Lecture 8: Gamma regression

Transcription

1 Lecture 8: Gamma regression Claudia Czado TU München c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

2 Overview Models with constant coefficient of variation Gamma regression: estimation and testing Gamma regression with weights c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

3 Motivation Linear models: V ar(y i ) = σ 2 constant Poisson models: V ar(y i ) = E(Y i ) = µ i non constant, linear Constant coefficient of variation: [V ar(y i )] 1/2 E(Y i ) = σ V ar(y i ) = σ 2 µ 2 i non constant, quadratic c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

4 Which transformation stablizes the variance if V ar(y i ) = σ 2 µ 2 i holds? Answer: Z i := log(y i ) Proof: According to a 2nd order Taylor expansion we have Z i = log(y i ) log(µ i ) + 1 µ i (Y i µ i ) 1 (Y 2µ 2 i µ i ) 2 i for σ 2 small E(Z i ) log(µ i ) 1 2µ 2 i = log(µ i ) 1 2 σ2 V ar(y i ) } {{ } =σ 2 µ 2 i According to a 1st order Taylor expansion we have E(Z i ) log(µ ( i ) ) E(Zi 2) E [log(µ i ) + 1 µ i (Y i µ i )] 2 ( ) ( = (log µ i ) 2 + 2E log(µ i ) 1 µ i (Y i µ i ) + E = (log µ i ) σ 2 µ 2 µ 2 i = (log µ i) 2 + σ 2 i V ar(z i ) (log µ i ) 2 + σ 2 (log µ i ) 2 = σ 2 For small σ 2 we have E(Z i ) log(µ i ) σ 2 /2 and V ar(z i ) σ 2 1 µ 2 i (Y i µ i ) 2 ) c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

5 Multiplicative error structure: Log normal model Assume Y i = e β 0 x β 1 i1 xβ p ip (1 + ɛ i ) } {{ } E(Y i )=:µ i where ɛ i := Y i E(Y i ) E(Y i ) E(ɛ i ) = 0, V ar(ɛ i ) = V ar(y i) E(Y i ) 2 =: σ 2 ln(y i ) = β 0 + β 1 ln x i β p ln x ip + ln(1 + ɛ i ) where E (ln(1 + ɛ i )) = E = E ( ) ln(1 + Y i E(Y i ) E(Y i ) ) ( ) ln( Y i E(Y i ) ) = E(ln(Y i )) ln E(Y } {{ } i ) = σ 2 /2 ln µ i σ2 2 and V ar (ln(1 + ɛ i )) = V ar (ln(y i ) ln E(Y i )) = V ar (ln(y i )) σ 2 c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

6 Consider ln Y i = β 0 + β 1 ln x i β p ln x ip σ2 2 + ɛ i, where ɛ i := ln(1 + ɛ i) + σ2 2 E(ɛ i ) 0, V ar(ɛ i ) σ2 If one assumes ɛ i N(0, σ2 ) i.i.d. and fits the log normal model ln Y i = α 0 + α 1 ln x i α p ln x ip + ɛ i then ˆα 1,..., ˆα p are consistent estimates for β 1,..., β p, but ˆα 0 is biased for β 0 with approx. bias σ 2 /2. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

7 Original scale: Gamma regression If one wants to work on original scale t β (ln x i ) µ i = E(Y i ) = e β 0x β 1 i1 xβ p ip = e } {{ } η i ln x i := (1, ln(x i1,..., ln(x ip )) t g(µ i ) = ln(µ i ). We need distribution for Y i 0 with E(Y i ) = µ i and V ar(y i ) = σ 2 µ 2 i. Such a distribution is the Gamma distribution, i.e. Y Gamma(ν, λ) with f Y (y) = λ Γ(ν) (λy)ν 1 e λy y 0 E(Y ) = ν λ = µ, V ar(y ) = ν = ( ) ν 2 1 λ 2 λ ν = 1 µ2. }{{} ν =σ 2 c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

8 Summary V ar(y i ) = σ 2 (E(Y i )) 2 µ i = E(Y i ) = e β 0x β 1 i1 xβ p ip Log normal model GLM with Y i Γ(ν, λ i ) (normal error on log scale) µ i = ν λ i σ 2 = 1 ν Linear model on log scale Gamma regression c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

9 Properties of the Gamma distribution 1) Mean parametrization µ = ν λ λ = ν µ f Y (y) = λ Γ(ν) (λy)ν 1 e λy = 1 Γ(ν) ( ν µ ) ( ν µ y ) ν 1 e ν µ y = 1 Γ(ν) ( ) ν νy µ e µ ν y 1 y f Y (y)dy = 1 Γ(ν) ( ) ν νy µ e µ ν y 1 y dy }{{} d(ln y) Gamma(µ, ν) Mean reparametrization c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

10 2) The form of the density is determined by ν <ν< ν>= ) Special cases : ν = 1 ν Skewed to the right Skewness exponential distribution normal distribution E(Y µ)3 σ 3 = = 2ν 1/2 0 for ν c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

11 Gamma regression as GLM For known ν in Y Gamma(µ, ν) the log likelihood is given by l(µ, ν, y) = (ν 1) log(y) ν µ y + ν log ν ν log µ log Γ(ν) = ν ( yµ log µ ) + c(y, ν) θ = 1/µ, b(θ) = log µ = log( 1/θ) = log( θ) b (θ) = ( 1) θ = 1/θ = µ expectation b (θ) = 1 θ 2 = µ 2 variance function a(φ) = φ, φ = 1/ν dispersion parameter c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

12 EDA for Gamma regression µ i = e xt i β log(µ i ) = x t i β a plot of x ij versus log(y i ) should be linear µ i = 1 x t i β 1 µ i = x t i β a plot of x ij versus 1/Y i should be linear c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

13 Example: Canadian Automobile Insurance Claims Source: Bailey, R.A. and Simon, LeRoy J. (1960). Two studies in automobile insurance. ASTIN Bulletin, Description: The data give the Canadian automobile insurance experience for policy years 1956 and 1957 as of June 30, The data includes virtually every insurance company operating in Canada and was collated by the Statistical Agency (Canadian Underwriters Association - Statistical Department) acting under instructions from the Superintendent of Insurance. The data given here is for private passenger automobile liability for non-farmers for all of Canada excluding Saskatchewan. The variable Merit measures the number of years since the last claim on the policy. The variable Class is a collation of age, sex, use and marital status. The variables Insured and Premium are two measures of the risk exposure of the insurance companies. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

14 Variable Description: Merit 3 licensed and accident free 3 years 2 licensed and accident free 2 years 1 licensed and accident free 1 year 0 all others Class 1 pleasure, no male operator < 25 2 pleasure, non-principal male operator < 25 3 business use 4 unmarried owner or principal operator< 25 5 married owner or principal operator < 25 c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

15 Variable Description(continued): Insured Earned car years Premium Earned premium in 1000 s (adjusted to what the premium would have been had all cars been written at 01 rates) Claims Number of claims Cost Total cost of the claim in 1000 s of dollars Variable of Interest is Cost. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

16 Data: > cancar.data Merit Class Insured Premium Claims Cost c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

17 Exploratory Data Analysis (unweighted case): Log Link: Merit Rating Class empirical log mean empirical log mean Merit Rating Class and Merit Class Merit and Class empirical log mean empirical log mean Merit Class No obvious functional relationships, treatment of covariates as factors might be appropriate. No strong interactions expected. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

18 Inverse Link: Exploratory Data Analysis (continued): Merit Rating Class empirical inverse mean empirical inverse mean Merit Rating Class and Merit Class Merit and Class empirical inverse mean empirical inverse mean Merit Class No obvious functional relationships, treatment of covariates as factors might be appropriate. Some interactions expected. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

19 Residual deviance in Gamma regression Y i Gamma(µ i, ν) independent µ i = e xt i β l(µ, ν, y) = n i=1 l(y, ν, y) = n i=1 i=1 { ν [ ] y i µ i ln(µ i ) } ln(γ(ν)) + ν ln(νy i ) ln(y i ) {ν [ 1 ln(y i )] ln(γ(ν)) + ν ln(νy i ) ln(y i )} 2 (l(ˆµ, [ ν, y) l(y, ν, y)) n ] = 2 ν( 1 ln(y i ) + y i ˆµ i + ln(ˆµ i )) i=1 = 2 n [ ( ) ] ν ln yi ˆµ i y i ˆµ i ˆµ i D(y, ˆµ) = 2 n i=1 [ ln ( yi ˆµ i ) ] y i ˆµ i ˆµ i saturated log likelihood deviance We have D(y, ˆµ) a φχ 2 n p where φ = 1/ν. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

20 Residual deviance test Y i Gamma(µ i, ν) independent µ i = e xt i β (Model (*)) Reject model (*) at level α if Here ˆφ is an estimate of φ. D(y, ˆµ) ˆφ > χ 2 n p,1 α c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

21 Partial deviance test Consider β = (β t 1, β t 2) t β 1 R p 1, β 2 R p 2 ˆµ 1 = fitted mean in reduced model with design X 2 ˆµ 2 = fitted mean in full model with design X 1 and X 2 ˆφ 1 = estimated dispersion parameter in reduced model ˆφ 2 = estimated dispersion parameter in full model The partial deviance test for H 0 : β 1 = 0 versus H 1 : β 1 0 is given by: Reject H 0 at level α D(y, ˆµ 1 ) ˆφ 1 D(y, ˆµ 2 ) ˆφ 2 > χ 2 p 1,1 α c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

22 Canonical link η i = θ i = 1 µ i Since we need µ i > 0 we need η i < 0, which gives restrictions on β. Therefore the canonical link is not often used. Most often the log link is used. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

23 Estimation of the dispersion parameter For the estimation of the dispersion parameter φ = σ 2 = 1/ν method of moments is used E(Y i ) = µ i V ar(y i ) = φµ 2 i i Z i = Y i µ i has expectation 1, variance φ and is i.i.d. ˆφ = 1 n p n i=1 ( ) 2 Yi ˆµ i 1 = 1 n p n i=1 ( ) 2 Yi ˆµ i ˆµ i since p parameters need to be estimated. Remark: Pearson χ 2 statistic for the gamma regression is given by ˆφ = χ 2 P /(n p). χ 2 P = n i=1 = n i=1 (Y i ˆµ i ) 2 V (ˆµ i ) ( ) 2 Yi ˆµ i ˆµ i V (ˆµ i ) = ˆµ 2 i c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

24 Relationship between gamma regression with log link and a linear model on the log scala For σ 2 = φ small we have V ar(log(y i )) σ 2, therefore we can use log Y i = x t i α + ɛ i ɛ i N(0, σ 2 ) i.i.d. linear model on log scala Cov( ˆα) = σ 2 (X t X) 1. For the gamma regression with Y i Γ(µ i, 1/σ 2 ) and µ i = exp{x t iβ} we have I(ˆβ)) = Fisher information = ˆβ a N(β, I(ˆβ) 1 ), ( E where ( )) 2 l(β) β= β β ˆβ. t c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

25 Here l(β) = 1 σ 2 n E i=1 l(β) β j = 1 σ 2 n i=1 2 l(β) n β s β j = 1 σ 2 ( ) 2 l(β) β s β j i=1 = 1 σ 2 n i=1 I(ˆβ) = 1 σ 2 X t X ( ) y µ i log µ i + const ( yi x ij e xt i β x ij ( ) y ix is x ij e xt i β ) j = 1,..., p x is x ij (independent of β) Cov(ˆβ) σ 2 (X t X) 1 = Cov( ˆα) Distinction between both models is difficult. If X corresponds to an orthogonal design, i.e. (X t X) 1 = I p, we have that the components of ˆα (ˆβ) are (asymptotically) independent. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

26 Gamma regression with weights Example: Y i = total claim size arising from n i claims in group i with covariates x i. Y i = n i j=1 Y ij ; Y ij = j th claim in group i Y s i := Y i /n i = average claim size in group i If Y ij Gamma(µ i, ν), independent, we have E(Y s i ) = 1 n i n i j=1 E(Y ij ) } {{ } =µ i = µ i V ar(y s i ) = 1 n i V ar(y i1 ) = 1 n i µ 2 i /ν = µ2 i /n iν. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

27 Since we have n i Y s i = n i j=1 Y ij Gamma(n i µ i, n i ν) Y s i Gamma(µ i, n i ν). This follows since m Y s i (t) := E ( e ty i s ) tn i Yi s = E(e n i 1 ) = (1 n i µ i n i ν t n i ) n i ν = 1 (1 µ i t n i ν )n i ν = m X(t) t, where X Gamma(µ i, n i ν). Therefore we need to introduce weights for a gamma regression for Yi s. Since E(Yi s) = µ i, the EDA for a weighted gamma regression does not change compared to an unweighted one. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

28 Exploratory Data Analysis (weighted case): Log Link: Merit Rating Class empirical log mean empirical log mean Merit Rating Class Class and Merit Merit and Class empirical log mean empirical log mean Merit Class c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

29 Exploratory Data Analysis (weighted case): Inverse Link: Merit Rating Class empirical inverse mean empirical inverse mean Merit Rating Class and Merit Class Merit and Class empirical inverse mean empirical inverse mean Merit Class c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

30 For estimation of the dispersion parameter σ 2 = φ = 1/ν in a weighted model we use ˆσ 2 = ˆφ = 1 n ( ) 2 Yi ˆµ i n i n p ˆµ i i=1 Residuals: E(Y s i ) = µ i Y i s µ i σ ni µ i = n i σ V ar(yi s) = µ2 i n i ν = µ2 i σ2 n i ( ) Y s i µ i µ i has zero expectation and unit variance i.e. standardized residuals can be defined by r s i := ni σ ( Y s i ) ˆµ i. ˆµ i c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

31 Example: Canadian Automobile Insurance Claims: Gamma Regression: Main Effects with Log Link > f.gamma.main_cost/claims ~ Merit + Class > r.gamma.log.main_glm(f.gamma.main, family = Gamma(link = "log"), weights = Claims) c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

32 Example: Gamma Regression: > summary(r.gamma.log.main,cor=f) Call: glm(formula = Cost/Claims ~ Merit + Class, family= Gamma(link = "log"), weights = Claims) Coefficients: Value Std. Error t value (Intercept) Merit Merit Merit Class Class Class Class (Dispersion Parameter for Gamma family taken to be 13 ) Null Deviance:1556 on 19 degrees of freedom Residual Deviance:157 on 12 degrees of freedom c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

33 Example: Gamma Regression (continued): To check the dispersion estimate, use the Pearson Chi Square Statistics. > sum(resid(r.gamma.log.main,type="pearson")^2) [1] 159 > 159/12 [1] 13 For the residual deviance test, we need to scale the deviance > 1-pchisq(156/13,12) [1] 0.45 A p-value of.45 shows no lack of fit. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

34 Main Effects with Inverse Link > f.gamma.main_cost/claims ~ Merit + Class > r.gamma.inverse.main_glm(f.gamma.main, family = Gamma, weights = Claims) > summary(r.gamma.inverse.main,cor=f) Coefficients: Value Std. Error t value (Intercept) Merit Merit Merit Class Class Class Class (Dispersion Parameter for Gamma family taken to be 14 ) Null Deviance:1556 on 19 degrees of freedom Residual Deviance:167 on 12 degrees of freedom c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

35 Example: Gamma Regression (continued): For the residual deviance test, we need to scale the deviance > 1-pchisq(167/14,12) [1] 0.45 A p-value of.45 shows no lack of fit. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

36 Log Link With Interaction > f.gamma.inter_cost/claims ~ Merit * Class > r.gamma.inter_glm(f.gamma.inter, family = Gamma, weights = Claims)) c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

37 Example: Gamma Regression (continued): > summary(r.gamma.log.inter,cor=f) Coefficients: Value Std. Error t value (Intercept) NA NA Merit NA NA Merit NA NA Merit NA NA Class NA NA Class NA NA Class NA NA Class NA NA Merit1Class NA NA Merit2Class NA NA Merit3Class NA NA Merit1Class NA NA Merit2Class NA NA Merit3Class NA NA Merit1Class NA NA Merit2Class NA NA Merit3Class NA NA Merit1Class NA NA Merit2Class NA NA Merit3Class NA NA (Dispersion Parameter for Gamma family taken to be NA ) c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

38 Example: Gamma Regression (continued): Null Deviance: 1556 on 19 degrees of freedom Residual Deviance: 0 on 0 degrees of freedom Number of Fisher Scoring Iterations: 1 This model is the saturated model since only one observation per cell. Therefore the dispersion estimate can no longer be estimated, since the Pearson Chi Square Statistics is 0. To fit a model with fixed dispersion, use > summary(r.gamma.log.inter,dispersion=1,cor=f) c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

39 Example: Gamma Regression (continued): Coefficients: Value Std. Error t value (Intercept) Merit Merit Merit Class Class Class Class Merit1Class Merit2Class Merit3Class Merit1Class Merit2Class Merit3Class Merit1Class Merit2Class Merit3Class Merit1Class Merit2Class Merit3Class (Dispersion Parameter for Gamma family taken to be 1 ) Null Deviance: 1556 on 19 degrees of freedom Residual Deviance: 0 on 0 degrees of freedom From this we see that certain interaction effects are strongly significant. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

40 Log Link With Some Interaction Terms > f.gamma.inter3_cost/claims ~ Merit + Class + I((Merit == 2) * ( Class == 2)) + I((Merit == 2) * (Class == 3)) + I((Merit == 1) * (Class == 4)) + I(( Merit == 3) * (Class == 4)) + I((Merit == 3) * (Class == 2)) + I((Merit == 2) * ( Class == 5)) > r.gamma.log.inter3_glm(f.gamma.inter3,family =Gamma(link="log"),weights=Claims) c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

41 Example: Gamma Regression (continued): > summary(r.gamma.log.inter3,cor=f) Call:glm(formula=Cost/Claims ~ Merit+Class+I(( Merit == 2) * (Class == 2)) + I((Merit == 2) * (Class == 3)) + I((Merit == 1) * ( Class == 4)) + I((Merit == 3) * (Class == 4)) + I((Merit == 3) * (Class == 2)) + I(( Merit == 2) * (Class == 5)), family = Gamma(link = "log"), weights = Claims) Deviance Residuals: Min 1Q Median 3Q Max e c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

42 Example: Gamma Regression (continued): Coefficients: Value Std. Error (Intercept) Merit Merit Merit Class Class Class Class I((Merit==2)*(Class==2)) I((Merit==2)*(Class==3)) I((Merit==1)*(Class==4)) I((Merit==3)*(Class==4)) I((Merit==3)*(Class==2)) I((Merit==2)*(Class==5)) c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

43 Example: Gamma Regression (continued): t value (Intercept) Merit1-6.6 Merit2-4.4 Merit3-8.6 Class2 1.3 Class3 2.7 Class4 7.2 Class5-4.1 I((Merit == 2) * (Class == 2)) 3.8 I((Merit == 2) * (Class == 3)) -4.0 I((Merit == 1) * (Class == 4)) 3.0 I((Merit == 3) * (Class == 4)) 3.6 I((Merit == 3) * (Class == 2)) 1.8 I((Merit == 2) * (Class == 5)) -1.9 (Dispersion Parameter for Gamma family taken to be 2.7 ) Null Deviance: 1556 on 19 degrees of freedom Residual Deviance: 16 on 6 degrees of freedom c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

44 Example: Gamma Regression (continued): > 1-pchisq(16/2.7,6) [1] 0.43 # Residual Deviance Test > 1-pchisq((156/13)-(16/2.7),6) [1] 0.41 # Partial Deviance Test We see that the joint effect of the interaction terms are not significant, since the partial deviance test yields a p-value of.41. A partial deviance test when only I((Merit == 2) * (Class == 2)) and I((Merit == 2) * (Class == 3)) are included gives a p-value of.39, thus showing that the interaction effects are not present in this data set. The same results are true when an inverse link is used instead. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

45 Standardized Residual Plots:: log.main inverse.main standardized residuals Index Difference between log link and inverse link is minimal. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

46 What needs to be done if one wants to use the linear model on the log scale? V ar(y s i ) = σ2 n i µ 2 i ( E(log(Yi s)) E log(µ i ) + 1 µ i (Yi s E [ (log(y s i ))2] = log(µ i ) 1 µ 2 i E ( V ar(yi s ) } {{ } µ 2 i σ2 n i [log(µ i ) + 1 µ i (Y s = [log(µ i )] 2 + V ar(y s i ) µ 2 i = (log µ i ) 2 + σ2 n i µ i) 1 µ 2 i σ = log(µ i ) i µ i)] 2 ) (Y s i µ i) 2 ) 2 2n i V ar(log(y s i )) (log µ i) 2 + σ2 n i (log µ i ) 2 = σ2 n i for σ 2 small linear model on the log scale needs also to be weighted c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

47 Log Normal Models: > f.lognormal.main_log(cost/claims)~merit + Class > r.lognormal.main_glm(f.lognormal.main, weights=claims) > summary(r.lognormal.main,cor=f) Call: glm(formula = log(cost/claims) ~ Merit + Class, weights = Claims) Deviance Residuals: Min 1Q Median 3Q Max c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

48 Log Normal Models (continued): Coefficients: Value Std. Error t value (Intercept) Merit Merit Merit Class Class Class Class (Dispersion Parameter for Gaussian family taken to be 13 ) Null Deviance:1515 on 19 degrees of freedom Residual Deviance:156 on 12 degrees of freedom c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

49 Log Normal Models (continued): > 1-pchisq(156/13,12) [1] 0.45 # Residual Deviance Test # Results when Splus function lm() is used: > summary(lm(f.lognormal.main,weights=claims),cor=f) Coefficients: Value Std. Error t value (Intercept) Merit Merit Merit Class Class Class Class Pr(> t ) (Intercept) Merit Merit Merit Class Class Class Class c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

50 Log Normal Models (continued): Residual standard error: 3.6 on 12 degrees of freedom Multiple R-Squared: 0.9 F-statistic: 15 on 7 and 12 degrees of freedom, the p-value is 4.7e-05 Both approaches (R-squared=.9, residual deviance p-value=.45) show no lack of fit. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

51 Fitted Costs per Claims: fitted Cost per Claim log.main inverse.main lognormal.main observed Index Difference between Models are minimal. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen

52 Fitted Cost: fitted Cost log.main inverse.main lognormal.main observed Index Difference between Models are minimal. c (Claudia Czado, TU Munich) ZFS/IMS Göttingen