Binomial Link Functions. Lori Murray, Phil Munz

Transcription

1 Bnomal Lnk Functons Lor Murray, Phl Munz

2 Bnomal Lnk Functons Logt Lnk functon: ( p) p ln 1 p Probt Lnk functon: ( p) 1 ( p) Complentary Log Log functon: ( p) ln( ln(1 p))

3 Motvatng Example A researcher s examnng beetle mortalty after 5 hours of exposure to carbon dsulphde, at varous levels of concentraton of the gas. Beetles were exposed to gaseous carbon dsulphde at varous concentratons (n mg/l) for fve hours (Blss, 1935) and the number of beetles klled were noted. The data are n the followng table:

4 Example (contnued) > beetle<-read.table("beetledata.txt",header=true) > head(beetle) Dose Num.Beetles Num.Klled > logtmodel<-glm(cbnd(num.klled,num.beetles-num.klled) ~ Dose, data = beetle, famly = bnomal) > summary(logtmodel) > probtmodel<-glm(cbnd(num.klled,num.beetles-num.klled) ~ Dose, data = beetle, famly = bnomal(lnk=probt)) > summary(probtmodel) > logmodel<-glm(cbnd(num.klled,num.beetles-num.klled) ~ Dose, data = beetle, famly = bnomal(lnk=cloglog)) > summary(logmodel)

5 Don t forget to plot the data!

6 LOGIT MODEL: Call: glm(formula = cbnd(num.klled, Num.Beetles - Num.Klled) ~ Dose, famly = bnomal, data = beetle) Devance Resduals: Mn 1Q Medan 3Q Max Coeffcents: Estmate Std. Error z value Pr(> z ) (Intercept) <2e-16 *** Dose <2e-16 *** --- Sgnf. codes: 0 *** ** 0.01 * (Dsperson parameter for bnomal famly taken to be 1) Null devance: on 7 degrees of freedom Resdual devance: on 6 degrees of freedom AIC: Number of Fsher Scorng teratons: 4

7 PROBIT MODEL: Call: glm(formula = cbnd(num.klled, Num.Beetles - Num.Klled) ~ Dose, famly = bnomal(lnk = probt), data = beetle) Devance Resduals: Mn 1Q Medan 3Q Max Coeffcents: Estmate Std. Error z value Pr(> z ) (Intercept) <2e-16 *** Dose <2e-16 *** --- Sgnf. codes: 0 *** ** 0.01 * (Dsperson parameter for bnomal famly taken to be 1) Null devance: on 7 degrees of freedom Resdual devance: on 6 degrees of freedom AIC: Number of Fsher Scorng teratons: 4

8 COMPLEMENTARY LOG-LOG MODEL: Call: glm(formula = cbnd(num.klled, Num.Beetles - Num.Klled) ~ Dose, famly = bnomal(lnk = cloglog), data = beetle) Devance Resduals: Mn 1Q Medan 3Q Max Coeffcents: Estmate Std. Error z value Pr(> z ) (Intercept) <2e-16 *** Dose <2e-16 *** --- Sgnf. codes: 0 *** ** 0.01 * (Dsperson parameter for bnomal famly taken to be 1) Null devance: on 7 degrees of freedom Resdual devance: on 6 degrees of freedom AIC: Number of Fsher Scorng teratons: 4

9 Example (contnued)

10 Bnomal Lnk Functons Dfferences n choce of lnk affect model and devance. Why have 3 lnk functons and what about them cause these dfferences. All models are wrong, but some are useful George Box

11 Dfferences n Lnk Functons

12 Dfferences n Lnk Functons Numercally, consder the specfc value of each functon correspondng to varous levels of p: p Logt Probt C Log Log

13 Devances Logt: Probt: C Log Log:, ˆ ;ˆ ˆ )ln ( ˆ ln 2 1 n p n y y n y n y n y y y D ˆ]} exp[ exp{ 1 ˆ ˆ) ( ˆ 1 ˆ ˆ ˆ T T x x x p x p e e p T T

14 Dfferences n Lnk Functons problowerlogt <- vector(length=1000) problowercloglog <-vector(length=1000) logtdevance <-vector(length=1000) probtdevance <-vector(length=1000) cloglogdevance <- vector(length=1000) problowerlogtclog <- vector(length=1000) for( n 1:1000){ } x <- rnorm(1000) y <- rbnom(n=1000, sze=1, prob=pnorm(x)) logtmodel <- glm(y~x, famly=bnomal(lnk="logt")) probtmodel <- glm(y~x, famly=bnomal(lnk="probt")) cloglogmodel <- glm(y~x, famly=bnomal(lnk="cloglog")) logtdevance[] <- devance(logtmodel) probtdevance[] <- devance(probtmodel) cloglogdevance[] <- devance(cloglogmodel) problowerlogt[] <- probtdevance[] < logtdevance[] problowercloglog[] <- probtdevance[] < cloglogdevance[] problowerlogtclog[] <- logtdevance[] < cloglogdevance[]

15 Dfferences n Lnk Functons >sum(problowerlogt)/1000 [1] > sum(problowercloglog)/1000 [1] >sum(problowerlogtclog)/1000 [1] Dfferences (last teraton): > devance(logtmodel) - devance(probtmodel) [1] > devance(cloglogmodel) - devance(probtmodel) [1] Consder the last teraton of the scrpt: Dev Probt Dev Logt Dev. cloglog

16 Orgns of the Bnomal Lnk Functons 1. Complementary log log lnk (1922) 2. Probt lnk (1933) 3. Logt lnk (1944)

17 Complementary log-log lnk (1922) R. A. Fsher, Englsh Statstcan Dluton assay 12.3 Descrbes an experment where a seres of dlutons were made of a sol or water sample to determne the presence or absence of some mcrobal contamnant. Used a cll transformaton and appled maxmum lkelhood estmaton.

18 Complementary log-log lnk (1922) Assume that dlutons are made n powers of 2, then after x dlutons the number of nfectve organsms, p x, per unt volume s p x = p 0 /2 x x = 0,1, where p 0 s the densty of nfectve organsms n the orgnal soluton (we wsh to estmate). The expected number of organsms on any plate s p x v, and the actual number of organsms follows a Posson dstrbuton wth ths parameter.

19 Complementary log-log lnk (1922) The probablty that a plate s nfected s π x = 1 exp { p x v} At dluton x we have, log log 1 π x = log v + log p x = log v + log p o x log 2 If at dluton x we have r nfected plates out of m, the observed proporton of nfected plates s y = r/m, and E(Y x) = π x A complementary log-log transformaton s log log 1 π x = α + βx

20 Probt lnk (1933/1934) John Gaddum was an Englsh pharmacologst who wrote a comprehensve report on the statstcal nterpretaton of boassay. Blss was largely self taught, worked wth Fsher, and eventually settled at Yale. Publshed 2 bref notes n Scence where he ntroduced the word probt (probablty unt).

21 Probt lnk (1933/1934) Blss uses an example of the effectveness of a pestcde to combat an nsect pest. Descrbes how a dosage-mortalty curve has an asymmetrcal S-shaped curve.

22 Probt lnk (1933/1934) Observaton that n many physologcal processes equal ncrements n response are produced when dose s ncreased by a constant proporton of the gven dosage, rather than by a constant amount. Blss proposed the same rule mght hold for toxcologcal processes, n whch case dosage would have to be plotted n logarthmc terms to show a unform ncrease n mortalty. Proposed to transform the percentage klled to a probt and then plot aganst the logarthm of the dose to acheve a straght lne.

23 Probt lnk (1933/1934) Transformaton by use of logarthms and probts.

24 Logt lnk (1944) Joseph Berkson was a medcal doctor and chef statstcan of the Mayo Clnc. Research was on statstcal methodology of boassay. Proposed the use of the logstc nstead of the normal probablty functon, conng the term logt by analogy to the probt of Blss.

25 Logt lnk (1944) Berkson gves several reasons for usng the logt The logstc functon s very close to the ntegrated normal curve. Snce t apples to a wde range of physochemcal phenomena, t may have a better theoretcal bass than the ntegrated normal curve. It s easer to handle statstcally. Intally the logt was regarded as nferor and dsreputable, snce t cannot be related to an underlyng normal dstrbuton of tolerance levels.

26 Logt lnk (1944) By the 1960s, Berkson s logt had ganed acceptance. The power of the logstc s analytcal propertes were startng to surface. By the 1970s, the logt takes the lead because t was now wdely used among many dscplnes.

27 Logt s Consdered the Default Lnk Advantages of Logt lnk functon: Leads to smpler mathematcs due to complexty of the standard normal CDF It s easer to nterpret (Log odds)

28 Fnal Remarks If the logt lnk s consdered the default lnk, why do we stll use probt and Complementary log log? Theoretcal Consderatons Influences by dscplnary tradton Economsts favour probt models Toxcologsts favour logt models Underlyng characterstcs of the data Complementary log log works best wth extremely skewed dstrbutons

29 References Berskon, Joseph. (1944). Applcaton of the Logstc Functon to Bo-Assay. Journal of the Amercan Statstcal Assocaton 39: Blss, C. I. (1934). The Method of Probts. Scence 79: Cramer, J.S. (2003). The orgns and development of the logt model. Workng Paper. Unversty of Amsterdam and Tnbergen Insttute, Amsterdam. Dobson, Annette J. (2002). Introducton to Generalzed Lnear Models. Chapman & Hall/CRC: Boca Raton.

30 References Fsher, R. A. (1944). On the Mathematcal Foundatons of Theoretcal Statstcs. Phlosophcal Transactons of the Royal Socety of London 222: Ftzmaurce, Lard and Ware. (2004). Appled Longtudnal Analyss. John Wley & Sons: New Jersey. McCullagh, P. and J. A. Nelder. (1983). Generalzed Lnear Models 2 nd Edton. Chapman and Hall: London.