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APPENDIX 3-1 SUMMARY EXPLANATION OF FORMULA CHARACTERISTICS

AND WEIGHTS

Section 509(a) of the National Affordable Housing Act (NAHA)

established a system of formula allocation of modernization funds to

HAs with 250 or more units.

Data Used for the Formula

This formula was developed using data derived from the following

sources:

1\. The Modernization Needs Study: National, Regional and Field

Office Estimates: Backlog of Modernization Needs, prepared by Abt

Associates, Inc., of Cambridge, Massachusetts, 1988 (\"the Abt

report\");

2\. Future Accrual of Capital Repair and Replacement Needs of

Public Housing, prepared by ICF, Inc., of Fairfax, Virginia, 1989 (\"

the ICF report\"); and

3\. HUD operational information systems on past uses of

modernization funds.

The basis for the formula established in NAHA is data on the

estimated modernization needs of public housing developments derived

from the inspections conducted as part of the study of the backlog of

modernization needs in public housing developments. This
development-level information was then tested against a wide variety of

characteristics of public housing developments to determine which

characteristics came closest to predicting what HUD already knew about

the developments from the inspection data.

While the modernization needs study was not designed to produce

development-level estimates of modernization need, but rather to

produce national and regional totals of this need, the method of

developing these estimates called for detailed inspections at a

scientifically selected sample of 996 public housing developments.

This method produced a wealth of data about individual developments,

enough data to permit the development of modernization backlog

estimates for each of these sampled developments.

The availability of this data for a large sample of individual

developments permits the use of an indirect method, based on the

characteristics of public housing developments, for determining the

relative modernization need of individual HAs. This indirect method

involves estimating statistical relationships between modernization

needs and the characteristics of a HA\'s inventory of developments.

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These relationships can be estimated from the information

provided in the Modernization Needs Study, the Study of Future Accrual

of Capital Repair and Replacement Needs, and from supplementary

information provided by HUD and the Census on the PHAs and the

communities in which they are located. These relationships are then

applied to a database containing development, HA, and community

characteristics for each HA with over 250 units to yield an estimate

of modernization needs for each of these HAs. These HA needs

estimates, when expressed as relative shares of total modernization

needs, provide indicators of relative need and allocation factors that

can be used to distribute funds in a formula-based funding system.

The Abt and ICF studies of modernization needs provide the

essential tools for estimates of HA need based on development

characteristics. First, they provide estimates of the backlog and

accrual need for 996 developments on the basis of standardized

inspection and modeling.

Second, Abt collected basic data about developments for these and

5,670 other developments in more than 950 HAs (including almost all

HAS with 500 or more units under management). Previous Abt and HUD

studies also provided contextual indicators of need at the HA and

community level to apply to all of the developments.

Third, the Abt and ICF studies provided sampling weights that

enable statistical relationships of need for the 996 developments to

be generalized to all developments with the same set of objective

indicators\--the Abt sampling frame of 6,670 developments as well as

other developments for which data are easily collected.

The formula estimate of needs for public housing developments was

developed using a statistical technique called multivariate

regression. Multivariate regression, a technique that HUD already

uses to help set HA allowable operating expenses under the Performance

Funding System (PFS), allows selection of a set of indicators which

maximizes the explanation (or \"fit\") of an independent measure. The

multivariate technique also suggests the explanatory contribution of

each indicator, individually and in combination with other indicators.

Selection Criteria for Indicators

Development characteristics had to meet several criteria in order

to be used as explanatory indicators in the final regression estimates

of backlog need and accrual need (each estimated separately). These

criteria were:

Believability. Previous studies or professional judgment related

the characteristic to some aspect of public housing need.

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Individual explanatory significance. The development

characteristic had to be significantly related (positively or

negatively), with 97.5 confidence, to the need measure\--in statistical

terms, a \"t-ratio\" above 2, or below minus 2, as appropriate.

Net explanatory impact. When added to other characteristics to

be used as indicators, the characteristic adds significantly to the

\"fit\" (for this formula, an increase in \"adjusted R-squared\" of at

least .25, when the maximum R-squared is 100).

Expected Sign. The characteristic is related to the estimated

measure in the expected direction (over different time periods, in the

case of the accrual measure).

Non-collinearity. The explanatory indicator cannot be explained

to a large degree by one or a set of other explanatory indicators

(operationalized by a R-squared of less than 75, where 100 is

maximum). When indicators are collinear, the indicator with the best

quality and rationale is chosen, or collinear indicators are combined.

Clarity. The characteristic has a clear-cut specification not

easily manipulated in data reporting.

Continuity. The indicator has as continuous a range of values as

possible, in order to avoid a notch effect.

Ease of collection. The characteristic has been collected, or

can be collected, in a standardized form from all HAs.

Indicators Selected for the Backlog Formula

Using the selection criteria just described, seven indicators

were chosen to estimate the backlog of modernization needs at public

housing developments to modernize them to the HUD modernization

standards and to ensure their long-term viability. These indicators

were presented in Appendix B of the Report to Congress on Alternative

Methods of Funding Public Housing Modernization, and are listed as the

backlog formula elements in Section 509(a) of NAHA.

The formula elements used in the backlog formula and their

assigned weights (i.e., formula coefficients) are as follows:

1\. Average number of bedrooms or unit in a development

(AVEBED), weighted at 2858.7. This indicator represents the mix of

elderly and family units in a development and represents the square

footage per unit that can require repair. A higher number should

yield a higher backlog per unit. Efficiencies are counted as zero

bedrooms for this indicator.

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2\. Large family units in a development (LFAM), weighted at

7295.7. This indicator represents very large families and a

concentration of children in a development, with consequent wear and

tear on the physical facilities. A higher number should yield a

higher backlog per unit. This indicator is the difference been the

average number of bedrooms per unit and 2.5, with values below zero

set to zero.

3\. High-rise family development (HRFAM), weighted at 5555.8.

This indicator represents the special problems, such as elevator

repair, of housing families with children in high-rise buildings. A

value of \"1\" is given for developments averaging at least 35 units per

building and averaging at least 1.5 bedrooms per unit or averaging

between 1.2 and 1.5 bedrooms per unit and having at least 100 units of

2 or more bedrooms. A value of \"1\" should yield higher backlog per

unit, relative to all other developments given a value of zero.

4\. Age of development (BLDAG), weighted at 206.5. This

indicator is the physical age of the development as of 1985 (the year

in which the backlog inspections were conducted), capped at 50 years

for acquired developments. It represents years of physical wear, and

sometimes, years of undermaintenance. A higher value should yield a

higher backlog per unit.

5\. Total family units in a large HA (FAMLPHA), weighted at

.433. This indicator is the number of units with 2 or more bedrooms in

an HA, with 5,000 deducted from that number. Values are capped at

zero and 15,000. It represents the complex social and physical

environment of housing low-income families in a concentrated manner in

a large HA. A higher number should yield a higher backlog per unit.

6\. Area cost index (MEANS), weighted at 27544.3. This indicator

is the R.S. Means index (the area cost index used in the Abt and ICF

studies), calibrated nationally at 1.00, with values expressed as the

index minus 1. It represents inter-area differences in the cost of

rehabilitating a given physical property. A higher value should yield

a higher backlog per unit.

7\. Severe population decline in the community (SPOPL), weighted

at 759.5. This indicator is a population loss from 1970 to 1980 in

excess of 12 percent for the community. When community population

loss is below 12 percent, or when the project is a project for the

elderly, the value is zero. It represents community and neighborhood

problems, such as abandonment of property, that can accentuate wear

and tear for developments with children. A higher value should yield

more backlog per unit. (Although data from the 1990 census are

available, this indicator is not updated to reflect the more recent

data, since it reflects conditions in a community in the years

preceding the 1985 development inspections on which the regression

equation is based.)

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In addition, an equation constant of 1412.9 is used to calibrate

these indicators for estimating backlog need per unit.

The following table shows the impact of each of these indicators

in estimating the backlog needs for three types of developments:

1\. FM\--a \"family\" development in a medium HA. It is assumed to

average 2 bedrooms per units; to be in a HA with 400 units of 2 or

more bedrooms; to be in an area with a R.S. Means index of .98 (so

that the Means indicator is -.02=.98-1.0); and to be in a community

whose population increased 1% from 1970 to 1980.

2\. EL\--an \"elderly\" development in a very large HA. It is

assumed to average one bedroom per unit and to be in an area with a

R.S. Means index of 1.02, so that .02=1.02-1.00. Although the HA for

this \"elderly\" development has 7000 family units and its area had a

population loss of 16 percent, the \"elderly\" development is still

given a value of zero on the FAMLPHA and SPOPL indicators. (Compare

this to the very large family development assumed to be in the same

PHA discussed below.)

3\. LFL\--A large family development in a very large HA. It is

assumed to average 3.0 bedrooms per unit (so the value for LFAM is .50

which is 3 minus 2.5); to be in a HA with 7000 units of 2 or more

bedrooms (so its indicator value for FAMLPHA is 2000=7000 minus 5000);

and to be in a community with a population loss of 16% from 1970 to

1980 (so its value for SPOPL is 4= 16-12).

The net effect on estimated need column in the table below is the

result of multiplying the regression coefficient by the respective

indicator values.

Table 1 \-- Estimating Backlog for Three Types of Developments

Net Effect on

Indicator Values Estimated Need

Regression Dev. Type: Dev. Type:

Indicator Coefficient FM EL LFL FM EL LFL

AVEBED 2858.7 2.0 1.0 3.0 5718 2859 8577

FAMPR N/A 100 0 300 N/A N/A N/A

LFAM 7295.7 0 0 .5 0 0 3648

HRFAM 5555.8 0 0 1.0 0 0 5556

BLDAG 206.5 20 10 30 4130 2065 6195

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FAMLPHA .433 0 0 2000 0 0 866

MEANS 27544.3 -.02 +.02 +.02 -551 551 551

SPOPL 759.5 0 0 4 0 0 3038

CONSTANT 1412.9 1413 1413 1413 1413 1413 1413

Predicted Need Per Unit: 10,710 6,888 28,844

Indicators Selected for the Accrual Formula

For this formula, accrual need is the 1986-1995 average of age-

related capital accrual needs as modeled by ICF from Abt data, and

updated to 1990 national totals. Using the selection criteria

discussed above, six indicators were chosen to estimate accrual need

per unit, and are incorporated as formula elements in Section 509 of

NAHA. The formula elements used in the accrual formula and their

assigned weights (i.e., formula coefficients) are as follows:

1\. Average number of bedrooms per unit (AVEBED), weighted at

100.1. This indicator represents the mix of elderly and family units

in a development, and the square footage per unit that can require

repair. A higher value should yield a higher accrual per unit.

Efficiencies are counted as zero bedrooms.

2\. Age of development (BLDAG), weighted at 10.4. This indicator

is the physical age of the development as of 1985, capped at 50 years

for acquired developments. It represents the stage of the useful life

of building systems and equipment. Up to a certain age, as modelled,

a higher value should yield a higher accrual per unit.

3\. Large family units in a development (LFAM), weighted at

356.7. This indicator represents very large families and a

concentration of children in a development, with consequent wear and

tear on the physical facilities. A higher number should yield a

higher accrual per unit. This indicator is the difference between the

average number of bedrooms per unit and 2.5, with values below zero

set to zero.

4\. Low-rise developments (LRIS), weighted at 87.1. This

indicator represents the likelihood of fewer economies of scale for

major systems, and the likelihood of larger square footage of floors,

walls, and roof covering when the number of bedrooms per unit is held

constant. It is the difference between five units per building and

the actual average number of units per building, e.g., a maximum value

of four for single-unit buildings. Values below zero are set to zero.

A higher value should yield a higher accrual per unit.

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5\. Cost index (MEANS), weighted at 679.1, This indicator is the

R.S. Means index (the area cost index used in the Abt and ICF

studies), calibrated nationally at 1.0, with values expressed as the

index minus 1.0. It represents inter-area differences in the cost of

rehabilitating a given physical property. A higher value should yield

a higher accrual per unit.

6\. HA total units (PHAUN), weighted at .0144. This indicator is

the total number of units in the HA, capped at 8000. It represents

the probable complexity of physical systems not captured by

development-specific indicators. A higher value should yield a higher

accrual per unit.

In addition, an equation constant of 602.1 is used to calibrate

these indicators to estimate accrual need per unit.

Table 2 shows the effect of these indicators in estimating

accrual need for the same three developments used to illustrate the

effects of the backlog indicators in Table 1, above.

Table 2 \-- Estimating Accrual for Three Types of Developments

Net Effect on

Indicator Values Estimated Need

Regression Dev. Type: Dev. Type:

Indicator Coefficient FM EL LFL FM EL LFL

1\. AVEBE 100.1 2.0 1.0 3.0 \$200 \$100 \$300

2\. BLDAG 10.4 20 10 30 205 104 312

3\. LFAM 356.7 0 0 .5 0 0 178

4\. LRIS 87.1 3 0 0 261 0 0

5\. MEANS 679.1 .02 +.02 +.02 -14 +14 +14

6\. PHAUN .0144 1000 8000 8000 14 115 115

Constant 602.1 602.1 602.1 602.1 602 602 602

Predicted Accrual Per Unit: \$1268 \$935

\$1521

Despite points of similarity, the selected indicators of accrual

differ in their specification, impact, and statistical fit from the

selected indicators of backlog need.

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Differences in the indicators of accrual and backlog needs simply

reflect differences in the measures of backlog and accrual needs that

are being estimated. Backlog and accrual are markedly different both

in concept and in the way they are measured. Backlog in concept is an

accumulation of need that can be expected to be higher, all else being

equal, in older or undermaintained developments. These types of

developments are usually found in larger HAs. Accrual in concept is

an incremental development modernization need that arises as systems

age, and begins to be high in developments of medium age. Many

medium-sized and small HAs have housing developments in this age

range. In practice, moreover, the accumulation of backlog needs in

some developments in larger HAs led to their partial abandonment by

both residents and management, thereby intensifying their backlog.

Backlog can mark the ravages of long, neglectful time, whereas accrual

is the slow decay of steady time.

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