.. Section title decorators for this document: ============== Document Title ============== Section Level 1 (#.0) +++++++++++++++++++++ Section Level 2 (#.#) --------------------- Section Level 3 (#.#.#) ~~~~~~~~~~~~~~~~~~~~~~~ Section Level 4 ^^^^^^^^^^^^^^^ Section Level 5 ''''''''''''''' The depth of each section level is determined by the order in which each decorator is encountered below. If you need an even deeper section level, just choose a new decorator symbol from the list here: https://docutils.sourceforge.io/docs/ref/rst/restructuredtext.html#sections And then add it to the list of decorators above. .. _2021_cause_alzheimers_presymptomatic_mci: ================================================================== Alzheimer's disease with presymptomatic and MCI stages (GBD 2021) ================================================================== .. contents:: :local: .. list-table:: Abbreviations :header-rows: 1 * - Abbreviation - Definition * - AD - Alzheimer's Disease * - BBBM - Blood-Based Biomarker * - CSU - Client Services Unit * - DW - Disability Weight * - FHS - Future Health Scenarios * - MCI - Mild Cognitive Impairment * - YLD - Years Lived with Disability * - YLL - Years of Life Lost Disease Overview ++++++++++++++++ GBD 2021 Modeling Strategy ++++++++++++++++++++++++++ The IHME dementia modelers use DisMod to estimate the prevalence and incidence of a "dementia envelope" comprising all types of dementia combined, and then they estimate what proportion of the envelope corresponds to each subtype of dementia. The proportions of dementia due to stroke, Parkinson's disease, Down's syndrome, and traumatic brain injury are attributed to those GBD causes, and the remaining dementia in the envelope is attributed to the GBD cause "Alzheimer's disease and other dementias" (cause ID 543). For further information, see the methods appendices and HUB page: * `Alzheimer's disease and other dementias in the GBD 2021 fatal methods appendix `_ * `Alzheimer's disease and other dementias in the GBD 2021 nonfatal methods appendix `_ * `Alzheimer's/Dementia HUB page `_ .. _ADOD_2021_fatal_methods_appendix: https://www.healthdata.org/gbd/methods-appendices-2021/alzheimers-disease-and-other-dementias .. _ADOD_2021_nonfatal_methods_appendix: https://www.healthdata.org/gbd/methods-appendices-2021/alzheimers-disease-and-other-dementias-0 .. _ADOD_HUB_page: https://hub.ihme.washington.edu/spaces/BIRDS/pages/123831566/Alzheimers+Dementia Restrictions ------------ The following table describes any restrictions in GBD 2021 on the effects of the cause "Alzheimer's disease and other dementias" (such as being only fatal or only nonfatal), as well as restrictions on the ages and sexes to which the cause applies. We also list the implied age restriction on YLDs for the MCI-AD state of the cause model below. .. list-table:: GBD 2021 Cause Restrictions :widths: 15 15 20 :header-rows: 1 * - Restriction Type - Value - Notes * - Male only - False - * - Female only - False - * - YLL only - False - * - YLD only - False - * - YLL age group start - 40 to 44 - age_group_id = 13 * - YLL age group end - 95 plus - age_group_id = 235 * - YLD age group start - * 40 to 44 for AD-dementia cause state * No *a priori* age restriction for MCI-AD cause state - * Restriction to age_group_id = 13 (40 to 44) for AD-dementia cause state is from GBD. However, due to simulation dynamics, it is possible for simulants to enter this state before age 40. * In practice, the age start for MCI-AD will be age_group_id = 10 (25 to 29) because we will be adding simulants at most 10.2 years before AD incidence (so 40 -- 10.2 = 29.8, in the 25-29 age group) * - YLD age group end - 95 plus - age_group_id = 235 Vivarium Modeling Strategy ++++++++++++++++++++++++++ For :ref:`Model 4 <2025_alzheimers_model_runs_table>` of the :ref:`CSU Alzheimer's simulation <2025_concept_model_vivarium_alzheimers>`, we will add two pre-dementia states to the Alzheimer's disease model. The model still functions similar to an SI model, but now there are multiple with-condition states, with unidirectional progression between them. In Model 4 we used the incidence and prevalence for GBD's "Alzheimer's disease and other dementias" (cause ID 543), so our numbers were inflated by the "other dementias" part. In :ref:`Model 5 <2025_alzheimers_model_runs_table>`, we remove the "other dementias" from the disease model. To do this, the dementia modelers recommended *not* using the published GBD data directly, but to start with the GBD 2023 "dementia envelope" data from DisMod, and multiply by proportion of the envelope due to Alzheimer's disease. The estimates of the proportions of the envelope due to each dementia subtype are unpublished as of September 2025, but the modelers shared a .csv file we can use as long as we don't expose the raw numbers. See the `Data Values and Sources`_ section below for details. Cause Model Diagram ------------------- .. graphviz:: digraph AlzheimersDisease { rankdir=LR; bbbm [label="BBBM-AD"] mci [label="MCI-AD"] ad [label="AD-dementia"] S -> bbbm [label="i_BBBM"] bbbm -> mci [label="i_MCI"] mci -> ad [label=i_AD] } .. list-table:: State Definitions :widths: 5 5 20 :header-rows: 1 * - State - State Name - Definition * - S - Susceptible - Simulant does not have Alzheimer's disease or any of its precursors * - BBBM-AD - Blood-Based-Biomarker-presymptomatic Alzheimer's Disease - Simulant has presymptomatic Alzheimer's disease that is detectable using blood-based biomarkers * - MCI-AD - Mild Cognitive Impairment due to Alzheimer's Disease - Simulant has mild cognitive impairment due to Alzheimer's disease * - AD-dementia - Alzheimer's Disease dementia - Simulant has mild, moderate, or severe dementia due to Alzheimer's disease * - Death (not pictured) - Death - Simulant has died .. list-table:: Transition Definitions :widths: 5 5 10 10 :header-rows: 1 * - Transition - Transition Name - Definition - Notes * - i_BBBM - BBBM incidence hazard - Incidence hazard of BBBM-AD - This will be equal to GBD's incidence rate of Alzheimer's disease and other dementias, but with the age group and year shifted backward by the average duration of the BBBM-AD and MCI-AD states combined, and inflated to account for deaths in those two states * - i_MCI - MCI incidence hazard - Incidence hazard of MCI due to AD - This will be a **time-dependent hazard rate**, depending on how long a simulant has been in the BBBM-AD state, not a constant hazard like we usually use * - i_AD - AD dementia incidence hazard - Incidence hazard of Alzheimer's disease dementia - We will define this as a constant hazard rate for simulants in MCI-AD * - m_X (not pictured) - Mortality hazard in state X - Total mortality hazard for simulants in cause state X - X is a variable representing an arbitrary cause state State and Transition Data ------------------------- The tables in this section describe the data needed for the cause model drawn in the `Cause Model Diagram`_ section above. The variables in the tables are defined in the the `Data Values and Sources`_ section below. The following tables describe the data for each state and transition if modeling only simulants with AD dementia or pre-dementia AD as described in the :ref:`Alzheimer's population model `: .. _2021_cause_alzheimers_presymptomatic_mci_state_data_table: .. list-table:: State data when modeling only simulants with AD dementia or pre-dementia AD :header-rows: 1 * - State - Initial prevalence - Entrance prevalence - Excess mortality rate - Disability weight * - S - 0 - 0 - 0 - 0 * - BBBM-AD - :math:`\Delta_\text{BBBM} / \Delta_\text{(all AD states)}` - 1 - 0 - 0 * - MCI-AD - :math:`\Delta_\text{MCI} / \Delta_\text{(all AD states)}` - 0 - 0 - :math:`\text{DW}_\text{MCI}` * - AD-dementia - :math:`\Delta_\text{AD} / \Delta_\text{(all AD states)}` - 0 - emr_c543 - :math:`\text{DW}_\text{c543}` **Note:** The variable :math:`\Delta_\textsf{X}` denotes the average duration in cause state X, as defined in the :ref:`data values and sources table below <2021_cause_alzheimers_presymptomatic_mci_data_sources_table>`. .. _2021_cause_alzheimers_presymptomatic_mci_transition_data_table: .. list-table:: Transition Data :header-rows: 1 * - Transition - Source State - Sink State - Value * - i_BBBM - S - BBBM-AD - Not explicitly used because we're not modeling susceptible simulants. Defined implicitly in the :ref:`Alzheimer's population model `, which computes how many simulants to add into the BBBM-AD state on each time step. * - i_MCI - BBBM-AD - MCI-AD - :math:`h_\text{MCI}(t - T_\text{BBBM})`, where :math:`t` is the current time in the simulation, and :math:`T_\text{BBBM}` is the time the simulant entered the BBBM-AD state. Adjusted in :ref:`intervention_hypothetical_alzheimers_treatment` scenario. * - i_AD - MCI-AD - AD - :math:`1 / \Delta_\text{MCI}` * - m_X - X - Death - acmr --- csmr_c543 + emr_X **Note:** :math:`h_\text{MCI}` is the time-dependent hazard function for transitioning into MCI-AD, defined in the :ref:`data values and sources table below <2021_cause_alzheimers_presymptomatic_mci_data_sources_table>`. Because i_MCI is defined in terms of a non-constant hazard function :math:`h_\text{MCI}`, simulants initialized into the BBBM-AD state will need to be assigned a value for :math:`T_\text{BBBM}` to determine how long they have been in that state. For simulants in BBBM-AD at time :math:`t=0`, assign :math:`T_\text{BBBM}` uniformly in the interval :math:`[-\Delta_\text{BBBM},\, 0]`. .. _alzheimers_cause_state_data_including_susceptible_note: .. attention:: If we model the entire population including susceptible simulants, the state data should be modified as follows. Define :math:`p_\textsf{X}` to be the prevalence of cause state X in the total population including susceptible simulants, and define :math:`p_\text{(all AD states)}` to be the sum of :math:`p_\textsf{X}` for the three AD cause states X. Then multiplying the prevalence of each AD state in the :ref:`above state data table <2021_cause_alzheimers_presymptomatic_mci_state_data_table>` by :math:`p_\text{(all AD states)}` gives the prevalence of that state in the entire population. Since we know that .. math:: \begin{align*} p_\text{AD} &= \text{prevalence_AD} \\ &= \text{prevalence_m24351} \times \text{proportion_AD}, \end{align*} the prevalence of AD dementia computed from GBD's dementia envelope (see :ref:`data values and sources table below <2021_cause_alzheimers_presymptomatic_mci_data_sources_table>`), we can solve to obtain .. math:: :label: prevalence_all_AD_states_eq p_\text{(all AD states)} = \frac{\Delta_\text{(all AD states)}}{\Delta_\text{AD}} \cdot \text{prevalence_AD} \quad\text{(for ages 40+)}. Note that since the GBD prevalence applies to a given demographic group, so does the formula for :math:`p_\text{(all AD states)}`. The above formula applies to age groups 40+ since this is where prevalence_AD and :math:`\Delta_\text{AD}` are nonzero. For ages 30--39, use the value of :math:`p_\text{(all AD states)}` for age group 40--44; for ages <30, set :math:`p_\text{(all AD states)} = 0`. The following state data table shows the resulting initial prevalences when modeling the total population, as well as the birth prevalences, which replace the entrance prevalences. The excess mortality rate and disability weight of each state remain the same. .. list-table:: State data when modeling entire population including susceptible simulants :header-rows: 1 * - State - Initial prevalence - Birth prevalence * - S - :math:`1 - p_\text{(all AD states)}` - 1 * - BBBM-AD - :math:`\frac{\Delta_\text{BBBM}}{\Delta_\text{(all AD states)}} \cdot p_\text{(all AD states)}` - 0 * - MCI-AD - :math:`\frac{\Delta_\text{MCI}}{\Delta_\text{(all AD states)}} \cdot p_\text{(all AD states)}` - 0 * - AD-dementia - :math:`\frac{\Delta_\text{AD}}{\Delta_\text{(all AD states)}} \cdot p_\text{(all AD states)}` - 0 .. note:: Although we will not need all the values in this table for Model 4, the value of :math:`p_\text{(all AD states)}` defined in :eq:`prevalence_all_AD_states_eq` **will be needed in order to compute the model scale and initialize the correct number of simulants in each demographic subgroup.** Note that in the notation on the :ref:`Alzheimer's population model page `, :math:`p_\text{(all AD states)}` refers to the prevalence within the entire population of a location, including all age groups and sexes. On the other hand, if we compute prevalence_AD for a specific demographic subgroup :math:`g` (e.g., a single age group and sex) and year :math:`t`, then :math:`p_\text{(all AD states)}` as computed in :eq:`prevalence_all_AD_states_eq` corresponds to :math:`p_{g,t}` on the Alzheimer's population model page. Data Values and Sources ----------------------- Unless otherwise noted, all data values depend on year, location, age group, and sex, as defined by GBD. The following paths on the cluster contain the data files listed in the table below: * :file:`population_agg.nc` and :file:`mortality_all.nc` from FHS team * :file:`squeezed_proportions_to_sim_sci.csv` from dementia modelers * :file:`all.hdf` disability weight file saved by Simulation Science team .. code-block:: bash # Data folder for Alzheimer's sim, including data from FHS team and # dementia modelers (see README.txt for data provenance) /mnt/team/simulation_science/pub/models/vivarium_csu_alzheimers/data # Disability weights saved by Simscience team: /mnt/team/simulation_science/costeffectiveness/auxiliary_data/GBD_2021/02_processed_data/disability_weight/sequela/all/all.hdf .. _2021_cause_alzheimers_presymptomatic_mci_data_sources_table: .. list-table:: Data values and sources :widths: 20 30 25 25 :header-rows: 1 * - Variable - Definition - Source or value - Notes * - proportion_AD - The proportion of the dementia envelope that is Alzheimer's disease dementia - :file:`squeezed_proportions_to_sim_sci.csv` - Point estimate stratified by age group and sex for ages 40+. Includes proportions for all subtypes of dementia --- filter to type_label == "Alzheimer's disease". **Note:** These estimates were provided by the dementia modelers and are not yet published, so they should not be stored directly in the Artifact or any other public location. * - prevalence_m24351 - Prevalence of GBD 2023 dementia envelope - get_draws( source="epi", gbd_id_type = "modelable_entity_id", gbd_id=24351, release_id=16, year_id=2023, measure_id=5 ) - The dementia envelope represents the combined prevalence all types of dementia. By contrast, the GBD cause "Alzheimer's disease and other dementias" (c543) does not include certain dementias that result from other modeled GBD causes. * - prevalence_AD - Prevalence of AD dementia in total population - prevalence_m24351 :math:`\times` proportion_AD - * - :math:`p_\textsf{X}` - Prevalence of cause state X in total population - Defined in the "Initial prevalence" column of the state data table in the :ref:`Attention box above ` - By definition, :math:`p_\text{AD} =` prevalence_AD, and :math:`p_\text{BBBM}` and :math:`p_\text{MCI}` are derived from this * - :math:`p_\text{(all AD states)}` - Prevalence of all stages of AD combined - Defined in :eq:`prevalence_all_AD_states_eq` above - Equals :math:`p_\text{BBBM} + p_\text{MCI} + p_\text{AD}` * - incidence_m24351 - Total-population incidence rate for GBD 2023 dementia envelope - get_draws( source="epi", gbd_id_type = "modelable_entity_id", gbd_id=24351, release_id=16, year_id=2023, measure_id=6 ) - Raw value from get_draws, different from susceptible-population incidence rate automatically calculated by Vivarium Inputs * - incidence_AD - Total-population incidence rate of AD dementia - incidence_m24351 :math:`\times` proportion_AD - Used in :ref:`AD population model ` to calculate BBBM-AD incidence. We are assuming the prevalence proportions can be applied to incidence. We are assuming the AD-dementia incidence rate is constant over time in each demographic group. * - acmr - All-cause mortality rate - :file:`mortality_all.nc` - Draw-level, age-specific forecasts from GBD 2021 Forecasting Capstone. See `Abie's population and mortality forecasts notebook`_ for a demonstration of how to load and transform the ``.nc`` file * - population_forecast - Forecasted average population during specified year - :file:`population_agg.nc` - Draw-level, age-specific forecasts from GBD 2021 Forecasting Capstone. Numerically equal to person-years. Used in :ref:`AD population model ` to calculate BBBM-AD incidence counts. See `Abie's population and mortality forecasts notebook`_ for a demonstration of how to load and transform the ``.nc`` file. * - :math:`\text{population}_{2021}` - Average population during the year 2021 - get_population - Point estimate. Used only for the calculation of csmr_c543 by Vivarium Inputs * - :math:`\text{deaths_c543}_{2021}` - Deaths from Alzheimer's disease and other dementias in 2021 - codcorrect - Used only for the calculation of csmr_c543 by Vivarium Inputs * - csmr_c543 - Cause-specific mortality rate for Alzheimer's disease and other dementias - :math:`\frac{\text{deaths_c543}_{2021}}{(\text{population}_{2021}) \cdot (\text{1 year})}` - Calculated automatically by Vivarium Inputs. Assumed to remain constant over time in each demographic group. * - :math:`\text{prevalence_c543}_{2021}` - Prevalence of Alzheimer's disease and other dementias in 2021 - como - Used only for calculation of emr_c543 by Vivarium Inputs * - emr_c543 - Excess mortality rate for Alzheimer's disease and other dementias - :math:`\frac{\text{csmr_c543}}{\text{prevalence_c543}_{2021}}` - Calculated automatically by Vivarium Inputs. Assumed to remain constant over time in each demographic group. * - emr_X - Excess mortality rate in cause state X - Values listed in "Excess mortality rate" column of :ref:`state data table above <2021_cause_alzheimers_presymptomatic_mci_state_data_table>` - * emr_S, emr_BBBM, emr_MCI, emr_AD * - m_X - Mortality hazard in cause state X - acmr --- csmr_c543 + emr_X - * m_S, m_BBBM, m_MCI, m_AD * See :ref:`Mortality Impacts ` section of cause model design page * - sequelae_c543 - Sequelae of Alzheimer's disease and other dementias - Set of 3 sequelae: s452, s453, s454 - Obtained from gbd_mapping. Sequela names are "Mild," "Moderate," or "Severe Alzheimer's disease and other dementias," respectively. Same for all years, locations, age groups, and sexes. * - :math:`\text{prevalence}_s` - Prevalence of sequela :math:`s` - como - * - :math:`\text{DW}_s` - Disability weight of sequela :math:`s` - :file:`all.hdf` disability weight file in our team's auxiliary data - Disability weights are stored as draws and do not vary by year, location, age group, or sex. For reference, the values are: - s452: 0.069 (0.046-0.099) - s453: 0.377 (0.252-0.508) - s454: 0.449 (0.304-0.595) * - :math:`\text{DW}_\text{c543}` - Average disability weight of AD-dementia - :math:`\sum_\limits{s\in \text{sequelae_c543}} \text{DW}_s \cdot \text{prevalence}_s` - Prevalence-weighted average disability weight over sequelae, computed automatically by Vivarium Inputs. Used to calculate YLDs. * - :math:`\text{DW}_\text{motor}` - Disability weight for health state "motor impairment, mild" - :file:`all.hdf` disability weight file in our team's auxiliary data - Disability weights are stored as draws and do not vary by year, location, age group, or sex. See `Abie's disability weight notebook`_ for details on pulling the correct value. * - :math:`\text{DW}_\text{motor+cog}` - Disability weight for health state "motor plus cognitive impairments, mild" - :file:`all.hdf` disability weight file in our team's auxiliary data - Disability weights are stored as draws and do not vary by year, location, age group, or sex. See `Abie's disability weight notebook`_ for details on pulling the correct value. * - :math:`\text{DW}_\text{MCI}` - Disability weight of mild cognitive impairment - :math:`\frac{\text{DW}_\text{motor+cog} - \text{DW}_\text{motor}} {1 - \text{DW}_\text{motor}}` - Disability weights are stored as draws and do not vary by location, age group, or sex. For reference, the value is * 0.021 (0.013, 0.032) Obtained by removing DW of "motor impairment, mild" from DW of "motor plus cognitive impairments, mild," at the draw level. See `Abie's disability weight notebook`_ for details, and see the :ref:`derivation below ` for further explanation. * - :math:`T_X` - The time at which a simulant enters the cause state :math:`X` - Determined within the simulation - Random variable for each simulant. :math:`T_\text{BBBM}` is used to determine how long a simulant has been in the BBBM-AD state, in order to compute the hazard rate of transitioning to MCI-AD at a given simulation time :math:`t`. * - :math:`D_\text{BBBM}` - Dwell time in cause state BBBM-AD - :math:`T_\text{MCI} - T_\text{BBBM}` - Random variable for each simulant, constructed implicitly through simulation dynamics to have approximately a `Weibull distribution`_ with shape parameter :math:`k` and scale parameter :math:`\lambda` * - :math:`k`, :math:`\lambda` - Shape and scale parameters, respectively, of Weibull distribution for :math:`D_\text{BBBM}` - * :math:`k = 1.22` * :math:`\lambda = 6.76` - Chosen to match client's specification for :math:`D_\text{BBBM}`: The probability of progression from BBBM-AD to MCI-AD is about 50% at 5 years and 80% at 10 years, corresponding to an average annual rate of progression of approximately 15% . Use the same parameters for all years, locations, age groups, and sexes. * - bbbm_dist - Python object representing the Weibull distribution for :math:`D_\text{BBBM}` - scipy.stats.weibull_min(k, scale=λ) - An instance of `SciPy's Weibull distribution class`_. * - :math:`h_\text{MCI}(t)` - Hazard function for transitioning into the MCI-AD state from BBBM-AD - * bbbm_dist.pdf(t) / bbbm_dist.sf(t), or * exp( bbbm_dist.logpdf(t) --- bbbm_dist.logsf(t) ), an equivalent expression that may help avoid underflow - Equal to :math:`\frac{k}{\lambda} \left(\frac{t}{\lambda}\right)^{k-1}`, but can also be computed as the ratio of the probability density function to the survival function, using the methods defined in `SciPy's Weibull distribution class`_ * - :math:`\Delta_\text{BBBM}` - Average duration of BBBM-presymptomatic AD in the absence of mortality - bbbm_dist.mean() - Equal to :math:`\lambda \Gamma(1 + 1/k)`, where :math:`\Gamma` is the `gamma function`_. Can be computed using `scipy.special.gamma`_, but using bbbm_dist.mean() is more general if we update the underlying distribution. Does not vary by year, location, age group, or sex. * - :math:`\Delta_\text{MCI}` - Average duration of MCI due to AD in the absence of mortality - 3.85 years - Obtained from Table 3 in `Potashman et al.`_, assuming a constant hazard rate of transitioning to AD-dementia. Corresponds to an annual conditional probability of 0.771 of staying in MCI-AD given that you don't die within one year, since :math:`\exp(-1 / 3.85) \approx 0.771`. Does not vary by year, location, age group, or sex. **Note:** The paper reports a 68.2% chance of staying in MCI and a 5.3% chance of returning to asymptomatic---these probabilities have been combined to get an annual probability of 73.5% of staying in MCI since our model assumes that a backwards transition is not possible. The conditional probability above is computed as :math:`0.771 = 0.735 / (1 - 0.047)` since the paper reports a 4.7% chance of dying within a year when starting in the MCI state. * - :math:`\Delta_\text{AD}` - Average duration of AD-dementia - * prevalence_AD / incidence_AD for ages 40+ * 0 for ages under 40 - Follows from the steady-state equation (prevalent cases) = (incident cases) x (average duration). Note that the denominator is the **raw total-population incidence rate from GBD**, not the susceptible-population incidence rate usually returned by Vivarium Inputs. This is because we want the total-population person-time in the denominators of prevalence and incidence to cancel out, leaving a ratio of counts. * - :math:`\Delta_\text{(all AD states)}` - Average duration of all stages of AD combined if there is no mortality in the BBBM-AD and MCI-AD stages - :math:`\Delta_\text{BBBM} + \Delta_\text{MCI} + \Delta_\text{AD}` - .. _Abie's population and mortality forecasts notebook: https://github.com/ihmeuw/vivarium_csu_alzheimers/blob/39fe76203a8031da7983bcb5d8824216a61b5d43/src/vivarium_csu_alzheimers/data/population_forecasts/2025_08_12a_alz_artifact_forecast_population_and_mortality.ipynb .. _Abie's disability weight notebook: https://github.com/ihmeuw/vivarium_research_alzheimers/blob/4d5dde0b74eb09ea997af7c2de88b81670ba7d61/2025_08_03a_alz_dw_explore.ipynb .. _gamma distribution: https://en.wikipedia.org/wiki/Gamma_distribution .. _Weibull distribution: https://en.wikipedia.org/wiki/Weibull_distribution .. _SciPy's gamma distribution class: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gamma.html .. _SciPy's Weibull distribution class: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.weibull_min.html .. _gamma function: https://en.wikipedia.org/wiki/Gamma_function .. _scipy.special.gamma: https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.gamma.html .. _Potashman et al.: https://doi.org/10.1007/s40120-021-00272-1 .. _alzheimers_mci_disability_weight_derivation: Deriving a disability weight for MCI ------------------------------------ .. todo:: Derive the formula for the disability weight of MCI, and include Abie's plot comparing DWs of various relevant health states.