Section A: Meyers, Taylor & Marshall Flashcards
Taylor: What are the results of Taylor’s Theorems?
Theorems 3.2 and 3.3 are similar to 3.1 about the ODP Mack model and mean:
- Forecasts from the ODP Mack and ODP Cross-Classified models are identical and the same as those from the chain ladder method despite the different formulations.
- We can get forecasts for the ODP Cross-Classified model without considering the model directly and working as if the model was an ODP Mack model.
Marshall: Preparing the Claims Portfolio:
Considerations for Splitting into Valuation Classes
- Preferable to split the claims portfolio into valuation classes the same as the split used to derive the central estimates
- This allows analyzed sources of uncertainty to be aligned with the central estimate analysis.
- If the valuation of the central estimate is at a granular level, it may make sense to do the quantitative analysis on aggregated valuation classes (which are more credible) and allocate results down.
Taylor: Tweedie Sub-Family
Expected Value & Variance
Taylor: ODP Cross-Classified Model GLM Representation (for a 4 x 4 triangle)
Taylor: The expected value and variance of an EDF:
E[Y] = ?
Var[Y] = ?
E[Y] = µ = b’(𝜃)
Var[Y] = a(ɸ)b’‘(𝜃)
Taylor: The Exponential Dispersion Family (EDF) is the family of distributions with probability density function (pdf) π(y;𝜃,ɸ)of the form:
ln(π(y;𝜃,ɸ)) = ?
ln(π(y;𝜃,ɸ)) = [y𝜃 - b(𝜃)] / a(ɸ) + c(y,ɸ)
y = value of observation Y
𝜃 = location parameter called canonical parameter
ɸ = dispersion parameter called scale parameter
b(•) = cumulant function which determines shape of distribution
ec(y,ɸ) = normalizing factor producing unit total mass for the distribution
Meyers: Bayesian Loss Models
Correlated Incremental Trend (CIT) Formulas
µw,d= ⍺w+ βd + 𝜏(w + d - 1)
Zw,d ~ lognormal(µw,d, σd)
IncLoss1,dsim ~ normal(Z1,d, 𝝳)
IncLossw,dsim ~ normal(Xw,d + ⍴(IncLossw-1,dsim - Zw-1,d)e𝜏,𝝳) for w>1
- LIT model is the same, but sets ⍴ = 0
Marshall: Calibrating balance scorecard scores to CoVs
Example Scale and Comments
Meyers: Non-Bayesian Models
Briefly describe how the ODP Bootstrap model on paid data performed in the Meyers paper?
- Histogram shows more low percentiles than expected
- p-p plot shows a slanted “U” curve
- K-S test fails at the 5% level for all the lines combined
Conclusion
The ODP Bootstrap model is biased high
Meyers: Describe three tests for uniformity for n predicted percentiles.
- Histogram - if percentiles are uniformly distributed, the bars should be of similar height
-
p-p Plot - sort the predicted percentiles into increasing order, then plot the expected percentiles on x-axis and sorted predicted on y axis.
- sorted predicted percentiles should follow the expected percentiles along the 45 degree line
- {ei} = 100*{1/(n+1), 2/(n+1),….,n/(n+1)}
-
K-S Test - reject hypothesis of uniformity if D>D* at 5% level of significance
- D = Max | pi - fi |
- Critical Value = 136 / √n
- fi = 100 * {1/n, 2/n,….,n/n} where n is the # of percentiles
- On a p-p plot, these critical value appear as 45 degree bands that run parallel to the line y=x. We reject the hypothesis of uniformity if the p-p plot lies outside the bands.
Marshall: What is the goal of External Benchmarking?
Goal - compare the reasonableness of CoVs and risk margins to those from external sources.
Should NOT be relied on instead of analysis, but may be useful when there is little information available for analysis; especially for independent risk.
Marshall: External Systemic Risk
Latent Claim Risk
Latent Claim Risk
- Negligible for most valuation line classes but material for some Workers Compensation and liability classes
- Low probability risk with very high potential severity (e.g. Asbestos)
- If the risk exposure is significant enough to be included in the central estimate valuation or capital adequacy modeling, then it should be examined more closely for setting risk margins
- Discuss with underwriters to identify sources of latent claim risk
Marshall: Preparing the Claims Portfolio:
Allocation of Valuation Classes to Claim Groups
- If different groups of claims within a valuation class have materially different uncertainty, they should be treated separately in the risk margin analysis
- e.g Splitting the Home portfolio between CAT and non-CAT claims, splitting non-CAT between liability and non-liability
- Balance the benefit gained and the practicality/cost
Marshall: External Systemic Risk
Claim Management Process Change Risk
Claim Management Process Change Risk
- Typically impacts all valuation classes with different levels of materiality
- Discuss current and future process changes with claims management
- Consider impact on reporting/payment patterns, closing and reopening rates, and case estimation
- Could do sensitivity testing of key valuation assumptions to help assess CoV for this category
- Likely to be more relevant for OSC than premium liabilities
Taylor: The standardized Pearson residuals are defined as follows:
RiP = ?
RiP = (Yi - Yhati) / σhat<strong>i</strong>
Check residual plots against each variable:
- Standardized Pearson residuals should be random about zero (unbiased) and have uniform dispersion (homoscedasticity)
Marshall: Independent Risk Assessment for Oustanding Claim Liabilities
Goal - a model is used to “fit away” past systemic episodes in order to analyze the residual volatility and get the CoVs for independent risk.
Methods Used
- Mack Method
- Bootstrapping
- Stochastic Chain Ladder
- GLM Techniques
- Bayesian Techniques
Marshall: External Systemic Risk
Event Risk
Event Risk
- Catastrophes have most material impact on Property
- Include in OSC liability if there are outstanding events
- Information to help quantify risk for premium liabilities:
- past experience from event risk - could create frequency/severity model and then adjust past experience for changes in portfolio/inflation/…
- CAT models which produce a range of possible outcomes for modelled events on an insurer’s book
- Reinsurance intermediaries
Marshall: External Sytemic Risk CoV Consolidation Formula
CoV2<em>external</em> = ∑CoVi2
- Formula should be similar to internal as some external risks are correlation (within the same class or between PL and CL)
Taylor: ODP Mack Model GLM Representation
Marshall: Summary of Risk Margin Analysis Framework
Marshall: Describe the internal benchmarking that should occur for the following source of risk:
Independent Risk
Portfolio Size: larger portfolios have lower CoVs due to lower volatility from random effects
Length of Claim Runoff: Longer tailed lines have higher CoVs due to more time for random effects to have an impact
Implications:
Long-tailed portfolios - premium liability CoV should be higher than claim liability CoV
Short-tailed portfolios - premium liability CoV should be lower than claim liability CoV
Marshall: What are the sources of Systemic Risk?
Internal Systemic Risk
Risks internal to the liability valuation process
→ Reflects the extent to which the actuarial valuation approach is an imperfect representation of complex, real-life process
External Systemic Risk
Risks external to the actuarial modeling process
→ Even if the model represent the current conditions well, future systemic trends may cause future experience to differ from current expectations
Taylor: What do you need to do to specify a GLM?
The selection of a GLM consists of the following:
- Selection of a cumulant function, controlling the model’s error distribution
- Selection of an index p, controlling the relationship between the models means and variance
- Selection of the covariates xiT
- the variables that explain ui
- Selection of link function, controlling the relationship between the mean ui and the associated covariates
Meyers: Interpreting Diagnostics
Light-Tail Model
Marshall: External Systemic Risk
Recovery Risk
Recovery Risk
- Likely to be insignificant except possible for Auto 3rd party recoveries
- Non-Reinsurance recovery rates can be analyzed to understand past systemic risks. Discuss potential risks with Claims management.
- Reinsurance Recoverability
- Systemic risk of recovery could be driven by CATs or falling asset returns
- Discuss with reinsurance management to identify possible scenarios, likelihood and potential impact
Marshall: Mechanical Hindsight Analysis Steps
Step 1 - apply the chain ladder method to the data as of the valuation date to calculate the current unpaid estimate.
Step 2 - iteratively remove a diagonal of data and apply the same method (and average age-to-age factors) to calculate unpaid estimates for prior valuation dates.
Step 3 - Compare the past estimates to the current estimate for the same accident years (similar to a runoff). The relevant payments made between the past valuation dates and the current valuation date should be added to the current unpaid estimate.
Taylor: What is a consquence of using Pearson residuals? How is this overcome?
One consequence of Pearson residuals is that they will reproduce any non-normality that exits in the observations (eg. skewed loss data).
Normally distributed residuals are more useful for model assessment, so its common to look at standardized deviance residuals when assessing a GLM
Taylor: Results of the Mack Model
Results of the Mack Model:
- The conventional chain ladder LDF estimators, fjhat are:
- Unbiased
- Minimum variance among estimators that are unbiased linear combinations of the age-to-age factors, fjhat
- The conventional chain ladder reserve estimator, Rkhat, is unbiased.
Taylor: Theorem 3.3 Regarding Cross-Classified Model
If the Theorem 3.2 assumptions hold and the fitted values of Yk,jhat and reserve estimates Rkhat are correct for bias, then they are MVUEs of Yk,j and Rk.
Marshall: External Systemic Risk
Legislative, Political and Claim Inflation Risk
Legislative, Political and Claim Inflation Risk
→ more material for long-tail classes and sub-risks are often correlated
Sub-groups include:
- Impact of recent legislation and court interpretations
- Potential future legislation with retrospective impacts
- Precedent setting in courts
- Changes in medical technology costs
- Change in legal costs
- Systemic shifts in large claim frequency/severity
Marshall: What is the goal of Internal Benchmarking?
Goal: check for internal consistency and reasonableness by comparing CoV
- between similar valuation lines
- between claim and premium liabilities within classes
Marshall: Describe the internal benchmarking that should occur for the following source of risk:
Internal Systemic Risk
Compare CoVs by valuation class:
- If template models are used for similar valuation classes, we would expected similar CoVs.
- i.e. classes with homogenous claim groups should have similar CoVs
- If similar valuation methodology is used on both short and long-tail classes, then we would expect a higher CoV for the long-tail class.
Meyers: Bayesian Paid Loss Models
Describe the Correlated Incremental Trend (CIT) model and how it performed on paid loss data analyzed in the paper.
Key Improvement
- CIT includes a calendar year trend for incremental paid loss
- CIT allows for correlation between accident years (like CCL)
- Histogram and p-p plot show more low percentiles than expected
- K-S test fails at the 5% level for all lines combined
Conclusion
The CIT model is biased high on paid losses.
Marshall: Independent Risk CoV consolidation Formula
CoVind = ?
σ2ind = ∑σi2wi2
CoVind = [(∑CoVi • weighti)2]1/2
- assume no correlation between Home/Auto (classes)
- weights sum up to 1 when looking at totals
CoVi = [∑(CoVi•weighti)2]1/2 / ∑wi
Marshall: Internal Systemic Risk
Balance Scorecard Approach
Goal - assess the adequacy of the modeling infrastructure and its ability to reflect and predict the underlying insurance process.
Steps:
- Conduct a qualitative assessment of the modeling infrastructure for each specification/parameter/data risk component (score between 1 and 5)
- Calculate a weighted average score using weights for each risk indicator that reflect the risks relative importance.
- Calibrate weighted average scores to a CoV scale for internal systemic risk.
Meyers: Bayesian Incurred Loss Models
Describe the Level Chain Ladder (LCL) model and how it performed on incurred loss data analyzed in the paper.
Key Improvement over Mack
- LCL uses random level parameters for each AY
µw,d = ⍺w + βd
Lossw,dsim ~ lognormal (µw,d, σd)
- Histogram and p-p plot show more high/low percentiles than expected
- K-S test fails at the 5% level for all lines combined
Conclusion
LCL is still too light in the tails, but better than Mack
Marshall: Briefly describe general hindsight analysis.
Goal - compare the past reserve estimates of liabilities against the latest view of the same liabilities to review the actual volatility in the past.
Considerations
- Valuation models may have changed from previous valuations
- May be different sources of external systemic risk in the future
- Less valuable for long-tail portfolio with serial correlations between consecutive valuations
Taylor: What does Taylor suggest for correcting for heteroscedasticity?
- if a residual plot exhibits heteroscedasticity, then weights can be used to correct it.
-
General Rule - assign weights that are inversely proportional to the variance of the residuals
- e.g. residuals above age 55 have double the standard deviation of those below age 55 so this means the residuals above age 55 have 4 times the variance than those below (weight would be 1/4)
Marshall: Describe Scenario Testing
Goal: identify scenarios (e.g. higher frequency or severity, loss ratios, etc.) that would result in the central estimate increasing to the same level as the risk-loaded actuarial central estimate.
- how key assumptions underlying the central estimate calculation would need to change in order to produce a central estimate equal to the risk-loaded actuarial central estimate
Marshall: Internal Systemic Risk CoV Consolidation Formula
CoVinternal = [(CoVi•wi)2 + (CoVj•wj)2 + 2⍴ijwiwjCoViCoVj]1/2
- same as the variance formula with correlation
Meyers: Briefly describe the reasons why models may not accurately predict distributions of actual outcome?
- Insurance losses are too dynamic to be fully reflected in a single stochastic model
- other models may prove to be a better fit
- Data used in the model might be missing key information
- difficult to make accurate estimates if data is missing (e.g. changes in underlying business, claims process or reinsurance)
Marshall: External Systemic Risk
Economic and Social Risk
Economic & Social Risk
- Standard inflation
- Unemployment
- GDP growth
- Interest rates
- Asset returns
- Fuel prices
- Changing driving patterns
- Systemic shifts in claim frequency for short-tail lines
Marshall: Describe the internal benchmarking that should occur for the following source of risk:
External Systemic Risk
CoVs should be higher for long-tail portfolios, except for event risk for Property and liability risk for Home.
Meyers:
Total Risk =
Var(X) = E𝜃[Var[X|𝜃]] + Var𝜃[E[X|𝜃]]
Total Risk = Process Risk + Parameter Risk
Meyers: Non-Bayesian Models
Briefly describe how the Mack model performed on the paid loss data in the Meyers paper?
- Histogram shows more low percentiles than expected
- p-p plot shows a slanted “U” curve
- K-S test fails at the 5% level for all the lines combined
Conclusion
Paid Mack model is biased high
Marhsall: Total Consolidated CoV Formula
CoVtotal
CoV<em>total</em> = (CoV<em>ind</em>2 + CoV<em>internal</em>2 + CoV<em>external</em>2)1/2
Marshall: Describe Sensitivity Testing
Goal: gain insights about the sensitivity of the final risk margin to key assumptions
Steps:
- Independently, vary key assumptions (the CoVs and correlations) and then monitor the impact on the risk margin
- Review key assumptions that have significant impact on the risk margin
Marshall: Why is both quantitative and qualitative analysis necessary to properly assess risk margins?
- Quantitative analysis can only reflect uncertainty in historical experience and can’t capture adequately all possible sources of future uncertainty
- Judgement is necessary to estimate future uncertainty
Taylor: Re-Normalized ODP Cross-Classified Model Parameters
Marshall: What sources of uncertainty can quantitative modeling best assess?
- Quantitative modeling is best for analyzing independent risk and past episodes of external systemic risk.
- Quantitative modeling must be supplemented with other qualitative or quantitative analysis to incorporate internal systemic risk and external systemic risk.
- Future external systemic risk may differ from past episodes
Marshall: What are the main sources of Internal Systemic Risk?
Specification Error
Error arising from an inability to build a model that fully represents the underlying insurance process.
Parameter Selection Error
Error that the model can’t adequately measure all predictors of claim cost outcomes or trends in predictors. May be more cost drivers than can be captured in the valuation model.
Data Error
Error from poor data or unavailability of data required for a credible valuation model.
Marshall: Risk Margin Formula
Lognormal Distribution
Risk Margin (%) = [e[µ+Zσ] / (w1 + w2)] - 1
Marshall: What are the sources of uncertainty?
Systemic Risk
Risks that are potentially common across valuation classes or claim groups
Independent Risk
Risks arising from randomness inherent in the insurance process
Taylor: Theorem 3.1 Regarding the EDF Mack Model
Theorem 3.1:
a. If the original Mack variance assumption also holds, then the MLEs of the fj parameters are the chain ladder LDF estimators, fjhat, and these are unbiased estimators.
b. If the model is restricted to the ODP Mack model and if the dispersion parmeters are just column dependent, ɸk,j = ɸj, then the weighted average chain ladder LDFs, fjhat, are MVUEs.
c. For the same conditions as (b), the cumulative loss estimates, Xk,jhat, and reserve estimates, Rkhat, are also MVUEs.
Meyers: Interpreting Diagnostics
Validated Model
Taylor: Describe Continuous Covariates.
Continuous covariates are predictors with continuous levels such as age
- represented by polynomial
- transformed these variables before they go into the model (e.g. apply a log transformation)
Meyers: Interpreting Diagnostics
Biased High Model
Meyers: What is parameter risk?
Parameter Risk
Variance due to the uncertainty in the parameters, reflected in the posterior distributions of the parameters
- Parameter risk represents the overwhelming majority of the total risk in the loss data sets that Meyers looked at.
Marshall: What are the correlation effects on internal systemic risks?
Assumed to be uncorrelated with independent risk and external systemic risk sources.
There is correlation between classes and between outstanding claim and premium liabilities due to:
- Same actuary effect - effect of common valuation models or approaches across different valuation lines
- Linkage between premium liability methodology and outcomes from outstanding claims valuation
Marshall: Why quantitative method might not be appropriate for assessing correlation effects
- Techniques tend to be complex and require substantial data
- Time/effort required may outweigh the benefits
- Correlations would be heavily influenced by past correlations
- Difficult to separate past correlation effects between independent risk and systemic risk or identify the effects of past systemic risks
- Internal systemic risk can’t be modelled with standard correlation risk techniques
- Results unlikely to be aligned with framework, which splits between independent, internal systemic, and external systemic risk
Taylor: What is the implication of Theroem 3.1?
This theorem means that the conventional chain ladder estimates and forecasts are optimal estimators (both MLE and MVUEs).
This is stronger implication than the original Mack model
→It shows the LDF estimators, fjhat, are minimum variance for all unbiased estimators, not just of the linear combinations f the age-to-age factors.
Meyers: Non-Bayesian Models
Briefly describe how the Mack model performed on the incurred loss data in the Meyers paper?
- Histogram shows more high/low percentiles than expected
- p-p plot shows a slanted “S” curve and the K-S test failed at the 5% level for all the lines combined
Conclusion
Incurred Mack model is too light in the tails
Taylor: Non-Parametric Mack Model Assumptions
- Accident years are stochastically independent
- For each AYk, the cumulative losses Xkj form a Markov chain
- For each accident year k and development peirod j:
- E[Xk,j+1|Xk,j] = fjXk,j
- Var[Xk,j+1|Xk,j] = σ2jXk,j
Meyers: What is process risk?
Process Risk:
Average variance of outcomes from the expected result
Marshall: External Systemic Risk:
List the Risk Categories
- Economic & Social Risks
- Legislative, Political Risks and Claim Inflation Risks
- Claim Management Process Change Risk
- Expense Risk
- Event Risk
- Latent Claim Risk
- Recovery Risk
Marshall: What are the correlation effects on external systemic risks?
Uncertainty from each risk category is assumed to be uncorrelated with independent risk, internal systemic risk and the uncertainty from other external systemic risk categories.
There is correlation between classes (home & auto) and between outstanding claim and premium liabilities from similar risk categories
- e.g. claims inflation risk across all long-tail portfolios
Marshall: Internal Systemic Risk
Potential Risk Indicators
Specification Error
- Number of independent models used
- Range of results produced by models
- Checks made on reasonableness of results
Parameter Selection Error
- Best predictors have been identified
- Best predictors are stable over time (or change due to process changes)
- Value of predictors used (close to best predictors)
Data Error
- Good knowledge of past process affecting predictors
- Extent, timeliness, consistency and reliability of information from business
- There are appropriate reconciliations and quality control for the data
Marshall: External Systemic Risk
Expense Risk
Expense Risk
- Generally a small contributor to external systemic risk
- Discuss with product/claims management to understand the drivers of policy maintenance and claim handling expenses
- Can do sensitivity testing around key drivers
- Relevant to Premium Liabilities (paper mentions only PL (unexpired portion of policies) not CL)
Taylor: GLM Deviance
D*(Y, Yhat) = 2(loglikihoodsaturated - logliklihoodModel)
The saturated model has a parameter for each observation so the model completely fits the observations.
Marshall: What are the sources of Independent Risk?
- Random component of parameter risk
- Randomness of the insurance process compromises ability to select appropriate parameters for valuation models
- Random component of process risk
- Pure effect of randomness of the insurance process
Taylor: How do you determine all the expected future increamental losses at once?
- Use a forecast design matrix (X)
Yhat* = µhat* = h-1(X*βhat)
Marshall: Calibrating balance scorecard scores to CoVs
Developing a Scale
- Scale should represent uncertainty of internal systemic risk ranging from worst to best practice
- Relative difference in performance of modeling infrastructures over time can give insight into uncertainty of poor vs. good modeling
- Minimum CoV for best practice is unlikely to be <5%
- the best models can’t completely model the insurance process
- Might have CoV>20% for single, aggregated model with limited data
Marshall: Risk Margin Formula
Normal Distribution
Risk Margin (%) = z•CoV<em>total</em>
Marshall: Framework for Assessing Risk Margin
Meyers: Interpreting Diagnostics
Heavy-Tail Model
Taylor: Describe Categorical Covariates.
Categorical Covariates are predictors with discrete levels
- e.g. state or province, gender, make or model of car
- each level is treated as dummy variable (values of 0 or 1)
- e.g. Alberta, Manitoba and Saskatchewan are 3 provinces in the dataset the model is fit to. If the policyholder lives in Alberta then the covariate would be 1 0 0
Marhsall: What are the correlation effects on independent risks?
Assumed to be uncorrelated with any other source of uncertainty either within or between valuation classes
Taylor: How can the scale parameter be estimated using deviance?
ɸhat = D*(Y,Yhat) / (n-p)
n = number of observations
p = number of parameters
Meyers: What is model risk?
Model Risk
The risk that we didn’t selected the “correct” model.
- Can think of model risk as a special case of parameter risk when we weight together different models. The weights used would be considered parameters.
- If the posterior distribution of the weights assigned to each model has significant variability then model risk exists.
- Quantification will show up in process risk portion of the total risk.
Meyers: Summary of Model Conclusions
Taylor: Parametric Mack Model Assumptions
Same assumptions as Mack Model except:
- Yk,j+1 | Xk,j ~ EDF(𝜃k,j, ɸk,j;a,b,c)
- Variance assumption is removed - Variance is driven by the selected EDF distribution
Taylor: Theorem 3.2 Regarding the Cross-Classified Model
For the cross-classified model, if the following assumptions hold:
- Yk,j is restricted to an ODP distribution
- The dispersion parameters are identical for all cells: ɸkj = ɸ
Then the MLE fitted values Yk,jhat are the same as those from the conventional chain ladder method.
Taylor: Cross-Classified Model Assumptions
- The random variables Yk,j are stochastically independent
- For each accident year k and development period j
- Yk,j+1 | Xk,j ~ EDF(𝜃k,j, ɸk,j; a,b,c)
- E[Yk,j] = ⍺kβj
- Σβj = 1
Meyers: Bayesian Incurred Loss Models
Describe the Correlated Chain Ladder (CCL) model and how it performed on incurred loss data analyzed in the paper.
Key Improvement over LCL
- CCL allows for correlation between accident years
µ1,d= ⍺1+ βd
µw,d= ⍺w+ βd+⍴(ln(Cw-1,d) - µw-1,d) for w>1
Lossw,dsim ~lognormal(µw,d, σd)
- ⍴ is generally positive which results in higher prediction variance than LCL
- Histogram and p-p plot show slightly more high/low percentiles than expected and the K-S test passes at the 5% level.
Conclusion
The CCL model is validated, but has mildly thin tails.
Note - since correlation & random level parameters is added to the model the variability increases and is a better fit compared to Mack and LCL.
Taylor: How are parameters of a GLM usually estimated?
Parameters of a GLM are often estimated using maximum likelihood estimation (MLE)
Meyers: Bayesian Paid Loss Models
Describe the Changing Settlement Rates (CSR) model and how it performed on paid loss data analyzed in the paper.
Key Improvement
- CSR parameter ɣ reflects changes to the claim settlement rate
µ<em>w,d</em>= ⍺w+ β<em>d</em> • (1-ɣ)<em>w-1</em>
Lossw,dsim ~ lognormal(µ<em>w,d</em>,σd)
- ɣ is generally positive, reflecting a speedup in payment pattern
- Histogram and p-p plot show CSR corrects the high bias or other Bayesian paid models and the K-S test passes at the 5% level of confidence
Conclusion
The CSR model is validated.
Taylor: Standardized Deviance Residuals
RiD = ?
RiD = sgn(Yi - Yhati)(di/ɸhat)0.5
di = the contribution of the ith ovservation to the deviance D*(Y,Yhat)
sgn = -1 if quantity is negative, 0 if quantity is zero, and 1 if quantity is +1
Taylor: ODP Cross-Classified Model Parameters Formulas
