Odds. Odds Ratio and Logistic Regression. Odds Ratio Review. Odds Ratio Review. Logistic Regression LR - 1

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1 Logistic Rgrssion Odds Ratio and Logistic Rgrssion Dr. Thomas Smotzr Odds If th robability of an vnt occurring is thn th robability against its occurrnc is -. Th odds in favor of th vnt ar /( - ) : At a rac track 4 : odds on a hors mans th robability of th hors losing is 4/5 and th robability of th hors winning is /5. 2 If th odds in favor of an vnt ar a : b thn th robability of th vnt occurring is a/(a + b) If th robability of a hors losing is 8/9 thn th robability of th hors winning is /9 so th odds ar 8 : Risk Factor (Bnzn) Odds Ratio Rviw Outcom (Brain Tumor) Ys (Cas) No (Control) Total Ys No 3 23 Total Risk Factor Odds Ratio Rviw Ys (Disas) Outcom No (No Disas) Total : robability of succss (disas) in row - : robability of failur (no disas) in row Th odds of gtting disas for th ol who wr xosd to th risk factor: Ys (Exosd) No (Unxosd) a c a + c b d b + d Total a + b c + d n 5 O + P[ disas xosd ] P[ disas xosd ] - P[ disas xosd ] P[ no disas xosd ] a a + c c a + c a c LR -

2 Logistic Rgrssion : robability of succss (disas) in row 2 : robability of failur (no disas) in row 2 Th odds of gtting disas for th ol who wr not xosd to th risk factor: P[ disas unxosd ] P[ disas unxosd ] O - P[ disas unxosd ] P[ no disas unxosd ] b b + d d b + d b d Th Odds Ratio of having brain tumor for ol who wr xosd to th risk factor vrsus not xosd: O+ OR θ O a ˆ [ a /( a + c)] /[ c /( a + c)] θ c [ b /( b + d)] /[ d /( b + d)] b d ad bc Intrrtation: Th odds of having brain tumor ar 35 tims highr for thos who xosd to bnzn than thos who wr not xosd to bnzn. 8 Confidnc Intrval for Odds Ratio: If θ >, thn th odds of succss ar highr for row (risk factor rsnt) than row 2 (risk factor not rsnt). For larg saml, th log of odds ratio, ln(θˆ) asymtotically a normal distribution., follows If θ <, thn th odds of succss ar lowr for row (risk factor rsnt) than row 2 (risk factor not rsnt). If θ, thn th odds of succss ar qual for row (risk factor rsnt) and row 2 (risk factor not rsnt). 9 Th ( α)% confidnc intrval stimat for th Log Odds Ratio is ± z α / 2 s * Th ( α)% confidnc intrval stimat for th Odds Ratio is zα 2 s* + zα / (, ) Confidnc Intrval for Odds Ratio: whr ad θˆ bc zα + zα (, ), standard rror of ln( θˆ ) is s* and a, b, c and d should not b zro a b c d Examl: (Brain tumor) Th 95% confidnc intrval stimat for th odds ratio is: z α / 2 s* ln(35).79, z ( ( z 5.96, s* α / 2 ln( ˆ) θ zα.79 8,, θˆ 3 + zα ad bc ) , 2 96 ) (9, 57) 2 LR - 2

3 Logistic Rgrssion (9, 57) This confidnc intrval dosn t covr. In fact, it covrs a rang that is gratr than. This mans that whn xosd to bnzn an individual is mor likly to hav a brain tumor than thos who wr not xosd to bnzn. Continuity Corrction: Th ( α)% confidnc intrval stimat for th Odds Ratio is zα + zα / 2 s* (, ) ˆ ( a + )( d + ) whr θ, standard rror of ln( θˆ ) ( b + )( c + ) is s* a b + c + d Examl: Risk Factors for Dngu Eidmics Data: Dants, Kooman and Addy t al., Dngu Eidmics on th Pacific Coast of Mxico, Intrnational Journal of Eidmiology 7 (988), Variabl Nam Idntification Ag Socioconomic Status Sctor Disas Status Saving Account Status -96 Dscrition Ag of rson (in yars) ur, 2 middl, 3 lowr Sctor in city: sctor 2 sctor 2 with disas, no disas has account, has no account. 5 With % of th data: Disas Status Ys, Y No, Y Sctor X, sctor X, sctor Sctor : ˆ 22/7, ˆ 22/7 95/7 Sctor 2: ˆ 35/79, ˆ 35/79 44/79 Odds ratio of having disas for sctor v.s. sctor 2: (22/95)/(35/44) 9 (Th odds of having disas living in sctor is around 3% of th odds 6 for living in sctor 2.) Estimat and comar roortions of rsons who contractd th disas by sctor, using first 5% of th data: Odds ratio of having disas for sctor v.s. sctor 2: (22/95)/(35/44) 9 (Th odds of having disas living in sctor is around 3% of th odds for living in sctor 2.) Sctor Disas Status Ys, Y No, Y X, sctor X, sctor Odds ratio of having disas for sctor 2 v.s. sctor : (35/44)/(22/95) 35 (Th odds of having disas living in sctor 2 is mor than 3 tims of th odds for living in sctor.) 7 Sctor : ˆ /59, ˆ 49/59 Sctor 2: ˆ 2/39, ˆ 8/39 Odds ratio of having disas for sctor v.s. sctor 2: (/49)/(2/8).75 (Th odds of having disas living in sctor is around 8% of th odds for living in sctor 2.) 8 LR - 3

4 Logistic Rgrssion SPSS Outut Count SECTOR Total SECTOR * DISEASE Crosstabulation Sctor (X) Sctor 2 (X2) Odds Ratio for SECTOR (Sctor (X) / Sctor 2 (X2)) For cohort DISEASE Ys (Y) For cohort DISEASE No (Y2) N of Valid Cass DISEASE Ys (Y) No (Y2) Total Risk Estimat 95% Confidnc Intrval Valu Lowr Ur Logistic Rgrssion Logistic rgrssion is a rgrssion mthod that can modl binary rsons variabl using both quantitativ and catgorical xlanatory variabls. This mthod can also b usd to rdict th robability of a binary outcom. Notation: Y is th rsons variabl, it taks on if disas rsnt and taks on if disas absnt. dnots th robability of succss i.., robability of y (disas rsnt) or P(Y X) 2 In Siml Linar Rgrssion. Rsons Variabl Intr-rution Tim Exlanatory Variabl Duration 2 22 In Siml Linar Rgrssion yˆ ˆ α + ˆ β x y Rgrssion Modls with Binary OutcomVariabl Y α + β X + ε Man of y at x Modls th man rsons x x 2 x 3 x 4 x LR - 4

5 Logistic Rgrssion Rgrssion Modls with Binary Outcom Variabl To modl th robability of succss: Sinc th outcom is ithr or, it can b modld by Brnoulli distribution. Y α + β X + ε, Y ~ Brnoulli Y Probability P(Y ) or P(Y X) P(Y ) or P(Y X) Can w fit a lin for and x? Problms. Non-normal rror trms: ε Rsons Variabl Exlanatory Variabl 3. Th man of Brnoulli distribution is th robability of succss: E[Y ] µ y x α + β X???? 2. Non-constant rror varianc: σ 2 (ε) 3. Constraints of rsons function: µ y x (W want to modl th robability of gtting, or {Y }, at a givn X.) 25 If fitting a lin, may xcd or bcom ngativ. Not good. 26 To modl th robability of succss: To modl th robability of succss: Fitting a Function 6 counts of s. Rsons Variabl. Rsons Variabl 4 counts of s Exlanatory Variabl 27 Exlanatory Variabl 28 To modl th robability of succss: Fitting a Logistic Function (Sigmoidal Curv) x f ( x) + x + x. Rsons Variabl + α + β x α + β x Exlanatory Variabl 29 3 LR - 5

6 Logistic Rgrssion f ( x) + x x + x Siml Logistic Rsons Function (Logistic Function) Th robability of succss (robability of Y givn x): + α + β x α + β x + α β x 3 Prortis: Sigmoidal (S-shad) Monotonic (Incrasing or dcrasing) Linarizabl (Logit transformation) 32 Diffrnt form of th logistic function: Logit Transformation (Logit link) Th natural logarithm of th odds in favor of succss at x: + x α β Diffrnt form of th logistic function: Odds Th odds in favor of succss (th odds in favor of Y) at x ar α + β x Solving for Logit rsons function (Th logarithm of th odds is linarly rlatd to x.) 33 + ( α + β x) ( α + β x) 34 Usful Statistics:. Prdictd Probability + ( α + β x) ( α + β x) Prdictd robability of succss: Us th maximum liklihood stimats Prdictd robability of succss for x : Prdictd robability of succss for x : + ( ˆ α + ˆ β x) ( ˆ α + ˆ β x) ( ˆ α + ˆ β ) + + ( ˆ α + ˆ β ) ( ˆ α + ˆ β ) ( ˆ α + ˆ β ) 35 Examl: Studying disas and sctors Modl: + x α β Sctor within city Catgorical Variabls Codings Sctor Sctor 2 Paramt di Frquncy () LR - 6

7 Logistic Rgrssion St a Variabls in th Equation B S.E. Wald df Sig. Ex(B) Lowr Ur SECTOR() a. Variabl(s) ntrd on st : SECTOR Maximum liklihood stimats: ˆ α 29, β ˆ Th rdictd robability of having disas can b calculatd with th rdiction quation ( 99 ( + Probability of gtting disas for ol from sctor : /( ) 8343 ˆ Probability of gtting disas for ol from sctor 2: 95.% C.I.for EXP(B) 34 x) x) 2. Estimatd Odds Th odds in favor of succss (th odds in favor of Y) at x is: ˆ α + ˆ β x ln( ) x ˆ /( ) Estimatd odds x Th odds of ol in sctor to hav th disas Th odds of ol in sctor 2 to hav th disas 3. Estimatd Paramtr β and Odds Ratio + x α β β chang of ln of odds for incrasing on unit of X, that is ( α + β ) α β x x If X is a binary variabl, thn β is th odds ratio and βˆ is th stimatd odds ratio for x v.s x 39 Odds ratio of gtting disas for ol from sctor v.s. sctor 2: Odds ratio βˆ Ex(B) in SPSS Th -valu of th tst for β (OR ) is. which is lss than. So sctor variabl is statistically significant. It suggsts that thr is som association btwn disas and sctor. Th 95% conf. intrval of odds ratio is (3, 53). Odds ratio of having disas for sctor 2 v.s. sctor would b th rcirocal of 9 and is 35, with 95% C.I. (, 64). 4 Studying disas and sctors, social con., ag, sav. Catgorical Variabls Codings Odds ratio of contracting disas for Ur v.s. Lowr Socio. Status is.757 Odds ratio of contracting disas for Middl v.s. Lowr Socio. Status is 3 Paramtr coding Socioconomic Status Sctor within city Ur Middl Lowr Sctor Sctor 2 Frquncy () (2) St a SECTOR() SCIOSTAT SCIOSTAT() SCIOSTAT(2) SAVINGS Variabls in th Equation 95.% C.I.for EXP(B) Lowr Ur B S.E. Wald df Sig. Ex(B) AGE a. Variabl(s) ntrd on st : SECTOR, SCIOSTAT, SAVINGS, AGE LR - 7

8 Logistic Rgrssion St a AGE SECTOR() a. Variabl(s) ntrd on st : AGE, SECTOR. Variabls in th Equation B S.E. Wald df Sig. Ex(B) Lowr Ur % C.I.for EXP(B) St a AGE SECTOR() a. Variabl(s) ntrd on st : AGE, SECTOR. Variabls in th Equation B S.E. Wald df Sig. Ex(B) Lowr Ur % C.I.for EXP(B) Odds ratio of gtting disas for sctor vrsus sctor 2 aftr adjustd for ag:.37 Odds ratio of gtting disas for sctor 2 vrsus sctor aftr adjustd for ag: 3 ( /.37) Th -valu is. for tsting th significanc of sctor variabl by controlling th ag variabl. Sinc -valu is lss than, thr is statistically significant association 43 btwn sctor and disas. Ag is also a significant variabl, by controlling th sctor variabl, with a -valu of. Ex(B)7 mans th odds of gtting disas incrass as th ag incrass. 44 St a AGE SECTOR() a. Variabl(s) ntrd on st : AGE, SECTOR. Variabls in th Equation 95.% C.I.for EXP(B) Lowr Ur B S.E. Wald df Sig. Ex(B) Hosmr-Lmshow Tst can b usd for tsting goodnss of fit. Hosmr and Lmshow Tst For a 25 yar old, controlling th sctor variabl, th odds ar x sctor + 7x25 for gtting th disas. St Chi-squar df Sig For a 2 yar old, controlling th sctor variabl, th odds ar x sctor + 7x2 for gtting th disas. -valu >, imlis a good fit. Th odds ratio would b x sctor + 7 x x sctor + 7 x 2 (*On can also tst for intractions btwn all th variabls.).964/ This mans that th odds of gtting disas for 25 yar-old is 5 tims highr than th 2 yar-old Examl: Low birth wight(y), arity or th first born child(x ), and smoking(x 2 ). Examining th association btwn low birth wight and arity and smoking variabls. Low Birth Wight Smoking Ys No Total Parity Ys Ys No Total Parity No Ys 39 5 No Total Modl without intraction: + x + 2 x2 α β β Variabl valus: Low Birth Wight: Low birth wight: y ; High birth wight: y. Parity: Ys: x ; No: x. Smoking: Ys: x 2 ; No: x 2. Parity Ys mans that th child was th first born of th mothr and Parity No mans it was not th first born of th mothr LR - 8

9 Logistic Rgrssion SPSS Outut St a PARITY SMOKE Variabls in th Equation B S.E. Wald df Sig. Ex(B) a. Variabl(s) ntrd on st : PARITY, SMOKE. Intrrtation: Smoking is significant in rdicting th robability of low birth wight baby with a -valu of. Parity is not significant with a -valu of.. Maximum liklihood stimats: α 373, βˆ 54, ˆ β ˆ Modl with intraction: St a PARITY SMOKE INTERACT + x α β + β x + β x x 2 Variabls in th Equation B S.E. Wald df Sig. Ex(B) a. Variabl(s) ntrd on st : PARITY, SMOKE, INTERACT. Intraction is insignificant (-valu.77). It inflatd th standard rror and can b liminatd from th modl. 5 LR - 9