models No. of obs = 4 Optmzaton : ML: Newton-Raphson Resdual df = 2 Scale param = 1 Devance = (1/df) Devance = Pearson = (1/df) Pearson = Varance functon: V(u) = u*(1-u/n) Lnk functon : g(u) = ln(u/(n-u)) Standard errors : OIM [Bnomal] [Logt] Log lkelhood = AIC = BIC = hd1 Coef. Std. Err. z P> z [95% Conf. Interval] snore _cons

18 Analyss of Ft - Devance Log Lkeldood - ก random component - ก logt Log Lkelhood = n ln(n ) n ln(n 0 0 ) nln(n) Devence = -2[n ln(n ) n ln(n 0 ) 0 nln(n)]

19 ก ก CHD Age chd Phat l ก Log Lkelhood, Devance constant - Devance (D) ก Log lkelhood - goodness of ft ก

20 Log Lkelhood = n ln(n ) n ln(n 0 = 43ln(43) + 57ln(57) 100ln(100) = = ) nln(n) Devence = -2[n ln(n ) 1 1 = -2( ) + n ln(n 0 ) 0 nln(n)] =

21 LogLkelhood πˆ ι = n y ln( πˆ = 1 = 1 ) + (1 y )ln(1 - ε (20) + ε (20) πˆ ) = LogLkelh ood = Devence = n 2 = 1 y ln( πˆ ) + (1 y )ln(1 - πˆ ) = -2( ) =

22 Model Statstcs Akake nformaton crteron (AIC) 2L(M k ) + 2p AIC = n ( ) + 2(2) AIC = = AIC better ft model

23 . glm chd age, famly(bnomal) lnk(logt) Iteraton 0: log lkelhood = Iteraton 1: log lkelhood = Iteraton 2: log lkelhood = Iteraton 3: log lkelhood = Generalzed lnear models No. of obs = 100 Optmzaton : ML: Newton-Raphson Resdual df = 98 Scale param = 1 Devance = (1/df) Devance = Pearson = (1/df) Pearson = Varance functon: V(u) = u*(1-u) [Bernoull] Lnk functon : g(u) = ln(u/(1-u)) [Logt] Standard errors : OIM Log lkelhood = AIC = BIC = chd Coef. Std. Err. z P> z [95% Conf. Interval] age _cons

24 Log lkelhood rato ft ก constant. glm chd, f(b) l(l) Iteraton 0: log lkelhood = Iteraton 1: log lkelhood = Iteraton 2: log lkelhood = Generalzed lnear models No. of obs = 100 Optmzaton : ML: Newton-Raphson Resdual df = 99 Scale param = 1 Devance = (1/df) Devance = Pearson = (1/df) Pearson = Varance functon: V(u) = u*(1-u) [Bernoull] Lnk functon : g(u) = ln(u/(1-u)) [Logt] Standard errors : OIM Log lkelhood = AIC = BIC = chd Coef. Std. Err. z P> z [95% Conf. Interval] _cons log lkelhood ก Devence = -2( ) =

25 ft ก constant age. glm chd age, f(b) l(l) Iteraton 0: log lkelhood = Iteraton 1: log lkelhood = Iteraton 2: log lkelhood = Iteraton 3: log lkelhood = Generalzed lnear models No. of obs = 100 Optmzaton : ML: Newton-Raphson Resdual df = 98 Scale param = 1 Devance = (1/df) Devance = Pearson = (1/df) Pearson = Varance functon: V(u) = u*(1-u) [Bernoull] Lnk functon : g(u) = ln(u/(1-u)) [Logt] Standard errors : OIM Log lkelhood = AIC = BIC = chd Coef. Std. Err. z P> z [95% Conf. Interval] age _cons

26 = varable the wthg lkelhood varable the wthout lkelhood 2ln G = = n 1 ) y (1 ) (1 y 0 n n 0 n 1 n n 1 n 2ln G πˆ πˆ [ ] [ ] = + + = n 1 nln(n) ) 0 )ln(n 0 (n ) 1 ln(n 1 n ) )ln(1 y (1 ) ln( y 2 G πˆ πˆ [ ] { } ln(100) 57ln(57) 43ln(43) G. = + =

27 . logt chd age Iteraton 0: log lkelhood = Iteraton 1: log lkelhood = Iteraton 2: log lkelhood = Iteraton 3: log lkelhood = Iteraton 4: log lkelhood = Logt estmates Number of obs = 100 LR ch2(1) = Prob > ch2 = Log lkelhood = Pseudo R2 = chd Coef. Std. Err. z P> z [95% Conf. Interval] age _cons

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