1. Demographic Analysis and Evaluation
Workshop on
Demographic Analysis and Evaluation 1

2. Age and Sex Structure: Correcting for Age Misreporting

3. Methods for Correcting Age Misreporting
While errors in an age-sex distribution are not always matters of age misreporting, this type of error is common for some populations. The analyst’s task is to distinguish (a) age misreporting over parts of the age distribution from (b) age misreporting over most or all of the age distribution from (c) age-and-sex-selective reporting errors from (d) irregularities in an age-sex distribution due to real demographic events.
Smoothing techniques have frequently been used for correcting data for age misreporting. 3

4. Age Misreporting and Smoothing - Introduction
In this lesson, we consider techniques for smoothing the population age distribution when we believe there are errors in age reporting.
What do we hope to accomplish by smoothing?
Reasons to smooth, and not to smooth
Types of smoothing methods:
Light vs. strong
Preserve vs. modify slightly population totals
Tips for deciding whether smoothing is needed and which method might be most appropriate
Also third key diff: which whole groups smoothed retained? (youngest/oldest) 4

5. Why Might We Want to Smooth?
Reasons to smooth:
When ages are misreported
Planning and policies that require accurate counts by age may be affected. Examples:
Children entering school system
Young males reaching military draft age
Distribution of older age public benefits
Flawed age-sex structures, when projected into the future, will also be flawed.
Flawed age counts used as denominators in demographic rates (e.g., child mortality) may bias those rates.
Adversely or positively affected: child marriage (girls); military (boys), positive
Other cases negative 5

6. Methods for Correcting Age Misreporting
Light versus Strong
Light (or “slight”) smoothing, which gently modifies irregularities in the age structure
Strong smoothing, which modifies most irregularities, and therefore more likely to modify features which may represent actual facts instead of errors
Light/slight vs. str bigger q
Both can erase
Spreadsheet:
AGESMTH 6

7. Smoothing Algorithms Compared
Light versus Strong
Smoothing techniques may be lighter or stronger depending on the formulas that construct the weighted averages:
Light Smoothing – formulas that give the greatest weight to what was reported for the age group in question and smallest weight to adjacent age groups
Strong Smoothing – formulas that give greater weight to adjacent age counts and/or over wider age intervals. Resulting pattern does not follow the contours of reported data as well as lighter smoothing. 7

8. Light Smoothing Formula8

9. Strong Smoothing Formula9

10. Smoothing Algorithms Compared10

11. Smoothing Algorithms Compared11

12. Smoothing Algorithms Compared12

13. Smoothing Algorithms Compared13

14. Methods for Correcting Age Misreporting
Light (or “slight”) smoothing, which gently modifies irregularities in the age structure:
Carrier-Farrag
Karup-King-Newton
Arriaga (“light” formula)
United Nations
Strong smoothing, which modifies most irregularities, and therefore more likely to modify features which may represent actual facts instead of errors
“Strong” Arriaga formula
Light/slight vs. str bigger q
Both can erase
Spreadsheet:
AGESMTH 14

15. Methods for Correcting Age Misreporting Preserve versus Modify Totals
Light (or “slight”) smoothing, which gently modifies irregularities in the age structure:
methods which preserve the enumerated population totals within each 10-year age group; and
(b) methods which modify the enumerated population totals.
Strong smoothing, which modifies most irregularities, is also more likely to modify features which may represent actual facts instead of errors.
Also modifies the enumerated population totals within 10-year age groups.
Grouped totals
Totals w/in ea 10 yr group
Even though strong; retains totals
Spreadsheet:
AGESMTH 15

16. Methods for Correcting Age Misreporting Preserve versus Modify Totals
Light (or “slight”) smoothing, which gently modifies irregularities in the age structure:
methods which preserve the enumerated population totals within each 10-year age group:
Carrier-Farrag, Karup-King-Newton, Arriaga (“light” formula)
(b) methods which modify the enumerated population totals in 10-year age groups: United Nations (and Arriaga’s strong smoothing formula)
Even though strong; retains totals
Spreadsheet:
AGESMTH 16

17. Methods for Correcting Age Misreporting Preserve versus Modify Totals
Strong smoothing, which modifies most irregularities, is also more likely to modify features which may represent actual facts instead of errors.
Also modifies the enumerated population totals in 10-year age groups (but not the overall population total):
“Strong” Arriaga formula
Even though strong; retains totals
Spreadsheet:
AGESMTH 17

18. The Carrier-Farrag Technique
The Carrier‑Farrag technique is based on the assumption that the relationship of a 5‑year age group to its constituent 10‑year age group is an average of similar relationships in three consecutive 10‑year age groups.
5Px+5 = 10Px / [1 + (10Px-10 / 10Px+10 )1/4 ] and
5Px = 10Px - 5Px+5
where:
5Px+5 represents the population at ages x+5 to x+9;
10Px represents the population at ages x to x+9; and
5Px represents the population at ages x to x+4.
Don’t look too closely/memorize formulas until you have a solid understanding of the bigger/real dem situation. (forest –trees) 18

19. The Karup-King-Newton Formula
The Karup‑King‑Newton formula assumes a quadratic relationship among each three consecutive 10-year age groups.
5Px = Px (10Px Px+10 ) and
5Px+5 = 10Px - 5Px
Where 5Px is the first of two 5-year age groups comprising a 10-year age group 10Px. 19

20. Arriaga’s Light Smoothing Formula
Arriaga’s formula assumes that a second degree polynomial passes by the midpoint of each three consecutive 10-year age groups and then integrates a 5-year age group. 20

21. Arriaga’s Light Smoothing Formula
When the 10-year age group to be separated is the central group of three, the following formulas (Arriaga, 1968) are used:
5Px+5 = (-10Px Px Px+10 ) / 24 and
5Px = 10Px - 5Px+5
where:
5Px+5 is the population ages x+5 to x+9;
10Px is the population ages x to x+9; and
5Px represents the population at ages x to x+4. 21

22. Arriaga’s Light Smoothing Formula
When the 10-year age group to be separated is an extreme age group (the youngest or the oldest), the formulas are different. For the youngest age group, the following formulas are used:
5Px+5 = (8 10Px Px Px+20 ) / 24 and
5Px = 10Px - 5Px+5
For the oldest age group, the coefficients are reversed:
5Px = (- 10Px Px Px) / 24 and
5Px+5 = 10Px - 5Px 22

23. The United Nations Formula
United Nations (Carrier and Farrag, 1959) developed the following formula:
5P'x = (1/16) (- 5Px Px Px + 4 5Px+5 - 5Px+10 )
where:
 5P'x represents the smoothed population ages
x to x+4. 23

24. Arriaga’s Strong Smoothing Formula
If more aggressive smoothing is desired (Arriaga, 1968), this can be achieved with the following formula:
 10P'x = (10Px Px + 10Px+10 ) / 4
 Where:
 10P'x represents the smoothed population
ages x to x+9. 24

25. There is no generalized solution for all populations.
What to Do?
There is no generalized solution for all populations.
The smoothing technique to be used will depend on the errors in the age and sex distributions, and so the age structure must be analyzed before deciding whether the smoothing should be strong or light.
While, as Arriaga and Associates (1994) note, differences in results across procedures are small, a decision to use strong smoothing should not be taken lightly.
Recognize that the whole age distribution need not be smoothed if only part is considered problematic. 25

26. What to Do?
Make a graph of the age and sex distributions before making any decision about whether or not smoothing is required and which formula or technique would be appropriate for the particular country's situation (Pyramid, Pyr2, GRPOP-YB).
In general, a regular saw-tooth pattern across successive age groups provide a good rationale for smoothing.
Comparisons among successive censuses and a knowledge of past trends of mortality, fertility, and migration will help in appraising the accuracy of the reported age and sex structure of the population. 26

27. To Smooth or Not to Smooth? - Cautions
Caution – Since strong smoothing may erase actual demographic history, a decision to use it should be considered very carefully.
Caution – Even if smoothing produces more plausible age distributions, it may not improve distortions in sex ratios by age (and vice versa).
Caution - The population age distribution may not need to be fully smoothed across all age groups if only part of it is considered problematic.
Caution – If underreporting exists at a particular age, instead of smoothing, one may need “filling.”
Filling w/special adj. 27