PubH 7405: REGRESSION ANALYSIS INTRODUCTION TO POISSON REGRESSION

Transcription

1 PubH 7405: REGRESSION ANALYSIS INTRODUCTION TO POISSON REGRESSION

2 PROTOTYPE EXAMPLE #1 We have data for 44 physcans workng n emergency medcne at a major hosptal system. The concern s about the number of complants each receved the prevous year. Why s t dfferent from physcan to physcan? Could t be explaned usng other factors? Data avalable consst of the number of patent vsts - and four covarates (the revenue, n dollars per hour; work load at the emergency servce, n hours; gender, Female/Male, and resdency tranng n emergency medcne, No/Yes.

3 Queston: Can we do Regresson usng the Normal Error Regresson Model? Possble Issue: The response, Number of Complants, s not on contnuous scale; t s a count of course, not normally dstrbuted.

4 PROTOTYPE EXAMPLE #2 Skn Cancer data for dfferent age groups n two metropoltan areas: Mnneapols-St. Paul and Dallas-Fort Worth: (1) Any age effect? If so, s t the same for the two ctes? (2) Any weather effect? (dfference between two ctes) If so, s t the same for all age groups? Skn Cancer Data Cty: Mnneapols-St. Paul Dallas-Ft. Worth Age Group Cases Populaton Cases Populaton , , , , , , , , , , , , , , , ,538

5 Questons are regresson-type questons (about man effects or margnal contrbutons and about possble effect modfcatons). & the same possble ssue: The Response or Dependent Varable, the Number of Skn Cancer Cases a count, s not on the contnuous scale and not normally dstrbuted.

6 In Regresson Analyss/Model, we usually mpose the condton that the Response Varable Y s on the contnuous scale maybe because of the popular Normal Error Model - not because Y s always on the contnuous scale. In a prevous lecture, the Dependent Varable of nterest was represented by an Bnary or Indcator Varable Y takng on values 0 and 1. The dstrbuton s Bernoull whch s a specal case of the Bnomal Dstrbuton. And we ntroduced Logstc Regresson.

7 We are now ntroducng a new form of regresson, the Posson Regresson, where the Response or Dependent Varable Y represents count data non-negatve ntegers.

8 RARE EVENTS It can be shown that the lmtng form of the bnomal dstrbuton, when n s ncreasngly large (n ) and π s ncreasngly small (π 0) whle θ = nπ (the mean) remans constant, s: Pr(X x θ e = x) = x! = p(x; θ) A random varable havng ths probablty functon s sad to have Posson Dstrbuton P(θ). For example, wth n = 48 and π =.05: b(x = 5; n, π) =.059 p(x = 5; θ) =.060 θ

9 The Posson Model s often used when the random varable X s supposed to represent the number of occurrences of some random event n an unt nterval of tme or space, or some volume of matter; numerous applcatons n health scences have been documented. For example, the number of vrus n a soluton, the number of defectve teeth per ndvdual, the number of focal lesons n vrology, the number of vctms of specfc dseases, the number of cancer deaths per household, the number of nfant deaths n certan localty durng a gven year, among others.

10 The mean and varance of the Posson Dstrbuton are: μ = θ σ 2 = θ (A very specal and strong characterstc where the varance s equal to the mean).

11 REGRESSION NEEDS The Posson model s often used when the random varable X s supposed to represent the number of occurrences of some random event n an nterval of tme or space, or some volume of matter and numerous applcatons n health scences have been documented. In some of these applcatons, one may be nterested n to see f the Posson-dstrbuted dependent varable Y can be predcted from or explaned by other varables. The other varables are called predctors, or explanatory or ndependent varables. For example, we may be nterested n the number of defectve teeth Y per ndvdual as a functon of gender and age of a chld, branch of toothpaste, and whether the famly has or does not have dental nsurance.

12 & refer to Example 1 (Number of complans) and Example 2 (Number of skn cancer cases).

13 POISSON REGRESSION MODEL When the dependent varable Y s assumed to follow a Posson dstrbuton wth mean θ; the Posson regresson model expresses ths mean as a functon of certan ndependent varables X 1, X 2,..., X k, n addton to the sze of the observaton unt from whch one obtaned the count of nterest. For example, f Y s the number of vrus n a soluton then the sze s the volume of the soluton; or f Y s the number of defectve teeth then the sze s the total number of teeth for that same ndvdual.

14 In our frame work, the dependent varable Y s assumed to follow a Posson dstrbuton; ts values y 's are avalable from n observaton unt whch s also characterzed by an ndependent varable X. For the observaton unt ( n), let s be the sze and x be the covarate value. The Posson regresson model assumes that the relatonshp between the mean of Y and the covarate X s descrbed by: E(Y ) = sλ(x ) = s exp(β 0 + β 1 x ) where λ(x ) s called the rsk of/for observaton unt (1 n).

15 The basc ratonale for usng the term rsk s the approxmaton of the Bnomal dstrbuton by the Posson dstrbuton. Recall that, when n goes to nfnty, π tends 0 whle θ = nπ remans constant, the bnomal dstrbuton B(n,π) can be approxmated by the Posson dstrbuton P(θ). The number n s the sze of the observaton unt; so the rato between the mean and the sze represents π (or λ(x) n the new model); that s the probablty or rsk (and, the rato of rsks s called the rsks rato or relatve rsk ).

16 Model wth Several Covarates Suppose we want to consder k covarates, X 1, X 2,, X k, smultaneously. The smple Posson regresson model of prevous secton can be easly generalzed and expressed as: E(Y ) = sλ(x1, x2,...,x = s exp(β + β x = s exp(β k j= 1 1 β k j ) + β x j ) 2 x β where λ(x j s) s called the rsk of/for observaton unt (1 n), x j s the value of the covarate X j measured from subject. 1 x k )

17 EXAMPLE The purpose of ths study was to examne the data for 44 physcans workng for an emergency at a major hosptal so as to determne whch of the followng four varables are related to the number of complants receved durng the prevous year. In addton to the number of complants, served as the dependent varable, data avalable consst of the number of vsts - whch serves as the sze for the observaton unt, the physcan - and four covarates. Table 6.2 presents the complete data set. For each of the 44 physcan there are two contnuous ndependent varables, the revenue (dollars per hour) and work load at the emergency servce (hours) and two bnary varables, gender (Female/Male) and resdency tranng n emergency servces (No/Yes).

18 No. of Vsts Complant Gender Resdency Revenue Hours Y F N M Y M N M N M N M Y M N M N F Y M N F N M N M N F N M Y M Y M N M Y M N M N F Y F Y F N M Y F N F Y M Y M N M Y F Y M N M N M N M N M Y F Y M N M N M Y M Y M N M Y F Y M

19 The nterpretaton or meanng of the Regresson Coeffcents could be seen as follows whch s smlar to the case of the normal Error Regresson Model.

20 MEASURE OF ASSOCIATION Consder the case of a bnary covarate X, say, representng an exposure (1 = exposed, 0 = not exposed). We have the followng: lnλ λ (exposed) λ (unexposed) β0 = β 0 + β = exp(β + lns 1 1 ) + lns f exposed(or x = 1) f unexposed (or x = 0) Ths quantty, represented by exp(β1), s the relatve rsk assocated wth the exposure.

21 Smlarly, we have for a contnuous covarate X and consder any value x of X, lnλ λ (X = x + 1) λ (X = x) = β = 0 β + β exp(β ) + β1x + lns f X = x (x + 1) + lns f X = x + 1 Ths quantty, represented by exp(β1), s the relatve rsk assocated wth one unt ncrease n the value of X, from x to (x+1).

22 ESTIMATION OF PARAMETERS Under the assumpton that Y s dstrbuted as Posson wth the above mean, the Lkelhood Functon s gven by: L(y; β) lnl(y; = β) n = 1 = [s { y lns lny! + y [β0 + β1x ] sexp[β 0 + β1x ]} λ(x )] y exp[ sλ(x y! from whch estmates of the two regresson coeffcents β 0 and β 1 can be obtaned by the Maxmum Lkelhood procedure. )]

23 EXAMPLE #1 Refer to the Emergency Servce data and suppose we want to nvestgate the relatonshp between the number of complants Y (adjusted for number of vsts) and resdency tranng X. It may be perceved that by havng tranng n the specalty a physcan would perform better and, therefore, less lkely to provoke complants. An applcaton of the smple Posson regresson analyss yelds: Varable Coeffcent St Error z-statstc p-value Intercept < No Resdency The result ndcates that the common percepton s almost true, that the relatonshp between the number of complants and no resdency tranng n emergency servce s margnally sgnfcant (p = ); the relatve rsk assocated wth no resdency tranng s: exp(.3041) = 1.36 Those wthout prevous tranng s 36 percent more lkely to receve the same number of complants as those who were traned n the specalty.

24 SAS SAMPLE a SAS program would nclude these nstructons: DATA EMERGENCY; INPUT VISITS CASES RESIDENCY; LN = LOG(VISITS); CARDS; (Data); PROC GENMOD DATA EMERGENCY; CLASS RESIDENCY; MODEL CASES = RESIDENCY/ DIST = POISSON LINK = LOG OFFSET = LN; where EMERGENCY s the name assgned to the data set, VISITS s the number of vsts, CASES s the number of complants (Y), and RESIDENCY (X) s the bnary covarate ndcatng whether the physcan receved resdency tranng n the specalty. The opton CLASS s used to declare that the covarate s categorcal.

25 Model wth Several Covarates Suppose we want to consder k covarates, X 1, X 2,, X k, smultaneously. The model s: E(Y ) = sλ(x1, x2,...,x = s exp(β + β x = s exp(β k j= 1 1 k β x j ) + β j ) 2 x β where λ(x j s) s called the rsk of/for observaton unt (1 n), x j s the value of the covarate X j measured from subject. L(y; β) lnl(y; = β) n = 1 = [s λ(x j y lns )] y exp[ s λ(x y lny!! + y [β 1 0 x j k + ) )] k j= 1 β j x j ] s exp[β 0 + k j= 1 β j x j ]

26 Also smlar to the smple regresson case, exp(β ) represents: () The Relatve Rsk assocated wth, say, an exposure f X s bnary (exposed or X =1 versus unexposed or X =0), or: () The Relatve Rsk due to one unt ncrease n the value of X f X s contnuous (X =x+1 versus X =x).

27 After estmates b s of regresson coeffcents β s and ther standard errors have been obtaned, a 95 percent confdence nterval for the log of the Relatve Rsk assocated wth the th factor s gven by: b ± SE(b ). These results are necessary n the effort to dentfy mportant rsk factors for the Posson outcome, the count.

28 Note: We form confdence ntervals of the regresson coeffcent before exponentatng (the endponts) to obtan relatve rsks.

29 Before such analyses are done, the problem and the data have to be examned carefully. If some of the varables are hghly correlated, then one or fewer of the correlated factors are lkely to be as good predctors as all of them; nformaton from other smlar studes also has to be ncorporated so as to drop some of these correlated explanatory varables. The uses of products, such as x 1 *x 2, and hgher power terms, such as x 12, may be necessary and can mprove the goodness-of-ft.

30 We are assumng a log-lnear regresson model n whch, for example, the Relatve Rsk due to one unt ncrease n the value of a contnuous X (X = x+1 versus X = x) s ndependent of x. Therefore, f ths lnearty seems to be volated, the ncorporaton of powers of X should be serously consdered. The use of products wll help n the nvestgaton of possble effect modfcatons.

31 There are no smple dagnostc tool, such as Scatter Dagram for Normal Error Regresson Model, for detectng lack of lnearty. One mght consder to nclude a quadratc term and see f t s sgnfcant.

32 And, fnally, the messy problem of mssng data; most packaged programs would delete a subject f one or more covarate values are mssng.

33 TESTING HYPOTHESES Once we have ft a multple Posson regresson model and obtaned estmates for the varous parameters of nterest, we want to answer questons about the contrbutons of varous factors to the predcton of the Posson-dstrbuted response varable. There are three types of such questons: () An overall test: taken collectvely, does the entre set of explanatory or ndependent varables contrbute sgnfcantly to the predcton of response? () Test for the value of a sngle factor: does the addton of one partcular varable of nterest add sgnfcantly to the predcton of response over and above that acheved by other ndependent varables? () Test for contrbuton of a group of varables: does the addton of a group of varables add sgnfcantly to the predcton of response over and above that acheved by other ndependent varables?

34 OVERALLL REGRESSION EFFECTS We frst consder the frst queston stated above concernng an overall test for a model contanng k factors. The null hypothess for ths test may be stated as: "all k ndependent varables consdered together do not explan the varaton n the response any more than the sze alone". In other words, H : β = β = = β = k 0

35 H : β = β =... = β = o 1 2 k 0 Snce: E(Y ) = sλ(x1, x2,..., x = s exp(β + β x = s exp(β k j= 1 If H 0 s true, the average/mean response has nothng to do wth the predctors. 1 β k j ) + β x j ) 2 x β k x k )

36 H0 : β1 = β2 = = βk = 0 Ths can be tested usng the Lkelhood Rato Ch-square test at k degrees of freedom: X 2 = 2{lnL k lnl 0 } where lnl k s the log lkelhood value for the model contanng all k covarates and lnl0 s the log lkelhood value for the model contanng only the ntercept. Computer packaged program, such as SAS PROC GENMOD, provdes these log lkelhood values but n separate runs.

37 If ths overall test s sgnfcant, t only means one or some of the regresson coeffcents (the slopes ) not equal to zero; t does not tell us whch coeffcents are not zero or whch factor or factors are related to the response. Ths s smlar to performng the oneway ANOVA F-test before checkng for parwse dfferences.

38 TESTING FOR SINGLE FACTOR The queston s: Does the addton of one partcular factor of nterest add sgnfcantly to the predcton of Dependent Varable over and above that acheved by other factors?. The Null Hypothess for ths test may stated as: "Factor X does not have any value added to the explan the varaton n Y-values over and above that acheved by other factors ". In other words, H : β = 0 0

39 The Null Hypothess s H Regardless of the number of varables n the model, one smple approach s usng b z = SE(b ) where b s the correspondng estmated regresson coeffcent for factor X and SE(b ) s the estmate of the standard error of b, both of whch are prnted by standard computer packaged programs such as SAS. In performng ths test, we refer the value of the z score to percentles of the standard normal dstrbuton; for example, we compare the absolute value of z to 1.96 for a two-sded test at the 5 percent level. Note: No use of t-dstrbuton here. : β 0 = 0

40 EXAMPLE #2 Refer to the data set on emergency servce; Usng all four covarates, we found that only the effect of work load (Hours) s sgnfcant at the 5 percent level. Varable Coeffcent St Error z-statstc p-value Intercept < No Resdency Female Revenue Hours

41 Gven a contnuous varable of nterest, one can ft a polynomal model and use ths type of test to check for lnearty. It can also be used to check for a sngle product representng an effect modfcaton. For example, to focus on the quadratc term of: E(Y ) = s λ(x ) = s exp(β 0 + β 1 Hour + β 2 Hour 2 )

42 TESTING FOR A GROUP OF VARABLES The queston s: Does the addton of a group of factors add sgnfcantly to the predcton of Y over and above that acheved by other factors? The Null Hypothess for ths test may stated as: "Factors {X +1, X +2,, X +m }, consdered together as a group, do not have any value added to the predcton of the Mean of Y that other factors are already ncluded n the model". In other words, H0 : β+ 1 = β+ 2 =... = β+ m = 0

43 Ths multple contrbuton test s often used to test whether a smlar group of varables, such as demographc characterstcs, s mportant for the predcton of the mean of Y; these varables have some trat n common. Another applcaton: collecton of powers and/or product terms. It s of nterest to assess powers & nteracton effects collectvely before consderng ndvdual nteracton terms n a model. It reduces the total number of tests & helps to provde better control of overall Type I error rates whch may be nflated due to multple testng.

44 The process can also be used to test for the contrbuton of a categorcal covarate whch s represented by several dummy varables.

45 EXAMPLE #3 In ths example, the dependent varable s the number of cases of skn cancer. Data nvolve only two covarates; age and locaton, both are categorcal. We use seven dummy varables to represent the eght age groups (wth 85+ age group beng the baselne) and one for locaton (wth Mnneapols-St. Paul as the baselne) Skn Cancer Data Cty: Mnneapols-St. Paul Dallas-Ft. Worth Age Group Cases Populaton Cases Populaton , , , , , , , , , , , , , , , ,538

46 SEQUENTIAL ADJUSTMENT In the type 3 analyss, we test the sgnfcance of the effect of each factor added to the model contanng all other factors lke n most common multple regresson analyses; that s to nvestgate the addtonal contrbuton of the factor to the explanaton of the dependent varable. Sometmes, however, we may be nterested n a herarchcal or sequental adjustment. For example, we focus on the quadratc term (adjusted) n addton to the regular term (unadjusted). Ths can be acheved usng PROC GENMOD by requestng the type 1 analyss opton

47 OVERDISPERSION The Posson s a very specal dstrbuton; ts mean µ and ts varance σ 2 are equal. If we use the varance-mean rato as a dsperson parameter then t s 1 n a standard Posson model, less than 1 n an under-dspersed model, and greater than 1 n an over-dspersed model. Over-dsperson s a common phenomenon n practce and t causes concerns because the mplcaton s serous; the analyss whch assumes the Posson model often under-estmates standard errors and, thus, wrongly nflates the level of sgnfcance.

48 MEASURING OVERDISPERSION After a Posson regresson model s ftted, dsperson s measured by the scaled devance or scaled Pearson ch-square; t s the devance or Pearson ch-square dvded by the degrees of freedom (number of observatons mnus number of parameters). The devance s defned as twce the dfference between the maxmum achevable log lkelhood and the log lkelhood at the maxmum lkelhood estmates of the regresson parameters.

49 EXAMPLE Refer to the data set on emergency servce wth all four covarates, we can see that both ndces are greater than 1 ndcatng an overdsperson. In ths example, we have a sample sze of 44 but fve degrees of freedom lost due to the estmaton of the fve regresson parameters, ncludng the ntercept. Crteron df Value Scaled Value Devance Pearson Ch-Square

50 FITTING OVERDISPERSED MODEL PROC GENMOD allows the specfcaton of a scale parameter to ft over-dspersed Posson regresson models. Instead of a varance equal to the mean, Var(Y) = µ t allows the varance functon to have a multplcatve over-dsperson factor ϕ (specfed by users): Var(Y) = ϕµ

51 EXAMPLE #4 Refer to the data set on emergency servce; Usng all four covarates, we have the followng results by fttng the regular Posson Model. Varable Coeffcent St Error z-statstc p-value Intercept < No Resdency Female Revenue Hours Crteron df Value Scaled Value Devance Pearson Ch-Square

52 The model could be ftted n the usual way, and the pont estmates of regresson coeffcent are not affected. The covarance matrx, however, s multpled by ϕ. There are two optons avalable for fttng over-dspersed models; the users can control ether the scaled devance (by specfyng DSCALE n the model statement) or the scaled Pearson ch-square (by specfyng PSCALE n the model statement). The value of the controlled ndex becomes 1; the value of the other s close to but may not be equal to 1.

53 Note: a SAS program would nclude ths nstructon: MODEL CASES = GENDER RESIDENCY REVENUE HOURS/ DIST = POISSON LINK = LOG OFFSET = LN(DSCALE);

54 NEW RESULTS Varable Coeffcent St Error z-statstc p-value Intercept < No Resdency Female Revenue Hours Crteron df Value Scaled Value Devance Pearson Ch-Square As compared to the results of the regular model, the pont estmates reman the same but the standard errors are larger; the effect of work load (Hours) s no longer sgnfcant at the 5 percent level.

55 STEPWISE REGRESSION In many applcatons, our major nterest s to dentfy mportant rsk factors. In other words, we wsh to dentfy from many avalable factors a small subset of factors that relate sgnfcantly to the outcome, e.g. the dsease under nvestgaton. In that dentfcaton process, of course, we wsh to avod a large Type I ( false postve ) error. In a regresson analyss, a Type I error corresponds to ncludng a predctor that has no real relatonshp to the outcome; such an ncluson can greatly confuse the nterpretaton of the regresson results. One popular procedure s Stepwse Regresson

56 Wth Posson Regresson, t s stll desrable and possble to perform stepwse regresson. Unfortunately, PROC GENMOD does not have an automatc opton; one has to run t many tmes and, at each step, choose to add or remove a varable manually.

57 Wth Posson Regresson, we lost these tools that we use wth NERM: R 2 and all coeffcents of partal determnaton (there are some substtutes but not as good) All graphs (Scatter dagram, all resdual plots, and Varable-added Plot) Least Squares method (but MLE s better) All other methods/tools are unchanged (test for sngle factors, stepwse, etc )

58 However, f Y s dstrbuted as Posson, try Y = Y; t both stablzes the varance and mproves normalty. Then you could get approxmate results usng PROC REG and the Normal Error Regresson Model (If Y s bnary, you would have no choce but Logstc Regresson).

59 Readngs & Exercses Readngs: A thorough readng of the text s secton 14.3 (pp ) s hghly recommended. Exercses: None

60 Due As Homework 22.1 Refer to dataset Emergency Servce, let Y = Number of Complants and four ndependent varables, X 1 = Gender, X 2 = Resdency, X 3 = Revenue, and X 4 = Hours; choose codng 0/1 for X1 and X2. a) Ft the Posson Regresson Model usng PROC GENMOD and confrm the results of Example #2. b) Usng the same data set, take the square root of Y and use as the new Dependent Varable and ft the Normal Error Regresson Model usng PROC REG; draw your conclusons on effects of covarates. c) Comment on the smlartes or dfferences between the two sets of results obtaned n (a) and (b).