Lecture 2: Poisson and logistic regression

Transcription

1 Dankmar Böhning Southampton Statistical Sciences Research Institute University of Southampton, UK S 3 RI, December 2014

2 introduction to Poisson regression application to the BELCAP study introduction to logistic regression confounding and effect modification comparing of different generalized regression models meta-analysis of BCG vaccine against tuberculosis

3 introduction to Poisson regression the Poisson distribution count data may follow such a distribution, at least approximately Examples: number of deaths, of diseased cases, of hospital admissions and so on... Y Po(µ): where µ > 0 P(Y = y) = µ y exp( µ)/y!

4 introduction to Poisson regression P(Y=y) y

5 introduction to Poisson regression but why not use a linear regression model? for a Poisson distribution we have E(Y ) = Var(Y ). This violates the constancy of variance assumption (for the conventional regression model) a conventional regression model assumes we are dealing with a normal distribution for the response Y, but the Poisson distribution may not look very normal the conventional regression model may give negative predicted means (negative counts are impossible!)

6 introduction to Poisson regression the Poisson regression model log E(Y i ) = log µ i = α + βx i the RHS of the above is called the linear predictor Y i Po(µ i ) this model is the log-linear model

7 introduction to Poisson regression the Poisson regression model can be written equivalently as log E(Y i ) = log µ i = α + βx i µ i = exp[α + βx i ] Hence it is clear that any fitted log-linear model will always give non-negative fitted values!

8 introduction to Poisson regression an interesting interpretation in the Poisson regression model suppose x represents a binary variable (yes/no, treatment present/not present) { 1 if person is in intervention group x = 0 otherwise log E(Y ) = log µ = α + βx x = 0: log µ no intervention = α + βx = α x = 1: log µ intervention = α + βx = α + β hence log µ intervention log µ no intervention = β

9 introduction to Poisson regression an interesting interpretation in the Poisson regression model hence or log µ intervention log µ no intervention = β µ intervention µ no intervention = exp(β) the coefficient exp(β) corresponds to the risk ratio comparing the mean risk in the treatment group to the mean risk in the control group

10 introduction to Poisson regression Poisson regression model for several covariates log E(Y i ) = α + β 1 x 1i + + β p x pi where x 1i,, x pi are the covariates of interest testing the effect of covariate x j is done by the size of the estimate ˆβ j of β j t j = ˆβ j s.e.( ˆβ j ) if t j > 1.96 covariate effect is significant

11 introduction to Poisson regression estimation of model parameters consider the likelihood (the probability for the observed data) L = n i=1 for model with p covariates: µ y i i exp( µ i )/y i! log µ i = α + β 1 x i1 + β 2 x i β p x ip finding parameter estimates by maximizing the likelihood L (or equivalently the log-likelihood log L) guiding principle: choosing the parameters that make the observed data the most likely

12 application to the BELCAP study The simple regression model for BELCAP with Y = DMFSe: log E(DMFSe i ) = α + β 1 OHE i + β 2 ALL 2i + β 4 ESD i + β 5 MW i + β 6 OHY i + β 7 DMFSb i { 1 if child i is in intervention OHE OHE i = 0 otherwise { 1 if child i is in intervention ALL ALL i = 0 otherwise

13 application to the BELCAP study analysis of BELCAP study using the Poisson regression model including the DMFS at baseline covariate ˆβj s.e.( ˆβ j ) t j P-value OHE ALL ESD MW OHY DMFSb

14 introduction to logistic regression Introduction to logistic regression Binary Outcome Y Y = { 1, Person diseased 0, Person healthy Probability that Outcome Y = 1 Pr(Y = 1) = p is probability for Y = 1

15 introduction to logistic regression Odds odds = p 1 p p = odds odds + 1 Examples p = 1/2 odds = 1 p = 1/4 odds = 1/3 p = 3/4 odds = 3/1 = 3

16 introduction to logistic regression Odds Ratio OR = odds( in exposure ) odds( in non-exposure ) = p 1/(1 p 1 ) p 0 /(1 p 0 ) Properties of odds ratio 0 < OR < OR = 1(p 1 = p 0 ) is reference value

17 introduction to logistic regression Examples risk = risk = { p 1 = 1/4 p 0 = 1/8 { p 1 = 1/100 p 0 = 1/1000 effect measure = eff. meas. = Fundamental Theorem of Epidemiology p 0 small OR RR { OR = p 1 /(1 p 1 ) p 0 /(1 p 0 ) = 1/3 1/7 = 2.33 RR = p 1 p 0 = 2 { OR = 1/99 1/999 = RR = p 1 p 0 = 10 benefit: OR is interpretable as RR which is easier to deal with

18 introduction to logistic regression A simple example: Radiation Exposure and Tumor Development cases non-cases E NE odds and OR odds for disease given exposure (in detail): 52/ /2872 = 52/2820 odds for disease given non-exposure (in detail): 6/ /5049 = 6/5043

19 introduction to logistic regression A simple example: Radiation Exposure and Tumor Development cases non-cases E NE OR odds ratio for disease (in detail): OR = 52/2820 6/5043 = = or, log OR = log = 2.74 for comparison RR = 52/2872 6/5049 = 15.24

20 introduction to logistic regression Logistic regression model for this simple situation where log p x 1 p x = α + βx p x = Pr(Y = 1 x) { 1, if exposure present x = 0, if exposure not present log px 1 p x is called the logit link that connects p x with the linear predictor

21 introduction to logistic regression benefits of the logistic regression model is feasible since whereas log p x = p x 1 p x = α + βx exp(α + βx) (0, 1) 1 + exp(α + βx) p x = α + βx is not feasible

22 introduction to logistic regression Interpretation of parameters α and β log p x 1 p x = α + βx now p 0 x = 0 : log 1 p 0 = α (1) p 1 x = 1 : log 1 p 1 = α + β (2) p 1 (2) (1) = log log = α + β α = β 1 p 1 1 p }{{ 0 } log p 1 1 p 1 p 0 1 p 0 =log OR p 0 log OR = β OR = e β

23 confounding and effect modification A simple illustration example cases non-cases E NE OR odds ratio: OR = =

24 confounding and effect modification stratified: Stratum 1: Stratum 2: cases non-cases E NE OR = = 1 cases non-cases E NE OR = = 1

25 confounding and effect modification Y E S freq

26 confounding and effect modification The logistic regression model for simple confounding where log p x 1 p x = α + βe + γs x = (E, S) is the covariate combination of exposure E and stratum S

27 confounding and effect modification in detail for stratum 1 now log p x 1 p x = α + βe + γs p 0,0 E = 0, S = 0 : log 1 p 0,0 = α (3) p 1,0 E = 1, S = 0 : log 1 p 1,0 = α + β (4) (4) (3) = log OR 1 = α + β α = β log OR = β OR = e β the log-odds ratio in the first stratum is β

28 confounding and effect modification in detail for stratum 2: now: log p x 1 p x = α + βe + γs p 0,1 E = 0, S = 1 : log 1 p 0,1 = α + γ (5) p 1,1 E = 1, S = 1 : log 1 p 1,1 = α + β + γ (6) (6) (5) = log OR 2 = α + β + γ α γ = β the log-odds ratio in the second stratum is β

29 confounding and effect modification important property of the confounding model: assumes the identical exposure effect in each stratum!

30 confounding and effect modification (crude analysis) Logistic regression Log likelihood = Y Odds Ratio Std. Err. [95% Conf. Interval] E (adjusted for confounder) Logistic regression Log likelihood = Y Odds Ratio Std. Err. [95% Conf. Interval] E S

31 confounding and effect modification A simple illustration example: passive smoking and lung cancer cases non-cases E NE OR odds ratio: OR = = 1.19

32 confounding and effect modification stratified: Stratum 1 (females): Stratum 2 (males): cases non-cases E NE OR = = 1.10 cases non-cases E NE OR = = 1.63

33 confounding and effect modification interpretation: effect changes from one stratum to the next stratum!

34 confounding and effect modification The logistic regression model for effect modification where log p x = α + βe + γs + (βγ) E S 1 p x }{{} effect modif. par. x = (E, S) is the covariate combination of exposure E and stratum S

35 confounding and effect modification in detail for stratum 1 now log p x 1 p x = α + βe + γs + (βγ)e S p 0,0 E = 0, S = 0 : log 1 p 0,0 = α (7) p 1,0 E = 1, S = 0 : log 1 p 1,0 = α + β (8) (8) (7) = log OR 1 = α + β α = β log OR = β OR = e β the log-odds ratio in the first stratum is β

36 confounding and effect modification in detail for stratum 2: log p x 1 p x = α + βe + γs + (βγ)e S now: p 0,1 E = 0, S = 1 : log 1 p 0,1 = α + γ (9) p 1,1 E = 1, S = 1 : log 1 p 1,1 = α + β + γ + (βγ) (10) (10) (9) = log OR 2 = α + β + γ + (βγ) α γ = β + (βγ) log OR = β OR = e β+(βγ) the log-odds ratio in the second stratum is β + (βγ)

37 confounding and effect modification important property of the effect modification model: effect modification model allows for different effects in the strata!

38 confounding and effect modification Data from passive smoking and LC example are as follows: Y E S ES freq

39 confounding and effect modification CRUDE EFFECT MODEL Logistic regression Log likelihood = Y Coef. Std. Err. z P> z E _cons

40 confounding and effect modification CONFOUNDING MODEL Logistic regression Log likelihood = Y Coef. Std. Err. z P> z E S _cons

41 confounding and effect modification EFFECT MODIFICATION MODEL Logistic regression Log likelihood = Y Coef. Std. Err. z P> z E S ES _cons

42 confounding and effect modification interpretation of crude effects model: log OR = OR = e = 1.19 interpretation of confounding model: log OR = OR = e = 1.24

43 confounding and effect modification interpretation of effect modification model: stratum 1: stratum 2: log OR 1 = OR 1 = e = 1.10 log OR 2 = OR 2 = e = 1.63

44 comparing of different generalized regression models Model evaluation in logistic regression: the likelihood approach: L = n i=1 is called the likelihood for models log p xi 1 p xi = p y i x i (1 p xi ) 1 y i { α + βe i + γs i + (βγ)e i S i, (M 1 ) α + βe i + γs i, (M 0 ) where M 1 is the effect modification model and M 0 is the confounding model

45 comparing of different generalized regression models Model evaluation in logistic regression using the likelihood ratio: let L(M 1 ) and L(M 0 ) be the likelihood for models M 1 and M 0 then LRT = 2 log L(M 1 ) 2 log L(M 0 ) = 2 log L(M 1) L(M 0 ) is called the likelihood ratio for models M 1 and M 0 and has a chi-square distribution with 1 df under M 0

46 comparing of different generalized regression models illustration for passive smoking and LC example: model log-likelihood LRT crude homogeneity effect modification note: for valid comparison on chi-square scale: models must be nested

47 comparing of different generalized regression models Model evaluation in more general: consider the likelihood L = n i=1 p y i x i (1 p xi ) 1 y i for a general model with p covariates: example: log p xi 1 p xi = α + β 1 x i1 + β 2 x i β p x ip (M 0 ) log p xi 1 p xi = α + β 1 AGE i + β 2 SEX i + β 3 ETS i

48 comparing of different generalized regression models Model evaluation in more general: example: log p xi 1 p xi = α + β 1 AGE i + β 2 SEX i + β 3 ETS i where these covariates can be mixed: quantitative, continuous such as AGE categorical binary (use 1/0 coding) such as SEX non-binary ordered or unordered categorical such as ETS (none, moderate, large)

49 comparing of different generalized regression models Model evaluation in more general: consider the likelihood L = n i=1 p y i x i (1 p xi ) 1 y i for model with additional k covariates: log p xi 1 p xi = α + β 1 x i1 + β 2 x i β p x ip +β p+1 x i,p β k+p x i,k+p (M 1 )

50 comparing of different generalized regression models Model evaluation in more general for our example: log p xi 1 p xi = α + β 1 AGE i + β 2 SEX i + β 3 ETS i +β 4 RADON i + β 5 AGE-HOUSE i

51 comparing of different generalized regression models Model evaluation using the likelihood ratio: again let L(M 1 ) and L(M 0 ) be the likelihood for models M 1 and M 0 then the likelihood ratio LRT = 2 log L(M 1 ) 2 log L(M 0 ) = 2 log L(M 1) L(M 0 ) has a chi-square distribution with k df under M 0

52 comparing of different generalized regression models Model evaluation for our example: then { M 0 : α + β 1 AGE i + β 2 SEX i + β 3 ETS i M 1 :...M β 4 RADON i + β 5 AGE-HOUSE i LRT = 2 log L(M 1) L(M 0 ) has under model M 0 a chi-square distribution with 2 df

53 comparing of different generalized regression models model evaluation for model assessment we will use criteria that compromise between model fit and model complexity Akaike information criterion Bayesian Information criterion AIC = 2 log L + 2k BIC = 2 log L + k log n where k is the number of parameters in the model and n is the number of clustered observations we seek a model for which AIC and/or BIC are small

54 meta-analysis of BCG vaccine against tuberculosis Meta-Analysis Meta-Analysis is a methodology for investigating the study results from several, independent studies with the purpose of an integrative analysis Meta-Analysis on BCG vaccine against tuberculosis Colditz et al. 1974, JAMA provide a meta-analysis to examine the efficacy of BCG vaccine against tuberculosis

55 meta-analysis of BCG vaccine against tuberculosis Data on the meta-analysis of BCG and TB the data contain the following details 13 studies each study contains: TB cases for BCG intervention number at risk for BCG intervention TB cases for control number at risk for control also two covariates are given: year of study and latitude expressed in degrees from equator latitude represents the variation in rainfall, humidity and environmental mycobacteria suspected of producing immunity against TB

56 meta-analysis of BCG vaccine against tuberculosis intervention control study year latitude TB cases total TB cases total

57 meta-analysis of BCG vaccine against tuberculosis Data analysis on the meta-analysis of BCG and TB these kind of data can be analyzed by taking TB case as disease occurrence response intervention as exposure study as confounder

58 meta-analysis of BCG vaccine against tuberculosis Log likelihood = Pseudo R2 = TB_Case Odds Ratio Std. Err. z P> z [95% Conf. Interval] Intervention _cons estat ic, n(13) Model Obs ll(null) ll(model) df AIC BIC Note: N=13 used in calculating BIC

59 meta-analysis of BCG vaccine against tuberculosis Logistic regression Number of obs = LR chi2(2) = Prob > chi2 = Log likelihood = Pseudo R2 = TB_Case Odds Ratio Std. Err. z P> z [95% Conf. Interval] Latitude Intervention _cons estat ic, n(13) Model Obs ll(null) ll(model) df AIC BIC Note: N=13 used in calculating BIC

60 meta-analysis of BCG vaccine against tuberculosis Logistic regression Number of obs = LR chi2(3) = Prob > chi2 = Log likelihood = Pseudo R2 = TB_Case Odds Ratio Std. Err. z P> z [95% Conf. Interval] Latitude Intervention Year _cons estat ic, n(13) Model Obs ll(null) ll(model) df AIC BIC Note: N=13 used in calculating BIC.

61 meta-analysis of BCG vaccine against tuberculosis model evaluation model log L AIC BIC intervention latitude year