David A. Swanson
Department of Sociology
University of California Riverside
Riverside, California 92521 USA
E-mail: [email protected]
&
Lucky Tedrow
Department of Sociology
Western Washington University
Bellingham, Washington 98225 USA
Email: [email protected]
THE TOP TEN REASONS TO USE THE COHORT CHANGE RATIO METHOD
1
OUTLINE
1. OVERVIEW
2. COHORT CHANGE RATIOS
3. THE TOP TEN REASONS TO USE
THE COHORT CHANGE RATIO
METHOD
2
OVERVIEW The use of cohort change ratios (CCRs) has a long
history in demography. It can be traced at least as far
back as 1911 when Hardy and Wyatt used cohort
change ratios for generating a population projection
they needed to assess the cost of the initial beginning
of what became the national health insurance
program in the UK.
Under the rubric of “Census Survival Ratios,” they have been used to estimate life expectancy (Swanson
& Tedrow 2012) and under the rubric of the “Hamilton-
Perry” method, to make population projections (Hamilton & Perry 1962).
3
OVERVIEW
Although CCRs have been around for at least 100 years
(Hardy & Wyatt,1911), we believe that many of the their
desirable features have been overlooked. We started
discovering (or more likely, “re-discovering”) these features because many projects on which we have
worked over the years called for the use of CCRs and
the more we used them, the more we learned about
their desirable characteristics. Thus, this presentation
covers ten of them. We are sure that there are more,
and, of course, there are less than desirable features,
but that discussion we will save for another time.
4
OVERVIEW
In this presentation, we are taking a cue from David
Letterman, the recently retired late night TV show
personality, whose “top ten” lists became a mainstay of his show. While his typical Top Ten List is far more
humorous than the one we will use in regard to cohort
change ratios, it is unavoidable that some humor may
seep in as we go along.
5
OVERVIEW
Before starting the list, which goes in reverse order
(from 10 to 1), a brief description of Cohort Change
Ratios (CCRs) is in order. It is grounded in their
typical use, which is in a population projection.
Keep in mind that due to time constraints the 10 items
I present will be summaries, lacking the details and
nuances, which are found in the citations or otherwise
available in the questions and answers segment
following the presentation.
6
COHORT CHANGE RATIOS
In general, a CCR can be described as follows:
nCCRx = nPx,t,i / nPx-k,t-k,i
where
nPx,t is the population aged x to x+n at time t for area i,
which is typically the most recent census
nPx-k,t-k is the population aged x-k to x-k+n at a
preceding point in time (t-k) for area i, which is typically
the 2nd most recent census
k is the number of years between the most recent
census at time t and the one preceding it at time t-k. 7
COHORT CHANGE RATIOS
The basic formula for projecting age cohorts is:
nPx+k,t+k,i = (nCCRx,i ) ×( nPx,t,i)
where
nPx+k,t+k is the projected population aged x+k to x+k+n
at time (t+k) for area i,
(nCCRx,i ) is the cohort change ration as described
earlier, and
nPx,t is the population aged x to x+n at the most
recent census (t) for area i. 8
COHORT CHANGE RATIOS
The preceding two steps, are widely known as the
Hamilton-Perry method, which projects a population
by age (and sex) from time (t) to time (t+k) using
CCRs computed from the two most recent censuses. It
consists of two steps. The first uses existing data to
develop CCRs and the second applies the CCRs to
the cohorts of the launch year population to move
them into the future.
9
COHORT CHANGE RATIOS
The second step can be repeated infinitely, with the
projected population serving as the launch population
for the next projection cycle. One also can calculate
CCRs over a period of time and measure trends in
them, which also can be used to modify expected
CCRs and generate projections.
10
COHORT CHANGE RATIOS
Given the nature of the CCRs, 10-14 is the youngest age
group for which projections can be made if there are 10
years between censuses. To project the populations aged
0-4 and 5-9, one can use the Child Woman Ratio (CWR), or
more generally a “Child Adult Ratio” (CAR). It does not require any data beyond the decennial census. For
projecting the population aged 0-4, CAR is defined as the
population aged 0-4 divided by the population aged 15-44.
For projecting the population aged 5-9, CAR is defined as
the population aged 5-9 divided by the population aged 20-49.*
*There are both other methods to obtain these age groups and other
“adult” age groups that could be used to define CAR.
11
12
Projections of the oldest open-ended age group differ
slightly from the CCR projections for the age groups beyond
between the age for which a CAR is needed and to oldest
closed age group. If, for example, the final closed age group
is 70-74, with 75+ as the terminal open-ended age group,
then calculations for the ∞CCR75,t require the summation of
the three oldest age groups to get the population age 75+ at
time t and the summation of the age groups that will yield
P65+ at time t-k:
∞CCR75,t,i = ∞P75+,t,i / ∞P65+,t-k,i.
The formula for projecting the population 75+ for the year
t+k is: ∞P75+,t+k,i = (∞CCR75+,t,i )× (∞P65+,t,i).
COHORT CHANGE RATIOS
THE TOP TEN REASONS
Now that you have an idea of Cohort Change Ratios
and an example of how they can be used, let’s turn to the list of top ten reasons to use them, starting with
reason no. 10.
13
REASON NO. 10
You only need two census counts of population by
age to generate a population forecast that can provide
age groups, as well as sex, race, and a host of other
ascribed and achieved characteristics.
Contrast this with the most widely used method used
to generate this information in a population projection,
the cohort-component method, which requires not only
a census count by age, but vital statistics data and
migration data.
14
REASON NO. 10
Here is an example of a forecast of Australia for 2011,
using CCRs from 2001-2006 and an ex post facto
evaluation of its accuracy.
The data are taken from the US Census Bureau’s International Data Base (U.S. Census Bureau 2010).
15
16
"Hamilton-Perry" Method Forecast for Australia, 2011
IDB 2001 IDB 2006 CCR
2011
FORECAST IDB 2011
Difference
(Forecast -
IDB)
Percent
Difference
Total
Population:
0 to 4
years 1,256,383 1,276,793 0.30003 1,323,139 1,332,208 -9,069 -0.68%
Total
Population:
5 to 9
years 1,323,286 1,288,608 0.29136 1,314,999 1,315,041 -42 0.00%
Total
Population:
10 to 14
years 1,321,780 1,362,668 1.02976 1,326,958 1,331,143 -4,185 -0.31%
Total
Population:
15 to 19
years 1,344,763 1,373,989 1.03950 1,416,492 1,436,980 -20,488 -1.43%
Total
Population:
20 to 24
years 1,356,820 1,432,510 1.06525 1,463,643 1,517,354 -53,711 -3.54%
Total
Population:
25 to 29
years 1,469,451 1,449,055 1.06798 1,529,890 1,566,147 -36,257 -2.32%
Total
Population:
30 to 34
years 1,474,809 1,541,150 1.04879 1,519,759 1,526,837 -7,078 -0.46%
Total
Population:
35 to 39
years 1,464,614 1,528,156 1.03617 1,596,897 1,596,007 890 0.06%
Total
Population:
40 to 44
years 1,464,549 1,498,947 1.02344 1,563,979 1,559,961 4,018 0.26%
Total
Population:
45 to 49
years 1,347,381 1,478,682 1.00965 1,513,412 1,506,274 7,138 0.47%
Total
Population:
50 to 54
years 1,294,439 1,344,797 0.99808 1,475,846 1,462,914 12,932 0.88%
Total
Population:
55 to 59
years 1,000,399 1,279,736 0.98864 1,329,522 1,318,117 11,405 0.87%
Total
Population:
60 to 64
years 798,899 980,026 0.97964 1,253,674 1,244,644 9,030 0.73%
Total
Population:
65 to 69
years 659,591 768,912 0.96246 943,240 938,567 4,673 0.50%
Total
Population:
70 to 74
years 616,109 613,067 0.92947 714,677 714,288 389 0.05%
Total
Population:
75 years
and over 1,100,984 1,272,376 0.74101 1,397,124 1,400,229 -3,105 -0.22%
Total
Population 19,294,257 20,489,472 1.06195 21,683,251 21,766,711 -83,460 -0.38%
Data are from the US Census Bureau's International Data
Base ( http://www.census.gov/population/international/data/idb/informationGateway.php)
REASON NO. 10
REASON NO. 9
You can not only run the CCR Method forward in time
as a forecast, but also in reverse: A backcast
17
REASON NO. 9
Here is an example of a backcast used to estimate the
size of the Native Hawaiian population in 1778, the
year of first European Contact. The backcast starts
with Reverse Cohort Change Ratios for 1920 &
1910,using US Census data and produces decennial
estimates from 1900 to 1770, with an interpolated
estimate for 1778.
Before showing you the backcast, here is a set of
estimates from a range of sources. There is a wide
range in the estimates.
18
19
Exhibit 1. Example Range of Estimates of the Total Population of Hawai’i in 1778.*
Number Source Citation
200,000 Captain Dixon, visit of 1787
Schmitt (1968: 20)
242,000 Bligh, with Cook, 1st Visit, 1778
Schmitt (1968: 20)
200,000-250,000 Schmitt, 1971 Schmitt (1971)
300,000 Schmitt & Zane 1977 Nordyke (1989: 173)
400,000 King, with Cook, 2nd Visit, 1779
Adams (1937: 1)
450,000 Hommon, 2008 Hommon (2008:53)
500,000 Officers with Cook, 1st Visit, 1778
Schmitt (1968: 19)
800,000 – 1,000,000 Stannard, 1989 Stannard (1989: 50)
*There are more, often expressed as opinions concerning the initial estimates by Bligh, Dixon, King, and other British Naval officers, but most are in the range shown above (see, e.g., Schmitt, 1968: 18-23.
REASON NO. 9
20
It is not surprising that uncertainty would surround
the number of Hawaiians, a pre-literate population,
at the time of first European contact in the year
1778. No known census of this population at that
time exists and without a full count, the only
recourse is to estimate the size of this population.
REASON NO. 9
21
As can be seen in Exhibit 1, the estimates range
from 200,000 to 1,000,000. The retrospective
estimates by Schmitt and Stannard, as well as
some of those provided by the first Europeans
known to have contacted the Hawaiians, are
informed by methods and data; others are much
more speculative (Schmitt 1968: 18-22).
REASON NO. 9
REASON NO. 9
Here is the results for the size of the Native Hawaiian
population in 1778, the year of first European Contact.
A major benefit of using the Reverse Cohort Change
Ratios as a backcast is that the method is based in
data, the process is both simple and transparent, and
the results can be replicated. These characteristics are
not found in the other methods.
22
23
YEAR ESTIMATED CENSUS COUNT*
1900 29,336 29,799
1890 33,457 34,436
1880 39,711 N/A
1870 48,579 N/A
1860 61,931 67,084**
1850 80,574 82,035
1840 110,948 N/A
1830 149,297 N/A
1820 200,018 N/A
1810 267,971 N/A
1800 359,010 N/A
1790 480,978 N/A
1780 644,383 N/A
1778*** 683,200 N/A
1770 863,302 N/A
* Source: Schmitt(1968).
** The 1860 census did not distinguish between
Native Hawaiians and Part-Hawaiians.
*** 683,200 = 863,302*(er*8
), where r = -0.02925 =
[ln(644,383/863,302)]/10
TABLE 1. TOTAL POPULATION OF NATIVE
HAWAIIANS (IN HAWAI'I): 1900 TO 1770.
REASON NO. 9
REASON NO. 8
You can do small area projections with the CCR
Method. Here is an ex post facto evaluation of a 2010
projection of census tract 101 in Clark County, Nevada
(Las Vegas).
24
25
REASON NO. 8
REASON NO. 7
You can also use it for any population for which cohort
data are available over time, including populations that
are institutionally or administratively defined – school
enrollment by grade, for example.
You can do this in two ways, directly and “embedded” within a CCR Generated forecast by age.
As an example of the first way, “directly,” one can use school enrollment by grade for 2013 and 2014 to
develop grade progression ratios to forecast enrollment
by grade for 2015 by applying them to 2014 data.
26
REASON NO. 7
Here is an ex post facto evaluation of the example of
the first way, “directly,” using the fall, 2011 and fall, 2012 enrollment by grade for the Los Angeles County
(California) School District to project the fall, 2013
enrollment by grade and compare it with the reported
fall, 2013 enrollment by grade.”
27
28
REASON NO. 7
REASON NO. 7
As an example of the second way, one can embed the
enrollment data by grade within the corresponding age
groups and then use ratios (or changes in them) to
generate the enrollment once the age data are
generated: A “CCR & Shift-Share Projection.”
Here is an ex post facto of an evaluation of a “CCR & Shift-Share Projection” for the Enrollment of the Memphis (Tennessee) School District in 2010.
29
30
REASON NO. 7
31
REASON NO. 7
32
REASON NO. 7
33
REASON NO. 7
REASON NO. 6
It can be used to estimate life expectancy (Swanson
and Tedrow 2012)
34
35
The United Nations (2002: 6) shows that using the census survival method, that
the expectation of life at age x can be computed as
ex = (Tx/l(n/2))/( lx/l(n/2)) = Tx / lx [1]
where
x = age
n = the width of the age groups (up to, but not including the
terminal, open-ended age group)
ex = life expectancy (average years remaining) at age x
Tx = Total person years remaining to persons age x
lx = number reaching age x l(n/2) = persons aged x to x+n are assumed to be concentrated at
the mid-point of the age group
and
l(x+2n/2)/l(x-n/2) = P2(x,n)/P1(x-n,n) [2]
where
P2(x,n) = the number of persons counted in the second census
in age group x to x+n
P1(x-n,n) = the number of persons counted in the first census
In age group x-n to x
REASON NO. 6
36
In general, then, the life-table probability of surviving from the mid-point of one
age group to the next (l(x+2n/2)/l(x-n/2) ) is approximated by the census survival ratio (P2(x,n)/P1(x-n,n)).
Continuing, the same United Nations Manual (2002: 5-6) shows that the
cumulative multiplication of the probabilities shown in [2] gives the conditional
survival schedule lx/l(n/2). From the conditional lx values given by [2] the conditional estimates of the number of person years lived in each age group
(nLx) can be calculated as
nLx/l(n/2) = (n/2) *[(lx/l(n/2) + l(x+n)/l(n/2)] [3]
where
nLx = number of person years lived in each age group
Given a value of Tx/l(n/2) for some initial old age x, the UN shows that
total remaining years expected at age x (Tx) values can be calculated as:
T(x-n)/l(n/2) = Tx/l(n/2) + n L(x-n)/l(n/2) [4]
REASON NO. 6
37
This leads us back to equation [1], so that the expectation of life at age x using the
United Nations (2002) approach is:
ex = (Tx/l(n/2))/( lx/l(n/2)) = Tx / lx
In the Swanson-Tedrow approach, note that when the radix of a life table is equal to 1
(l0 = 1.00) life expectancy at birth can be computed directly from the expression:
e0 = S0 + (S0*S1) + (S0*S1*S2) +,...,+(S0*S1*S2,...,*Sx) [5] where
e0 = life expectancy at birth
S0 = survivorship from t=0 (e.g., birth) to t=1(e.g., age 1)
S1 = survivorship from t=1 (e.g., age 1) to t=2(e.g., age 2)
and so on through Sx and Sx = 1Lx/ 1L(x-n)
Equation [5] is set up for single year age groups. However, we can generalize it to
other age groups: nSx = nLx/ nL(x-n), so that
e0 = nS0 + (nS0*nS1) + (nS0*nS1*nS2) +,...,+(nS0*nS1*nS2,...,* nSx) [5.a]
As equation [5] and equation [5.a] both imply, the fundamental life table function is
inherent in our method. That is via the nSx values, we have nqx values.
REASON NO. 6
38
As equation [5] and equation [5.a] both imply, the fundamental life table function is inherent in this method. That is via the nSx values, we have nqx values. Recall, following,
e.g., Smith, Tayman, and Swanson (2013: 177) and using notation from equation [2], a
CCR can be generally defined as:
nCCRx = P2(x,n) /P1(x-n,n) [6]
The Swanson-Tedrow approach is the result of combining, on the one hand, either
equation [5] or [5.a] for computing life expectancy with, on the other hand, equation [6] in
order to estimate ex. Starting with
nSx = nLx/ nL(x-n) ≈ P2(x,n) /P1(x-n,n) [7]
We have, as shown in equation [5.a]
e0 = nS0 + (nS0*nS1) + (nS0*nS1*nS2) +,...,+(nS0*nS1*nS2,...,* nSx)
REASON NO. 6
39
As is the case with the more involved United Nations (2002) approach, this
approach will only work for populations for which migration is negligible, but there
are many areas of interest around the world where this is the case, or approximately
so (United Nations, 2002). The world as a whole meets this requirement. Countries with negligible migration include North Korea and Burma, among others. Other such
populations are found in the historical record - the former Soviet Union, Albania from
1950 to 1980, and the Peoples Republic of China from 1950 through 1970, for
example. Still others may be defined by race and ethnicity or other 'rules' of
membership (e.g., Indigenous Populations in Australia and Canada, Native Hawaiians; native-born populations).
Broadly speaking, the method can be applied to any population subject to renewal
through a single increment (birth) and extinction through a single decrement (death),
where there are at least two successive census counts that provide the population
by age. We also note that unlike the UN method, the approach we take can be used
to yield estimates of life expectancy at birth. Moreover, like the UN approach, this
one is not subject to the limitations imposed by stationary or even stable population
requirements.
REASON NO. 6
40
Source of Life Expectancy Estimate/Year 1975-80 1980-85 1985-90 1990-95 1995-2000 2000-2005 2005-2010
Life expectancy at birth (US Census, 2010) 54 56 56 59 61 63 65
Estimated Life Expectancy from CCRs 49.99 52.47 55.96 56.97 60.09 60.82 60.99
Comparison of Life Expectancy Estimates For
Burma Calculated from Cohort Change Ratios
(during each period), 1975-80 to 2005-10 as shown in
Equation [7] and Equation[5.a] with estimated
values available from the US Census Bureau (2010).
REASON NO. 6
REASON NO. 5
It provides formal demography enthusiasts with an
efficient numerical means of generating stable
populations, incorporating both sexes as well as
migration.
Because CCRs are always greater than zero, they can
be used in a Leslie Matrix that is guaranteed to
generate a population that converges to a stable form.
41
Here is an example of Australia converging to stability
using the CCRs from 2001-2006 and a launch from
2006 described earlier as input into a Leslie Matrix
along with fertility data.
Using this approach with the input data, the population
of Australia will converge to a stable form in about 500
years.
42
REASON NO. 5
43
REASON NO. 5
44
REASON NO. 5
45
The CCR approach simply takes the cohort change ratios found at
a given point in time and holds them constant until the population
reaches stability. In terms of our implementation of this approach
within the Leslie Matrix framework, this also means we hold the
initial ASFRs constant as well.
To determine when a population has reached stability, the well-
known “Index of Dissimilarity” is employed as an “Index of Stability” (S).4 The index is defined as:
S = {0.5* ∑│(npx/∑nPx)t+y - (npx/∑nPx)t │}. [11]
where y = number of years between census counts/projection cycles x = age n = width of the age group (in years)
t = year
REASON NO. 5
46
0.00000
0.00500
0.01000
0.01500
0.02000
0.02500
0.03000
0 200 400 600
STA
BIL
ITY
IN
DE
X
N OF YEARS
AUSTRALIA: PATH TO STABILITY
REASON NO. 5
REASON NO. 4
It is a great method for doing multi-race population
projections. All you need are two census counts. With
the cohort-component method, one needs, a census
count, vital statistics data, and migration data. The
vital statistics data may not be compatible with census
data and the migration data are difficult to obtain.
47
48
REASON NO. 4
REASON NO. 4
Given that the US census only started counting multi-
race people in 2000, the backcasting capacity of the
CCR method (per the example of the Native Hawaiian
Population in 1778) can be used to estimate the
population at an earlier date, such as 1990. In the US,
it would be very difficult to construct a pre-2000
estimate of a given multi-race population in the
absence of a CCR backcast.
49
50
REASON NO. 4
REASON NO. 3
Within a regression approach, the CCR method can
generate formal measures of uncertainty for
projections by age (and sex, race, and other
characteristics, both ascribed and achieved) and for
the total population
51
REGRESSION-ESTIMATED CCRs
The Hamilton-Perry Method is deterministic. However, we also
know that population forecasting is subject to uncertainty since we
do not precisely know the future components making up the
fundamental equation. So, the question is how to introduce an
element of statistical uncertainty into a method that is inherently
deterministic. One answer is found by employing regression
techniques to forecast CCRs and their intervals.
52
REASON NO. 3
REGRESSION-ESTIMATED CCRs
Recall that nCCRx,t = nPx,t / nPx-k,t-k.
From this, we can define the CCR for the preceding census period
as nCCRx,t-k = nPx,t-k / nPx-k,t-2k.
We then construct a regression model with nCCRx,t as the
dependent variable and nCCRx,t-k as the independent variable.
For age groups 0-4, 5-9, and the terminal open-ended age group
that the dependent and independent observations follow the
equations provided earlier.
53
REASON NO. 3
REGRESSION-ESTIMATED CCRs
Given this adjustment, we estimate the CCRs at time (t)
by:
nECCRx,t = a + b × nCCRx,t-k.
We then multiply the regression-estimated CCR and the
corresponding population by age at time (t) to forecast
the CCR at time (t+k):
nCCRx,t+k = nECCRx,t × nPx,t.
54
REASON NO. 3
Table A2.1 Ratios, 1980-1990 and 1990-2000 and Projected Population 2010, Minnesota
Ratiosa
Population 1990-2000 2010
Age 1980 1990 2000 1980-1990 Observed Estimatedb Population
0 to 4 307,249 336,800 329,594 1.09618 0.97860 1.11501 367,501
5 to 9 296,295 345,840 355,894 1.16722 1.02907 1.17641 418,677
10 to 14 333,378 313,297 374,995 1.01968 1.11341 1.04890 345,711
15 to 19 399,818 297,609 374,362 1.00443 1.08247 1.03572 368,607
20 to 24 393,566 316,046 322,483 0.94801 1.02932 0.98696 370,105
25 to 29 363,435 381,759 319,826 0.95483 1.07465 0.99286 371,689
30 to 34 313,104 397,984 353,312 1.01123 1.11791 1.04160 335,898
35 to 39 246,356 361,274 412,490 0.99405 1.08050 1.02675 328,381
40 to 44 202,860 304,810 411,692 0.97351 1.03444 1.00900 356,492
45 to 49 187,051 237,050 364,247 0.96223 1.00823 0.99925 412,181
50 to 54 193,199 191,410 301,449 0.94356 0.98897 0.98312 404,743
55 to 59 189,457 173,066 226,857 0.92523 0.95700 0.96727 352,325
60 to 64 170,638 171,220 178,012 0.88624 0.93000 0.93358 281,427
65 to 69 149,114 160,036 153,169 0.84471 0.88503 0.89769 203,647
70 to 74 121,034 134,486 142,656 0.78814 0.83317 0.84880 151,097
75+ 209,416 252,412 298,441 0.52634 0.54566 0.62254 369,954
Total 4,075,970 4,375,099 4,919,479 5,438,435
a Ages 0-4 = P0-4,t / P0-4,t-10.
Ages 5-9 = P5-9,t / P5-9,t-10.
Ages 10-74 = Px+10,t / Px,t-10.
Ages 75+ = P75+,t / P65+,t-10.
b Based on the regression equation, 0.1676667 + (0.8644256 × Ratios1980-1990)
c Ages 0-4 = Est.1990-2000 Ratio0-4 × P0-4,2000.
Ages 5-9 = Est.1990-2000 Ratio5-9 × P5-9,2000.
Ages 10-14 = Est.1990-2000 CCRx × Px-10,2000.
Ages 75+ = Est.1990-2000 CCR75+ × P65+,2000.
55
REASON NO. 3
MEASURING UNCERTAINTY
When the prediction from a regression equation is derived from an observed data value, we call the resulting value of a “fitted value.” This is not a forecast as the actual value of a predictor variable is used in the calculation. When values of the predictor variable are not part of the data used to estimate the model, the resulting prediction is a forecast.
Assuming that the regression errors are normally distributed, an approximate 95% forecast interval (also called a prediction interval) associated with this
forecast is given by Hyndman and Athanasopoulos, Chapter 4,2012) as.
56
REASON NO. 3
Jeff Tayman and I (Swanson and Tayman 2014)
developed and tested this regression-based approach
for developing 66% forecast intervals for age-group
forecasts made using the Hamilton-Perry Method. To
evaluate this method, we used 16 age groups (0-4, 5-
9,…, 70-74, 75+) taken from a sample of four states
(one from each census region in the United States) and
nine ex post facto tests, one for each census from 1930
to 2010. This yielded 576 observations for which we
could see if the forecast interval for a given age group in
a given census year contains the census count for the
same age group. 57
REASON NO. 3
EVALUATION OF THIS APPROACH
DATA
We constructed CCRs over two successive decennial periods
(e.g., 1910-1920/1900-1910) over the entire period, using
regression to estimate the CCR in the numerator from the CCR in
the denominator.
We then used the regression-based estimate of the CCR of the
“current period” (e.g., 1910-1920) to forecast the CCRs to the
next period, the “launch year” (e.g., 1920-1930) and developed
forecast intervals around these forecasted CCRs, which are then
translated into the forecasted age groups for the “target year” (e.g., 1930).
The forecast intervals are then examined to see if they contain
the census age groups for the target year. 58
REASON NO. 3
RESULTS
Table 2 provides a summary of the results for all four states at
each of the nine census test points. The table shows the number
of times (out of 16) that the 66% forecast interval contained the
corresponding census number for a given age group. If the
forecast intervals provide a valid measure of uncertainty, they will
contain approximately 11 of the 16 observed population counts.
The table also shows percent of the counts falling within the
forecast intervals for all target years for each state (144 intervals),
the percent falling within all states for each target year (64
intervals), and the single percent falling within all states for all
target years (576 intervals).
59
REASON NO. 3
Table 2. Number of Population Counts Falling within the 66% Forecast Intervals by State and Target Year
Target Year Georgia Minnesota New Jersey Washington Total
Percent (N/64)
1930 9 12 8 13 42 67%
1940 3 5 11 12 31 48%
1950 10 14 4 3 31 47%
1960 13 14 14 8 49 86%
1970 6 12 14 13 45 77%
1980 7 12 12 10 41 67%
1990 13 14 14 14 55 83%
2000 8 15 14 15 52 81%
2010 7 15 15 14 51 81%
Total 76 113 106 102 397
Percent 53% 78% 74% 71% 69%
Percent Percent Percent Percent Percent
60
REASON NO. 3
RESULTS
RESULTS
Table 3 contains a summary of the results by age group across all
of the nine census target years and the four states. The table
shows the number of times (out of 36) that the 66% forecast
interval contained the corresponding census number for a given
age group. If the forecast intervals provide a valid measure of
uncertainty, they will contain approximately 24 of the 36 observed
population counts.
In general, Table shows that forecast intervals capture the
population count at least 66 percent of the time for age groups 10-
14, 15-19, 20-24 and 40-44 through 75+.
61
REASON NO. 3
RESULTS
For age groups 0-4 and 5-9, the forecast intervals only
encompass the population counts 25 percent of time.
For age group 30-34, the count is encompassed 53
percent of the time while for age group 25-29, it is 58
percent of the time. The population counts are captured
by the forecast intervals 61 percent of the time for age
group 35-39.
62
REASON NO. 3
Table 3. Number of Population Counts Falling within the 66% Forecast Interval by Age Group
Age Number Percent
(N/36)
0 to 4 9 25%
5 to 9 9 25%
10 to 14 26 72%
15 to 19 27 75%
20 to 24 24 67%
25 to 29 21 58%
30 to 34 19 53%
35 to 39 22 61%
40 to 44 26 72%
45 to 49 28 78%
50 to 54 30 83%
55 to 59 31 86%
60 to 64 30 83%
65 to 69 31 86%
70 to 74 33 92%
75+ 31 86%
Total 397 69%
63
REASON NO. 3 RESULTS
DISCUSSION
Overall, the 66 percent intervals contain their
corresponding census age groups in 397 cases, which
represents 69 percent of the 576 total observations. In
terms of the nine census target years, the overall results
show that in five of them (1960, 1970, 1990, 2000, and
2010) the forecast intervals contain the census age
groups substantially more than 66 percent of the time. In
two target years (1930 and 1980), the intervals contain
the census age groups 67 percent of the time.
64
REASON NO. 3
DISCUSSION
In the remaining two target years, 1940 and 1950, the
intervals contain the census age groups 48 percent and
47 percent of the time, respectively. The 1940 test point
encompasses the economic boom experienced in the
1920s and the economic depression during the 1930s
and the large scale “baby bust” associated with it. The 1950 point encompasses the depression and baby bust
period of the 1930s and the economic recovery
stimulated by World War II and the initial part of the large scale “baby boom” from 1946 to 1950.
65
REASON NO. 3
DISCUSSION
In regard to Table 3 and the summary of results by age
group, it should not be surprising that the cohort
change method is better able to capture older age
groups than the very youngest since births are not part
of a cohort change ratio. In addition, migration likely
comes into play in that the population in the two
youngest age groups
(0-4 and 5-9) would be moving with their parents, who
are likely to be in age groups 25-29, 30-34, and 35-39,
the other age groups for which the forecast intervals
encompassed the population counts less than 66
percent of the time. 66
REASON NO. 3
DISCUSSION
Overall, we found that these effects are consistent with theory regarding migration in that those who tend to move are less socially integrated into communities than those who tend not to move and that as adults age, community social integration tends to increase (Goldscheider 1978). Finally, as shown at the bottom of Table 3, the intervals capture the population count 69 percent of the time (397 out of 576), which matches the summary for Table 2.
67
REASON NO. 3
REASON NO. 2
It is a re-expression of the fundamental demographic
equation. As such, it is embedded within formal
demographic theory as well as social demographic
theory (e.g., Easterlin’s hypothesis)
68
69
The Cohort Change Ratio Method (expressed in terms of a forecast, aka “The Hamilton-Perry Method) and the
Fundamental Demographic Equation
It is useful to recall that any quantitative approach to forecasting
is constrained to satisfy various mathematical identities (Land
1986). In regard to population forecasting, an approach should
ideally satisfy demographic accounting identities, which is
summarized in the fundamental demographic equation:
Pt = P0 + Births – Deaths + In-migrants – Out-migrants.
[A.1]
REASON NO. 2
70
That is, the population at some time in the future, Pt, must be
equal to the population at an earlier time, P0, plus the births and
in-migrants (to include both domestic and international migrants)
and less the deaths and out-migrants (to include both domestic
and international migrants) that occur between time 0 and time
t. The cohort-component method of population projection
satisfies the fundamental equation. As we show here, the
Hamilton-Perry Method also satisfies the fundamental
demographic equation.
Vaupel and Yashin (1985) also argue that a demographic
forecasting method needs to be consistent with the fundamental
demographic equation in order to minimize the potential errors
associated with hidden heterogeneity.
REASON NO. 2
71
nCCRx, t = nPx, t / nPx-k, t-k. [A.2]
where,
nPx, t is the population aged x to x+n at the most recent census (t),
nPx-k, t-k is the population aged x-k to x-k+n at the 2nd most recent
census (t-k), and
k is the number of years between the most recent census at time t
and the one preceding it at time t-k.
REASON NO. 2
72
The basic formula for the second step, moving the cohorts of a
population into the future is:
nPx+k, t+k = nCCRx, t × nPx, t [A.3]
where,
nPx+k, t+k is the population aged x+k to x+k+n at time t+k,
and
nCCRx, t and nPx, t are as defined in equation [A.2].
REASON NO. 2
73
To show the Hamilton-Perry Method satisfies the fundamental
demographic equation, we restate equation [A.2] using the terms in
equation [A.1]:
Pt+k = Pt + B – D + I – O
where,
Pt = Population at time t (the launch year),
Pt+k = Population at time t+k (the projection year),
B = Births between time t and t+k,
D = Deaths between time t and t+k,
I = In-migrants between time t and t+k, and
O = Out-migrants between time t and t+k,
then,
nCCRx,t = (nPx-k,t-k + B – D + I – O )/ nPx-k,t-k [A.3]
REASON NO. 2
74
Since we can also express equation [A.3] in terms of equation[A.1]:
nPx+k,t+k = ((nPx-k,t-k + B – D + I – O) / (nPx-k,t-k)) × ( nPx,t) [A.4]
where x+k >= 10,
then, nCCRxt = (nPx-k,t-k – D + I – i) / nPx-k,t-k, and since
N = I – O, where x+k ≥ 10, we have
nCCRx,t = (nPx-k,t-k – D + N) /(nPx-k,t-k). [A.5]
Equations [A.3], [A.4], and [A.5] show that the Hamilton-Perry
Method is not only consistent with the fundamental demographic
equation, but also closely related to the cohort-component method.
The Hamilton-Perry Method simply expresses the individual
components of change—births, deaths, and migration—in terms of
CCRs. As such, it satisfies the fundamental demographic equation.
REASON NO. 2
REASON NO. 1
The number 1 reason to use a Cohort Change Ratio
Method is that it is easy to explain and operate.
In a court case Swanson was engaged with that
involved population and enrollment projections, it made
an economist serving as the opposition’s expert witness grumble on the stand that his children could understand
and operate it.
The US Federal Judge hearing the case understood the
method and how it operated as well. The side the
economist was representing lost the case.
http://www.commercialappeal.com/news/wendi-c-thomas-municipal-schools-fight-makes-on
75
The Top Ten Reasons to use Cohort Change Ratios
76
10. You only need two census counts to generate a forecast
9. You only need two census counts to generate a backcast
8. You can generate small area forecasts with them
7. You can use them with institutional or administrative data
6. You can use them to estimate life expectancy
5. You can use them as a approach to Stable Population Theory
4. You can use them to forecast or backcast a multi-race population
3. You can use them in a regression-based approach to generate formal
measures of forecast uncertainty
2. In terms of forecasting & backcasting, they satisfy the Fundamental Demographic Equation
1. They are easy to use and explain
Works Cited in this Presentation
Adams, R. (1937). Interracial Marriage in Hawaii. New York, NY:
McMillan.
Goldscheider, C. (1978). Modernization, migration, and
urbanization. Paris, France: International Union for the Scientific
Study of Population.
Hamilton, C.H. and J. Perry. (1962). A short method for projecting
population by age from one decennial census to another. Social
Forces 41: 163-170.
Hardy, G.F. and F. B. Wyatt. (1911). Report of the actuaries in
relation to the scheme of insurance against sickness,
disablement, etc. Journal of the Institute of Actuaries XLV: 406-
443.
77
Works Cited in this Presentation
Hommon, R. (2008). Watershed testing the limited land
hypothesis. Pp. 1-92 in T. Dye (ed.) Research Designs for
Hawaiian Archaeology: Agriculture, Astronomy, and
Architecture. Honolulu: Society for Hawaiian Archaeology.
Hyndman, R. Y. and G. Athanasopoulos. (2012) Forecasting:
Principles and Practice (online at http://otexts.com/fpp/ ).
Land, K. (1986) Methods for national population
forecasts: A review. Journal of the American Statistical
Association 81: 888-901.
78
Works Cited in this Presentation
Nordyke, E. (1989). The Peopling of Hawai’i, 2ndEdition. Honolulu: University of Hawai’i Press.
Schmitt, R. (1968). Demographic Statistics of Hawaii,
1778-1965. Honolulu: University of Hawai’i Press.
Schmitt, R. (1971). New Estimates of the Pre-Censal
Population of Hawaii. Journal of the Polynesian
Society. 80, 237- 243.
79
Works Cited in this Presentation
Smith, S.K., J. Tayman, and D. A. Swanson. (2013). A
practitioner’s guide to state and local population projections.
Dordrecht, The Netherlands: Springer.
Stannard, D. (1989). Before the Horror: The Population of
Hawai’i on the Eve of Western Contact. Honolulu: University
of Hawai’i Press.
80
Works Cited in this Presentation
Swanson, D. and J. Tayman. (2014). “Measuring Uncertainty in Population Forecasts: A New Approach.” pp. 203-215 in
Proceedings of the 6th EUROSTAT/UNECE Work Session on
Demographic Projections. Rome, Italy: National Institute of
Statistics.
Swanson, D., A. Schlottmann, and R. Schmidt. (2010).
Forecasting the population of census tracts by age and sex:
An example of the Hamilton–Perry method in action.
Population Research and Policy Review 29: 47-63.
81
Works Cited in this Presentation
Swanson, D., L. Tedrow, and J. Baker.(forthcoming).
“Exploring Stable Population Concepts from the Perspective of Cohort Change Ratios: Estimating the
Time to Stability and Intrinsic r from Initial Information
and Components of Change.” Chapter 12 in R. Schoen (Ed.) Dynamic Demographic Analysis,
Springer B.V. Press. Dordrecht, Heidelberg, London,
and New York.
82
Works Cited in this Presentation
Swanson, D. A. and L. M. Tedrow (2012). “Using Cohort Change Ratios to Estimate Life Expectancy in
Populations with Negligible Migration: A New Approach.” Canadian Studies in Population 39: 83-90.
United Nations. (2002). Methods for Estimating Adult
Mortality. New York, NY: Population Division, United
Nations.
83
Works Cited in this Presentation
United States Census Bureau. (2010). World
Population Statistics, available at
http://www.census.gov/ipc/www/idb/informationGate
way.php
Vaupel, J, and A. Yashin (1985). Heterogeneity's
ruses: Some surprising effects of selection on
population dynamics. The American Statistician 39
(August): 176-185.
84
