Data Sources:

Census

- Census data collection

 2 forms

- De Jure: Based on where you normally reside

- De Facto: Based on wherever you happen to be at the time of the census

        - Strengths: time gap, simplicity, local level info

       - Weaknesses: expensive, coverage errors, classification errors, reporting errors

Data Sources:

 Vital statistics

Vital statistics include information on births, deaths, marriages, and divorces

- Events recorded as they occur

                - Strengths: continuous collection, covers everyone (in theory)

                - Weaknesses: Limited information, delayed reporting

Data Sources:

Household registration surveys

Household registration surveys 

   - Maintained in only a few countries

                        - Strengths: Full coverage

                        - Weaknesses: Typically not available, limited information

Data Sources:

Sample surveys

Sample surveys

                        - Cross-sectional versus longitudinal

                        - Retrospective versus prospective

- Strengths: Large amount of information

- Weaknesses: Non-response, problems magnified as one looks at smaller areas or smaller groups

- Demographers’ most basic task is to assess the quality of their data

Rate: number of occurrences / person-years of exposure to risk of occurrence

- It is often better to express demographic processes in terms of rates rather than in absolute numbers because rates consider both the size of the population and the amount of time over which an event can occur

Ratio: number of occurrences / number of people

        - Doesn’t account for duration of exposure

 Probability: number of events / initial population at risk

- A rate is different because it takes into account the people that exited the population at risk during the specified period of time

- We often use the mid-period or mid-year population to estimate the total number of person-years lived during a period

Recovering Person-Years Lived between 0 and T

- When we have daily counts of the population:

  - PY(2009, 2010) =ΣNi*1/365

 - This is the ideal option

- When we know N(0) and N(T) we have two options:

1  Assume constant growth, in which case we know that (N(T) – N(0)) / PY(0,T) = ln(N(T) / N(0)) / T and we can solve for PY(0,T)

2. Assume linear growth, in which case PY(0,T) = [N(0) – N(T)) / 2]*T

When we only know the mid-period population:

- Assume that the population grows linearly, in which case PY(0,T) = [(N(0) – N(T)) / 2]*T = N(T/2) *T, where N(T/2) is the mid-period population

- In other words, the mid-period population will be an accurate estimation of the person-years lived during the period if the population grows linearly

- If the population grows (or shrinks) exponentially, the mid-period population will always underestimate the actual number of person-years lived

- If we want to estimate population growth that is compounded at non-instantaneous intervals, we just use the following equation:

            - N(T) = N(0)*[1 + r/j]^j*t

- j is the number of times per year compounding occurs, and T is the number of years in the interval

- If we want to use this equation to estimate the growth rate from observed population size:

            r = j((N(T) / N(0))^1/j*t – 1)                                  

Incidence rate:

Prevalence rate:

Point prevalence

- Measures the number of cases at a single point in time

- Used when calculating prevalence of chronic illnesses, because extending the period of measurement beyond a day would have little effect on total number of cases

Period prevalence

-Measures the number of cases occurring over a longer period of observation, such as a month

- Used when calculating acute illnesses that only last days or weeks

Relative risk:

Incidence for people exposed to some risk factor divided by incidence for people not exposed to risk factor

_nq_x =

=(n* _nm_x) / [1+(n - _na_x)* _nm_x]

Period life tables

indicate what would happen to a cohort if it were subject throughout its life to the mortality conditions for that period

- In other words, it is a life table for a synthetic, or hypothetical, cohort

- The key to period life table construction is to convert _nM_x  to _nq_x

- This conversion typically derives from the relation between _nm_x  and _nq_x in an actual cohort

Cohort  life tables

When data are available, constructing life tables for cohorts can be very simple

However, cohort data may be unavailable, outdated, or incomplete

 Direct observation

- However, this information is rare, and even if it is available it is not always ideal because _na_x values are influenced by the age distribution of the population within the interval

Graduation from the _nm_x function

- The level and slope of the _nm_x function give clues about _na_x

- Although we don’t know the slope of _nm_x within an age interval, we typically know it across intervals

- Given the same _nm_x, _na_x will be greater when the slope is steeper as this indicates that mortality risk rises with age (both within and across intervals)

- Given the same slope, _na_x will be smaller for the group with higher mortality because more die out early

Borrowing _na_x from another population

- We can borrow the values from another population if there’s reason to think that _nm_x is similar to the population for which _na_x is known

Using rules of thumb

- One rule of thumb is that except for infancy and sometimes age 1, use  _na_x = n/2, which tells us that _nq_x = 2*n* _nm_x / 2 + n* _nm_x

- Any amount of error will be trivial if data are arranged by single years of age

- Another rule of thumb is to assume that death rates are constant in the interval, which implies that _na_x = n + (1/ _nm_x) – (n / 1 – e^{-n* nmx})

- With this method _na_x will be less than n/2 because when a death rate is constant, the number of deaths at a moment is proportional to the number of survivors, and this declines throughout the interval

Consequently, this rule actually works between for applications other than mortality, because we know that the mortality curve tends to be upward sloping after age 30

Which method is best depends on data quality and demographic conditions

The _na_x function in the in the very young ages

- Life expectancy estimates are most sensitive to procedures used in the very young, high mortality ages

- The lower the level of mortality, the more heavily infant deaths are concentrated at the earliest ages of infancy; the prenatal and perinatal environments dominate relative to the post-natal environment

- In their model West life tables, Coale and Demeny (1983) provide a set of _na_x values to use in the 0-1 and 1-4 age intervals based on the observed values of _nm_x in the population

- Traditionally, the open ended interval has begun at age 85, but in developed countries today about 50% of women survive to age 85 which has increased the need for 5 year intervals beyond age 85

- Best to use an open ended interval at an age to which only a small fraction of the population survives

Coale & Demeny 1983 

Stationary population

3 assumptions of a stationary population

                        1. Age-specific death rates are constant over time

                        2. Flow of births is constant over time

                        3. Net migration is zero at all ages

We can use life-table notation to describe various functions of the stationary population

CBR = #births/total size of population = l₀ / T₀ = 1 / e₀

This tells use that in a stationary population CBR = CDR = 1 / e₀

- An analogous relationship exists for any age x and above

- # of persons above age x is constant, so # if persons arriving at age x each year (l_x) much equal the number of persons dying above age x that year  

- Every population can form the basis of a model stationary population; i.e. the population that would emerge eventually if age-specific mortality rates remained constant, births were constant, and there was no migration

- More generally, any “population” has a set of attrition rates that describe the process of leaving the population and that can be arranged by duration of membership

-In a stationary population, the mean at age death equals life expectancy at birth and

_nN_x = _nL_x

In a population that’s not stationary the mean age of persons dying in a period can be very different from life expectancy at birth because the same _nm_x are being applied to two age structures that are very different from each other (the age structure of the life table (_nL_x) versus the age structure of the population (_nN_x))

- For example, if births are growing over time, then the actual age distribution is younger than the age structure of the population implied by current mortality rates

- As a result, the ratio of persons aged 1 to persons aged 30 in the actual population will be higher than in the stationary population

Consequently, the mean age of people dying will be lower than life expectancy at birth because more people will be dying at younger ages since there simply are more people at younger ages

The life table displays mortality functions only for discrete age intervals, however intuitively mortality is a continuous process

- The force of mortality (μ(x)) represent the death rate at a moment in time

            - μ(x) = limit as n-> 0   _nd_x / _nL_x = lim as n->0 _nm_x

- From this we can derive the following equation:

            - l(z) = l(y) *e ^-∫^z_y μ(x)dx

- This tells us that the proportionate change in cohort size between y and z is a function of the force of mortality between those ages

- Furthermore, what matters in the sum of the forces of mortality, not their order

            - All life table functions can be expressed in terms of l(x) and μ(x)

            - Cohort versus population death rates

- For a cohort, _nm_x equals a weighted average of the force of the mortality function between ages x and x+n, with the weights being supplied by the l(a) function (the number of survivors in the cohort at age a between x and x+n)

- In the population, _nM_x equals a weighted average of the force of the mortality function with the weights supplied by the N(a) function (the number of the people in the population age a between x and x+n)

- Even when a cohort and a population have exactly the same force of mortality function in the interval x to x+n, we cannot be sure that _nm_x = _nM_x

There are only two conditions were we can be certain that _nm_x = _nM_x

1. μ(x) is constant in the interval x to x+n, in which case the weights won’t matter

2. N(a) is proportion to l(a) throughout x to x+n

- This condition will apply whenever N(a) is stationary throughout the interval

- In many cases neither of these conditions will hold, therefore we may want to think twice before plugging in _nM_x values for a hypothetical cohort

- The most common source of problems is positive growth rates, becauseof increasing births or declining mortality

- This results in a population being younger than a hypothetical cohort and _nm_x will exceed _nM_x because the latter is biased downward by a youthful age structure

Standard methods of life table calculating life table estimates arose from a series of important developments

-Most dealt with the estimation of probabilities from rates

- A rate reflects the number of events divided by the total number of person-years at risk of experiencing the event over the interval (by age)

- A probability reflects the number of events divided by the total population at risk of experiencing the event at the start of the interval (by age)

- If we only have repeated cross-sections (i.e. prevalence rates of being in a given state) we can use Sullivan’s (1971) method

Logic of Sullivan’s method

- Allocate _nL_x values from the period life table across categories of interest

- Calculate T_x and e_x values using standard methodology

- Difference is that each of these values is now state-specific

- e(i)_x = T(i)_x / l_x ; i.e. the number of years one can expect to be in state ‘i’ above age x

- T must be specific to ‘i’ but l may or may not be

To use the Sullivan method, we need:

- An existing life table representing mortality (tells us _nL_x and l_x)

- Cross-sectional prevalence estimates of characteristics of interest

Issues with Sullivan method

- Assumptions about point-in-time estimates extending across years

- Comparability of cross-sectional estimates over time and space

- Variation in response to different types of data collection

- When using the Sullivan method authors often calculate variance estimates of the parameters of interest

- To do this, we use the variance of the prevalence rates to estimate the overall variance of the calculated life table values

- When comparing two population, calculate a z score to see if the difference in the characteristics of interest is statistically significant

- Purpose of book is to outline the history of population growth and to understand the mechanisms that have contributed to population growth, stagnation, or decline

Population size can be viewed as a proxy for prosperity

The size of the human species varies relatively slowly compared to other species

The growth potential of a population is a fxn of

1) # of births per woman and

2) life expectancy at birth

- Increasing life expectancy even beyond reproductive age might have biological effects (example: helping to care for the younger generation)

- High life expectancy and low TFR or low life expectancy and high TFR may produce = population growth rates

            - (20) Population growth is limited by the availability of resources

- The Neolithic revolution (i.e. the development of agriculture) -> increased potential for human growth

- The Industrial Revolution had a similar effect

- One environmental factor that continues to check population growth is degradation stemming from technology

- (30) Forces that constrain population growth include climate, disease, land, energy, food, space, and settlement patterns

- Choices that affect population growth include nuptiality, fertility, and migration

- (33) There exists a debate over why population growth occurred in the Neolithic period

- The classic hypothesis states that increased nutrition -> decreased mortality

-The alternative hypothesis states that mortality actually increased due to less varied diet and increased transmission of diseases, but that fertility increased by a larger amount

What were the effects of various historical “shocks” on population growth?

- The Black plague greatly reduced the size of the European population; took nearly 400 years for the population to rebound to pre-plague levels

- Smallpox brought by Spanish invaders to the New World came close to decimating many indigenous populations

          Why did mortality decline between 1750 and 1850?

- McKeown hypothesizes that it was due to increased nutrition, but this is debatable