Parameters of Household Composition as Demographic Measures

ABRAHAM AKKERMAN

PARAMETERS OF HOUSEHOLD COMPOSITION ASDEMOGRAPHIC MEASURES

(Accepted 12 January 2004)

ABSTRACT. Cross-sectional data, such as Census statistics, enable the re-enact-ment of household lifecourse through the construction of the household compo-sition matrix, a tabulation of persons in households by their age and by the ageof their corresponding household-heads. Household lifecourse is represented inthe household composition matrix somewhat analogously to survivorship in alife-table in demography. A measure of household lifecourse is the average house-hold size, specific to age of household-head. Associated with the age-specifichousehold size is the age-interval 0–4, which yields average number of childrenpresent in households, also by age of head. Trajectories of re-enacted householdlifecourses for Phoenix and for the State of Arizona are depicted here to track thegamma probability density function. Through this relationship also the associationbetween household size, children per household, and fertility emerges. To theextent that housing conditions or tenure impact average household size, or otheraspects of household composition, fertility in particular is discerned as a housing-related demographic attribute of households. Household size and headship ratio,both specific to age of head, are here shown to be analytically related to thehousehold composition matrix, their product yielding the age-specific headshipcoefficient. As a measure incorporating parameters of households and dwellers,thus also characterizing occupied dwelling units, the headship coefficient emergesas a demographic indicator of housing in a community.

INTRODUCTION

Demographic convention carried over from practice associated withlarge, human or other biological populations, usually presents agegroups separately for males and females. Inclusion of gender repre-sentation in a population’s age distribution has its historical rootsalso in the consideration of measurement of fertility linked tothe female reproductive age interval (Karmel, 1947; Yerushalmy,1943), and measurement in demography, accordingly, has been mostoften associated with gauges applied to age groups differentiatedby sex. The formal expression of change in a population, where

© Springer 2005Social Indicators Research (2005) 70: 151–183

ABRAHAM AKKERMAN

migration is negligible, has been pursued, therefore, through theapplication of fertility and survival rates specific to age-sex groups(see, e.g., Pressat, 1978, pp. 8–19). But whereas the application ofdemographic rates to homogenous groupings has a common-senseappeal in large populations, in smaller populations the conventionalapplication of demographic rates to undersized age-sex groups isquestionable (Congdon, 1994; cf. Ahlburg et al., 1998). Even moreproblematic in smaller populations is the calculation of rates, due toirregular variations or erratic demographic changes throughout thesmall communities in question (Dean and MacNab, 2001; Lunn etal., 1998; Tayman and Swanson, 1999).

Gender becomes a concern as a household headship attribute inhousing tenure (Peach, 1999), albeit without a structured consider-ation of age. Furthermore, functional inquiries into smaller humancommunities such as cities or neighborhoods utilize gender in agedistributions mainly in regard to policing (Dodge et al., 1999), health(Hama, 1998) and social welfare services (McBarnette, 1987). Forthe delivery of many other human or physical services to urbanor rural communities, additional attributes are often sought toconjoin an age-distribution, but gender is usually not one of them(Laws, 1994; Nelson and Dueker, 1990). For small-area populationanalysis, more often than not, age distribution by gender has oftenbeen found of peripheral use (Aickin et al., 1991; Lunn et al., 1998;Rees, 1994; Smith, 1996; but see also Geronimus et al., 1996).

In the measurement of fertility, Weinstein et al. (1990) hadquestioned the traditional application of age-sex specific fertilityrates to female (or male) age groups, pointing out that kinship asmuch as gender, is immanent to human reproduction. More recentlyInaba (1995) and Suzuki (2001) have suggested augmenting tradi-tional fertility rates by reproduction gauges based upon formalizedkin linkages. But whereas demographic distributions by selectedattributes offer a convenient form for the depiction of populations,neither age-distribution nor age-specific rates have sufficient facilityto discern formal kinship types among persons. A proposition couldbe made, in fact, that the tendency in descriptive demographyto measure change by relating to population as an aggregate ofindividuals with a common characteristic, such as age, rather thandepicting demographic change as attributable to the family or the

152

PARAMETERS OF HOUSEHOLD COMPOSITION

household, has been one of the main reasons for the absenceof a comprehensive demographic reference system for smallerpopulations (Murdock and Hoque, 1995). Such contention couldbe possibly made for large populations as well (cf., for example,Van Imhoff and Post, 1997). Although gender incorporation in therepresentation of age-distribution has been commonly interpreted ascompensating for lack of an explicit formulation of kinship relation-ships in fertility analysis (e.g., David and Sanderson, 1990), sugges-tions from several quarters for shifting demographic methodologyconcerning fertility measures towards a familial context have beenheard with increasing pertinence (Bongaarts, 1999; Mace, 1998;Sobel and Arminger, 1992; Van Imhoff and Post, 1997; Wilson,1999). In a broader scope, Astone et al. (1999) have pointed out theneed to integrate household parameters as well as the measurementof household change within mainstream concepts of populationresearch. Such a call is particularly timely in the analysis of smallerpopulations.

As demographic entities, households (or, for that matter, familiesor any other human organizations), however, cannot be perceivedwithin the comfort of groupings based upon uniform characteristics,since each person in a household usually belongs to a differentdemographic grouping, such as a different age group. Within ahousehold, different age groups of individuals, in particular, cannotbe even expected to correspond to the age distribution of thepopulation at large. Yet the composition of households by age ofhousehold-heads and by age of their other household members hassome particularly useful attributes. A tabular representation of agedistribution of members within households by age of heads hasbeen shown to constitute a linear relationship with the overall age-distribution of the population at large (Akkerman, 1996, 2000).The age-specific household size is shown here to track the gammaprobability density function, and in the product with the age-specificheadship ratio to yield a demographic measure of housing, theage-specific headship coefficient.

153

ABRAHAM AKKERMAN

REPRESENTATION OF AGE IN HOUSEHOLD DISTRIBUTIONS

Due to the heterogeneity in the demographic makeup of house-holds, household heads (referred to also as ‘householders’) oftenrepresent households or families in a census or a survey, as house-hold markers (Smith, 1992).1 This, of course, does not resolve theproblem of representing the heterogeneity itself within households.The notion of the householder, however, provides an important,even if obvious, one-to-one correspondence between the “marked”individuals, households, and occupied dwelling units.

The distinction between household heads and householdmembers in an age distribution is best facilitated by the dichotomousdivision by household-status (“head” vs. “member”). Unlike theconventional dichotomous representation of age by sex, the consid-eration of age by household-status yields three age-distributionsof interest. Each of the three distributions refers to persons inhouseholds by age:

Distribution, k, of household heads by age;Distribution, b, of persons in households by age of their house-hold heads;Distribution, w, of persons in households by age.

Only the third distribution, w, corresponds to the standardage-distribution of a population. Whereas much of conventionaldemographic analysis surrounds the distribution w, usually furtherdivided into males and females (such as in Table I or Figure 1,for Phoenix and Arizona), very little attention has been paid to theage-distributions k and b. Persons in distribution b are perceived asdwellers who are grouped according to the age of their respectivehouseholders, rather than according to their own age. Tables II andIII, for Phoenix and for Arizona respectively, show for both popula-tions the distribution k of householders by age and the distribution(b − k) of household members by age of their household heads.In all cases the distributions k and (b − k) are assumed to bemutually exclusive, thus forming the population of all dwellers ina community.2

The immediate benefit of considering the household distribution,k, and the household-member distribution, (b − k), in contrast tothe conventional age distribution, w, is that a lesser number of age

154


TABLE I

Fertility rates (Arizona) and age-sex distribution, Phoenix and Arizona, 1990

Phoenix Arizona Live births/

Age Male Female Total Male Female Total 1000 women

i ASFR(i)

0–4 42764 41085 83849 149642 143217 292859

5–9 38835 37299 76134 143978 137755 281733

10–14 34435 33010 67445 132160 126204 258364 1.5

15–19 35205 33336 68541 134271 126651 260922 75.6

20–24 39896 37136 77032 145249 134672 279921 143.4

25–29 49427 47718 97145 162127 155921 318048 133.7

30–34 47791 47044 94835 159973 156878 316851 84.8

35–39 41093 40965 82058 141870 140895 282765 33.9

40–44 35538 35742 71280 122852 123091 245943 6.2

45–49 26795 27883 54678 94547 98024 192571 0.2

50–54 20485 21923 42408 75955 80990 156945

55–59 17974 19626 37600 70240 76418 146658

60–64 16279 18893 35172 70890 81984 152874

65–69 14982 18031 33013 72823 87340 160163

70–74 11340 13779 25119 58748 71133 129881

75–79 7856 10518 18374 40594 54175 94769

80–84 4309 6579 10888 22303 33941 56244

85–89 1910 3556 5466 9215 17107 26322

90+ 674 1692 2366 3254 8141 11395

487589 495814 983403 1810691 1854537 3665228

Sources:1. U.S. Bureau of the Census, 1990 census of population, READEX C3.223/6:990 CP-1-4, pp. 76–171, Oct 92 – 19395. Department of Commerce, Wash-ington, D.C.2. Arizona Health Status and Vital Statistics, 1999 Annual Report, ArizonaDepartment of Health Services.

groups is required to represent dwellers. The youngest age groups,up to some r-th age group, in the distributions k and b, are always0, and in a routine demographic depiction, such as in Figures 2 and3, their representation can be omitted. While in the conventionalrepresentation n age groups are required, in depicting the householdage distributions, k and b (or b − k), only n − r age groups arerequired, with r being the number of age groups to which headship

155

ABRAHAM AKKERMAN

Figure 1. Age-sex distribution, Phoenix, 1990.

Figure 2. Households and household-members by age of householder, Phoenix,1990.

does not apply. On average, therefore, the (n − r) age groups of thedistribution b can be expected to be larger than the n age groups ofthe conventional age distribution, w. In small populations a decreasein the number of age groups and increase in the size of age groups(without changing their age intervals) must be welcome.

In the populations on hand the distributions are estimates fromstatistical samples of family-households in the Integrated PublicUse Microdata Series (IPUMS, 1997) of the 1990 U.S. Census, forPhoenix and Arizona, available through the University of Minnesota

156


Figure 3. Households and household-members by age of householder, Arizona,1990.

Population Center (Sobek and Ruggles, 1999).3 For consistency thedistributions in Tables II and III, as well as in all other representa-tions hereafter, utilize the IPUMS as the sole data source for bothPhoenix and the State of Arizona.

THE OLDEST-YOUNG IN THE AGE-DISTRIBUTION OFPOPULATION AND HOUSEHOLDS

The first age group of the age distribution, w, are children in theyoungest age interval. As opposed to cross-sectional age-specificfertility rates, which constitute measurement over a single intervalof time and age (of mother, usually), a census record can providea snapshot, at a single point in time, of children present in house-holds, according to their affiliation with household heads, by age.Thus, in contrast with the customary cross-sectional fertility byage-specific rates, attached to individual women by age, the censusrecord of children in households has been also interpreted as repre-senting the household recruitment of the young by the heads’ age(Rychtaríková and Akkerman, 2003). The notion of recruitment isconfined neither to a particular marital status of household-personsnor to the youngest age group. The first several age groups in a

157

ABRAHAM AKKERMAN

TAB

LE

II

Dw

elle

rs,h

ouse

hold

s,ho

useh

old-

mem

bers

and

asso

ciat

edra

tios

byag

eof

hous

ehol

der,

Phoe

nix,

1990

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Age

ofH

ouse

hold

sH

ouse

hold

erH

ouse

hold

-Pe

rson

sin

Hea

dshi

pA

vera

geho

use-

Hea

dshi

p

hous

ehol

der

boun

dm

embe

rsho

useh

olds

rate

hold

size

coef

ficie

nt

jk(

j)b(

j)b(

j)−

k(j)

w(i

)h(

j)s(

j)h(

j)∗s(

j)

0–4

00

064

016

01

0

5–9

00

066

979

01

0

10–1

40

00

5420

30

10

15–1

911

1813

8126

350

352

0.02

21.

235

0.02

7

20–2

414

118

2640

112

283

4600

00.

307

1.87

00.

574

25–2

939

806

9955

559

749

7575

40.

525

2.50

11.

314

30–3

442

692

1256

0082

908

7348

50.

581

2.94

21.

709

35–3

937

869

1115

6273

693

6587

50.

575

2.94

61.

694

40–4

439

312

1203

7381

061

6646

70.

591

3.06

21.

811

45–4

932

136

8744

255

306

4955

10.

649

2.72

11.

765

158


TAB

LE

II

Con

tinue

d

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Age

ofH

ouse

hold

sH

ouse

hold

erH

ouse

hold

-Pe

rson

sin

Hea

dshi

pA

vera

geho

use-

Hea

dshi

p

hous

ehol

der

boun

dm

embe

rsho

useh

olds

rate

hold

size

coef

ficie

nt

jk(

j)b(

j)b(

j)−

k(j)

w(i

)h(

j)s(

j)h(

j)∗s(

j)

50–5

422

100

5430

032

200

4049

70.

546

2.45

71.

341

55–5

920

709

4435

923

650

3339

10.

620

2.14

21.

328

60–6

420

007

3883

418

827

3144

90.

636

1.94

11.

235

65–6

919

552

3452

914

977

2984

90.

655

1.76

61.

157

70–7

417

238

2694

397

0524

598

0.70

11.

563

1.09

5

75–7

912

077

1783

857

6117

321

0.69

71.

477

1.03

0

80–8

471

1198

1327

0295

250.

747

1.38

01.

030

85–8

937

0547

5010

4543

470.

852

1.28

21.

093

90+

1560

1782

222

1801

0.86

61.

142

0.98

9

3311

1080

5460

4743

5080

5460

0.41

12.

433

1

Sour

ce:I

PUM

S,19

97.

159

ABRAHAM AKKERMAN

TAB

LE

III

Dw

elle

rs,h

ouse

hold

s,ho

useh

old-

mem

bers

and

asso

ciat

edra

tios

byag

eof

hous

ehol

der,

Ari

zona

,199

0

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Age

ofH

ouse

hold

sH

ouse

hold

erH

ouse

hold

-Pe

rson

sin

Hea

dshi

pA

vera

geho

use-

Hea

dshi

p

hous

ehol

der

boun

dm

embe

rsho

useh

olds

rate

hold

size

coef

ficie

nt

jk(

j)b(

j)b(

j)−

k(j)

w(i

)h(

j)s(

j)h(

j)∗s(

j)

0–4

00

022

7764

01

0

5–9

00

023

8569

01

0

10–1

40

00

2251

500

10

15–1

942

6464

4321

7918

7323

0.02

31.

511

0.03

4

20–2

446

111

9217

646

065

1617

600.

285

1.99

90.

570

25–2

912

1524

3049

0418

3380

2369

380.

513

2.50

91.

287

30–3

413

7605

4115

7727

3972

2479

910.

555

2.99

11.

660

35–3

913

4147

4480

5131

3904

2395

420.

560

3.34

01.

870

40–4

412

8960

4130

5928

4099

2204

160.

585

3.20

31.

874

45–4

910

2856

2982

8219

5426

1757

980.

585

2.90

01.

697

160


TAB

LE

III

Con

tinue

d

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Age

ofH

ouse

hold

sH

ouse

hold

erH

ouse

hold

-Pe

rson

sin

Hea

dshi

pA

vera

geho

use-

Hea

dshi

p

hous

ehol

der

boun

dm

embe

rsho

useh

olds

rate

hold

size

coef

ficie

nt

50–5

483

655

2117

3112

8076

1491

320.

561

2.53

11.

420

55–5

975

595

1676

7092

075

1275

680.

593

2.21

81.

314

60–6

487

425

1768

6189

436

1418

910.

616

2.02

31.

246

65–6

990

870

1632

9372

423

1479

360.

614

1.79

71.

104

70–7

484

955

1395

8154

626

1258

930.

675

1.64

31.

109

75–7

963

206

1003

0837

102

8760

80.

721

1.58

71.

145

80–8

439

910

5838

818

478

5210

90.

766

1.46

31.

121

85–8

916

432

2116

447

3219

164

0.85

71.

288

1.10

4

90+

6032

7341

1309

8276

0.72

91.

217

0.88

7

1223

547

3020

828.

6817

9728

230

2082

8.68

0.40

52.

469

1

Sour

ce:I

PUM

S,19

97.

161

ABRAHAM AKKERMAN

population’s age distribution, in fact, could be considered as beingsubject to household recruitment of the young.

The first r age groups in the population are not amenable tohousehold headship, usually due to institutionalized social sanction,and thus the age group within which headship can commence is (r +1). The first r age groups are referred to as age groups of the young,and the r-th age group, then, is the oldest young. In the case of 5-year age groups, such as those in Tables II and III, r = 3 might beaccepted as the standard number for the age groups of the young.Commencing with the age group, r + 1, persons up and throughoutthe last age group, n, such as 85+ or 90+, are formally able to attainor maintain household headship. Accordingly, the age groups (r +1), . . . , n, are referred to as the householder age groups.

The number of households whose heads are in age group j is kj ,j = 1, . . . , n. Referring to the household-head as a marker proxyfor his or her respective household, the distribution k of household-heads by age is, clearly, also the distribution of households (or ofoccupied dwelling units) by age of the household head, as in TablesII and III. The age distribution, k, of household heads is, therefore,referred to as household distribution, and since the number of house-holds with household heads in the first r age groups is always zero,the first r age groups are considered empty, as in columns (b), TableII and Table III, for Phoenix and for the State of Arizona, 1990.

Consistent with the distribution k, the first r age groups in thedistribution b (persons in households by age of their householdheads) are also empty, as shown in columns (c), Tables II andIII. The distribution b is the account of dwellers bound to house-holders, the former shown according to their householders’ age. Ina population where the number of persons living outside householdsis negligible, the age distribution of persons throughout all house-holds approximates the age-distribution of the entire population.The number of dwellers in age group i is wi , i = 1, . . . , n, and theirage-distribution is presented in a column vector, w, as in Tables IIand III, column (e). Among the three household distributions, onlythe distribution w has no empty age groups between 1 and n.

162


INTRAHOUSEHOLD DISTRIBUTION ANDTHE HEADSHIP COEFFICIENT

The three age distributions are interrelated. The distributions k andb are related through the average household size, sj , specific to theage j of household heads (j = r + 1, . . . , n):

sj = bj/kj . (1)

The age-specific average household size, sj , in turn can be alsoexpressed as the sum total of ratios of persons in different agegroups, per householder in age group j (j = r + 1, . . . , n). Thus, avalue aij is defined as the average number of members in age groupi per household whose head is in age group j (i = 1, . . . , n; j =r + 1, . . . , n). Similarly, the value (ajj + 1) denotes the averagenumber of persons per household who are in the same age group jas the household head. All values aij and (ajj + 1) can be ordered asentries in columns so that each column j represents the intrahouse-hold distribution corresponding to household heads in age groupj(j = r + 1, r + 2, . . . , n).

The ordered set of (n − r) column vectors, each showing intra-household distribution aij (i = 1, . . . , n), corresponding to each agegroup j of household-heads (j = r + 1, . . . , n), yields a table of (n − r)columns and n rows. The entries aij and (ajj + 1) are referred to ashousehold composition ratios, and the set of (n − r) column vectors,as in the rightmost part of Tables IV and V, is referred to as thehousehold composition table (Rychtaríková and Akkerman, 2003).The off-diagonal entries aij of the household composition tableshow the average number of members in age group i per householderin age group j (i = 1, . . . , n; j = r + 1, . . . , n). The diagonal entriesof the table are (ajj + 1), showing the average number of personsper household in the same age group as the householder. For anyage group j of householders, j > r, the sum of the entries aij (alongwith the diagonal entry, ajj + 1) in each column j yields the averagehousehold size, sj , for householders in age groups j (Akkerman,1996).

The ordering of the household composition ratios in columnvectors of age distribution facilitates the representation of relationsbetween household members and householders. The intrahouseholddistribution by age addresses a problem that is somewhat reverse

163

ABRAHAM AKKERMAN

TAB

LE

IV

Hou

seho

ldco

mpo

siti

onm

atri

x,Ph

oeni

x,19

90

Hou

seho

ldco

mpo

siti

onta

ble

0–4

5–9

10–1

415

–19

20–2

425

–29

30–3

435

–39

40–4

445

–49

50–5

455

–59

60–6

465

–69

70–7

475

–79

80–8

485

–89

90+

0–4

1.00

00.

000

0.00

00.

105

0.42

40.

543

0.45

10.

265

0.13

40.

036

0.01

90.

005

0.00

30.

000

0.00

00.

000

0.00

00.

000

0.00

0

5–9

0.00

01.

000

0.00

00.

000

0.11

10.

324

0.49

50.

446

0.22

80.

137

0.01

40.

029

0.00

50.

006

0.00

00.

000

0.00

00.

000

0.00

0

10–1

40.

000

0.00

01.

000

0.00

00.

022

0.04

60.

231

0.37

40.

442

0.19

80.

076

0.08

10.

047

0.00

00.

000

0.00

00.

000

0.00

00.

000

15–1

90.

000

0.00

00.

000

1.00

20.

037

0.03

40.

058

0.22

00.

397

0.38

10.

224

0.10

40.

065

0.01

50.

000

0.00

00.

000

0.00

00.

000

20–2

40.

000

0.00

00.

000

0.00

01.

140

0.12

40.

035

0.03

30.

135

0.25

80.

168

0.08

20.

079

0.06

50.

013

0.00

40.

000

0.02

50.

000

25–2

90.

000

0.00

00.

000

0.12

80.

065

1.28

50.

194

0.06

50.

052

0.08

70.

168

0.07

70.

064

0.05

60.

016

0.00

00.

000

0.00

00.

000

30–3

40.

000

0.00

00.

000

0.00

00.

027

0.06

41.

350

0.17

30.

050

0.03

70.

036

0.05

00.

027

0.02

10.

007

0.02

60.

000

0.00

00.

000

35–3

90.

000

0.00

00.

000

0.00

00.

000

0.04

00.

065

1.27

30.

204

0.06

50.

033

0.01

70.

048

0.03

10.

025

0.00

00.

016

0.00

00.

000

40–4

40.

000

0.00

00.

000

0.00

00.

005

0.01

00.

022

0.05

91.

329

0.19

60.

092

0.04

50.

018

0.02

70.

008

0.02

40.

000

0.00

00.

000

45–4

90.

000

0.00

00.

000

0.00

00.

021

0.01

00.

012

0.00

50.

055

1.21

60.

185

0.05

10.

032

0.02

40.

014

0.01

00.

042

0.00

00.

000

164


TAB

LE

IV

Con

tinue

d

Hou

seho

ldco

mpo

siti

onta

ble

0–4

5–9

10–1

415

–19

20–2

425

–29

30–3

435

–39

40–4

445

–49

50–5

455

–59

60–6

465

–69

70–7

475

–79

80–8

485

–89

90+

50–5

40.

000

0.00

00.

000

0.00

00.

013

0.01

50.

009

0.01

10.

011

0.05

41.

364

0.18

50.

083

0.01

10.

018

0.00

90.

049

0.03

50.

000

55–5

90.

000

0.00

00.

000

0.00

00.

000

0.00

20.

002

0.00

60.

000

0.00

30.

044

1.27

90.

191

0.05

50.

022

0.00

00.

007

0.00

00.

075

60–6

40.

000

0.00

00.

000

0.00

00.

000

0.00

40.

005

0.00

00.

010

0.00

70.

016

0.05

11.

205

0.20

10.

049

0.00

60.

000

0.02

50.

000

65–6

90.

000

0.00

00.

000

0.00

00.

005

0.00

00.

000

0.00

50.

007

0.01

00.

002

0.03

40.

038

1.20

50.

174

0.05

30.

029

0.02

10.

000

70–7

40.

000

0.00

00.

000

0.00

00.

000

0.00

00.

006

0.00

80.

002

0.02

30.

004

0.01

40.

012

0.02

71.

156

0.14

90.

049

0.00

00.

000

75–7

90.

000

0.00

00.

000

0.00

00.

000

0.00

00.

005

0.00

30.

000

0.00

30.

012

0.00

00.

003

0.00

00.

054

1.18

20.

139

0.10

20.

000

80–8

40.

000

0.00

00.

000

0.00

00.

000

0.00

00.

002

0.00

00.

006

0.01

00.

000

0.03

00.

000

0.01

30.

000

0.01

41.

049

0.07

40.

067

85–8

90.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

80.

015

0.00

90.

000

0.00

00.

000

1.00

00.

000

90+

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

60.

000

0.00

70.

000

0.00

00.

000

1.00

0

s(j)

1.00

01.

000

1.00

01.

235

1.87

02.

501

2.94

22.

946

3.06

22.

721

2.45

72.

142

1.94

11.

766

1.56

31.

477

1.38

01.

282

1.14

2

(2.4

33)

Sour

ce:I

PUM

S,19

97.

165

ABRAHAM AKKERMAN

TAB

LE

V

Hou

seho

ldco

mpo

siti

onm

atri

x,A

rizo

na,1

990

Hou

seho

ldco

mpo

siti

onta

ble

0–4

5–9

10–1

415

–19

20–2

425

–29

30–3

435

–39

40–4

445

–49

50–5

455

–59

60–6

465

–69

70–7

475

–79

80–8

485

–89

90+

0–4

1.00

00.

000

0.00

00.

338

0.43

10.

563

0.50

50.

337

0.13

10.

038

0.01

30.

011

0.00

70.

000

0.00

00.

000

0.00

00.

000

0.00

0

5–9

0.00

01.

000

0.00

00.

000

0.12

10.

294

0.49

20.

537

0.28

80.

132

0.04

00.

026

0.01

30.

003

0.00

10.

000

0.00

00.

000

0.00

0

10–1

40.

000

0.00

01.

000

0.00

00.

020

0.05

70.

274

0.52

40.

503

0.28

10.

099

0.05

20.

027

0.00

60.

002

0.00

20.

003

0.00

00.

000

15–1

90.

000

0.00

00.

000

1.01

50.

041

0.01

80.

052

0.22

60.

439

0.45

70.

260

0.11

70.

066

0.00

80.

005

0.00

30.

003

0.00

00.

000

20–2

40.

000

0.00

00.

000

0.08

21.

248

0.13

60.

036

0.02

70.

120

0.25

60.

193

0.12

20.

075

0.03

90.

012

0.00

40.

000

0.01

30.

000

25–2

90.

000

0.00

00.

000

0.03

40.

081

1.30

10.

190

0.06

40.

032

0.06

20.

138

0.09

30.

068

0.04

30.

013

0.00

30.

000

0.00

00.

000

30–3

40.

000

0.00

00.

000

0.00

00.

023

0.07

81.

324

0.22

20.

056

0.02

70.

040

0.04

70.

055

0.01

70.

013

0.01

60.

000

0.00

70.

000

35–3

90.

000

0.00

00.

000

0.02

40.

000

0.03

00.

077

1.30

60.

214

0.07

30.

040

0.03

60.

036

0.02

90.

020

0.01

50.

010

0.00

00.

000

40–4

40.

000

0.00

00.

000

0.01

80.

016

0.00

80.

019

0.05

81.

323

0.20

90.

074

0.04

50.

018

0.02

20.

010

0.02

40.

014

0.00

20.

000

45–4

90.

000

0.00

00.

000

0.00

00.

011

0.00

60.

006

0.01

70.

055

1.26

50.

238

0.06

90.

043

0.01

80.

022

0.01

00.

025

0.01

40.

000

166


TAB

LE

V

Con

tinue

d

Hou

seho

ldco

mpo

siti

onta

ble

0–4

5–9

10–1

415

–19

20–2

425

–29

30–3

435

–39

40–4

445

–49

50–5

455

–59

60–6

465

–69

70–7

475

–79

80–8

485

–89

90+

50–5

40.

000

0.00

00.

000

0.00

00.

004

0.00

70.

004

0.00

60.

018

0.05

31.

300

0.23

50.

081

0.02

30.

018

0.00

70.

022

0.00

90.

047

55–5

90.

000

0.00

00.

000

0.00

00.

000

0.00

70.

002

0.00

30.

001

0.00

70.

051

1.25

30.

210

0.06

40.

011

0.00

70.

011

0.00

60.

019

60–6

40.

000

0.00

00.

000

0.00

00.

000

0.00

30.

002

0.00

30.

006

0.00

50.

016

0.04

91.

227

0.21

40.

065

0.02

50.

007

0.01

60.

026

65–6

90.

000

0.00

00.

000

0.00

00.

003

0.00

10.

001

0.00

40.

009

0.00

60.

007

0.02

60.

058

1.23

90.

226

0.06

70.

026

0.03

30.

000

70–7

40.

000

0.00

00.

000

0.00

00.

000

0.00

00.

006

0.00

40.

003

0.01

40.

010

0.00

80.

011

0.04

91.

173

0.19

90.

082

0.01

70.

011

75–7

90.

000

0.00

00.

000

0.00

00.

000

0.00

00.

001

0.00

20.

002

0.00

70.

007

0.00

70.

003

0.01

00.

040

1.15

80.

150

0.06

30.

054

80–8

40.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

002

0.00

70.

003

0.01

70.

007

0.00

60.

006

0.04

41.

082

0.09

90.

056

85–8

90.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

001

0.00

00.

000

0.00

40.

007

0.00

30.

001

0.00

30.

028

1.00

00.

004

90+

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

001

0.00

20.

001

0.01

10.

004

0.00

50.

000

0.00

00.

009

1.00

0

s(j)

1.00

01.

000

1.00

01.

511

1.99

92.

509

2.99

13.

340

3.20

32.

900

2.53

12.

218

2.02

31.

797

1.64

31.

587

1.46

31.

288

1.21

7

(2.4

69)

Sour

ce:I

PUM

S,19

97.

167

ABRAHAM AKKERMAN

to the issue of intrahousehold resource allocation in economics.Intrahousehold resource allocation provides an alternative to tradi-tional economic approach where households are considered singularunits, as if they were indivisible entities (Chiappori, 1997). Indemography, it has been pointed out (Van Imhoff et al., 1995,pp. 345–351), the problem is conceptually opposite, in that nouniversally accepted theoretical framework exists to account forindividuals as affiliated persons within households. Implicit in theconventional methodology of “large population” demography is thetacit assumption that persons are unaffiliated, detached individualsas if households did not exist (cf. also critique by Inaba, 1995).Intrahousehold distribution counters this by facilitating a tabularrepresentation of household composition through a chosen attribute,age in the present case.

While representing the mutual affiliation between householdersand members, household composition ratios also link the threedistributions, k, b, and w, in an analytical bond. This bond becomesevident through examination of the rudimentary relations holdingbetween the three distributions, k, b, and w. Much as the age specifichousehold size sj defines the formal relationship (1) between thedweller distribution, b, and the age distribution, k, of householders,so also analogous arithmetic relations holding between the other twopairs of distributions yield meaningful household parameters.

The household distribution, k, and the age distribution, w, yieldthe age-specific headship ratios, hj (j = r + 1, . . . , n), defined(Murphy, 1991) as

hj = kj/wj . (2)

The remaining relation between the age distribution, w, and thedweller distribution, b, constitutes a linear dependency upon (1) and(2). Since each element bj of b is,

bj = kj × sj ,

it also follows immediately that

bj/wj = hj × sj . (3)

Both hj and sj relate to the number, kj , of household heads by age ina population, and the value in (3) could be therefore designated cj ,cj = hj × sj , and referred to as the age-specific headship coefficient.

168


FERTILITY AND RECRUITMENT OF THE YOUNG

The overall headship ratio, h, and the average household size, s, arethe inverse values of each other, and they yield the overall head-ship coefficient that equals 1. Since the headship coefficient, cj , isdefined by average household size and by the headship ratio, allparameters specific to age, it signifies a conceptual link betweena community’s demography and its housing. Households representoccupied dwelling units, on the one hand, while on the other handthe coefficients cj are also measures of household composition inthe community.

As an analytic concept, household composition has been repre-sented in a tabular format particularly suitable to demographicanalysis and forecasting. The household composition table, withadditional r columns prefixed according to a specified convention,yields the household composition matrix. The convention calls forall entries in the prefixed columns to be set to zero, except for thediagonal entries in the prefixed columns, which are set each to 1(Akkerman, 1980, 1996). The household composition matrix is adescriptive device of household population structure that involvesa linear relationship between household heads and household-persons. The matrix facilitates a linear transformation from the agedistribution k of household heads, onto the age distribution w ofpersons in households:

Ak = w. (4)

Tables IV and V show the household composition matrices forPhoenix and for the State of Arizona, 1990, respectively. The rela-tionship (4) is exemplified by columns (b) and (e) of Tables II andIII, with the respective household composition matrices in Tables IVand V. Possibly the main advantage in the tabular representation ofhousehold composition rests in the specific consideration of house-hold lifecourse, in its link to home-ownership, crowding or otheraspects of housing behavior (Myers, 1999).

The lifecourse of households is related to a similitude betweenthe trajectories of household recruitment and household composi-tion, with the headship coefficients cj emerging to be of represen-tative significance. Normalizing the household composition matrixby dividing each entry aij or (ajj + 1) by sj , results in a proba-

169

ABRAHAM AKKERMAN

bility matrix N such that each column-total of N yields 1. It followsimmediately that the probability matrix, N, constitutes a linear trans-formation from the age distribution b onto the age distributionw,

Nb = w, (5)

with probabilities pij (i, j = 1, . . . , n) making up the entries of N.Each entry pij is the conditional probability for a person in a house-hold, whose head is in age group j, to be in age group i(i, j = 1, . . . ,

n). Tables VI and VII show the normalized household compositionmatrices for Phoenix and for Arizona, 1990, with columns (c) and(e) in Tables II and III illustrating the relationship (5). Within thecontext of the probability matrix N the values bj and wj , as entriesof column vectors in the matrix relationship (5), also define theheadship coefficients cj in (3), cj = bj /wj , and thus also the dotproduct,

n∑

j=1

cj =n∑

j=1

bj/wj (j = 1, . . . , n).

Of further interest in the matrix N, are the entries p1j , each standingfor the probability for a member of a household whose head is inage group j, to be in the youngest age group, age group 1. Theprobability p1j is a proportion of children 0–4 to householders inage group j(j = r + 1, . . . , n). The probabilities p1j , therefore,provide a cross-sectional indication of recruitment of children withreference to the age of householder. Figure 4 shows the trajectoryof the household recruitment probabilities, p1j , for Phoenix and theState of Arizona as compared with conventional female age-specificfertility rates for the State of Arizona.

The observed affinity of household recruitment with a proba-bility density function in Figure 4 relates to the representation ofhousehold recruitment as a set of conditional probabilities in age-intervals of the young. As a measure of household recruitment, theconditional probability p1j of the youngest, 0–4, to be affiliated witha householder in age group j, could be seen as an extension to thenotion of enumerated own-children, introduced in demography forinstances where conventional fertility rates are not attainable (Cho,1971; Pressat, 1978, pp. 89). Consistent with Cho’s early approach

170


(also Cho et al., 1986), the notion of household recruitment couldbe further expanded to include the entire age-interval of the young,0–14. The age-groups 0–4, 0–9, and 0–14 as represented in thehousehold composition matrix could be seen, in the same vein, asthe net result of recruitment of the young by households, includinglive births, deaths of children, and child adoption and abandonment.

TRAJECTORIES OF HOUSEHOLD COMPOSITION

Bongaarts (2001) has pointed out that household size and fertilityare closely related. Other recent research has shown that conven-tional age-specific measures of fertility are invariably linked withfamily or household lifecourse (El-Khorazaty, 1997) and withhousing tenure (Paydarfar, 1995). Figures 5 and 6 show the 1990household composition ranges for age groups 0–4, 0–9, 0–14, andaverage household size, respectively, for Phoenix and for Arizona.The similarity in trajectories in Figures 4, 5 and 6 corroboratescorrespondence between fertility rates and ratios of children presentin households. Trajectories of average household size, and of thehousehold composition range 0–4 in Figures 5 and 6, show also afair similarity between Phoenix and Arizona.

Viewing the values sj according to age j of householders yieldsa trajectory of age-specific average household size, as shown inFigures 5 and 6 for Phoenix and for the State of Arizona, 1990.Both Figures suggest, along with Figure 4, that also the trajec-tories of age-specific average household size and of age-specificfertility rates are closely related. The results in the three figures showthat household composition provides the means to further explorethe proposition that fertility should be regarded jointly with other,housing related, considerations such as presence of older or othersiblings (Myers and Doyle, 1990).

The further notion of relationship between nuptiality and fertility,as elements of household formation (e.g., Haines, 1990), leads tothe suggestion that household recruitment might follow a statis-tical or probabilistic function linked in the past with marriage andfertility patterns. The trajectory of age-specific fertility in a popula-tion usually follows the trajectory of age-specific rates of firstmarriage (e.g., Pressat, 1978, pp. 74–79, 92–97), and same statistical

171

ABRAHAM AKKERMANTA

BL

EV

I

Nor

mal

ized

hous

ehol

dco

mpo

siti

onfo

rP

hoen

ix,1

990

0–4

5–9

10–1

415

–19

20–2

425

–29

30–3

435

–39

40–4

445

–49

50–5

455

–59

60–6

465

–69

70–7

475

–79

80–8

485

–89

90+

0–4

1.00

00.

000

0.00

00.

085

0.22

70.

217

0.15

30.

090

0.04

40.

013

0.00

80.

002

0.00

20.

000

0.00

00.

000

0.00

00.

000

0.00

05–

90.

000

1.00

00.

000

0.00

00.

059

0.13

00.

168

0.15

10.

074

0.05

00.

006

0.01

40.

003

0.00

30.

000

0.00

00.

000

0.00

00.

000

10–1

40.

000

0.00

01.

000

0.00

00.

012

0.01

80.

079

0.12

70.

144

0.07

30.

031

0.03

80.

024

0.00

00.

000

0.00

00.

000

0.00

00.

000

15–1

90.

000

0.00

00.

000

0.81

10.

020

0.01

40.

020

0.07

50.

130

0.14

00.

091

0.04

90.

033

0.00

80.

000

0.00

00.

000

0.00

00.

000

20–2

40.

000

0.00

00.

000

0.00

00.

610

0.05

00.

012

0.01

10.

044

0.09

50.

068

0.03

80.

041

0.03

70.

008

0.00

30.

000

0.02

00.

000

25–2

90.

000

0.00

00.

000

0.10

40.

035

0.51

40.

066

0.02

20.

017

0.03

20.

068

0.03

60.

033

0.03

20.

010

0.00

00.

000

0.00

00.

000

30–3

40.

000

0.00

00.

000

0.00

00.

014

0.02

60.

459

0.05

90.

016

0.01

40.

015

0.02

30.

014

0.01

20.

004

0.01

80.

000

0.00

00.

000

35–3

90.

000

0.00

00.

000

0.00

00.

000

0.01

60.

022

0.43

20.

067

0.02

40.

013

0.00

80.

025

0.01

80.

016

0.00

00.

012

0.00

00.

000

40–4

40.

000

0.00

00.

000

0.00

00.

003

0.00

40.

007

0.02

00.

434

0.07

20.

037

0.02

10.

009

0.01

50.

005

0.01

60.

000

0.00

00.

000

45–4

90.

000

0.00

00.

000

0.00

00.

011

0.00

40.

004

0.00

20.

018

0.44

70.

075

0.02

40.

016

0.01

40.

009

0.00

70.

030

0.00

00.

000

50–5

40.

000

0.00

00.

000

0.00

00.

007

0.00

60.

003

0.00

40.

004

0.02

00.

555

0.08

60.

043

0.00

60.

012

0.00

60.

036

0.02

70.

000

55–5

90.

000

0.00

00.

000

0.00

00.

000

0.00

10.

001

0.00

20.

000

0.00

10.

018

0.59

70.

098

0.03

10.

014

0.00

00.

005

0.00

00.

066

60–6

40.

000

0.00

00.

000

0.00

00.

000

0.00

20.

002

0.00

00.

003

0.00

30.

007

0.02

40.

621

0.11

40.

031

0.00

40.

000

0.02

00.

000

65–6

90.

000

0.00

00.

000

0.00

00.

003

0.00

00.

000

0.00

20.

002

0.00

40.

001

0.01

60.

020

0.68

20.

111

0.03

60.

021

0.01

60.

000

70–7

40.

000

0.00

00.

000

0.00

00.

000

0.00

00.

002

0.00

30.

001

0.00

80.

002

0.00

70.

006

0.01

50.

740

0.10

10.

036

0.00

00.

000

75–7

90.

000

0.00

00.

000

0.00

00.

000

0.00

00.

002

0.00

10.

000

0.00

10.

005

0.00

00.

002

0.00

00.

035

0.80

00.

101

0.08

00.

000

80–8

40.

000

0.00

00.

000

0.00

00.

000

0.00

00.

001

0.00

00.

002

0.00

40.

000

0.01

40.

000

0.00

70.

000

0.00

90.

760

0.05

80.

059

85–8

90.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

40.

008

0.00

50.

000

0.00

00.

000

0.78

00.

000

90+

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

30.

000

0.00

40.

000

0.00

00.

000

0.87

6

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

0

Sour

ce:T

able

IV.

172

PARAMETERS OF HOUSEHOLD COMPOSITIONTA

BL

EV

II

Nor

mal

ized

hous

ehol

dco

mpo

siti

onfo

rA

rizo

na,1

990

0–4

5–9

10–1

415

–19

20–2

425

–29

30–3

435

–39

40–4

445

–49

50–5

455

–59

60–6

465

–69

70–7

475

–79

80–8

485

–89

90+

0–4

1.00

00.

000

0.00

00.

224

0.21

60.

224

0.16

90.

101

0.04

10.

013

0.00

50.

005

0.00

30.

000

0.00

00.

000

0.00

00.

000

0.00

05–

90.

000

1.00

00.

000

0.00

00.

061

0.11

70.

164

0.16

10.

090

0.04

60.

016

0.01

20.

006

0.00

20.

001

0.00

00.

000

0.00

00.

000

10–1

40.

000

0.00

01.

000

0.00

00.

010

0.02

30.

092

0.15

70.

157

0.09

70.

039

0.02

30.

013

0.00

30.

001

0.00

10.

002

0.00

00.

000

15–1

90.

000

0.00

00.

000

0.67

20.

021

0.00

70.

017

0.06

80.

137

0.15

80.

103

0.05

30.

033

0.00

40.

003

0.00

20.

002

0.00

00.

000

20–2

40.

000

0.00

00.

000

0.05

40.

624

0.05

40.

012

0.00

80.

037

0.08

80.

076

0.05

50.

037

0.02

20.

007

0.00

30.

000

0.01

00.

000

25–2

90.

000

0.00

00.

000

0.02

30.

041

0.51

90.

064

0.01

90.

010

0.02

10.

055

0.04

20.

034

0.02

40.

008

0.00

20.

000

0.00

00.

000

30–3

40.

000

0.00

00.

000

0.00

00.

012

0.03

10.

443

0.06

60.

017

0.00

90.

016

0.02

10.

027

0.00

90.

008

0.01

00.

000

0.00

50.

000

35–3

90.

000

0.00

00.

000

0.01

60.

000

0.01

20.

026

0.39

10.

067

0.02

50.

016

0.01

60.

018

0.01

60.

012

0.00

90.

007

0.00

00.

000

40–4

40.

000

0.00

00.

000

0.01

20.

008

0.00

30.

006

0.01

70.

413

0.07

20.

029

0.02

00.

009

0.01

20.

006

0.01

50.

010

0.00

20.

000

45–4

90.

000

0.00

00.

000

0.00

00.

006

0.00

20.

002

0.00

50.

017

0.43

60.

094

0.03

10.

021

0.01

00.

013

0.00

60.

017

0.01

10.

000

50–5

40.

000

0.00

00.

000

0.00

00.

002

0.00

30.

001

0.00

20.

006

0.01

80.

514

0.10

60.

040

0.01

30.

011

0.00

40.

015

0.00

70.

039

55–5

90.

000

0.00

00.

000

0.00

00.

000

0.00

30.

001

0.00

10.

000

0.00

20.

020

0.56

50.

104

0.03

60.

007

0.00

40.

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0.00

50.

016

60–6

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000

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00.

000

0.00

00.

000

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001

0.00

10.

002

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0.02

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607

0.11

90.

040

0.01

60.

005

0.01

20.

021

65–6

90.

000

0.00

00.

000

0.00

00.

002

0.00

00.

000

0.00

10.

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0.00

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003

0.01

20.

029

0.68

90.

138

0.04

20.

018

0.02

60.

000

70–7

40.

000

0.00

00.

000

0.00

00.

000

0.00

00.

002

0.00

10.

001

0.00

50.

004

0.00

40.

005

0.02

70.

714

0.12

50.

056

0.01

30.

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75–7

90.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

10.

001

0.00

20.

003

0.00

30.

001

0.00

60.

024

0.73

00.

103

0.04

90.

044

80–8

40.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

001

0.00

20.

001

0.00

80.

003

0.00

30.

004

0.02

80.

740

0.07

70.

046

85–8

90.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

20.

003

0.00

20.

001

0.00

20.

019

0.77

60.

003

90+

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

00.

000

0.00

10.

000

0.00

50.

002

0.00

30.

000

0.00

00.

007

0.82

2

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

01.

000

1.00

0

Sour

ce:T

able

V.

173

ABRAHAM AKKERMAN

Figure 4. Age-specific female fertility rates (Arizona, 1990), conditional proba-bilities to belong in age group 0–4 (Phoenix and Arizona, 1990) and gammaprobability density function.

Figure 5. Age-ranges of the young and average household size, Phoenix, 1990.

or probability functions have often approximated the correspondingcurves of both trajectories. One of the more common approxima-tions for age-specific rates of both first marriage and fertility hasbeen the gamma probability density function (Coale and McNeil,1972; Hoem et al., 1981; cf. also Frejka and Calot, 2001).

The curve of the gamma probability density function in Figure4 has its scale parameter selected as 1, and its shape parameterselected as (s + 1), where s is the approximate observed averagehousehold size, for Phoenix or Arizona. Since the scale parameter is

174


Figure 6. Age-ranges of the young and average household size, Arizona, 1990.

1, the gamma function is shortened to the reduced standard gammaprobability density function,

f (x) = xse−x/�(s + 1), (6)

where, in Figure 4 and in Equation (6), s = 2.4 (the average house-hold size for Phoenix or Arizona, from Tables IV and V), and xattains the discrete quantities 1, 2, 3 . . . in correspondence to valuesof age at the beginning of 5-year householder age-intervals, 15, 20,25, . . . , 90, factored by 5.

As shown in Figure 4, the standard gamma function (6) appearsto provide a good fit to trajectories of the normalized recruitmentvalues p1j for both Phoenix and Arizona. The observation that thereduced standard gamma function could provide a fit to a recruit-ment measure that expresses relation between household composi-tion and age distribution, signals a step towards the interpretation offertility as a structural demographic aspect of housing.

The advantage of the relationship between household composi-tion and recruitment becomes evident particularly for smallerpopulations. Age-specific fertility rates for Phoenix, 1990, wouldhave to be calculated for exceedingly small numbers within eachage group (as in Table I), and an unreliably small numbers ofcorresponding live births. On the other hand, household compositionfor Phoenix appears to render information that would have beendifficult to attain through conventional, age-specific fertility ratesfor the relatively small age groups of Phoenix.

175

ABRAHAM AKKERMAN

HOUSEHOLD COMPOSITION AS A PROCESSUAL CONCEPT

Bongaarts (1987) had incorporated a generalized concept of thelife-table within a formulation of change in household composition.Household composition, as presented in Tables IV and V, could toobe viewed as a processual, longitudinal concept. By this reasoning,average number of persons per household along the horizontaldimension of householder age groups in each of the tables, is inter-preted as a function of householder’s age. Age-specific householdsize emerges thus as an indicator of household lifecourse, measuredagainst the age of householders.

Based upon these considerations household composition extendsbeyond its description as a relational demographic structure at asingle point in time. The household composition table, in fact,can be viewed somewhat analogously to the period life-table indemography. The life-table shows sequential survival pattern in atheoretical cohort of persons from their births at a single interval oftime, up until the end of life of the last person in the cohort, aftern intervals. In the period life-table, however, the survival pattern isnot an observational record of a longitudinal follow-up of newbornsthroughout their lives. Rather, the theoretical (stationary) populationof the period life-table has a survival pattern that is derived, duringa single interval of time, from observed proportions of personsin different age groups, surviving throughout their current age-intervals. Cohort survival of the stationary population in the periodlife-table is reflected in a period change represented by a column ofstationary age distribution. Household composition can be viewed,analogously, as a net-indicator of household formation, change andattrition over time.

The period life-table and the household composition table informexisting demographic structures at a single point in time: The life-table references, at any single point in time, the stationary popula-tion age distribution (identical to the survival pattern of age cohortsin this theoretical population); the household-composition table, too,describes the intrahousehold age distribution of household-personswithin the average household at a single point in time. And just asthe life-table entails a survival pattern during the entire lifecourseof a cohort, so too household composition (such as Tables IV andV) could be viewed as displaying the net result of ageing, reproduc-

176


tion, as well as household formation and attrition over the entirelifecourse of the average household.

Direct observation, of course, does not produce an averagehousehold, any more than it shows a stationary population. Never-theless, the theoretical consideration of the average household is asimportant in small populations as the consideration of a stationarypopulation is in mainstream demography. The ageing of a theoret-ical cohort in a life-table occurs through attrition, due to mortality,so that a cohort gradually decreases in size until it vanishes, oncepast the last age-interval. In a life-table the survivorship of thestationary population is shown vertically from the top of the age-distribution column to its very bottom. In contrast, in the tabularnotion of household composition, ageing and progressive householdaffiliation of the theoretical average household person is shownalong the diagonal direction of the household composition table.Due to individuals leaving and joining households, the household-affiliation of persons changes, and thus the entries aij of householdcomposition, such as those in Tables IV and V, are the net-results ofoverall household dynamics in the population.

SUMMARY AND CONCLUSION

The dichotomy between household heads and household membersaffiliated with household heads facilitates representation of house-hold affiliation status in an age-distribution. Thus, kj is the numberof household heads in age group j, and bj , is the number of dwellersaffiliated with household-heads in age group j. This also resultsin the formulation of intrahousehold distribution of persons byage. The household composition table constitutes the full range ofintrahousehold distribution applied across all age groups of house-holders. A summary measure of intrahousehold distribution, on theother hand, is the average household size specific to the age ofhouseholder.

Somewhat similar to cohort survival in a stationary populationof the life table, the age-specific household size in the householdcomposition table can be interpreted as a cross-sectional measure ofhousehold lifecourse. Such interpretation applies also to each age-range within the household composition. While in the life-table a

177

ABRAHAM AKKERMAN

vertical follow-up of cohort entries points to cohort survival overtime, in the household composition matrix a diagonal follow-upof intrahousehold distribution entries points to the net result overtime of survival, children-present in households, and householdformation and attrition,.

The formulation of household affiliation in an age-distributioncomes also in the wake of attempts to question the conventionalapplication of age-specific fertility rates in smaller populations, onthe one hand, and efforts to link fertility with housing conditionsor tenure, on the other. While application of age-specific fertilityrates to large populations has been the standard, formulation offertility measures in smaller populations has been proposed on thebasis of kinship or formal household settings (e.g., Suzuki, 2001).The household composition table is consistent with this effort in itsrepresentation of recruitment. The first several rows constitute a setof ratios of children per householder by age. Each of these ratiosis interpreted as a measure of recruitment, somewhat analogous toCho’s measure of fertility by enumerated own-children (e.g., Cho,1971). The advantage of this consideration is that the age-specificmeasure of children per householder is also related to intrahouse-hold distribution, and thus also to average household size by age ofhead.

Linkages between fertility and housing have been identified overthe last several decades both at the empirical level (e.g., Hohm,1984; Lapkoff, 1994), as well as in formalized, quantifiable contexts(Myers, 1990; Paydarfar, 1995). The advantage in the reinterpreta-tion of fertility as household recruitment of the young is underscoredby the similitude of household recruitment trajectories with thestandard gamma probability density function. The tabular represen-tation associated with household structure thus provides the basis fora conceptual framework in which community demography, reflectedin household composition ratios and in recruitment ratios in partic-ular, is linked to shelter. The measure emerging from this contextis the age-specific headship coefficient, defined as, hj × sj , wherehj is the conventional age-specific headship ratio (i.e., proportion ofhousehold heads within the population of an age group) and sj isthe average household size by age of head. Since the measure, asthe present study shows, is equivalent to the proportion between the

178


population of all dwellers affiliated with household heads in a givenage group, and the size of the age group, the age-specific headshipcoefficient links directly population with housing.

Unfortunately, household composition data are still not com-monly available from published official government statistics. TheIPUMS, made publicly accessible in 1997, provides a unique sourcefor comprehensive demographic and kinship information throughsampled household records extracted from U.S. censuses (see alsoRuggles et al., 2000). Presently, however, the IPUMS is the onlyeasily available source anywhere for the compilation of house-hold composition tables. Based upon the utilization of the IPUMS,national statistical agencies will, hopefully, give thought to theproduction of household composition tables and their publication.

ACKNOWLEDGMENTS

Revised from a paper presented at the Conference of Association ofCollegiate Schools of Planning, 8–11 November, 2001, Cleveland,Ohio. I am grateful to Ronald Lee, Tim Miller and Aaron Gullickfor their guidance through the IPUMS during my stay at the Depart-ment of Demography, University of California, Berkeley. Researchassistance was provided by Zachariah Akkerman.

NOTES

1 Consistent with a U.S. census definition (Smith, 1992), a household head ora householder is a person who is “marked” as the household’s reference person,and by definition, there is only one such person per household. Throughout thispaper the terms household head and householder will be used interchangeably,both denoting the same individual and household status.2 In a dweller population, such as the majority of the resident populationthroughout a city, each household can be thus perceived as having one andonly one householder and none, one or more members; a member cannot be ahouseholder, and vice versa; and any dweller who is not a member is a house-holder.3 The IPUMS consists of twenty-five high-precision samples of the Americanpopulation drawn from the thirteen federal censuses from 1850 to 1990. Some ofthese samples had been created independently for purposes of past research, whileothers were created specifically for the IPUMS. The twenty-five samples are said

179

ABRAHAM AKKERMAN

to collectively comprise the richest source of quantitative information on long-term changes in the American population (Gardner et al., 1999; Ruggles et al.,2000). In order to facilitate analysis of social and economic change the IPUMSassigns uniform codes across all the samples. In the particular case of the presentstudy, however, only the 1990 sample has been utilized.

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Department of GeographyUniversity of Saskatchewan9 Campus DriveSaskatoon, SK S7N 5A5CanadaE-mail: [email protected]

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Parameters of Household Composition as Demographic Measures