4.3.1.1 Age and Gender

Age is defined as the number of full years of age at the start of the study. It has been computed as 2009 minus the year of birth. In Cox proportional hazards model, we will also have a parameter t that represents time from the study start. The age at any moment can then be computed as $A + t$ , where A is the age at the study start. Having A as a constant simplifies the analysis.

Gender is recorded as a binary value: 1 – for Male, 2 – for Female, as recorded at birth.

4.3.1.2 IMD

IMD takes an integer value from 1 to 10 and is computed based on the postcode recorded at the study start. For individual records, the data item “IMD decile” is missing for nearly 58% of the total sample and nearly 61% of the general sample. In these cases, the missing value is imputed with the IMD for the individual’s registered GP. The remaining missing values (291 cases – 171 of which have the IMD recorded as 0 and 120 for which it is recorded as NaN) are allocated a value 5, which is a middle value for the index.

We acknowledge that the proportion of missing individual-level IMD data is substantial and that GP catchment areas are geographically broader than individual postcodes. However, excluding these records would result in a loss of over half the study population, drastically reducing statistical power and likely introducing significant selection bias, as data missingness is rarely random. Consequently, we utilised the GP-level IMD as a pragmatic proxy, operating on the reasonable assumption that patients generally register with a GP practice in close proximity to their residence, thereby sharing similar socio-economic characteristics.

4.3.1.3 Smoking Status

Files Smoke_ent_*.txt contain information about smoker status (1 – Yes, 2 – No, 3 – Ex smoker). From Smoke_ent_*.txt we received $12,511,090$ records. After removing records with value 0 “Data Not Entered” we have $12,509,309$ records. After removing duplicate records, we have $11,068,831$ records.

These files should be accompanied by Smoke_Clin_*.txt to identify dates. There were 10,714 records with missing event dates. These dates were substituted by system date, which is the date the record was made in the database.

In addition, files Smoke_Int_*.txt and Smoke_test_*.txt present smoking information using medcodes. ICD-10 codes and medcodes for smoker status are presented in Table 35. The total number of records from files Smoke_Int_*.txt, Smoke_test_*.txt and Smoke_diag.txt is 16,248. After removing all of records with medcodes unrelated to smoking status we have 4,107 records.

In the total data set we have 75,727 individuals, or $6.7{\rm{\% }}$ , without smoker status recorded. In the general sample, this applies to 50,080 individuals, or $21.6{\rm{\% }}$ . The results of the analysis of the randomness of missingness are presented in Table 36. They show that values are not missing completely at random. This means that we cannot remove records with missing values, and so we should impute the data. The imputation method we used was to set the value “Non-smoker” for records without smoker status. This approach is based on the heuristic of clinical coding practices, where positive risk factors (such as smoking) are actively recorded by clinicians, whereas the absence of a record is frequently used to implicitly denote a negative (non-smoking) status.

4.3.2.1 Body Mass Index (BMI)

There are two sources of BMI observation data. Information was extracted from both sources, merged, cleaned and then analysed for missing data bias.

The first source is any table containing data with medcodes specified in Tables 28 and 29. In total, there are 489,756 records with medcodes from Tables 28 to 29, which represent 4% of all BMI measurements. For the 277 observations with an interval code and no corresponding event date ( $0.05{\rm{\% }}$ ), the system date is substituted. That is, the date the information was entered into CPRD, which is, for those cases where both dates are known, within a few days of the event date.

The second source of BMI measurement observations is within the “Additional Clinical details” table of the CPRD data alongside the date of the observation as found in the clinical information file. There are $14,317,076$ records in this table, which is 96% of all BMI measurements. Each record contains patid (ID of individual), adid (ID of measurement) and the value of the measurement. These records were filtered to remove a) those with no measurement value (308,392 measurements, $2.5{\rm{\% }}$ ) and b) any duplicate with the same combination of (patid, adid) ( $1,824,379$ duplicates, 15%).

Joining the two sources provides a file with a time series of BMI measurement observations. Individuals have from 1 to a maximum of 1,508 observations within the study period. Records with obvious data input errors such as negative, nil, very small/large values (16,221 records, $0.13{\rm{\% }}$ ) are removed. After this, there are $12,193,099$ BMI measurement observations for 978,380 individuals. We call this the “BMI total sample”.

BMI is known to be recorded and missing in a biased way. This is because it is more likely for BMI to be recorded where health is poor or there is a diagnosis that requires more frequent interaction with the health system or management of weight (Nicholson et al., Reference Nicholson, Aveyard, Bankhead, Hamilton, Hobbs and Lay-Flurrie2019). For the analysis of missing data, a given individual is considered to have a known value where there is at least one BMI measurement, otherwise it is missing. This disregards the lack of recording frequency that is expected for healthier lives. While this approach does not distinguish between the clinical context of individuals with differing health needs, it offers a pragmatic framework for handling large-scale, routinely collected data. It is important to acknowledge that the presence of a single BMI measurement may reflect varying levels of data quality: for instance, in individuals with chronic conditions such as diabetes, infrequent BMI recording may indicate incomplete data, whereas in healthier individuals, infrequent contact with healthcare services may reasonably result in fewer recordings. Although this introduces some bias, treating the presence of at least one BMI value as a known measurement provides a consistent and transparent definition of observed versus missing data.

Table 30 presents the results of the relationship between missingness of BMI and diagnosed diseases of interest in the BMI total sample. For most diseases of interest, the missingness of BMI is not random, as would be expected based on clinical practices. This aligns with prior expectations, since conditions such as diabetes, cardiovascular disease and obesity are known to prompt more regular weight and BMI monitoring as part of ongoing management and risk assessment. Conversely, there are only three diseases – Epilepsy, AIDS and Multiple Sclerosis – for which the association with BMI missingness is statistically insignificant. This also seems reasonable, as these conditions may not routinely require weight monitoring as part of standard care pathways. One further condition, Learning Disability, shows a small p-value suggestive of statistical dependence but a very small effect size, indicating minimal practical impact. This may reflect heterogeneous care patterns in this group, or variability in healthcare engagement and data recording practices. Overall, the observed pattern of associations largely corresponds with clinical expectations and known drivers of BMI measurement in routine care.

For the general sample of 250,000 individuals, there are 769,757 BMI measurements for 162,238 individuals (65%). Therefore, only 65% of the samples have at least one BMI measurement observation. Table 31 shows the results of missing data analysis where the ${\chi ^2}$ independence test shows that there are only 4 attributes for which we do not have enough evidence to reject hypothesis about independence with confidence level 99%. There is only one attribute (diagnosed stroke) which has significant p-value ( $0.758$ ).

In conclusion, the missingness of BMI is highly correlated with age, sex and co-morbidities. This means that removing records with missing values is not appropriate and therefore imputing the missing data would be preferable. For this analysis, missing BMI measurements are imputed, based on the “nearest-neighbour” approach with regard to characteristics age, gender and co-morbidities. Specifically, we used the 1NN method, in which the imputed value corresponds to the actual BMI of the nearest neighbour, rather than an average or moving average of several nearest neighbours. This preserves the granularity of real observations while ensuring that imputations reflect biologically and clinically plausible values from similar individuals. This method is considered suitable in this context because these characteristics are strong predictors of BMI and are consistently recorded with high completeness in the data set.

4.3.2.2 Blood Pressure (BP)

Blood pressure measurements are found within the additional file named BP_medval_*.txt, which contains over 32 million observations. These records were filtered to remove those with a) no measurement value (544,923 records, $1.7{\rm{\% }}$ ) and b) any duplicate with the same patient ID ( $3,875,362$ records, 12%). Where the diastolic field was greater than the systolic field (61,286 records, $0.2{\rm{\% }}$ ), the values were reversed as this is a typical input error. The systolic and diastolic fields were reviewed for reasonableness (systolic range [50,300], diastolic range [30,120]). Where a record had values outside reasonable range, the record was removed ( $281,831 + 201,426$ records, $1.5{\rm{\% }}$ ). For the 475 cases with a missing event date, the date was substituted by system date with the same logic as per BMI.

After the above data treatment, there are over 27 million BP measurement observations that correspond to individuals in the data, hence leaving the remaining observations without BP observations: $7.6{\rm{\% }}$ (85,651) individuals in the total data set and 26% (65,286) individuals in the general sample. Table 31a shows that the analysis of the randomness of the missing data is not completely at random, as expected. Therefore, it is not appropriate to remove records with missing values and the preferred treatment is to impute the data. The imputation method adopted was to take the mean value from individuals with the same gender and the same (or similar, i.e., closest in the Hamming distance) set of co-morbidities.

4.3.2.3 Cholesterol Level (CL)

The majority of cholesterol measurement observation data is found in the “Test” tables and some results are also found in the “Referral” tables. The data can be filtered specifically for cholesterol tests using readcodes and medcodes for cholesterol tests that are presented in Table 33.

The following values are measured during a CL test with the normal levels presented in Table 32.

• • HDL (high-density lipoprotein) – the higher the better.

• • Non-HDL (which includes LDL) – the lower the better. This group includes IDL, VLDL and lipoprotein.

• • Total cholesterol (TC) or serum cholesterol – this is the total amount of cholesterol in the blood and includes both HDL and non-HDL cholesterol.

• • Triglycerides – the lower the better.

• • The TC to HDL ratio (TC:HDL) – the lower the better.

Under extreme conditions (e.g., homozygote with defective genes), TC can be above 1,000 mg/dL which corresponds to 26 mmol/L (Kattah et al., Reference Kattah, Gómez, Gutiérrez, Puerto, Moreno-Pallares, Jaramillo and Mendivil2019). Therefore, any case where the observation is above 40 mmol/L within the data is assumed to be an incorrect coding of units. That is, the true unit is mg/dL and as such, it is converted to mmol/L.Footnote ⁶

For the 256 cases with a missing event date, the date was substituted by system date with the same logic as for BMI. In total there are over 12 million CL measurement observations. After removing duplicates, there are over 10 million observations of 903,041 individuals, about 75% of all individuals in the total data set. For the general sample, cholesterol measurements are observed for about 33% individuals. Therefore, 67% of individuals do not have any CL measurements observations. Results of analysis of randomness of missingness (see Table 34) show that values are missed not completely at random. Therefore, it is not appropriate to remove records with missing values and the preferred treatment is to impute the data. The imputation method used is 1NN in space of age, gender and co-morbidities - this is the same method as we used for BMI.

4.3.2.4 Blood Glucose (Sugar) Level (HbA1c)

HbA1c information can be found mainly in table “Test” within the HES data. The result of HbA1c measurements can be presented either as interval or as scalar data. To extract measurements of HbA1c, the medcodes presented in Table 38 are used. There are 100,064 measurement observations with interval data among $13,930,560$ observations from the “Test” table.

For those records with a missing event date, the date of data registration is taken. This is considered appropriate because, based on available records with both values present, the date of measurement and the registration date typically occur within a few days of each other. Therefore, using the registration date provides a reasonable approximation of the measurement time with minimal impact on the temporal alignment of the data.

The units of measurements for all individuals and for the general sample individuals are presented in Tables 39 and 40, respectively. As we can see from these Tables, there are many units of measurements used, and the intervals of reasonable values in these units have non-empty intersections. Therefore, if the unit of measurement is not given, it is impossible to guess it based on the value. For this reason, measurements with some units such as “No Data Entered”, “No unit”, etc., were removed. When the unit is given, we convert the measurements to ${\rm{\% }}$ (A1c), which is one of the standard representations.Footnote ⁷

Extreme value observations were treated as incorrect data inputs and the observations removed. The sample data contains 10 records with HbA1c less than 2%, and 190 records with values greater than 20%. These values fall outside the ranges for reasonableness, because a HbA1c value of $3.2{\rm{\% }}$ is considered as pathologically low (Joob & Wiwanitkit, Reference Joob and Wiwanitkit2018); while, on the other hand, at values greater than 10% an individual “needs injectable therapy” (American Diabetes Association, 2017); and values exceeding 17% have been observed by medics only in rare cases (Petznick, Reference Petznick2011). The removal of these 200 outlier records ( $0.0014{\rm{\% }}$ of the $13.9$ million total observations) is not considered to materially affect the results, given the minimal proportion and their likely erroneous nature. These exclusions help to preserve the overall quality and reliability of the data set without introducing meaningful bias.

There are 1,522,020 measurement observations with a date recorded but a measurement value missing. In these cases, the observations are removed.

In total, HbA1c has been measured for about 61% of individuals in the total sample. That is, 39% of individuals do not have any HbA1c measurement observation data. This varies significantly across samples. In the diabetes sample, about 10% of individuals have missing HbA1c value. In the general sample, the missingness of HbA1c is 85%.

It is anticipated that this measure is missing for those without a diabetes diagnosis because HbA1c is not measured where medics do not consider an individual at risk of or showing signs of diabetes. Therefore the analysis of missing data is approached separately for the two groups.

Results of analysis of randomness of missingness show that, for some variables, missingness is not completely at random (see Table 41). Therefore, it is not appropriate to remove records with missing values and the preferred treatment is to impute the data. For individuals without a diabetes diagnosis for whom HbA1c has never been measured, a value within the “normal” range is imputed (say, 5%).Footnote ⁸

For individuals with a diabetes diagnosis, it is anticipated that HbA1c is missing without a significant bias by age (at least for adults) and gender (Weykamp, Reference Weykamp2013). This prior expectation is supported by our data. Hence, we focus the analysis on the dependence on co-morbidities. For every individual P with an unknown HbA1c value, a search is performed across all individuals ${P_t}$ in the diabetes sample such that (i) HbA1c is known and (ii) the Hamming distance between the co-morbidities of P and ${P_t}$ is less than k, where k is the minimal value such that the number of nearest neighbours is at least 10. The imputed HbA1c value for P is then taken as the average HbA1c of its nearest neighbours. While we acknowledge that this method has some limitations, for example, it does not incorporate the temporal sequence of HbA1c observations, this approach allows us to preserve co-morbidity-related structure in the data, which is particularly relevant given the strong clinical association between certain conditions (e.g., cardiovascular disease, renal impairment) and glycaemic control.

4.3.2.5 Diagnosed Diseases of Interest

Information is required regarding diseases (co-morbidities) that are expected to have a significant effect on the mortality risk. The list of diseases has been constructed by analysing the literature and in consultation with DSG and colleagues from Leicester Diabetes Research Centre. The resulting list of diseases is presented in Section 4.1.

The objective of the research is to differentiate mortality risk for individuals with diabetes. For example, those with coronary heart disease may have a higher mortality risk when there is also a diabetes diagnosis present compared to absence. Therefore, variables are coded to flag diabetes status and if any other disease is present.

Diagnosed diseases can be found in the CPRD data and in the HES linked data.

In the CPRD data, diagnosed diseases are encoded in the form of “medcodes” and “readcodes”; the codes relevant to diseases listed in Section 4.1 are summarised in Table 45. If any of the non-diabetes codes are present in the CPRD data for an individual, a corresponding disease status variable is set equal to 1.

A diabetes status variable is coded based on a list of diabetes-related medcodes/readcodes as described by Tate et al. (Reference Tate, Dungey, Glew, Beloff, Williams and Williams2017). This list is used here because it was freely available, and using it as a starting point significantly sped up the analysis.

The codes are split into three groups to define diabetes diagnosis status:

• • 127 codes that are clearly related to Type 1 diabetes (D1T) – see Table 42.

• • 124 codes that are clearly related to Type 2 diabetes (D2T) – see Table 43.

• • 106 “unspecified” codes from which it is not possible to identify the Type of diabetes – see Table 44.

For each individual (patID) within the data, two variables are defined for modelling purposes based on whether items from the diabetes specific list of medcodes/readcodes are present or not before the end of the study period:

• • ${d_1} = 1$ if Type 1 diabetes has ever been diagnosed, ${d_1} = 0$ otherwise.

• • ${d_2} = 1$ if Type 2 diabetes has ever been diagnosed, ${d_2} = 0$ otherwise.

• • For individuals with unspecified codes, these are classified as follows:

  • • If ${d_1} = {d_2} = 1$ , unspecified codes are disregarded.

  • • If ${d_1} = 1$ and ${d_2} = 0$ , unspecified codes are classified as Type 1.

  • • If ${d_1} = 0$ and ${d_2} = 1$ , unspecified codes are classified as Type 2.

  • • If ${d_1} = {d_2} = 0$ , the classification is age dependent. If the first code is below age 28, set ${d_1} = 1$ and ${d_2} = 0$ , otherwise set ${d_1} = 0$ and ${d_2} = 1$ .

Classification of the unspecified codes by age is considered appropriate because Type 1 Diabetes is usually diagnosed earlier in life compared to Type 2 (Thomas et al., Reference Thomas, Jones, Weedon, Shields, Oram and Hattersley2018). The distribution of first diagnosis within the total sample by Type of Diabetes is presented in Figure 2. It can be seen that this also holds within the diabetes sample; for individuals with age of first diagnosis of diabetes less than 28 it is more likely to be Type 1 diabetes and for first diagnosis of diabetes at age 28 or more it is more likely to be Type 2 diabetes.

In the HES data, diagnosed diseases are encoded using ICD codes instead of medcodes/readcodes. At the time of the research, it was not possible to find a clear list of ICD codes corresponding to each disease as in Tables 42–45. To utilise the HES data, the ICD codes have been converted to medcodes/readcodes. Then we identify the diagnosis as per the process outlined above and summarised in Tables 42–45.

For the conversion, a dictionary from file ICD10.DBF from the “Clinical Terminology Browser” was used (the only version released on 19.03.2018) available online at https://isd.digital.nhs.uk/trud3/user/guest/group/0/pack/9/subpack/8/releases. It was developed to translate readcodes to ICD10 and there may be several readcodes for one ICD10 code. There are 14,702 different ICD10 codes covering 116,374 readcodes in the file. Unfortunately, there are 458 ICD10 codes used in the HES data within the sample that are not defined in this dictionary. In these cases, a nearest match is used to identify the diagnosis. For any given code, the ICD code with the same four symbols that could be found in the dictionary ICD code is used and then the corresponding row is used for decoding; otherwise the ICD code with the same three-symbol code is used.

For every disease, the date of the first record with the corresponding ICD, readcode, or medcode is used as the date of diagnosis. In effect, all records with the same disease at a later date are ignored. Some records are missing a date of diagnosis. However, all these cases contain a date of inclusion of information into the system (known as “system date”). Therefore, where the diagnosis date is missing the system date is used. Adopting this approach reduces the $0.6{\rm{\% }}$ of records in the total sample with missing date of diagnosis to nil.

6.2.2.1 Age

Age is calculated as the number of whole years at the study start and assigned the risk factor ${z_0}$ in the model. This approach uses exact single-year age rather than broader age bands. This reflects standard practice in life insurance and actuarial modelling, where mortality assumptions are typically constructed on a per-year-of-age basis.

Using single-year age allows for more granular modelling of the age-related variation in mortality, which is known to change markedly even across adjacent years, particularly at older ages. Given the size and richness of the data set, there is sufficient statistical power to estimate risk at this fine level of resolution without the need to aggregate into coarser categories. This avoids potential loss of information and smoothing over important inflection points in the age–mortality relationship that could influence model accuracy and underwriting decisions. Moreover, insurers routinely rely on single-age mortality tables (NLT), and using a consistent age definition enhances the interpretability and practical relevance of the model outputs for actuarial and underwriting audiences.

In the model, the hazard rate (e.g., force of mortality) depends on age as:

$$\lambda \left( {t,z} \right)\sim {e^{{\beta _0}{z_0}}},$$

where ${z_0}$ is a function of age. To determine ${z_0}$ , the force of mortality as it varies with age must be assessed. For example, the force of mortality by age for males is presented in Figure 11. It can be seen that the force of mortality is almost linear in logarithmic coordinates. This suggests that ${z_0} = $ age is already a reasonable model.

However, the fit to data can be improved by assuming that ${z_0}$ is a piecewise-linear function of age. For example, the manual splitting of age interval [17–100] into three intervals of [17–27], [28–73] and [74–100] significantly improves the fit to data – see Figure 12. In order to identify intervals with the best accuracy automatic segmentation is used. The results of this segmentation are presented in Table 72, Figure 13 and Figure 14. Applying the Elbow rule to Figure 13 suggests that only 2 segments should be used. That is, ${z_0}$ is given by the formula:

(4) $${z_0} = \left\{ {\matrix{ {k \cdot \left( {{\rm{age}} - {z^{\rm{*}}}} \right) + {z^{\rm{*}}},} \hfill & {{\rm{if\;age}} \le {z^{\rm{*}}}} \hfill \cr {{\rm{age}},} \hfill & {{\rm{if\;age}} \ge {z^{\rm{*}}},} \hfill \cr } } \right.$$

where ${z^{\rm{*}}}$ and k are fitted to the partitioned data, for example, for males and females separately. As indicated in Table 72, the optimal break point ${z^{\rm{*}}}$ is 50 for males and 65 for females. The optimal value of k is $0.6510$ for males and $0.7335$ for females.

6.2.2.2 Body Mass Index (BMI)

The latest observation of the BMI measurement at the study start is transformed into covariates to improve model simplicity. The ranges employed are in line with those adopted by the NHS (NHS, 2023b). However, the groupings are altered to improve model accuracy: specifically, the obese range was subdivided into the 3 classes shown below, as it significantly improved model accuracy. The following risk factor transformation is introduced:

• • Underweight: ${z_{b1}} = 1$ if BMI $ \lt 18.5$ , and ${z_{b1}} = 0$ otherwise.

• • Overweight: ${z_{b2}} = 1$ if BMI $ \geq\!25.0$ , and ${z_{b2}} = 0$ otherwise.

• • Obese Class 1: ${z_{b3}} = 1$ if BMI $ \geq \!30.0$ , and ${z_{b3}} = 0$ otherwise.

• • Obese Class 2: ${z_{b4}} = 1$ if BMI $ \geq 35.0$ , and ${z_{b4}} = 0$ otherwise.

• • Obese Class 3: ${z_{b5}} = 1$ if BMI $ \geq 40.0$ , and ${z_{b5}} = 0$ otherwise.

For example, an individual with BMI 32.5 has a BMI which is both $ \geq 25.0$ and $ \geq 30.0$ and so is described by a vector $\left( {{z_{b1}},{z_{b2}},{z_{b3}},{z_{b4}},{z_{b5}}} \right) = \left( {0,1,1,0,0} \right)$ . Therefore, coordinates are not independent as, for example, if ${z_{b3}} = 1$ then ${z_{b2}} = 1$ .

This structure reflects a cumulative threshold-based encoding that mirrors the way underwriters and actuaries typically think about BMI risk categories – each level of obesity builds on the previous. This encoding captures the escalating nature of risk associated with higher BMI thresholds, which is especially useful in underwriting, where risk loading increases progressively across weight categories.

6.2.2.3 Blood Pressure (BP)

Measurements for systolic blood pressure (SBP) and diastolic blood pressure (DBP) are used at the latest observation as at study start. The risk factor transformation introduced is shown below and is considered in line with NHS ranges used (NHS, 2023a):

• • Low blood pressure: ${z_{p1}} = 1$ if either SBP $ \le 90$ or DBP $ \le 60$ . Otherwise ${z_{p1}} = 0$ .

• • Pre-hypertension: ${z_{p2}} = 1$ if either SBP $ \geq 120$ or DBP $ \geq 80$ . Otherwise ${z_{p2}} = 0$ .

• • High BP (Stage 1): ${z_{p3}} = 1$ if either SBP $ \geq 140$ or DBP $ \geq 90$ . Otherwise ${z_{p3}} = 0$ .

• • High BP (Stage 2): ${z_{p4}} = 1$ if either SBP $ \geq 160$ or DBP $ \geq 100$ . Otherwise ${z_{p4}} = 0$ .

In terms of insurance underwriting, BP is routinely assessed at point of application, and risk assessment typically involves identifying whether an individual exceeds particular thresholds that align with these exact categories. Many underwriting manuals apply stepwise premium loadings based on these categories, and reinsurer rating guides often treat elevated BP levels as risk multipliers, particularly in the presence of co-morbid conditions like diabetes or obesity.

6.2.2.4 Cholesterol Levels (CL)

Measurements for Cholesterol levels (CL) are used at the latest observation as at study start. The risk factor transformation introduced is shown below and is informed/based on widely used clinical guidelines (Superdrug Online Doctor, 2023):

• • High CL: ${z_{c1}} = 1$ if CL $ \gt 5$ mmol/L. Otherwise ${z_{c1}} = 0$ .

• • Very high CL: ${z_{c2}} = 1$ if CL $ \gt 6.5$ mmol/L. Otherwise ${z_{c2}} = 0$ .

• • Extremely high CL: ${z_{c3}} = 1$ if CL $ \gt 7.8$ mmol/L. Otherwise ${z_{c3}} = 0$ .

In the underwriting context, elevated cholesterol is a known risk factor and often triggers further review or a premium loading. While insurers may use more comprehensive lipid panels, TC remains a common and interpretable metric. The use of the most recent value at the time of underwriting is standard practice, as underwriters make risk assessments based on an individual’s current health status, not historical averages or trends. The cumulative structure of the indicators (e.g., someone with CL = $8.0$ mmol/L will satisfy all three conditions) reflects the increasing risk load and provides flexibility in the model to capture marginal effects across ranges.

6.2.2.5 Smoking Status

Smoking status is based on the categorisation defined at the study start. The risk factor transformation introduced is shown below and is informed by epidemiological and actuarial research demonstrating the differential mortality risks associated with current, former and never smokers (Doll et al., Reference Doll, Peto, Boreham and Sutherland2004).

• • Ever smoked: ${z_{s1}} = 1$ if an individual ever smoked. Otherwise ${z_{s1}} = 0$ .

• • Current smoker: ${z_{s2}} = 1$ if an individual is a current smoker. Otherwise ${z_{s2}} = 0$ .

That is to say, at the study start, $\left( {{z_{s1}},{z_{s2}}} \right) = \left( {0,0} \right)$ is an individual that never smoked, $\left( {{z_{s1}},{z_{s2}}} \right) = \left( {1,0} \right)$ an individual who is an ex-smoker and $\left( {{z_{s1}},{z_{s2}}} \right) = \left( {1,1} \right)$ an individual who is a smoker.

This approach is reasonable and aligns well with underwriting practices, where smoking status is a key and typically static data point collected at application. While smoking behaviour can change over time, underwriting generally relies on smoking status reported or verified at the point of underwriting, which is assumed to be constant for pricing and risk classification purposes.

6.2.2.6 Index of Multiple Deprivation (IMD)

The IMD is taken as at the study start. The risk factor is transformed using 4 variables set to either a value of 0 or 1.

• • ${z_{md1}} = 1$ if IMD level is at least 3, and ${z_{md1}} = 0$ otherwise.

• • ${z_{md2}} = 1$ if IMD level is at least 5, and ${z_{md2}} = 0$ otherwise.

• • ${z_{md3}} = 1$ if IMD level is at least 7, and ${z_{md3}} = 0$ otherwise.

• • ${z_{md4}} = 1$ if IMD level is at least 9, and ${z_{md4}} = 0$ otherwise.

In other words, IMD levels 1 and 2 are coded as $\left( {0,0,0,0} \right)$ , IMD 3 and 4 as $\left( {1,0,0,0} \right)$ , IMD 5 and 6 as $\left( {1,1,0,0} \right)$ , IMD 7 and 8 as $\left( {1,1,1,0} \right)$ and IMD 9 and 10 as $\left( {1,1,1,1} \right)$ .

From an underwriting perspective, the IMD is assumed to be a static data point, reflecting the individual’s health or risk profile at the point of underwriting (i.e., study start). This is consistent with insurance risk assessment practices where data used for pricing or risk classification is typically fixed at application. Although individuals’ health status may change over time, underwriting decisions usually cannot account for such longitudinal variations. Additionally, using this binary stepwise transformation avoids potential issues with sparse data if the IMD were used as a continuous variable or in very granular categories. It also simplifies interpretation for underwriters and actuaries, who can associate specific IMD threshold levels with corresponding risk adjustments.

6.2.2.7 Diabetes Disease Status

The diabetes disease status is encoded based on diagnoses as at the start of the study. A later diagnosis is not considered relevant because this approach mimics the real-world underwriting process in insurance. In insurance underwriting, the assessment of mortality risk and premium setting is based on the information available at the time of application, which includes the individual’s current health conditions and medical history up to that date. Later developments or diagnoses occurring after the policy issue date are generally not factored into the initial risk classification, as underwriting decisions and pricing are fixed at inception. The transformation of this data item is:

• • ${z_{D1}} = 1$ if an individual has been diagnosed with Type 1 diabetes, and ${z_{D1}} = 0$ otherwise.

• • ${z_{D2}} = 1$ if an individual has been diagnosed with Type 2 diabetes, and ${z_{D2}} = 0$ otherwise.

For example, an individual diagnosed with Type 2 diabetes in June 2010 has ${z_{D1}}$ and ${z_{D2}}$ of $\left( {0,0} \right)$ .

It is considered reasonable to model Type 1 and Type 2 diabetes separately when estimating mortality risk because these conditions differ significantly in their pathophysiology, typical age of onset, progression, complications and associated mortality patterns. Numerous epidemiological studies have demonstrated statistically significant differences in mortality risk between individuals with Type 1 and Type 2 diabetes, see, for example, Genuth et al. (Reference Genuth, Palmer and Nathan2021).

6.2.2.8 Duration since Diabetes Diagnosis

This risk factor is only considered for individuals with diabetes. For an individual with Type i diabetes, $i = 1,2$ , it measures the duration (in years) since the Type i diabetes was diagnosed. This is computed by the formula “2009 – calendar year of the diagnosis” given the study start of 01/01/2010. For example, an individual diagnosed in June 2009 with Type 2 diabetes has a duration since diagnosis of $2009 - 2009 = 0$ . Duration therefore increases in whole numbers from 0.

Including duration since diagnosis as a risk factor is reasonable and important for mortality risk modelling because the length of time an individual has lived with diabetes is strongly associated with the progression of complications and overall mortality risk (Wright et al., Reference Wright, Suarez-Ortegon, Read, Kontopantelis, Buchan, Emsley, Sattar, Ashcroft, Wild and Rutter2020).

From an underwriting perspective, while duration since diagnosis may not always be explicitly recorded, it is often estimated or indirectly assessed through medical history or reported diagnosis dates. Incorporating this information aligns with underwriting practices that seek to assess not only the presence of a condition but also its chronicity and severity, which influences risk classification and pricing decisions. It is assumed that duration as a static variable at study start is consistent with underwriting timelines, where risk is assessed based on information available at application. This approach balances model complexity and practical utility, providing a meaningful metric to enhance mortality risk prediction for individuals with diabetes.

6.2.2.9 Blood Glucose (sugar) Level (HbA1c)

Measurements for blood glucose (sugar) level (HbA1c) are used at the latest observation as at study start. The risk factor is transformed using the ranges defined by guidelines such as those from the American Diabetes Association (ADA) and the World Health Organization (WHO):

• • Pre-diabetes: ${z_{h1}} = 1$ if HbA1c $ \geq 6.0{\rm{\% }}$ . Otherwise ${z_{h1}} = 0$ .

• • Diabetes: ${z_{h2}} = 1$ if HbA1c $ \geq 6.5{\rm{\% }}$ . Otherwise ${z_{h2}} = 0$ .

• • Diabetes high: ${z_{h3}} = 1$ if HbA1c $ \geq 7.5{\rm{\% }}$ . Otherwise ${z_{h3}} = 0$ .

• • Diabetes very high: ${z_{h4}} = 1$ if HbA1c $ \geq 9.0{\rm{\% }}$ . Otherwise ${z_{h4}} = 0$ .

These variables define certain HbA1c ranges. For example, an HbA1c $ \lt 6.0\% $ is defined by $\left( {0,0,0,0} \right)$ , an HbA1c between $6.0$ and $6.5$ by $\left( {1,0,0,0} \right)$ , an HbA1c between $6.5$ and $7.5$ as $\left( {1,1,0,0} \right)$ , and so on.

From an underwriting perspective, HbA1c is a key biomarker routinely assessed and recorded in medical underwriting questionnaires or clinical tests. Using the latest available value at study start reflects standard underwriting practice, where the most recent medical data available at application is used for risk classification. Although HbA1c can fluctuate, the latest measurement provides a practical snapshot of glycemic control and risk status. The cumulative binary coding captures clinically meaningful cutoffs while maintaining simplicity and interpretability for underwriting decisions. It balances granularity with data availability and aligns well with mortality risk modelling objectives focused on diabetes-related risk stratification.

6.2.2.10 Disease Status

For each of the diseases listed in Section 4.1, ${z_i}$ is set to 1 if disease i has ever been diagnosed at the study start and ${z_i} = 0$ otherwise. Diagnoses do occur during the study period. To keep vector z constant for each individual, the following strategy is introduced:

• • If disease i has never been diagnosed then ${z_i} = 0$ .

• • If disease i has been diagnosed before the study start, then ${z_i} = 1$ .

• • If disease i has been diagnosed x years after the study start, then we split the record into two pseudo-individual records: one with ${z_i} = 0$ for x years and one with ${z_i} = 1$ for the remaining $10 - x$ years.

6.2.2.11 Pairwise Interaction of Risk Factors

For each pair $\left( {i,j} \right)$ of risk factors discussed above, a new risk factor is introduced ${z_{ij}} = {z_i}{z_j}$ . For example, if each ${z_i}$ and ${z_j}$ takes values in $\left\{ {0,1} \right\}$ , then ${z_{ij}} \in \left\{ {0,1} \right\}$ , and ${z_{ij}} = 1$ if and only if ${z_i} = {z_j} = 1$ .

Higher order interactions are not considered because on balance the working group prioritised having coefficient value outputs which could be easily interpreted.

6.2.3.1 Risk Factor Selection: Backward and Forward Feature Selection

The objective of this step is to build Model A by selecting meaningful non-disease and non-interaction risk factors, independent of sex, that have significant influence on the mortality of individuals with diabetes.

Model A includes all risk factors listed in Section 6.2.2, apart from the disease status and interaction risk factors. As the data is not partitioned, a risk factor of sex is introduced to account for variation. The model is fit on the GS data. The resulting coefficients are presented in Table 76, alongside the standard errors (SE) and p-values, where the p-value measures the significance of the risk factor and a higher p-value indicates a lower significance. Risk factors with a p-value greater than 0.01 are considered insignificant. We use a 1% threshold rather than the more conventional 5% because the data set is very large, so a 5% cutoff would retain many effects that are statistically detectable but too weak to be useful in a parsimonious underwriting model. The stricter threshold therefore helps reduce over-fitting and focuses the model on the most robust signals. Table 76 therefore contains 7 insignificant attributes. Risk factor selection can be reviewed by starting with the full list of 20 risk factors (as above) and removing those considered insignificant via the backward feature selection (BFS) process. The process of BFS (or dimensionality reduction) removes variables with the greatest p-values one-by-one until there is at least one statistically insignificant variable. The results are presented in Table 77. The model resulting from this process is presented in the right side of Table 76. Assuming that the baseline hazard rate is given by the exponential model (8) described below, the computed parameters are:

• • $a = 0.0552$ , $b = - 12.3610$ for the full model (before exclusion of insignificant risk factors).

• • $a = 0.0553$ , $b = - 12.3236$ for the reduced model (after exclusion of insignificant risk factors).

Correlation matrix between risk factors of the full model is presented in Table 78. The removed risk factors are strongly correlated with the remaining risk factors.

Table 76 shows some risk factors with negative coefficients, that is, the presence of the corresponding risk factor decreases the baseline force of mortality. Most risk factors with negative coefficients in the model also exhibit a moderate to strong positive correlation with age. This suggests that their apparent protective effect may be confounded by age: these factors become more common in older individuals, but age itself is already a strong predictor of increased mortality. Thus, when age is controlled for in the model, these variables may appear to lower risk simply because they are positively associated with age, not necessarily because they are truly protective. However, the variable Cl3 is an exception. Despite having a negligible correlation with age ( $0.032$ ), it retains a significant negative coefficient. This finding aligns with the “Cholesterol Paradox” often observed in observational studies of chronic diseases (Wang et al., Reference Wang, Chang, Wei, Liu, Tzeng and Chiu2023). In this context, the inverse association between cholesterol and mortality is typically attributed to reverse causality, where low CLs serve as a marker for underlying frailty, malnutrition, or chronic inflammation; rather than higher CLs conferring a direct biological benefit.

To understand the meaning of a negative coefficient (see Table 76), a direct estimation of the force of mortality for age bands is calculated using the GS data (we do not have enough data for mortality estimation for each year separately). Results are presented in Tables 79–84. As you can see, in all cases negative coefficients were explained by the supporting data. In other words, the observed negative dependence is really present in the data, it is not an artefact of the model.

Alternatively, FFS can be employed; starting with one attribute (e.g., age), and adding the next “most significant” (measured by p-value) risk factor one-by-one, until there are no more significant risk factors. The results are presented in Tables 85 and 86.

The models resulting from BFS and FFS are both the same, which provides confidence in the robustness and stability of the selected set of risk factors. This convergence suggests that the final feature set is not overly sensitive to the choice of selection method and likely reflects genuinely important predictors within the data.

6.2.3.2 Risk Factor Selection: Industry Standard Risk Classification

In Model B and the final output models, attributes describing age, IMD, BMI, BP, smoker status and CL are classified as “industry standard” and therefore must be included in the model, irrespective of their significance. There are other risk factors commonly used in the industry that are not included in our models. For example, family medical history, occupation and physical activity levels, are often considered in traditional underwriting are omitted due to data availability constraints.

The exclusion of these factors presents both benefits and limitations. On the one hand, the streamlined model may be easier to implement and explain, reducing complexity and potential biases introduced by subjective or self-reported data. On the other hand, it may compromise predictive accuracy, especially for edge cases where lifestyle or family medical history plays a crucial role. It is not possible, at this time, to test the impact, however, as the data is not recorded within the available data.

6.2.3.3 Risk Factor Selection: Disease Significance Considered by Sex

Model B partitions the data by sex and considers all risk factors from Section 6.2.2, excluding only the disease pairwise interactions.

As explained in Section 6.2.2, diseases are diagnosed during the study period, and, to keep attribute vector z constant for each individual, we split the records of such individuals into two pseudo-individual records: one without this disease from the beginning of the study to the date of diagnosis, and another one with this disease from the date of diagnosis. In case of several different diagnoses, an individual record can be split into more than two records. After such splitting of general sample linked to ONS, we obtained a data set with 157,143 records.

Table 87 presents the resulting Cox coefficients. There are 12 negative coefficients. A comparison of Cox coefficients for models A and B (e.g., with and without diseases) is presented in Table 88. A notable change is that a cholesterol observation in the “high” range (i.e., Cl) has a significant negative coefficient value in Model A (without disease status) to a significant positive coefficient value in Model B (with diseases). This suggests that the Cholesterol paradox is diminished somewhat after allowing for disease.

Risk factors with a p-value greater than 1% can be considered “insignificant” but may still be included where these are considered “industry standard risk classification”. The 1% cutoff is used deliberately here because, with a data set of this size, a 5% threshold would retain many effects that are statistically detectable but too small to be useful in a practical underwriting model. With the objective of minimising “over-fit” of the data, other insignificant attributes can be removed one-by-one until all the remaining attributes are significant. This process is undertaken on the partitioned data, for example, for males and females separately. The Cox coefficients for the models fit to the partitioned data are presented in Tables 89 and 90, for males and females respectively. In terms of statistical significance, some notable differences emerge due to sex partitioning and disease status. Several covariates demonstrate significance in one sex but not the other, highlighting underlying biological and epidemiological variations. Specifically, two male-specific diseases were found to be statistically significant for males but not for females. Conversely, seven female-specific diseases are significant only in the female cohort. The observed variation underscores the reasonableness and necessity of sex-specific modelling approaches. Sex-based biological differences (e.g., hormonal influences, genetic expression) and social determinants (e.g., healthcare access and behaviour) contribute to distinct patterns of morbidity and mortality. Incorporating these nuances into survival models enhances both statistical robustness and appropriate understanding of risk.

6.2.3.4 Risk Factor Selection: Pairwise Interactions Considered by Sex

To form models with interactions we consider all individual attributes and all possible pairs of attributes. Initially, we have 59 individual attributes and $59\cdot58/2 = 1,711$ pairs. However, some “interactions” are duplicates,Footnote ¹⁰ and were therefore removed. We also removed interactions for which there are less than 100 individuals in the data set. After this, 1,616 attributes remain for males and 1,573 for females.

Fitting Cox coefficients for this number of attributes is not possible in one go due to insufficient memory, even on a supercomputer. Therefore, the following procedure is employed:

1. (i) Start with “industry standard risk factors”, that is, attributes describing age, IMD, BMI, BP, smoker status and CL.

2. (ii) Add one of several other attributes (e.g., diseases attributes or interaction attributes).

3. (iii) Compute Cox coefficients of the resulting model.

4. (iv) Remove insignificant attributes, that is, with p-value greater than 0.01. If an “industry standard risk factor” attribute is insignificant, it can be removed unless it is the last attribute in the corresponding group. That is, the age attribute is protected and cannot be removed. Out of 4 IMD attributes, at least one should stay, and so on.

5. (v) Go to step (ii) and repeat the process, until all attributes are considered.

The result of the procedure described depends on the order in which we add the attributes in step (ii). To make the procedure reproducible, we must fix the order. To establish the order in which the attributes are added in step (ii), we have used Lasso (least absolute shrinkage and selection operator) method. This method is based on minimising sum-of-squares error. It has been originally developed for regression analysis and has been adapted for the Cox model (Tibshirani, Reference Tibshirani1997). It has been implemented in the glmnet package for R. The main advantage of this method is that it works much faster than the procedure (i)–(v) above and produces a ranking of the attributes. The disadvantage is that the resulting ranking is inaccurate (some significant interactions are ranked low by Lasso and vice versa) and therefore cannot be used alone to build the final model. However, it can be used to form the initial order of attributes before applying the main procedure (i)–(v).

For female and male models, the resulting list of attributes and their Cox coefficients and p-values are presented in Tables 93 and 94, respectively.

6.3.2.1 Time-Homogeneous Model

We next consider models in which ${\lambda _0}\left( t \right)$ is a continuous function. We first consider a time-homogeneous model, which is based on the following assumption

• • Consider two individuals A and B with the same set of attributes ${z_1}, \ldots ,{z_m}$ but different ages. Consider two different time moments ${t_1}$ and ${t_2}$ . Assume that the age of A at time ${t_1}$ is equal to the age of B at time ${t_2}$ , and this age is greater than ${z^{\rm{*}}}$ defined in (4). Then the force of mortality of A at ${t_1}$ is equal to the force of mortality of B at ${t_2}$ .

In other words, the assumption states that the probability of death depends on risk factors and current age, but not on calendar year. For example, if individual A was born in 1940 and individual B in 1945, and these individuals are otherwise identical, then the probability of death of A in 2010 (age 70) is the same as the probability of death of B in 2015 (also age 70).

From (2), the mortality rate of A at ${t_1}$ is:

$${\lambda _0}\left( {{t_1}} \right){e^{{\beta _0}{z_0}\left( A \right)}} \cdot {\rm{exp}}\left( {\mathop \sum \limits_{i = 1}^m {\beta _i}{z_i}} \right),$$

while the mortality rate of B at ${t_2}$ is:

$${\lambda _0}\left( {{t_2}} \right){e^{{\beta _0}{z_0}\left( B \right)}} \cdot {\rm{exp}}\left( {\mathop \sum \limits_{i = 1}^m {\beta _i}{z_i}} \right),$$

where ${z_0}\left( A \right)$ and ${z_0}\left( B \right)$ are the values of attribute ${z_0}$ for A and B, respectively.

Hence, the time-homogeneity assumption implies that

$${\lambda _0}\left( {{t_1}} \right){e^{{\beta _0}{z_0}\left( A \right)}} = {\lambda _0}\left( {{t_2}} \right){e^{{\beta _0}{z_0}\left( B \right)}}$$

or equivalently,

$${{{\lambda _0}\left( {{t_1}} \right)} \over {{\lambda _0}\left( {{t_2}} \right)}} = {e^{{\beta _0}\left( {{z_0}\left( B \right) - {z_0}\left( A \right)} \right)}}.$$

Because the age is greater than ${z^{\rm{*}}}$ , (4) implies that ${z_0}\left( A \right)$ and ${z_0}\left( B \right)$ are the ages of A and B, respectively, at the start of the study. The difference in age is therefore, ${z_0}\left( B \right) - {z_0}\left( A \right)$ . Because the age of A at time ${t_1}$ is equal to the age of B at time ${t_2}$ , this implies that

$${z_0}\left( B \right) - {z_0}\left( A \right) = {t_1} - {t_2}.$$

Hence,

$${{{\lambda _0}\left( {{t_1}} \right)} \over {{\lambda _0}\left( {{t_2}} \right)}} = {e^{{\beta _0}\left( {{t_1} - {t_2}} \right)}}.$$

The only function that satisfies this equality for every pair of real numbers ${t_1},{t_2}$ is the function

(7) $${\lambda _0}\left( t \right) = {e^{{\beta _0}t + b}}$$

for some parameter b. The constant b is then estimated by fitting (7) to the annual values ${\lambda _j}$ , $j = 2010, \ldots ,2019$ , obtained from (6).

6.3.2.2 Exponential Model

The only disadvantage of a time-homogeneous model is that it ignores the improvement of medicine over time. In reality, a 50-year-old individual with cancer has a higher chance to survive in 2020 than in 1970. To incorporate this effect, we can adjust the model as:

$${\lambda _0}\left( t \right) = {e^{\left( {{\beta _0} - \varepsilon } \right)t + b}},$$

where $\varepsilon $ is the term responsible for medicine improvement. Denoting $a = {\beta _0} - \varepsilon $ , we obtain:

(8) $${\lambda _0}\left( t \right) = {e^{at + b}}.$$

Parameters a and b can then be found to best fit the data ${\lambda _j}$ , $j = 2010, \ldots ,2019$ found from (6). We will call (8) the exponential model for baseline hazard rate.

Obviously, we may also shift the numbering of years, so that t is the time (measured in years) from study start. Then t ranges in $\left[ {0,10} \right)$ for the study period.

In both the homogeneous model (7) and exponential model (8) the hazard rate (1) has the form:

$$\lambda \left( t \right) = {A_z}{e^{\gamma t}},$$

where ${A_z}$ is some constant that depends on vector z of attributes and $\gamma $ is equal to ${\beta _0}$ and a in homogeneous and exponential models, respectively.

Then the probability $S\left( {{t_0},{t_1}} \right)$ to survive from time ${t_0}$ to time ${t_1}$ is:

(9) $$S\left( {{t_0},{t_1}} \right) = {\rm{exp}}\left( { - \mathop \int \nolimits_{{t_0}}^{{t_1}} \lambda \left( t \right)dt} \right) = {\rm{exp}}\left( { - \mathop \int \nolimits_{{t_0}}^{{t_1}} {A_z}{e^{\gamma t}}dt} \right) = {\rm{exp}}\left( { - {{{A_z}} \over \gamma }\left( {{e^{\gamma {t_1}}} - {e^{\gamma {t_0}}}} \right)} \right).$$

Once we compute the survival probability, we can easily compute any other quantities of interest. For example, the expected survival time at time ${t_0}$ is given by:

(10) $$E\left( {{t_0}} \right) = \mathop \int \nolimits_0^\infty S\left( {{t_0},{t_0} + t} \right){\rm{\;}}dt = \mathop \int \nolimits_0^\infty {\rm{exp}}\left( { - {{{A_z}{e^{\gamma {t_0}}}} \over \gamma }\left( {{e^{\gamma t}} - 1} \right)} \right)dt.$$

Having fit the model, the survival or death probability for a given set of inputs can be found using formula (9). From this, an estimate of life expectancy can be found using formula (10).

6.3.3.1 Coefficient Values & Hazard Rate: General Sample

In order to compare the risk of diabetics to a “standard” individual, a probability of death for the general population is required. The mortality models are fit to GS.

HbA1c attributes are not included in the model fit because most individuals in the GS will have no HbA1c measurements given there is no diabetes diagnosis or prior investigations. All other risk factors, as discussed in Section 6.2.2 are included, and data is partitioned by sex. The age attribute ${z_0}$ is given by (4) with ${z^{\rm{*}}} = 50$ and $k = 0.6510$ for males and ${z^{\rm{*}}} = 65$ and $k = 0.7335$ for females.

First the Cox coefficient values are solved for, then the baseline force of mortality, and finally the probability of death can be computed for a given set of input information. For the female model, the resulting list of attributes and their Cox coefficients and p-values are presented in Table 93. The baseline hazard rate ${\lambda _0}\left( t \right)$ is computed, as per Section 6.3.2, via the homogeneous model (7) and exponential model (8). For the homogeneous model (7), the value of parameter b that best fits the data is $4.0430 \times {10^{ - 7}}$ . For the exponential model, the resulting baseline hazard rate is:

$${\lambda _0}\left( t \right) = {\rm{exp}}\left( {0.0616t - 14.3738} \right) = A{\rm{exp}}\left( {0.0616t} \right),$$

where $A = {e^{ - 14.3738}}$ . The probability to survive from time ${t_0}$ to time ${t_1}$ is then given by (9):

$${\rm{exp}}\left( { - {{{A_z}} \over {0.0616}}\left( {{e^{0.0616{t_1}}} - {e^{0.0616{t_0}}}} \right)} \right),$$

where ${A_z}$ is a constant that depends on the vector z of the attribute values.

For male model, the resulting list of attributes and their Cox coefficients and p-values are presented in Table 94. For the homogeneous model with the baseline hazard rate (7), the value of parameter b that best fits the data is $3.8460 \times {10^{ - 6}}$ . For the exponential model, the baseline hazard rate (8) is given by:

$${\lambda _0}\left( t \right) = {\rm{exp}}\left( {0.0442t - 12.1609} \right),$$

while the probability to survive from time ${t_0}$ to time ${t_1}$ is given by (9):

$${\rm{exp}}\left( { - {{{A_z}} \over {0.0442}}\left( {{e^{0.0442{t_1}}} - {e^{0.0442{t_0}}}} \right)} \right).$$

6.3.3.2 Coefficient Values & Hazard Rate: Type 1 Diabetes

Next, the DS data is used to study the force of mortality of individuals with Type 1 diabetes. All risk factors in Section 6.2.2 are included. Recall that the age attribute ${z_0}$ is given by (4) with ${z^{\rm{*}}} = 50$ and $k = 0.6510$ for males and ${z^{\rm{*}}} = 65$ and $k = 0.7335$ for females.

Models are generated in turn with the risk factor selection, as described in Section 6.2.3. The resulting Cox coefficients for males and females are presented in Table 95 for industry standard risk classifications, Table 96 for industry standard risk classifications and disease status. A comparison of Cox coefficients in these models is presented in Table 97 for the female model and in Table 98 for male models.

For the final model including pairwise interactions, Tables 99 and 100 provide a list of risk factors, the Cox coefficient value and p-value for females and for males, respectively.

For the homogeneous model (7) baseline hazard rate, the value of parameter b that best fits the data is $2.1956 \times {10^{ - 5}}$ for females and $1.5983 \times {10^{ - 5}}$ for males. For the exponential model (8), the baseline hazard rate is:

$${\lambda _0}\left( t \right) = {\rm{exp}}\left( {0.0447t - 8.9282} \right)$$

for females and

$${\lambda _0}\left( t \right) = {\rm{exp}}\left( {0.0416t - 9.0859} \right)$$

for males. The probability to survive from time ${t_0}$ to time ${t_1}$ can then be computed by (9).

6.3.3.3 Coefficient Values & Hazard Rate: Type 2 Diabetes

Next, the DS data is used to study the force of mortality of individuals with Type 2 diabetes. All risk factors in Section 6.2.2 are included. A similar process is undertaken: fit to industry standard risk classifications, then include disease status, then pairwise interactions. The results are presented for male and female model fits in Tables 101 and 102, and a comparison for females in Table 103 and for males in Table 104. Tables 105 and 106 include the resulting list of attributes and the Cox coefficient values and p-values for females and for males, respectively.

For the homogeneous model (7) baseline hazard rate, the value of parameter b that best fits the data is $1.0148 \times {10^{ - 6}}$ for females and $2.8654 \times {10^{ - 6}}$ for males. For the exponential model (8), the baseline hazard rate is:

$${\lambda _0}\left( t \right) = {\rm{exp}}\left( {0.0783t - 11.6993} \right)$$

for females and:

$${\lambda _0}\left( t \right) = {\rm{exp}}\left( {0.0657t - 10.6501} \right)$$

for males. The probability to survive from time ${t_0}$ to time ${t_1}$ can then be computed by (9).

The coefficients for exponential and homogeneous baseline hazard rates for different models are summarised in Table 107.