CAS Exam MAS-II Fall 2019, Question 1
Study guide: Exam MAS-II
In the supplemental material, you have been given a case study, "Warranty Payments Case Study". You have also been asked to evaluate three statements regarding the results of the study.
The relevant output from the case study is reproduced below. The log of individual warranty payments for one product is modelled with linear mixed-effects models in R's nlme package, using payment lag (months since sale, 1 to 12) and month of sale as fixed effects.
Study set-up, as the case study describes it:
The product is sold in 100 different stores, but only three stores are included in the sample. Those three stores are under the variable Store and are denoted by: Store_1, Store_2 and Store_3.
Model I — payment lag as a fixed effect, store as a random effect, constant variance:
## Linear mixed-effects model fit by REML
## Data: Repair
## AIC BIC logLik
## 191545.6 191580.3 -95768.8
##
## Random effects:
## Formula: ~1 | Store
## (Intercept) Residual
## StdDev: 0.2302979 2.220299
##
## Fixed effects: Log_Payment ~ Pay_Lag
## Value Std.Error DF t-value p-value
## (Intercept) 0.6518274 0.13489914 43196 4.83196 0
## Pay_Lag 0.2986713 0.00309451 43196 96.51643 0
Model III — payment lag and month sold as fixed effects, store as a random effect, variance a function of payment lag using the exponential variance function:
wgt_function_1 <- varExp(form = ~Pay_Lag/2)
Model_III <- lme (Log_Payment ~ Pay_Lag + Sale_Month, data =Repair,
random = ~1|Store,
weights=wgt_function_1)
summary(Model_III)
## Linear mixed-effects model fit by REML
## Data: Repair
## AIC BIC logLik
## 181627.8 181766.6 -90797.89
##
## Random effects:
## Formula: ~1 | Store
## (Intercept) Residual
## StdDev: 0.02249708 1.012199
##
## Variance function:
## Structure: Exponential of variance covariate
## Formula: ~Pay_Lag/2
## Parameter estimates:
## expon
## 0.2061607
## Fixed effects: Log_Payment ~ Pay_Lag + Sale_Month
Model IV — the same fixed and random effects, with variance a function of payment lag using the power transform of payment time:
wgt_function_3 <- varPower(form =~Pay_Lag)
Model_IV <- lme (Log_Payment ~ Pay_Lag + Sale_Month, data =Repair,
random = ~1|Store,
weights=wgt_function_3)
summary(Model_IV)
## Linear mixed-effects model fit by REML
## Data: Repair
## AIC BIC logLik
## 181693.1 181831.8 -90830.53
##
## Random effects:
## Formula: ~1 | Store
## (Intercept) Residual
## StdDev: 0.01945993 0.8774883
##
## Variance function:
## Structure: Power of variance covariate
## Formula: ~Pay_Lag
## Parameter estimates:
## power
## 0.4884623
## Fixed effects: Log_Payment ~ Pay_Lag + Sale_Month
- I. Model I is a marginal model.
- II. Placing the Store variable in the random effects category is appropriate since we only have a subset of possible results.
- III. Model III is a nested model of Model IV.
Determine which of the above statements are true.
- A) I only
- B) II only
- C) III only
- D) I, II and III
- E) The answer is not given by (A), (B), (C), or (D)
Answer this question and see the worked solution