# The number of observed cases The number of parameters. ith case of the dichotomous dependent variable. the ith case of the jth parameter

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1 LOGISTIC REGRESSION Notato Model Logstc regresso regresses a dchotomous depedet varable o a set of depedet varables. Several methods are mplemeted for selectg the depedet varables. The followg otato s used throughout ths chapter uless otherwse stated: p y The umber of observed cases The umber of parameters vector wth elemet y, the observed value of the th case of the dchotomous depedet varable X p matrx wth elemet x j, the observed value of the th case of the jth parameter β w l L I p vector wth elemet β j, the coeffcet for the jth parameter vector wth elemet w, the weght for the th case Lkelhood fucto Log lkelhood fucto Iformato matrx The lear logstc model assumes a dchotomous depedet varable Y wth probablty π, where for the th case, 6 exp η π + exp η or 6,

2 LOGISTIC REGRESSION l π - π η X β Hece, the lkelhood fucto l for observatos y, K, y, wth probabltes π, K, π ad case weghts w, K, w, ca be wrtte as wy l º π - π 6 6 w y - It follows that the logarthm of l s 6 Ê L l l w y l π + w - y l -π ad the dervatve of L wth respect to β j s * X j L L Ê w y -π x β j 6 j Maxmum Lkelhood Estmates (MLE) The maxmum lkelhood estmates for β satsfy the followg equatos 6 0 Ê w y - p \$ xj where x 0 for, K,., for the jth parameter Note the followg: () A Newto-Raphso type algorthm s used to obta the MLEs. Covergece ca be based o (a) Absolute dfferece for the parameter estmates betwee the teratos (b) Percet dfferece the log-lkelhood fucto betwee successve teratos (c) Maxmum umber of teratos specfed

3 LOGISTIC REGRESSION 3 () Durg the teratos, f p\$ ( p - \$ ) s smaller tha 0 8 for all cases, the loglkelhood fucto s very close to zero. I ths stuato, terato stops ad the message All predcted values are ether or 0 s ssued. After the maxmum lkelhood estmates β \$ are obtaed, the asymptotc covarace matrx s estmated by I, the verse of the formato matrx I, where! I - E L β β W Dag w, K, w, j " # \$ # XWVX \$, π π6 K π π6 η 7, η7 V\$ Dag \$ - \$,, \$ - \$, exp \$ \$ π + exp \$ ad \$ X β \$. η B Stepwse Methods of Selectg Varables Several methods are avalable for selectg depedet varables. Wth the forced etry method, ay varable the varable lst s etered to the model. There are two stepwse methods: forward ad backward. The stepwse methods ca use ether the Wald statstc, the lkelhood rato, or a codtoal algorthm for varable removal. For both stepwse methods, the score statstc s used to select varables for etry to the model. Three statstcs used later are defed as follows: Score Statstc The score statstc s calculated for each varable ot the model to determe whether the varable should eter the model. Assume that there are r varables, amely, α, K, α r the model ad r varables, γ, K, γ r, ot the model. The score statstc for γ s defed as

4 4 LOGISTIC REGRESSION S L B * 4 γ 9, f γ s ot a categorcal varable. If γ s a categorcal varable wth m categores, t s coverted to a m 6-dmeso dummy vector. Deote these ew m varables as ~ γ, K, ~ γ + -. The score statstc for γ s the S L~ B L * * γ, ~ γ 4 9 m 4 9 ad the m m m where L * ~ * * γ L~ γ, K, L~ γ ,,,, B A A A A wth A A A α V\$ α ~, α V\$, γ ~ ~, γ V\$ γ, ~ ~ 6 6 matrx B, s whch α s the desg matrx for varables α, K, α r ad γ ~ s the desg matrx for dummy varables ~ γ,, ~ K γ + m -. Note that α cotas a colum of oes uless the costat term s excluded from η. Based o the MLEs for the parameters the model, V s estmated by V \$ Dag \$ π - \$ π,, \$ π - \$ 6 K π 6B. The asymptotc dstrbuto of the score statstc s a ch-square wth degrees of freedom equal to the umber of varables volved. Note the followg: () If the model s through the org ad there are o varables the model, B, s defed by A, ad V \$ s equal to 4 I. () If B, s ot postve defte, the score statstc ad resdual ch-square statstc are set to be zero.

5 LOGISTIC REGRESSION 5 Wald Statstc The Wald statstc s calculated for the varables the model to determe whether a varable should be removed. If the th varable s ot categorcal, the Wald statstc s defed by Wald \$ β \$ σ \$ β If t s a categorcal varable, the Wald statstc s computed as follows: Let \$ β be the vector of maxmum lkelhood estmates assocated wth the m dummy varables, ad C the asymptotc covarace matrx for \$ β. The Wald statstc s - Wald β\$ C β\$ The asymptotc dstrbuto of the Wald statstc s ch-square wth degrees of freedom equal to the umber of parameters estmated. Lkelhood Rato (LR) Statstc Codtoal Statstc The LR statstc s defed as two tmes the log of the rato of the lkelhood fuctos of two models evaluated at ther MLEs. The LR statstc s used to determe f a varable should be removed from the model. Assume that there are r varables the curret model whch s referred to as a full model. Based o the MLEs of the full model, l(full) s calculated. For each of the varables removed from the full model oe at a tme, MLEs are computed ad the lkelhood fucto l(reduced) s calculated. The LR statstc s the defed as 6 l 7 l reduced LR L reduced L full lfull6 6 6 LR s asymptotcally ch-square dstrbuted wth degrees of freedom equal to the dfferece betwee the umbers of parameters estmated the two models. The codtoal statstc s also computed for every varable the model. The formula for the codtoal statstc s the same as the LR statstc except that the

6 6 LOGISTIC REGRESSION parameter estmates for each reduced model are codtoal estmates, ot MLEs. The codtoal estmates are defed as follows. Let β \$ \$,, \$ 4β K β r 9 be the MLE for the r varables the model ad C be the asymptotc covarace matrx for β \$. If varable x s removed from the model, the codtoal estmate for the parameters left the model gve β \$ s ~ β β\$ β\$ c c where β \$ s the MLE for the parameter(s) assocated wth x ad β \$ 6 s \$ β wth \$β 6 removed, c s the covarace betwee β \$ 6 ad \$ β 6, ad c s the covarace of β \$. The the codtoal statstc s computed by 4β6 9 6 ~ 4 9 s the log lkelhood fucto evaluated at β \$ ~ L L full where L β6 Stepwse Algorthms 6. Forward Stepwse (FSTEP) () If FSTEP s the frst method requested, estmate the parameter ad lkelhood fucto for the tal model. Otherwse, the fal model from the prevous method s the tal model for FSTEP. Obta the ecessary formato: MLEs of the parameters for the curret model, predcted probablty \$π, lkelhood fucto for the curret model, ad so o. () Based o the MLEs of the curret model, calculate the score statstc for every varable elgble for cluso ad fd ts sgfcace. (3) Choose the varable wth the smallest sgfcace. If that sgfcace s less tha the probablty for a varable to eter, the go to step 4; otherwse, stop FSTEP. (4) Update the curret model by addg a ew varable. If ths results a model whch has already bee evaluated, stop FSTEP. (5) Calculate LR or Wald statstc or codtoal statstc for each varable the curret model. The calculate ts correspodg sgfcace. (6) Choose the varable wth the largest sgfcace. If that sgfcace s less tha the probablty for varable removal, the go back to step ; otherwse, f

7 LOGISTIC REGRESSION 7 the curret model wth the varable deleted s the same as a prevous model, stop FSTEP; otherwse, go to the ext step. (7) Modfy the curret model by removg the varable wth the largest sgfcace from the prevous model. Estmate the parameters for the modfed model ad go back to step 5. Backward Stepwse (BSTEP) () Estmate the parameters for the full model whch cludes the fal model from prevous method ad all elgble varables. Oly varables lsted o the BSTEP varable lst are elgble for etry ad removal. Let the curret model be the full model. () Based o the MLEs of the curret model, calculate the LR or Wald statstc or codtoal statstc for every varable the model ad fd ts sgfcace. (3) Choose the varable wth the largest sgfcace. If that sgfcace s less tha the probablty for a varable removal, the go to step 5; otherwse, f the curret model wthout the varable wth the largest sgfcace s the same as the prevous model, stop BSTEP; otherwse, go to the ext step. (4) Modfy the curret model by removg the varable wth the largest sgfcace from the model. Estmate the parameters for the modfed model ad go back to step. (5) Check to see ay elgble varable s ot the model. If there s oe, stop BSTEP; otherwse, go to the ext step. (6) Based o the MLEs of the curret model, calculate the score statstc for every varable ot the model ad fd ts sgfcace. (7) Choose the varable wth the smallest sgfcace. If that sgfcace s less tha the probablty for varable etry, the go to the ext step; otherwse, stop BSTEP. (8) Add the varable wth the smallest sgfcace to the curret model. If the model s ot the same as ay prevous models, estmate the parameters for the ew model ad go back to step ; otherwse, stop BSTEP.

8 8 LOGISTIC REGRESSION Statstcs Ital Model Iformato If β 0 s ot cluded the model, the predcted probablty s estmated to be 0.5 for all cases ad the log lkelhood fucto L6 0 s 6 6 L 0 Wl W wth W Ê w. If β 0 s cluded the model, the predcted probablty s estmated as Ê wy W \$π 0 ad β 0 s estmated by \$ β l 0 \$ π 0 - \$ π 0 wth asymptotc stadard error estmated by \$ σ \$ β 0 W \$ π \$ 0 - π06 The log lkelhood fucto s! \$ π 0 L6 0 W \$ π 0 l 06 - \$ 0 + l - \$ π. π " \$ #

9 LOGISTIC REGRESSION 9 Model Iformato (a) - Log Lkelhood The followg statstcs are computed f a stepwse method s specfed. Ê wy l \$ π + w - y l - \$ π (b) Model Ch-Square (c) Block Ch-Square (d) Improvemet Ch-Square (log lkelhood fucto for curret model - log lkelhood fucto for tal model) The tal model cotas a costat f t s the model; otherwse, the model has o terms. The degrees of freedom for the model ch-square statstc s equal to the dfferece betwee the umbers of parameters estmated each of the two models. If the degrees of freedom s zero, the model ch-square s ot computed. (log lkelhood fucto for curret model - log lkelhood fucto for the fal model from the prevous method.) The degrees of freedom for the block ch-square statstc s equal to the dfferece betwee the umbers of parameters estmated each of the two models. (e) Goodess of Ft (log lkelhood fucto for curret model - log lkelhood fucto for the model from the last step ) The degrees of freedom for the mprovemet ch-square statstc s equal to the dfferece betwee the umbers of parameters estmated each of the two models. Ê 6 6 w y - \$ π \$ π - \$ π (f) Cox ad Sell s R (Cox ad Sell, 989; Nagelkerke, 99)

10 0 LOGISTIC REGRESSION R CS l 0 - ( ) l ( \$ β) W where l 49 \$ β s the lkelhood of the curret model ad l(0) s the lkelhood of the tal model; that s, l 0 Wlog 05. f the costat s ot cluded the model; \$ log \$ / - \$ 6B+ log - \$ 6 l W π o π o π o π o f the costat s cluded the model, where \$ Ê wy / W. π o (g) Nagelkerke s R (Nagelkerke, 98) RN RCS /max4rcs9 where max R 4 9 < 6A / W l 0 CS -. Hosmer-Lemeshow Goodess-of-Ft Statstc The test statstc s obtaed by applyg a ch-square test o a g cotgecy table. The cotgecy table s costructed by cross-classfyg the dchotomous depedet varable wth a groupg varable (wth g groups) whch groups are formed by parttog the predcted probabltes usg the percetles of the predcted evet probablty. I the calculato, approxmately 0 groups are used (g 0). The correspodg groups are ofte referred to as the decles of rsk (Hosmer ad Lemeshow, 989). If the values of depedet varables for observato ad are the same, observato ad ' are sad to be the same block. Whe oe or more blocks occur wth the same decle, the blocks are assged to ths same group. Moreover, observatos the same block are ot dvded whe they are placed to groups. Ths strategy may result fewer tha 0 groups (that s, g 0 ) ad cosequetly, fewer degrees of freedom. Suppose that there are Q blocks, ad the qth block has m q umber of observatos, q, K, Q. Moreover, suppose that the kth group (k, K, g) s composed of the q th,, q k th blocks of observatos. The the total umber of q k observatos the kth group s sk mj. The total observed frequecy of q evets (that s, Y ) the kth group, call t O k, s the total umber of observatos the kth group wth Y. Let E k be the total expected frequecy of

11 LOGISTIC REGRESSION the evet (that s, Y ) the kth group; the E k s gve by E k sk ξ k, where ξ k s the average predcted evet probablty for the kth group. ξ qk k m j\$ π j / s k q The Hosmer-Lemeshow goodess-of-ft statstc s computed as χ g Ok Ek HL ( ) E k ξ k ( k) The p value s gve by Pr χ χ HL dstrbuted wth degrees of freedom 0g where χ s the ch-square statstc Iformato for the Varables Not the Equato For each of the varables ot the equato, the score statstc s calculated alog wth the assocated degrees of freedom, sgfcace ad partal R. Let X be a varable ot curretly the model ad S the score statstc. The partal R s defed by Partal_ R % & K ' K 0 S df L tal 6 f S > df otherwse where df s the degrees of freedom assocated wth S, ad Ltal6 s the log lkelhood fucto for the tal model. The resdual Ch-Square prted for the varables ot the equato s defed as 4 9 RCS Lγ B L * * γ where L * γ 4L * γ, K, L * γ r 9.

12 LOGISTIC REGRESSION Iformato for the Varables the Equato For each of the varables the equato, the MLE of the Beta coeffcets s calculated alog wth the stadard errors, Wald statstcs, degrees of freedom, sgfcaces, ad partal R. If X s ot a categorcal varable curretly the equato, the partal R s computed as Partal_ R % & K ' K Wald sg \$ 4β9 Ltal6 0 f Wald > otherwse If X s a categorcal varable wth m categores, the partal R s the Partal_ R % & K ' K 0 6 Wald m f Wald > m Ltal6 6 otherwse Casewse Statstcs (a) Idvdual Devace The devace of the th case, G, s defed as G % &K 'K y l \$ π + - y l - \$ π f y > \$ π y l \$ π + - y l - \$ π otherwse (b) Leverage The leverage of the th case, h, s the th dagoal elemet of the matrx 4 9 V\$ XXCVX \$ XV \$ where

13 LOGISTIC REGRESSION K B V \$ Dag \$ π - \$ π,, \$ π - \$ π (c) Studetzed Resdual ~ G G h (d) Logt Resdual e~ e \$ π - \$ π 6 where e (e) Stadardzed Resdual y - \$π. z e - π6 \$ π \$ (f) Cook s Dstace D z h h (g) DFBETA Let Dβ be the chage of the coeffcet estmates from the deleto of case. It s computed as Dβ XCVX \$ X e - h

14 4 LOGISTIC REGRESSION (h) Predcted Group If \$ π 05,. predcted group group whch y. Note the followg: For the uselected cases wth omssg values for the depedet varables ~ the aalyss, the leverage 49 h s computed as ~ \$ Vh h h + Vh \$ where 4 9 h V\$ X X CVX \$ X For the uselected cases, the Cook s dstace ad DFBETA are calculated based o ~ h.

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