VHA MODEL REVIEW by

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THESIS S...DVHA MODEL REVIEW

by

Michele L. Williams

March 1990

Thesis Advisor: Laura Johnson

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VHA MODEL REVIEW

12. PERSONAL AUTHOR(S)WILLIAMS, Michele L.

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Master's Thesis FROM TO _ 1990 March 30816 SUPPLEMENTARY NOTATION The views expressed in this thesis are those of the

author and do not reflect the official policy or position of the Depart-ment of Defense or the U.S. Government

17 COSATI CODES 18 SUBJECT TERMS (Continue on reverse if necessary and identify by block number)

FIELD GPOUP SUB-GROUP VHA, Regression models, median rent, weighted

least squares, Analysis of covariance, BAQ

19 ABSTRACT (Continue on reverse of necessary and identify by block number)

A regression model is used to predict median rents by the Office of

the Secretary of Defense (OSD) to find variable housing allowance (VHA)

as a supplement to Basic Allowance for Quarters (BAQ). These allowancesare made for service members in the continental United States. It isthis model that is reviewed in this thesis. Median rental data takenfrom the annual VHA survey is used to test this model. From thisanalysis, the mocel indicates lack of fit, invalid assumptions andperhaps not even a "reasonable" approach. A more sensible approach isused to propose two other regression models.

These models are a Weighted Regression Model which, like the currentmodel, predicts medians; and an Analysis of Covariance model which

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#19 (Continued)

predicts or analyzes the mean rent. More reasonable predictions ofmedian and mean rent are indicated by these two models respectively.

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VHA Model Review

by

Michele L. WilliamsLieutenant, United States Naval Reserve

BSBA, University of Denver, 1980

Submitted in partial fulfillment of therequirements for the degree of

MASTER OF SCIENCE IN OPERATIONS RESEARCH

from the

NAVAL POSTGRADUATE SCHOOL

March 1990

Author: . 6 ;6 )Michele L. Williams

Approved By: II-/ auraD/ohnson, Thesis Advisor

Donald P. Ga r, Second Reader

feter ue, Chairman, Department ofOperations Research

iii

ABSTRACT

A regression model. is used by the Office of the Secretary

of Defense (OSD) to predict median rents so as to find variable

housing Rllowance (VHA) as a supplement to Basic Allowance for

Quarters (BAQ). These allowances are made for service members

in the continental United States. It is this model that is

reviewed in this thesis. Median rental data taken from the

annual VHA survey are used to test this model. From this

analysis, the model indicates lack of fit, invalid assumptions

and perhaps not even a reasonable'2Yapproach. A more sensible

approach is used to propose two other regression models.

These models are a Weighted Regression Model which, like

the current model, predicts medians; and an Analysis of

Covariance model which predicts or analyzes the mean rent.

More reasonable predictions of median and mean rent are

indicated by these two models respectively.

iv/

iv

THESIS DISCLAIMER

The reader is cautioned that computer programs developed in

this research may not have been exercised for all cases of

interest. While every effort has been made, within the time

available, to ensure that the programs are free of computa-

tional and logic errors, they cannot be considered validated.

Any application of these programs without additional verifica-

tion is at the risk of the user.

v

TABLE OF CONTENTS

I. INTRODUCTION ..................................... 1

A. BACKGROUND ................................... 1

B. CURRENT VHA COMPUTATIONAL PROCESS .............. 2

C. PROPOSED PLAN TO UPDATE VHA COMPUTATIONALPROCESS ...................................... 5

D. DATA DESCRIPTION ............................. 7

E. PROBLEMS WITH THE DATA ....................... 9

F. PURPOSE OF THESIS ............................ 9

II. ANALYSIS PROCEDURES .............................. 11

A. ORDINARY LEAST SQUARES REGRESSION .............. 11

B. INITIAL MODELS TESTED USING ORDINARY LEASTSQUARES REGRESSION ........................... 16

C. WEI.,HTED LEAST SQUARES REGRESSION .............. 19

D. ANALYSIS OF COVARIANCE MODEL ................... 21

E. CROSS VALIDATION TECHNIQUES ..................... 22

III. ANALYSIS ......................................... 23

A. ANALYSIS OF CURRENT MODEL ....................... 23

B. ANALYSIS OF PROPOSED MODEL ...................... 27

C. ANALYSIS OF WEIGHTED LEAST SQUARES MODEL ..... 30

D. ANALYSIS OF THE ANALYSIS OF COVARIANCEMODEL ........................................ 32

IV. CONCLUSIONS AND RECOMMENDATIONS ..................... 35

APPENDIX A SCATTER AND RESIDUAL PLOTS ............. 38

APPENDIX B SAS PROGRAM EXAMPLE ................... 111

vi

APPENDIX C TABLES 1 - 14........................... 118

L IIST OF REFERENCES.................................. 298

INITIAL DISTRIBUTION LIST........................... 299

vii

I. INTRODUCTION

A. BACKGROUND

VHA, Variable Housing Allowance, is a supplement to the

BAQ, Basic Allowance for Quarters, paid to service members who

live in private housing in the United States. VHA is designed

to aid the service member who is assigned to a "high cost area"

of the United States where the median monthly cost of housing

for a person in the same grade or dependency status exceeds 80%

of the national median for members in the same rank or

dependency status [Ref. 1:p. 2-1]. VHA is computed from the

following equation [Ref. 1:p. 2-2]:

VHA = local median housing costs - 80 % of the natibonal (1)by paygrade and marital median housing coststatus by paygrade and

marital status.

The law specifies that each member's VHA allowance will be

determined by the actual housing costs currently paid by the

service member [Ref. 1:p. 2-2]. VHA rates are computed by the

Per Diem Travel and Transportation Allowance Committee Staff,

a subset of the Office of the Secretary of Defense (OSD), with

the aid of the Defense Manpower Data Center, DMDC. The basic

process by which the rates are computed is as follows:

1. Distinct areas in which military members reside aredetermined.

2. Proper sample sizes are determined.

3. Survey samples of housing costs are taken, edited andmedian rents are computed for each category of paygrade,house type, number of bedrooms, and marital status.

1

4. Preliminary VHA rates for each area and dependency statusare computed by determining an estimated median rent foreach category using the GPX program which utilizesvarious regression analysis techniques and smoothingprocedures. (GPX is the name of the model developed byOSD.)

5. Preliminary VHA rates are reviewed to ensure that the V

rates determined by GPX are in line with the costguidelines set by Congress.

B. CURRENT VHA COMPUTATIONAL PROCESS

The computation of preliminary VHA rates for each area

(MHA - military housing area), paygrade, and dependency status

has developed into an extremely complicated process. Once the

median rents are computed for each category of house type,

number of bedrooms, paygrade, and marital status, a count of

the number of median rents per category is taken [Ref. 1:p. 2-

56]. If the number of counts in each category for a particular

MHA is too small then larger sample sizes are obtained by

incorporating median rent information from the same category

from a close, in geographic terms, MHA. (Ref. 1:p. 2-58] This

information, taken from these close MHA's is then weighted.

The closer, in terms of miles, this MHA is to the original MHA

the more weight is placed on the information from that MHA.

[Ref. 1:p. 2-59] A new vector of median rents, incorporating

the information from the geographically close MHAs and

dimensioned by the four categories above is calculated. [Ref.

1:p. 2-59] The underlying reason for finding this vector of

median rents is to find the underlying relationship between

the total pay of a military member and the amount of rent a

2

military member will pay [Ret. 1:p. 2-60]. Let Pijkl = the total

pay for a person in the ith paygrade, in the jth dependency

status who has 'k' number of bedrooms in his or >her home and

an 'I' type oC home. Let Tijkl equal the median rent for

military members in that same group. Then the current

regression model in use is:

1 A + B + Eijkl (2)

Tijkl Pijkl

where rijkl is the error term. Standard linear Regression

techniques are use to est-mate A and B which assume the error

is normally distributed, homoscedastic, and with mean zero.

This in turn means that the distribution of inverted median

rent is normal and homoscedastic. It is not clear that these

assumptions are in any sense "reasonable". In fact if medians

tend to be normal, then the inverse will certainly not be

normal. Let A and B denote the regression estimates of A and

B, respectively. The estimates A and B are used to determine

the estimated median rents, Rijkl through the equation

Rijkl ( Pijkl (3)

(A + B Pijkl)

where Rijk, and Pijkl denote the rent and total pay, respectively,

for paygrade, marital status, number of bedrooms and house type

[Ref. 1:p. 2-60]. Generally, a separate A and B are determined

for the enlisted, company grade officers, and field grade

officer ranks. Thus a separate Rijkl is computed for each one

3

of these three ranks of military personnel. Rijkl is then sed

to determine owner equivalency median rents. Owner equivalency

rents are the rent fig- es assigned to a military member who

owns and does not rent his or her residence. Costs assignedV

to owners are thought not to be appropriate for use in

calculatir VHA since intangible benefits accrue to owners and

not to renters. These owner equivalency median rents are

weighted according to population percentage of owners and are

then incorporated into the vector of median rents [Ref. 1:p.

2-61]. This new vector of median rents, including both owner

and renter information, still has four dimensions and must then

be aggregated to the paygrade and dependency status level.

[Ref. l:p. 2-61] After this aggregation, a further smoothing

process and a denormalization process, the VHA rate multipliers

are finally computed by dividing by a weighted average of BAQ

rates [Ref. 1:p. 2-63]. These multipliers are checked and if

an inversion exists, which for example, is when paygrade 02

receives less VHA than paygrade 01, then additional smoothing

across paygrades will take place. If inversions still exist

after the smoothing process has taken place then the entire

computation of VHA multiplier rates begins again from the point

where data from close, in geographic terms, MHAs is used [Ref.

1:p. 2-64]. Median rent information is then taken from these

MHA's and the entire process is run again and again, up to 11

more times until the rate inversions cease to exist. If after

11 more times an inversion still exists then the GPX program

4

aborts and an inversion in the total population data is

assumed. [Ref. 1:p. 2-64)

C. PROPOSED PLAN TO UPDATE VHA COMPUTATIONAL PROCESS

In an effort to get away from the geographical weighting

of data from close proximity MHA's and in an attempt to

simplify the process of computing VHA rates, the Per Diem

Committee is investigating a new method for computing VHA

rates. Under this "new" plan, survey data from each MHA is

placed into various costing bands based on county rental data

from HUD (Department of Housing and Urban Development) in the

following manner. From each county in the United States, HUD

has data for the average rental costs in that county. A

military housing area is placed into a costing band with other

military housing areas which have the same average rental

costs. Therefore if the computed average rental cost for MHA

A is $260.00 and the median rental cost for MHA B is also

$260.00, MHA A and MHA B would be placed in the same costing

band. The computed median rent figure used in this "new"

process is a single figure found by taking a weighted average

of rental costs, based on number of bedrooms and house type,

from the national military population. For example, if 10% of

the national military population resides in one bedroom

apartments, the average rental cost of one bedroom apartments

for that MHA accounts for 10% of the total average rental cost

figure for that county. Initially the bands will be broken

into groups of $45.00 increments. The costing bands begin at

5

an average rental cost of $260.00 and continue up to $890.00.

There is one further ccsting band which accounts for the

extremely high average rental cost areas such as Alaska which

are so far above all of the other areas in terms of cost. Thus

there are a total of 15 different costing bands including the

"high" costing band. The idea behind grouping military housing

areas together which have similar average rental costs is to

provide more data points to reliably predict median rental

costs per paygrade and dependency status based on the survey

data. Also using an "outside", other than military, source to

group the data provides a small means of getting away from the

military raising its own VHA rates. The "intent of VHA is not

to reimburse the military member for what he or she pays for

housing costs but to enable the military person to live in

adequate housing in whichever area he or she is assigned"' .

The costing bands will be used for two major purposes. One

purpose is, through the use of an appropriate regression model,

to determine owner equivalency housing costs, and the other

purpose is to provide housing cost data when there is

insufficient data in a category to determine a median rent for

that category. Once this needed data is found it will be

incorporated back into the MHA data, and then, within the MHA,

a median rent figure will be computed for each paygrade and

dependency status. This figure will then be utilized in the

congressionally mandated equation, (1), local median rent - 80%

K From a conversation with Debra Davis, DMDC., June 1989.

6

of national median rental cost, to determine the VHA rates for

that MHA. Of course these VHA rates are then subject to

budgetary constraints and congressional approval.

D. DATA DESCRIPTION

The data used to determine VHA rates come from data

collected from military members who participate in the VHA

Survey. The VHA Survey is taken every other year. The data

collected from the survey are kept by the Defe-e Manpower Data

Center which is the repository for all of the data used in the

VHA calculations. The data used in the VHA process consist of

raw survey data taken from each military housing area, and

contain information such as what type of house a military

member lives in, whether it is a single family home, townhouse,

apartment, or mobile home, how many bedrooms the house

contains, whether or not the military member has any dependents

or whether he or she shares the housing costs with another

military member, and the paygrade and service of the military

member. Also contained in the data for each military person

who participates in the survey is the rental cost, utility

costs, and maintenance cost of the housing. Other items such

as social security numbers, whether the member rents or owns

the housing, and other miscellaneous information are also part

of each data record for that particular military person.

The data used in this analysis and taken from the 1989

survey, consist of the paygrade (El-09) and dependency status,

having dependents, single, or single and sharing, of the

7

military member. In addition, the total housing cost for that

member which consists of the rent plus the maintenance cost

plus the utility and insurance costs is used. Further

information on the living space for the individual is also

needed, such as the number of bedrooms (1-4), and the type of

living space, detached house, townhouse type, apartment, and

or mobile home. Additionally, total pay (basic pay + BAQ) has

to be associated with each military member's dependency status

and paygrade in order to perform the regression analysis.

These raw data are edited to reflect only true rental costs not

ownership costs. Thus one data record used in this analysis

consists of information regarding paygrade, house type, number

of bedrooms, dependency status, total housing costs, and total

pay.

From this initial set of data one median rent for each

category of house type, number of bedrooms, marital status,

and paygrade is then computed. Thus data for an individual

costing band which might have consisted of over 50,000 records

is reduced to a data set which contains a maximum of 1104

records which reflects all of the possible combinations of

paygrade, house type, number of bedrooms and dependency status.

SAS was used to extract and edit the raw data, match total

pay to paygrade and dependency status, and compute a median

rent figure for each category of paygrade, dependency status,

number of bedrooms, and house type. (An example of this

program can be found in Appendix B.)

8

E. PROBLEMS WITH THE DATA

There is one major problem associated with the data used

in the VHA computational process. The data used does not

include data from the military members who are in paygrades E5

and above and who share a residence with another person. These

data, which might provide further information and might enable

a more reliable estimate of median rents for a MHA, to be

computed, are not being used. This is a policy decision. This

is a major problem in the computation of VHA rates because one

of the basic reasons for the existence of the "costing band"

idea and one of the major problems associated with the current

manner in which VHA rates are calculated, is the sparsity of

data. This policy decision essentially throws away what could

be valuable and informative data and is contradictory to the

purpose of finding "good" estimates of median rents.

F. PURPOSE OF THESIS

The main purpose of this thesis will be to test the

validity of the currently used regression model equation (2).

The data in its newly proposed format of costing bands will be

used. If the current regression model is not found to be

adequate then the second goal of this thesis is to suggest a

better, more sensible model which will more accurately predict

total housing costs for each costing band. Thus this thesis

will basically consist of two different types of analyses and

will analyze the MHA data from two vantage points. Since there

is no explanation as to why an inverse of rent is predicted

9

linearly by the inverse of pay (equation 2) a more sensible

regression model will be examined to explain the relationship

between total rent and total pay.

A secondary goal of this thesis will be to test the current

and any proposed regression models not only with the data that

is currently assigned to each costing band but also with

fifteen other costing bands comprising of data from the

original costing band plus data from the military members who

are E5 and above who share housing with another person. Thus

thirty costing bands will be formed and a comparison of the

regression models using the data from the original costing

bands and data from the "new" costing bands will be made. This

is important because it may show that the regression models are

better able to predict housing costs with the added information

and this in turn will provide better, more accurate VHA rates.

10

II. ANALYSIS PROCEDURES

A. ORDINARY LEAST SQUARES REGRESSION

Most of the analysis performed in this thesis employs

simple linear regression (ordinary least squares) to test the

various postulated models.

In ordinary least squares regression, a linear model,

Y; = B + BX: + e; (4)

is used to find the relationship between the Xi's (independent

variables) and the Y ,s (dependent variables). The random error

component is denoted by e: and assumed to be normally

distributed independent random variables with mean zero and

constant variance, a2. This relationship as described by B.

and B. is used to further predict or estimate other Y's. Since

B and B, are fixed and unknown, b0 and bI, are used to denote

the estimates of their values [Ref. 2:p. 11]. With the

utilization of these estimators the least squares regression

fitted values are described by [Ref. 2:p. 11],

Y = b0 + b:Xi . (5)

The values for b0 and b, are determined by minimizing

n nS( - B0 - BX.) 2 (6)i=1 i=l

By differentiating this equation first with respect to B, and

then with respect to B,, and then by setting these results

equal to zero and solving for B and B,, the values for b and

b. are found by setting the solution for B, equal to b and B,

11

equal to bi. [Ref. 2:p. 13] The rationale behind this

minimization process is to ensure that the predicted ith value

is as "close" as possible (in Euclidean vertical distance) to

the actual ith value for all i. If the model (4) is correct

these estimates have minimum variance among all unbiased v

estimates. [Ref. 2:p.14] Utilizing the method above, the

value for b0 [Ref. 2:p. 14] is

given by

bo= Y - bI (7)

and the value for bi [Ref. 2:p. 13) is given by

n.E (xi - ) )(Yi - Y)b -= 1= 1 (8)

n2Z (x i - X)2.

i=1

The sum of the residuals squared divided by the number of

observations, n, minus two is given by

n 2Z (Yi- Y )

s 2 = i= 1 1 (9)(n-2)

and represents the unbiased estimator of the variance about

the regression o2 . [Ref. 2:p. 21) if the model is correct. If

a postulated model (i.e., the conditional variance of y givenx) is the true model then c 2 = 02Y. [Ref. 2:p. 23) Thus s 2 is

an estimate of o2 if the model is correct. [Ref. 2:p. 23)

The basic assumptions of ordinary least squares regression

are:

1. E(ei) = 0, V(e i ) =a .

2. ej and ej are uncorrelated, Cov(e i , e4)=O.

12

3. ei is a normally ditributed random variable with meanzero and variance o. Thus the ei's are independent.

4. E(YJX) = a + bX, the conditional expectation of Y givenX is linear in X.

If assumptions 1 and 2 hold then ordinary least squares

provides the best minimum variance linear unbiased estimates

of the B, and B1. [Ref. 2:p. 87] If all of the above

assumptions hold then b0 and b, are the maximum likelihood

estimates of B, and B1 and s is an unbiased estimate of a2.

[Ref. 2:p. 88]

If the residuals are normally distributed it is then

possible to use the F and t tests to test the significance of

the regression and to test the individual null hypotheses that

B, equals 0 or that B1 equals 0. If the null hypothesis is not

rejected and the values for B, and B1 are not deemed different

from zero then, of course, there is no significant linear

relationship between the independent variables and the

dependent variables. The t test statistic is

n(b.-0) (Z (X -X)2)ti= (10)

and has a student's t distribution with n-2 degrees of freedom.

[Ref. 2:p. 26] The F test statistic tests the overall

significance of the regression. The F test statistic is

F b {Z (Xi - R)(Yi- (11)

s2

13

and has 1 and n-2 degrees of freedom. [Ref. 2:p. 32]

The R2 value measures the "proportion of total variation

about the mean Y explained by the regression". [Ref. 2:p. 33]

R2 is the sum of squares due to regression divided by the total

sum of squares, corrected for the mean Y and is denoted by

n -z (_

R= i= (12)nZ (Y. Yi=l

Values for R2 fall between 0 and 1. The closer the value of

R2 is to 1 the better the regression equation explains the

variation of the data about Y.

The "residuals contain all available information on the way

in which the fitted model fails to properly explain the

observed variation in the dependent variable Y" [Ref. 2:p. 34].

Thus careful examination of the residuals will provide

indications as to the adequacy of the proposed model. A

graphic examination of the residuals may provide an indication

that the model is systematically deficient. Also utilizing a

lack of fit test may indicate that the model appears to be

inadequate.

The lack of fit test breaks the residual sum of squares

into the mean square due to lack of fit, MSL, and the mean

2square due to pure error, s [Ref. 2:p. 37] The MS,

estimates a2 if the model is correct and o2 plus a bias term if

the model is inadequate. The value for so2 estimates o2. [Ref.

14

2:p. 37) The lack of fit test compares the F ratio MS,/se2 with

the 100(1-a)% point of an F distribution with (nr - ne) and ne

degrees of freedom where nr equals the number of degrees of

freedom associated with the residual sum of squares and ne

equals the number of degrees of freedom associated with the

pure error sum of squares. If the comparison is significant

(i.e., the F ratio is greater than the tabled F value) this

then serves as an indication that the model is inadequate [Ref.

2:p. 37]. If the test is not significant (i.e., the F ratio

value is less than the tabled F value), this is an indication

that "there appears to be no reason to doubt the adequacy of

the model and both pure error and lack of fit mean squares can

be used as estimates of o2'. [Ref. 2 :p. 37]

By graphically examining the residuals, a scatter plot of

the e4's versus the Yi's will give an indication as to whether

or not the assumptions of normality, homoscedasticity and

linearity of ordinary least squares have been violated. If the

proposed model is correct, the resulting residuals should

indicate that these assumptions hold. [Ref. 2:p. 141) If the

model is correct a plot of the residuals versus the fitted

values should take the shape of a horizontal band as shown in

Figure 2.1 below [Ref. 2:p. 145]. If the plot of the residuals

takes the shape of a funnel as shown in Figure 2.2 below [Ref.

2:p. 146], the variance, o2, is not constant and is increasing

with x, which indicates the need either for weighted least

15

squares or a transformation on the observations Yj before

performing a regression analysis. (Ref. 2:p. 147]

-x

Figure 2.1 Satisfactory Residual Plot(Ref. 2:p. 145]

y

x

Figure 2.2 Unsatisfactory Funnel-Shaped Residual Plot[Ref. 2:p. 146]

B. INITIAL MODELS TESTED USING ORDINARY LEAST SQUARES

REGRESSION

The first step in this analysis was to test the model

currently in use, equation (2), to see if it could be used to

predict median rents for each of the thirty costing bands.

The model was tested under several different conditions.

First, the model was run using all of the available data in

each costing band. Next the data was divided by marital status

16

and within each costing band the model was tested using all of

the data for those military personnel with dependents and then

the model was tested using all of the data for those military

personnel without dependents. The model was tested under

another condition in which the data was broken down further by

paygrades into enlisted, paygrades 1-9, company grade officers,

paygrades 10-19, and field grade officers, paygrades 20-23.

Thus the model was tested within each costing band according

to groupings of the data consisting of enlisted personnel,

company grade officers, and field grade officerc Finally the

current model was tested within each costing band by grouping

the data by a combination of dependency status and paygrade

categories. In this case the data in each costing band was

first broken into groups by dependency status and within each

dependency group, the data was further broken into categories

of enlisted, company grade officer and field grade officer.

For each of the above mentioned conditions in which the

model was tested, the data was plotted 1/T. versus 1/P k, the

model was tested using Ordinary Least Squares regression

procedures, the residuals were plotted versus the fitted values

of the median rents, Tik ' and the residuals were tested for

normality. (These results are given in the next chapter.)

After reviewing the results of the regression procedures,

the initial model did not seem to adequately describe the

relationship between total pay and median rental ccsts nor did

it serve as an adequate predictor of fitted values for median

17

rental costs since the assumptions of least squares regression

were violated. Evidence of this, includes low R2 values, non-

normality of the residuals, unequal variance of the data, and

an indication of significant lack of fit. This, along with

cross-validation results are explained in detail in the

analysis portion of this thesis. Therefore a new model was

postulated. The new model was

Tijkl = PijkiA + B + (13)

in which all of the variables have the same meaning as in the

previous model. The only difference was that the total pay and

median rental cost vectors were not inverted. This model was

tested in all of the same conditions as the initial model. In

other words the model was first tested using all of the data.

The data was then broken into groups by dependency status and

the regression was run again. The data was next broken into

groups by paygrade and ordinary least squares regression was

used to test the model using this data. Finally the data was

broken into groups by a combination of both by paygrade and by

dependency status and the model was again tested.

The results of the regression analysis testing this model

again indicated that a systematic deficiency in the model

existed; namely that the residuals exhibited a tendency towards

nonconstant variance and that the residuals were not normally

distributed. The nonconstant variance is explainable by the

fact that different medians from different population sizes

will have different variances. Thus a weighted least squares

approach was attempted in conjunction with this model.

18

C. WEIGHTED LEAST SQUARES REGRESSION

If a postulated model has been tested using ordinary least

squares procedures and examination of the residuals shows a

nonconstant variance, a need for some type of transformation

on Y is necessary. This transformation will change the ei's

so that the assumptions of ordinary least squares regression

will hold. [Ref. 2:p. 147] Generally a nonconstant variance

among the residuals indicates that some of the observations are

"less reliable" than others. 'Ref. 2:p. 108] In this case the

e's are normally distributed with mean 0 and variance o,2

instead of o2. Thus the ei's have variance of vio2. To combat

this nonconstant variance term, via2 , the entire regression

equation

Y, = b0 + bX; + ei (14)

is multiplied by the weight, vi"2/2 Thus the regression

equation becomes

Yi= b0 + b1 Xi + ei (15)

Then E(e./v,i)= 0 and the V(e/v-7) E(ei2/vi) = vo2/v = 0.

Thus ei//V-i' N(0,o 2). Therefore the assumptions of ordinary

least squares will now hold and ordinary least squares

procedures may now be applied to the transformed regression

equation.

Evidence of nonconstant variance was seen in the residual

plots after OLS regression was applied using the model (13)

for most of the costing bands. This implies, as stated above,

that some of the observations were less reliable than others.

19

Intuitively this makes sense in this problem since each

observation represents a median cost and not an individual

cost. Thus some observations represent the median of 20 or 30

data points while other observations represent the median of

only 5 data points. This makes the median of only five data

points "less reliable" than the median of a data point which

represents 20 or 30 data points.

In order to transform the model into one in which the

assumptions of ordinary least squares holds a weight vi-1 2 must

be found. In this case the necessary weight is i/si where

Si 1.25 Ri (16)

1. 35f

This is the Gaussian-based approximation (Kendall and Stuart,

1967) of the standard deviation of the median. [Ref. 3:p. 16]

Ri equals the interquartile range for the ith subset of data

and ni equals the number of data points comprising that median.

The reason for this is that if x is N (p,o) then the median is

N(pC -a). From the normal table, for normal distributions,

S2n

IQR = 1.35o thus

S IQR - 1.25 Ri (17)

21.35 1.35

Since the variance of ej = oj2 and since s is an estimate

of oi if we transform the ei's into ei/s the variance of ei/s i

should approximate 1. The variance of the transformed ei's is

now estimated to be one and is thus approximately constant.

Accordingly, the predictor will have more neatly constant

20

variance. Therefore this assumption of ordinary least squares

hold and OLS regression procedures are more appropriately

performed on the transformed model.

D. ANALYSIS OF COVARIANCE MODEL

The results of using a weighted least squares approach

with the transformed model, equation (15), indicated that this

was more sensible than using ordinary least squares, however,

another approach also seemed plausible. Analysis of Covariance

(ANCOVA) was used in which the grand mean rental cost is

adjusted within each group of paygrade, number of bedrooms and

house type by the rental cost which is determined by these

factors. Thus the ANCOVA model would becomeYijk o X0o Xijkiik + eijk (18)

~ijk XE ij +'

in which the X0B0 term is the grand mean, the Xi kBijk term is the

total pay for each group of number of bedrooms and house type.

The Y;,k term would represent rental cost for each ith person

dimensioned by jth type of house and the kth number of bedrooms

in the house. This model differs from the previous model in

that instead of using medians of total pay within groups of

paygrade, house type, bedrooms, and dependency status to

predict median rent, the model used the total pay of each

individual person in a costing band and the deviations caused

by differences in house type and number of bedrooms to predict

rent. Thus, in this case, total pay becomes the continuous

variable and house type and number of bedrooms become the

categorical term. Paygrade and Dependency status were not used

as class variables in this model since total pay adequately

21

reflected their values. Their inclusion would cause

collinearity to exist among the variables and the regression

estimates would then be biased.

E. CROSS VALIDATION TECHNIQUES

Since the weighted least squares approach with the model

(15) and the ANCOVA approach (18) using all the data, not the

median data, were thought to be the most sensible, a cross

validation technique was used in each case to test the

parameter estimates and the models. For the weighted least

squares model half of the data was used to determine regression

coefficients and these coefficients were then used with the

other half of the data to calculate new fitted values. These

values were then compared to the actual observed values to find

estimates of slope and intercept. The equation

n 2E (Y; - Y) (19)i=1l

is the residual sum of squares. These values for sum of the

squares of the residuals were compared for each half of the

data within each of the thirty costing bands for the weighted

least squares model. For the ANCOVA model, no provision in SAS

was available for the above described cross validation so the

data for each costing band was randomly divided in half and the

parameter estimates of the coefficients and its standard error

for each half of the data were compared (See results in

Analysis chapter).

22

III. ANALYSIS

A. ANALYSIS OF CURRENT MODEL

The current model, equation (2), was run using OLS

regression procedures with the data from the thirty costing

bands, fifteen of which contained data as specified by the Per

Diem Committee and fifteen which contained the additional data

obtained from those military members who are in paygrades E5

and above and who share their residences. The results of the

regression analysis indicated that this model was suspicious

in that it did not adequately fit the data, and would therefore

perhaps not produce an adequate prediction of median rent based

on total pay.

Initially the current model, equation (2), was run using

all of the available data within each costing band. The data

was plotted, median rent versus total pay, for each costing

band. A spread in the variance of the data was seen and in

some instances a curve was present, indicating a nonlinear,

instead of linear type of relationship (See Appendix A). The

regression analysis results as seen in Table 1 (See Appendix

C) showed that in twenty-three out of twenty-eight cases the

model had a significant lack of fit. (The data from the other

two costing bands contain only two data points and regression

analysis is not valid in these two cases.) The residual plots

from each of these regressions also exhibited evidence of

nonconstant variance which was a further indication that the

23

model was inadequate. (These residual plots can be seen in

Appendix A.) The regression results from the costing bands

which did not exhibit a significant lack of fit did, however,

have residuals which had a nonconstant variance and were not

normally distributed. Also the R2 values in each of these

cases were extremely low (less than .32) which again served as

an indication that the model only explained at most a third of

the variance.

The data within each of the thirty costing bands was then

broken into two groups according to dependency status. The

"zero" group within each costing band contained the data from

those military members who had dependents, and the "one" group

contained the data from those military members who claimed no

dependents. The regression model, equation (2), was run again

using these new groupings of the data. The results of the

regression analysis again indicated that this model was

entirely inappropriate. Although there was not one case of

significant lack of fit, the residual analysis of the data, as

seen in Table 2 (Appendix C), from twenty-six out of twenty-

eight of the costing bands, illustrated that the residuals were

not normally distributed. The residual plots (Appendix A)

again show nonconstant variance. Two costing bands, the "zero"

labeled data from both costing bands 510 and 512, while

indicating that the residuals were normally distributed and had

constant variance, not showing significant lack of fit, and

according to the F test for significance of the regression

24

exhibiting evidence of a significant regression, had low R2

values of less than .500 which indicates a lot of unexplained

variance. In this instance, with the data broken into groups

by dependency status, the model again was inadequate.

Next the data within each of the thirty costing bands was

broken into groups according to paygrade. Paygrade 1 consisted

of the data from military members who are in paygrades El to

E9. Paygrade 2 consisted of the data from military members who

are in paygrades W1-W4, 01E-03E, and 01-03. Paygrade 3

consisted of the data from military members in paygrades 04-

07. Data from paygrades 08 and above are included in the data

for paygrade 07. The model, equation (2), was again tested

usi:-g this data. With the data from the costing bands broken

into groups in this manner there were 84 different cases in

which the model was tested. In fifty out of eighty-four cases,

as can be seen in Table 3 (Appendix C), a significant lack of

fit was found. Of those thirty four cases where there was not

a significant lack of fit, twenty eight of them had residuals

which were not normally distributed and had residual plots

which showed evidence of nonconstant variance. The six cases

which showed no evidence of lack of fit, and which had

residuals which were normally distributed, namely costing band

632 paygrade 3, costing band 530 paygrade 2, costing band 590

paygrade 2, costing band 570 paygrade 3, costing band 650

paygrade 3, and costing band 510 paygrade 2, all had R2 values

less than .330. Thus once again there was strong evidence that

25

even in this case where the data was broken into groupings

according to paygrade the model was inadequate.

To further ensure that the model was tested under all

appropriate conditions, the data was broken into groups first

by dependency status and then further broken into groups by

paygrade. Thus the data from each costing band was broken into

"zero" or "one" groups as defined previously. The "zero" or

"one" groups were then broken into further groupings according

to paygrade. Thus the "zero" group, for example, was broken

into three further groups, paygrade 1, paygrade 2, and paygrade

3 also as previously defined. Therefore each of the twenty

eight costing bands now has two dependency status' and within

each dependency status three paygrades associated with it.

Thus the model was tested using 168 different sets of data.

The results of the regression analysis, using each of these

different data sets, can be seen in Table 4 (Appendix C). At

an alpha level of .05 three out of the 168 data sets showed

significant lack of fit. Of those data sets which did not show

a significant lack of fit 105 had residuals which were not

normally distributed and which had residual plots which

exhibited nonconstant variance. Of those remaining sixty sets

of data which show no significant lack of fit and normally

distributed residuals, nineteen of them did not have

significant overall regressions according to the F test at an

alpha level of .05. Of the remaining forty-one data sets which

did not show significant lack of fit, which had normally

26

distributed residuals and residual plots showing constant

variance (Appendix A), and which had regressions which were

significant according to the F test, all had R2 values which

were less than .440. In fact all but four of these remaining

data sets had R2 values which were less than .220. Thus this

analysis indicates once again that the original model was

woefully inadequate and that in none of the cases where the

data was broken into groups according to dependency status, or

by paygrade, or by a combination of both would this model

adequately predict median rent based on total pay. An adequate

model would be one in which there was no lack of fit, the

assumptions of Least Squares Regression would hold, and the R2

values would be high indicating that the model explains the

variance of the data.

B. ANALYSIS OF PROPOSED MODEL

The proposed model, equation (13), was tested using the

same data from the thirty costing bands as was used to test the

current model, equation (2). The results of the regression

analysis indicated that in certain cases the use of this model

may be more adequate in predicting median rent from total pay;

however it must be used with caution.

This model, equation (13), was also tested using the same

groupings of data as used in testing the current model,

equation (2). Initially, the model was tested using all of the

data within each costing band. As in the previous model median

rent versus total pay was plotted. The plots indicated an

27

increase in variance but indicated a strong linear relation-

ship. The results of the regression analysis showed that in

all twenty-eight instances, see Table 5, a significant lack of

fit was evidenced. Next the data within each costing band was

broken into groups by dependency status. The data was labeled

with a zero if the military member had dependents and the data

was labeled with a one if the military member had no dependents

or had no dependents and was sharing his or her residence. The

plots of median rent versus total pay for each costing band

indicated an even stronger linear relationship than in the

original plots but they still exhibited evidence of unequal

variance. The results of the regression analysis, see Table

6, showed that in eight out of fifty-six cases a significant

lack of fit was evidenced. Of the remaining forty-eight cases

twelve of these had residuals which were not normally

distributed. The residual plots of these data sets showed that

nonconstant variance was present. The residual plots of the

thirty-six cases which did not have significant lack of fit,

which had residuals which were normally distributed, and which

were significant regressions at the alpha level .05, also

showed some evidence of nonconstant variance. Also, the R2

values were in the .4 to .5 range with the highest a value of

.55. These R2 values are lower than the ones obtained with the

use of the Weighted Least Squares model, seen in the next

section, whose purpose is to reduce or eliminate the

nonconstant variance of the residuals. Thus prediction was

28

worse for residuals with more variance. See Appendix A. The

data within each costing band was next broken into groups by

paygrade. This procedure was the same as the one used in

testing the current model, paygrade 1 reflected paygrades El-

E9, paygrade 2 reflected paygrades WI-W4, O1E-03E, and 01-03,

and paygrade 3 reflected paygrades 04-07 with paygrades 08-

010 included in paygrade 07. When the data was broken into

these groups there were many more, fifty-six out of eighty-

four, see Table 7 (Appendix C), cases of significant lack of

fit. Also because of few data points within each group, the

overall regressions in many instances were not significant.

Finally the data was broken into groups first by dependency

status and then by paygrade. The results of the regression

analysis indicated that while there were only eight cases of

significant lack of fit, see Table 8 (Appendix C), out of one

hundred and sixty-eight, thirty had residuals which were not

normally distributed and because of few data points within each

group, some of the data sets did not have significant

regressions, at the .05 alpha level. Of the regressions on the

data sets which did fulfill all of the criteria the R2 values

were low. Thus the model best predicted median rents from total

pay when the data was divided by dependency status, however,

this model must be viewed as possibly inaccurate since the

residual plots indicated evidence of nonconstant variance, and

a better model would predict points in an unbiased fashion.

29

C. ANALYSIS OF WEIGHTED LEAST SQUARES MODEL

Analysis of the Weighted Least Squares Model, equation

(15), with Yi = median rent and X = total pay for the ith

group, was conducted in the same manner as that of the current

model, equation (2), and that of the proposed mnodel, equation

(13). The only difference here was that initially the data

were randomly divided into two sections in order to use cross

validation procedures to compare the sum of the squares of the

residuals of the first division of data to the sum of the

squares of the errors of the second division of data in which

the parameter estimates from the first set of data were used

to compute the predicted values for the second set of data.

Thus the Weighted Least Squares model was first tested using

one half of all of the data available within each costing band,

next the model was tested by the half of the data which had

been divided into groups by dependency status, then the model

was tested by the half of the data which had been broken into

groups by paygrade within each costing band, and finally the

model was tested with half of the data which had been broken

first into groups according to dependency status and then by

paygrade.

The results of the regression analysis using half of all

of the data within each costing band showed (see Table 9,

Appendix C) that a significant lack of fit existed for each

costing band. When the data was broken into divisions by

dependency status the regression analysis results, see Table

30

10 (Appendix C), showed that seventeen out of fifty-six cases

exhibited significant lack of fit and that nine out of the

thirty nine remaining cases did not have normally distributed

residuals. Three out of the remaining thirty cases did not

have regressions which were significant overall and of the

remaining twenty seven cases in which all statistical criteria

were met, the R2 values were typically between .44 and .75.

There was no evidence of nonconstant variance in the residual

plots and they seemed to appear to have been normally

distributed in most cases.

When the data was broken into groups by paygrade, only

twenty-five out of a possible eighty four cases, see Table 11

(Appendix C), met all of the criteria of successful regression

in that they did not have significant lack of fit, their

residuals were normally distributed, and their regressions were

significant at the .05 alpha level. The R2 values, however,

ranged from very low to a high of .73. Again the residual

plots appeared to indicate a fairly normal distribution with

little evidence of nonconstint variance.

The results of the regression analysis, when the data was

broken into groups both according to dependency status and

paygrade, see Table 12, showed that better than half, 93 out

of 168, met the criteria for a successful regression and had

R2 values ranging mostly between .4 and .65. There were

however, very few data points in some categories, thus these

results must be viewed with suspicion. The statistics for lack

31

of fit, normality of the residuals, and overall significance

of the regression all might have been affected by this small

number of data points. Therefore this model using a weighted

least squares approach, equation (15), performed best when the

data within each costing band was divided according to

dependency status.

The cross validation technique used here proved to be

unsuccessful since only the sum of squares of the residuals

(SSR) term were compared, see Table 13 (Appendix C), in the

case where all of the data was used within each costing band.

The differences between the SSR for the first group of data and

the data with predicted values found by employing the parameter

estimates from the first set of data for each costing band were

quite large. This could be due to the lack of fit which was

found or due to the fact that the second group generally had

several more data points than the first group. Either of these

two factors or a combination of both might have accounted for

these tremendous differences.

D. ANALYSIS OF THE ANALYSIS OF COVARIANCE MODEL

The results of the regression analysis on the ANCOVA model

indicated that this model may be the best model discussed thus

far for use in predicting rent based on total pay (see Table

14, Appendix C). All of the regressions were significant and

had R2 values ranging from .42 to .58 with few values above or

below these numbers. The residup! plots, normal plots, and

stem and leaf diagrams indicated that the residuals were

32

normally distributed (See Appendix C). The significance levels

of the normal statistic used to test the normality of the

residuals, however, did not, in most cases, indicate that the

residuals were normally distributed. However the residuals

were fairly symmetric and the sample size was quite large,

therefore the model should be fairly robust to the lack of

normal fit. The residual plots showed the fairly typical box-

like pattern of randomly distributed data. The stem and leaf

and normal plots supported a fairly good defense for the

normality of the residuals.

In the case of several of the costing bands there did not

appear to be a significant difference in the least squares

means of the rent pertaining to different house types and

different number of bedrooms. This was particularly true

between house types 1 and 2 (single family home and townhouse)

and also between house types 3 and 4 (apartment or mobile

homes). In some costing bands there also appeared to be no

significant difference between the least square means of rent

predominantly in the case between 3 and 4 bedrooms and less

predominantly with 1 and 2 numbers of bedrooms. This

indicates, that, when there is not a significant difference

between the least squares means between two different types of

housing or two residences with different numbers of bedrooms,

either of the parameter estimates of two types of housing or

number of bedrooms may be used to predict rent. Thus the

ANCOVA model which predicted rent based on the total pay

33

associated with number of bedrooms and house type may not have

been completely correct in these cases since the mean amount

of rent associated with each type of house or number of

bedrooms may not have been different.

The cross validation technique used here, since GLM does

not provide a vehicle to compute the Sum of Squares of the

Residuals from previously calculated parameter estimates, was

one in which the data was randomly divided into two sections

and after the ANCOVA model was run on both sets of data, the

coefficient of the slope parameter estimate and its standard

error were compared. A comparison of the slope parameter and

its standard error between the two sections of data from each

costing band revealed that the model was not at serious fault

since in both of the sections of the data the slope parameter

estimates were very close and the standard errors were small

and similar (See Table 14).

34

IV. CONCLUSIONS AND RECOMMENDATIONS

The purpose of this thesis was to test and validate the

current model, equation (2), to see if it could effectively be

used to predict rent based on total pay from the survey data

which had been arranged in a newly devised, simplified format.

If the current model was deemed invalid or suspicious, then the

second purpose of this thesis, was to propose a better, more

sensible model which would adequately predict rent based on

total pay.

There are two major conclusions from the analysis conta4ned

in this thesis. The first conclusion is that the current

model, equation (2), should not be used to predict median rents

in each paygrade and dependency status when the data is divided

into costing bands in the manner previously described. This

conclusion is justified by the results of the regression

analysis which show that this model is inadequate and may not

accurately predict median rent. The second conclusion is that

both the weighted least squares model and the ANCOVA model are

possible alternative models for use in predicting rent based

on total pay. They are shown to be at least as reasonable as

the current model, if not better. The ANCOVA model may be

preferable for predicting mean rather than a median rent. Also

the ANCOVA model may be preferable if the model is used to

determine owner equivalency rents. If a median rent figure

must be used in the congressionally mandated formula for the

35

computation of VHA the weighted least squares model is

preferable.

The secondary purpose of this thesis was to determine if

the data from military personnel in paygrades E5 and above who

share housing should be used or discarded since these data had

been previously discarded on the basis of a policy decision

without any statistical backing. Curiously enough, there seems

to be no systematic difference across all of the models

investigated in relation to the addition of this data. In some

instances when regression analysis results from the same two

costing bands, one which contained the additional data and one

which did not contain the additional data, were compared, lack

of fit was affected. Also in some cases the significance of

the regression would be affected, or in some cases the R2

values would go up or down. Thus there was no instance in

which, for example, all of the R2 values would go up or all of

the significance of regression statistics would suddenly

increase or decrease for a certain model. The important

consideration here was that the additional data did affect R'

values; it did affect the lack of fit, significance value

statistics, and the normality of residuals. Thus while the

additional data did not have a systematic effect, it did have

an effect and this aspect should not go overlooked when a

decision is made whether or r:t to include these data when VHA

rates are actually calculated.

36

There are several recommendations for further analysis.

First, the way in which the data is broken into costing bands

must be investigated. Perhaps a better method or a different

dollar figure could be used to divide the data into costing

bands. If a different method is used and the data contained

in each costing band is different, analysis of each of the

regression models discussed in this paper must be redone. If

the data is put into different costing bands other than the

ones used in this thesis, the models discussed may be more or

less accurate predictors of median rent. In either case the

original data must be investigated and natural breaks in the

data must be discovered in order to achieve the best placement

of data into costing bands. A second area which requires

further analysis concerns the ANCOVA model. The data, before

testing the ANCOVA model, should be divided into groups either

by dependency status or by paygrade. A better fit of the

regression model may be accomplished in either case. Other

models should also be investigated as possible solutions to the

problem. Perhaps instead of the weighted least squares,

another transformation on the data could be devised which may

provide a better model. Since there is an indication of non-

normal errors, perhaps GLIM (Generalized Linear Models) could

be used for more accurate prediction [Ref. 4]. Further

Analysis and other models should still be investigated as

possible predictors of median rents for the VHA.

37

APPENDIX A. SCATTER AND RESIDUAL PLOTS

A. USING DATA SET 540 AS AN EXAMPLE, SCATTER AND RESIDUAL PLOTS

FOR THE CURRENT MODEL.

PLOT OF 0MCCST TOTP .ECE6D: A I OBS. B O 2 OBS. ETC.

009

2226. A B

2.203_A BAA

AA A1 AA A

-0 A A

0 00B A2.0. A C A A A A ABAB A A A AB A BA

A B A AB A A A AA A A B

Ar . A Ae 40A i/ aOntB.2/B A C A B A

S ABA A A B A B A A

A A C 2 2AAl A l AC A ABAB BA A AB

A AA *B Al fEAB0

- 000 A A B

0000

Figure 1. Data Set 540 1/Median Rent vs. 1/Total Pay.

38

PLOT OF RESIZ':X.STHT CEND: A L 25. B 2 0BS. EC.

208

AA

':6 A

AA

? A

A A

A A A

F~~~~~~ A~tBA. *B

A A

AA

_ C ....... ..... ........ ..

- b z - B BFge.a SB 54.Asdul C A Acd u

9B A B B

0012 405 "is ).30 0024* 0223 33 030 0033 0036 2.389 3.PREDICTED VALLE

Figure 2. Data Set 540. Residuals vs. Predicted Values.

39

NSHR-0

F Or C :X-5:7:07 'SCROD: A 2BS. B : BS. E::,

CA

A A BA A

C AA BC CA8 A B 1 A 8

oc:A ABECS002 B A

A A A B A9B B

*C. A A

C 0050

000

Figure 3. Data Set 540.Dependency Status '0'.

1/Median Rent vs. 1/Total Pay.

40

.4BHR* I

CFCM0CBCS :CP LEGEND: A 1 CBS. b 08B. TC.

:9

A A

A A

0.004 A AA A A aA A A

A B A A AA

AC I A b A DA B' , -aBA S B A A B A 5 A

S A2A S DO D DAF r .S A ta b A 4A

B. BC .A S ,BB A A ' A A A 3aBn B A A C B A BtA

C A AB A A AAA A A A

3:11

C 00

0.30001 0.0003 0.0003 0 0004 0.000B 0.00 000 0.0006 .0009 20010 C01

ITO?



41

PLOT OF RESIDt!MCSThT LEGEND. A I O0 . B O 0BS. ETC.

3 003.5

0,00,0

)0035

3.0030

,.0025

30015 A4A

A A

0.000A A A A

A A A A* A B A0 0000 A A

A A A A CA A A I A .B A A A S A

B C A BAB 5 :B A AA CA VS BA CA A A

3. 0000 .................. .A ........ B-B . B-AB .... B .......... ......... A... .............. .B B

A BA AB AA AAA B A A 0 A At A A A A A A A

-0.3005 A A AA A B A 5 a

AC A A A

-0.0010

-0. 0013

-0. 0020

FPUOCTU VA3.Ut


Residuals vs. Predicted Values.

42

NSHB: ],

r tS::oMCsm ;r£OoN Do A : B. B : CBS. ETC.

:308

0C6 A

3 33A

D.003

A

-? 000 -A_1A- - - - - - -- - - - - -- - - - - -

B A:C.A A 1A b A A 19

AA A A B

A000 A S B

• 0,003

-A 004 A

-0.005 B. C- B--

PRD..I CMID VAL UE

Figure 6. Data Set 540.

Dependency Status ' 1' .Residuals vs. Predicted Values.

43

P7'CT CF IMCOST0ITOTP SOCEND. A I CBS. B 2 CBS. ETC.

AA

AA

B SA

000. A A

B AA B A AB

A AA A A AG

003 A B A A A I AA A A - A B S C C ~ B :A B ,.B AS 2

0 002 BA C A E B A B B E B A0 A C 3 B B

IA - SBA AA B

: O

0 o060 ................ .. ............. ................................ ................................................. ".. 66i6PC. 000325 0.S0000 0.000475 0.0005B0 0.00025 0.000700 0,000575 .000850 0 000925 C C:10O C 00106;

irma

Figure 7. Data Set 540.Paygrade '1'.


44

2 '36

23..

A

AA

A A A A AAA A A

A A

ITOT?


l/Median Rent vs. l/Total Pay.

45

F AMOT T END .A 5 !S. E.C.

0 0CZ:

0 020 A

A A

AAA b*

A

AAA A *

O 0010 A A0 000150 00012 lo0 10 0.000092 0.00 0 o 0 000225 660i0 000252 0.000270 co0as ;00]¢s

ITOT?


1/Median Rent vs. l/Total Pay.

46

?L.:- :-csRES' ' SH i BS. B * .B5. ETC.

--,1 --

3 A

00 .. .... .... ...... ........... ... .... ..." -........ ..... .. ....... ....... i...... ......... ......... ...... .. ... .. ....

A B

- 0i. 0 0020 b. 0022 0 002", 0 0026 ; 0028 1. 0030 0 0032 D,003., o 0036 . O : .

PR(EDI CTED VALUE

Figure 10. Data Set 540.Paygrade '11.


47

.F M:ts: F. E12;W. A 280B. B C55 E-:

A

aAS:C36 AA A I

A

SA A A A

5 A

A2 ;2 AA

2 2 A002 . . A A

A A

A A A AA

0 0002 AA

B AA2 33A A

A A0A A•

A A A A

A A A

-'20002 AB00 AA

-0 0 '. 00i80G .. . . ...... 0--0---0 .......02 00 0 0 7 0 2 G 0

PIZDICTX VAI.XE


Paygrade 121.Residuals vs. Predicted Values.

48

?3 3 8:-:T- 0ESD0'!S147 3EED A- 'B5. B 2 . E-C.

)2z8

0016

301.

A.00-

AAAA

A

330 2*

A AAA

A A

-33306 *

* 0 0003C. 00CO 0 0012 75 0,CSS 0035 .001425 3 01500 300375 001650 0. 001725 3080 037

PIEDICTD VALUE



49

- 2- ST-:TCTP -':E' A B . Z B E7C

AA

AA A A

A A' A

... .. ...o.. .. .. .. ...... .57 6 i............. ........ .... ........ 7 6 6 .... 2 6 g i .... 6 ~ ~ 6. .. .......;.... ...66 6........... ........

A020 A 0027 3 D030 ,002 -1

Figure 13. Data Set 540.Dependency Status '0' and Paygrade 11

l/Median Rent vs. l/Total Pay.

50

.... . 3m 2~m m m

71.- : :MOS -:T7F --END A 'BS. - B .!C

* AA A

A AB

A AA

A A,

0.0017 A AA

A A

A A

1013 -A A A

0 00030 0.00033 ,00036 _ 00039 o 0' 0* 0* 03 03 03 .06ITOTP

Figure 14. Data Set 540.Dependency Status '0' and Paygrade '2'.


51

SHRIC P-S.3

C-:F9 A

D. COC

001920

- 001'.

€A

,'. CCO

0.0019

0.0010

C.0012 A

A

A00020 A A0.0011

0,0010 A0. 000116 C 000190 0.000204 0 O00213 0 000222 0 000231 020004 0 2 000249 0.000215 0. 000250 0 000076

ITOTP



52

..CT CF RESI'-IMCSIHT --CEND: A " BS. B 2 CBS. Z'C.

C. 002

20030

n,

0 000 A=

A A

SAA B0000 A

A

A SB A A

B.Gcoo ..... . . . . . ...... B ..... . . . A ..... A.... ... ....... .0.0............B...................

A AAA AAA C A A-COCOA A B

A BA

200010

0 0020CO00160 OSO 009 OB 0 .00205 0200220 2 00235 2 00250 C.00265 2 00280 C 09

PREDICTED VALUE



53

010:~ 00H % : A - .010. B -BS. E7'

3 0015

0 009

A

o0002

- 0007

0.000o 006

C. 0005-

A A

0,0001 8

A A

........................... ......---- .................................... ---....... ......... ... ....... ......... ...........0.0001 AAB

A

0 A Ba

A

BA A A A-0 0004. BA AA

A A

•00000 A A AA B

.0,0006 A

-0 0007 A0.00170 010017 0.00171 0.0019 00.00156 0 0010 i 0002 000198 01002002 0C206

PREIICaED VALUE



54

-LCT -F RES:D :-.CSTTiS B ^BS. T::

02

G005

A

00010032

2 2002 A

C 3004

A *

-o0005

-2 0006

-0. 0008

0.00120 C0012'. J0010 0.00132 0.00136 0 00140 C 00141 .04 0,00152 0 00156 -. 00160 '0016Z

PMIDOCTED VALVI



55

,LC !5 :-C7P -EMEN BS5. 1 LB3. E7

006 3

A A

B

B^

A A

AA

00,0 A A

-0000.3000325 02000400 0.000O475 0. 000350 0.000625 . 3000 0.000775 0.00COI0 C0009,15 50.300 2.00,65i

,TOT?



56

. 3036

0 2032

0 0330

A A

2 AA

A

A

F AA A

A A

A

2.0018 A A

AA

0 00" A *

-O3C14

A

20010

0,0000

tTOT?


l/Median Rent vs. 1/Total Pay.

57

FLOT '; B B5. E7:

1 0036

-003'.

0. 0,030

0030

O 0C28

'4 0.0-024

D 0022

0 0020

A0,0016

0.0014

A

012 A

J 00100.0001 .00060 . .08i0 0 .0001 .000 0 0.00ii..6060ii6 . . 6 ii0 66 6 .006i . . 0680ii6 . 0 8

ITOT?



58

308

S007

: 06

:0.

i : : : ...*

3 A00. . . ------------------------------ --------- ------------ -------- ----- -----------A...............

A B A

-0000 Is0 A

A A

00.

-0 005

0.0020 0-0023 0.0026 a 0029 0032 0.0035 0 0036 01.30*0 0.0044 1.7 0 300

PUD0C10D VA'



59

ASKR, P0-r'.C1 -F RES.: V1-M000 500N . OBB*2 6. E::

2. 0C -

2 0226 A

)6 1x

A 0 A

" 200. AA

2 2000 A A

B

^.O^

..0000 ............ . . .....................................................

~AA A AA

AAA, A

- 0002 - A AA

A A A

AA

-0.0006A

-0.0008

-0. 00100. 002000 ,0020005 0 002000 0. 002226 0 002300 002376 0 00040,0 2 002000

ILOEICTED VALUE


Residuals vs. Predict - Values.

60

P*-'- OF RES::.:-:oCT :-,~ -BS. B 2 :BS. ET:

3008

B B'002. A

0006

0008

.0,000

0.00135 0.0.' 00153 0.0016z J.00171 &1 8 -00109 00198 OCZC7 00

PREDI CTED VALUE



61

B. USING DATA SET 540 AS AN EXAMPLE SCATTER PLOTS AND RESIDUALPLOTS FOR THE PROPOSED MODEL.

?LP T :F M CST'TOTP 2EGEND A DBS B - CBS. ETC.

-630 A

:.00

:8@0

.Zoo

I 00

A2200 A

BA AS *A A

go

A * A A A A AXAC A A AA A A

AA B1AAA A A AA A ADB B A A BA

600 B A A A A B 0C0 BAc0A A CCA S BE ?p A B B ?Ab A A B A

AB B A~I AA AA A A C o* XACA 1~~ACA A A AABBA 3

FAAB B BE AB A A.0 B B A: B A AA -Z ACE BA B A AAIAB; 20 E A 'B AB I A A A BA

BBA" SA.B A A A B,SAA; A A A A

A Ac: A:00 A

Al

5O0 1530 :500 3000 COO CB 500 .000 -B0B 5000 5B00 "000 ABAC,

TOT?

Figure 25. Data Set 540. Median Rent vs. Total Pay.

62

PL C O DF - ',.DD1CSTH O..GCND A : B. . : BS. E7C.

'-000

* AA A B

AAA 7 A A A

AB AB 8 A AA AAADA E AA~j A

A A,-0 B .AA ADOC CA A A A A.... .- - -- D-BB-A$ - -C---BB ..... ....................AZ := :B: :,E SA- . DA B CABAl A AD B A

A A A C .AAB B ADF .. C BAAA A B BBBD A XA B A A

A0 A A A A*BA A A

.1000

PIEDOC'ED VALUE


63

%SHR G 0

F* o7: MCB1:7n LEGENID. A BS. B Z BS. ETC.

AA A

AA

A A A

A A A

BAA A A

8 1 1 5

7 A

-' 'A A BAA!0B A BAAB A A A A

A B A A A AA 3C A I A AIB

zO A A A A B A'AB A A ~ A

A A B A - A BAA B A G A A

A AA - A A A

A A A

AA

i.000 1400 3.S0 21.00 Z600 3000 3400 3$00 4,200 46 00 5000 5.00 5O

TOT?


Median Rent vs. Total Pay.

64

NSHB 1FL-T OF iTCSCT0CfTP ZOO;D: A BS. 0 CBS. ETC.

:600 A

2I400

i:00

000

A000 0AA

A A B600 B A C AA B 3I : I b A

A BA B A A A A

A B B B EA w 5 BA A B-00 A B E 0 BA A

A A AA C 0 20 B B 5 A A

A 60 0 B A SB B B AAA A A

AC B

Iooo0 ..................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

500 1000 00 00 00 00 00 O 0 00 00 0 0



65

NSHR-0PLOT OF RE£5:- CSTH LEGEND, A 0 03S. B 2 OBS. ET:

-20AA

A A

A AA A

A:50 A A

A A AA A A

A AA A 3 AB A A A B A

S A A A A AA B A

A3 A A A A A

A IA A A A A A

A 3 A A

B• B A l A AA

A B A A A A

A A A A

AA.. AD A.. A AA B A A A A AA

- A B- A A A A

A A A:A A A A

A A A-A0 A A A

AA

2150AA A A A AA

A A

-A00 A A

AAAA

-350•

B0 360 380 4B0 420 440 460 480 500 52B 540 560 B80 600 620 640 660 6a0 700 72B 740 760 230o $ 22

UIZDCTED YALUZ



66

NSH= IPLOT OF RESID" -CSTHT :ECEND A I BES. B C 2BS. ET.

:50

2SA

:5 A A A ~ A0 2*.A AA A *B A A AA

'A A A A AA A=A 2 A A A A A ?

..... A-- .... - --A--"B. --... B 'A... B .. ............... ..... ................

SB A A B

D . A A A A A-'P. A

-500

-'!00

.1000

PItIOCTtD VALUE



67

?LOT OF m3S0Tc0 p 1OCT02 A :BS, -'B5. ETC

2-00

2:^0

L00

SA A

A- AA AB b A BB600 B5 A A A

A B A B B A B

SA B A

A oA § I 0 2 B A A-A B - B A AA20 :B : B B B Ab B AA

A B :A A A A B AB AA

,3j A A A A

AC B

TOT?



68

PLOZ OF MCDS'!Nflp LEGEND: A I 0B5. 3 2 0 . ETC.

A

BA A

AA

AA

A AA A

AAA

A A A

2 .~A A

N^A A LA

A A ANAN

A A4 A A-0A A

0 A

A A A 44 AA A-AO A A AA AA

SA AA

A B A

:A

2000 1200 2400 1300 1800 2000 20 240 200 JID 80 2200 2200 :42012000 TOTP



69

?~3

:T F tCCS7'TTP L0ENE A AS. B BS. ETC

A AA A

AA

AA

A

!C A A

AA

A A

0 A

A

0C

A

450

A

200

3500 3700 3900 .400 .300 ASO 400 A700 000 000 5300 5500 5700 0900 6000 6i300 9500

TOTF



70

r::T OF , rESf DMCSTlT LEGND A •I CBS. B 2 -3S. E--,

AA

i:"S

A. AC A C B B A C B B AA : CS A B B A CB.-.................--..... . .. . .. . ....... ..... D.... . . . . . B"

A-- .. . . . . . . .. . . . . . B B- B B- -- - -A- -

-: . A ,

- *53

tIEDCTD VAUL



71

?G:

7C FRS:: sC51.7 '-CEND A -,25. B :!S. T

A

A

:22 *A

A AB

AAA

AAA A

~AA A 5

B A A

AA

B A AAAa

AA

.330Li . ....0 L. ... .. . .. . . .500 .. . 0. 0 . 30 . . 0 o 00 2. . .

PKI D VALUE



72

F FES!: M I T. "S-. A C ZBS. 5 B !S. E--

AA

*8

AA

-150

-2 0

-70 A

-200

-250 AA

tIEOXCTED VALUE



73

P^

'-C, -F ':-:: A OBO B -:8S. E7C

A A

00 A A

A I A

A A

:55

.00:soo .250 .5O .00 . 0 100 :500 X0 5.00 IB00 2800 .Q;5 :0 .-50

TOT?



74

'F T:: 7p L N :B. B 2 . I

2A A

A A

A0 A

30000?

Figure 38. Data Set 540.Depf-dency Status '0' and Dependency Status '2'.


75

NSHB'C PC-S

U :S-TC'P ZC;N" 1 CBS. B ZBS. 7

'CO

200

TOT?

Figure 39. Data Set 540.Dependency Status lot and Paygrade '3'.


76

'.T OF RES- .C5TIW :ZN A . BS, a ^B25. ETC.

- 8 - AA

AA

P AA

SA A A

* AA

-A

A

AA

^A

PI8K0 C , 2 VALL6



77

BA

"50

00 6A

AA

A A

A A

S A

0*

.:50- A

o250

500 55 510 55 3520 !.5 530 535 5 B 545 550 555 565 5,7B 0 57i BSB 390j;6 595 600'PUDICTID VAL!E

Figure 41. Data Set 540.Dependency Status'0' and Paygrade '2'.


78

53-.R-O PC 23

CF RES: " STHT -Z. ED: A I OBS. B : 3S. E-..

A

] .................................................................... ........................................... . . . . . .

ARA

220

-300•

50


Dependency Status '0' and Paygrade '3'.Residuals vs. Predicted Values.

79

?LOT Of MCCST T.P :;ENr A C BIS. I -BS. £7Z

:200

:40

z:0C

:Uc

0 .OO

800

000 - A60A A

B A* A A 02

A C -C A

A AC A

50 0 0 00 0C.00 .700 .900 220 .00 00 70 0oar



80

IS~ z I

.20 AF AC0O0 2N 3.3 221

630 B

A

690 A

55.0 - A A

A AAA

A2 50 A

F^

AA * A

AA

0 .0

AABA

AA

A A

00 ABA A A

300A A

A

A A A A B A

A A A360 3A

300 A A

300

270

TOT?



81

,:Snz- I PC 3

P?..T or CCS T3TP :END A - IOS, B 8 2 CBS. :C

B O A

500

?A

A2 .

Boo

5330

A

600

500

A

A0

A A22O 300 0 3700 ..00 .100 30300 .700 .900 0100 0300 0000 3700 .9.0 .7,0 6300 600

TO" P



82

'SAR 2 22A

sOF RESID 3T.. ETC.

-250 - B A *

- .*.A A A

"] 6 ' 6 " -"--"--"'-"----6 ------ -- --- -------"---"'---"-- "A ."'...".."'.." ....' ..." ...." ..." .."' ..

A 3 F BVB UCE A

- 33

1000200 30 30 30 34 30 *0 30 31 30 30 43 40 30 44 ~0 60 3B AI 40 00 31 30 303A

PIDOC50VLU

Fiue 6 at et50Deednc ttu 1 ndPyrae0'

Reiuasv. rditdVaus

.. .... .... .... .... .... . ..... ..... .... .... .... .... .... .... .... ...

7'-' F RE:D'Ms-1A -END3 A OS. B :?5. E

A.A

A

A A

AA

A

o .............. 1 ............................... ....................................... ........................ . . . . . . .B

A

A AA A

30 A

AA

-560

PKDICTED VAL.UE

Figure 47. Data Set 540.Dependency Status Ill and Paygrade '2'.


84

P : F R-S::- MS7H7 -r: . B5. B

L -1C

-100

* 150

-200 A A

-300

-350 . .

520 530 540 550 560 570 Soo 590 600 610 6:0 630 6,0 650 60 60 60 40 0 0 1 0 * 7 6PXEVI rD VALUE



85

C. USING DATA SET 540 AS AN EXAMPLE, SCATTER PLOTS AND RESIDUALPLOTS FOR THE WEIGHTED LEAST SQUARES MODEL.

E^E: A I 01S. B *2 CBS. 7:

.50C

1500

:250

500

A A A A

AB B A BA B * A nAA A A B * A B b...

BA bA A A A * A A -a AB A

A A050 A A

-500

- "50

.1000

PREDICTED VALUE


86

7,So 9

L:A A

0 AB B

A A

AAA A

A BAA

A A A

AC

A A BA A A A

-C A AA A " AA.A A B A AA

3 A A A A A200 A

A

AA

A

1000 IsO0 2600 300 AM0 -zoo 0 600 i00 5ooTOTP



87

:80

2630

A

500 A AA A A A

b A A A A A AA A AA A At 5 AF00igure 5 A A b A

B - B At AB A A A A A

A -A A A

tot

Figure 51. Data Set 540.Dependency Status '1.


88

SA

A A

AA

AM

V A A A A - A

AAAAAA

A

A AA

A A

AA

ZOO

-230

-30A AA



89

7F ~ ~ -ESS. a z -Is. t::

':00

000

I A

A AA A A A A*A A A A A A

A A A B SA....... ..-. A -..A,-A- - - - - - -a-. A ----------- -------------- 8--- ----

AR Ab A A AA A A- - - - - - - - - -

AA A A AA A

-200* b A A A

260 2180 300 3.0 3.0 360 380 400 .00 ..0 _60 f5 50 50 55 6 8 0 60 60PUDICTED VALSI

Figure 53. Data Set 540.Dependency Status 1'.


90

?LCT OF ZCOSTTOTP . END. A OBS B * 2 BS. E7C.

:.00

Xc0o

N ;4CC

:130

BOO

600 A A A BA A A A A A

B B A A A AA A A A

AA A B B C B tB AAACO0 B AE A A A A A

A BA B A A A a AAB B A B B A A AA AA CA B A A

AOB A AAA

BAt

aB0 COBS ZS00 0455 .I0B .850 2555 2000 2400 2600 UB00 3000 200



91

PLZT CF CC5"T Tp LZ ND A ± BS. B Z CBS. ETC

30C

02. A

too

60

650

6:0 A

- A AAA

AA A

A A

AB

A AA A A A

-oo AA A

A A

A A A A

AAA A

A A AA A A

AAA A

300

230

i000 ---- 0400 .000 - -00 2000 200 mo0 2600 2800 3000 3:00 3400

Carp



92

-C0 OF MCOST-TCTP LEGENC A BS. B 2 I BS. ETC.

-CC

350iOC A

A350

6'3

A

SSJ A

0 620

AA!50

500 A

A

-00 A

350 A

300

:50

200

3000 3200 3400 3600 3800 4000 .200 -400 4600 .800 5000 3200 5.00

TOT?



93

CFT ? 0ESI-'CS-H5T 2ECE4 A 20. 5 2 CBS.E .

!-0

2:5:

2 ::3

A:50 A5 A

A B A A A AA A A A A

A b I S A A * A A..... .... A.I-C..........

A A S B 05 AAA A A A A

AAA 0

-500

-50

1000

300 320 340 360 380 .30 *20 .0 .60 .0 500 520 50 563 80 600

PIDICD V*341



94

PLOT OF RES:-'MCSTMT U0202 A : coS. S OBS. ETC

4A AA

A AA

S CD- A-

A A

A A

A A A A AA A A B

- 00 AA

A A A

SAAAA A A

-200

-050A

-300

360 370 380 390 400 410 420 430 -- 0 450 460 Al0 480 490 500 510 520 530 540 550 560 570 580 590 600

?lOICTfl 0VALUE



95

I C7 F RES:: "!CSTH -EE . 0 S '2 5 25S. ETC.

A

A

* CO •

A

A A

.300

AA

AA

-300•

550 '60 570 580 590 600 610 620 630 640 650 660 i'0 680 690 700 710 720 730 740 750 756 770 580 790

PREDITED VA.UZ



96

CF ~ :!C-77 :BS B : BS :ss.

A

A AA

530

.5AS A AA

A

BA

A5 A

0 A

230A

:50

A

sea0 2:00 1600 1600 :S00 :000 3:00 z'00 2600 2100 3000 3200 540



97

P-'7 F 07:7- P : 1 :BS, B BS. ETZ

A A

AA

-50

-:0 AA A

290 AA

360 A

10

050 .500 2000 0000 0000 2500 A 00 0500 :600 0700 I00 00 OSC 000 5000 -s:oo *500300?

Figure 61. Data Set 540.Dependency Status '0' and Dependency Status '2'.


98



99

7F ' E,:Z :;E Z B B 2?S.Z

- SO- A"......... 7 ..... ...... .. ..... ....... a..... 6..... . ;6..... " a..... ...... ,6..... 6 ..... Z ..... ; ..... ... ..... ;, "

-30

PfDlICTtD VJ.U



100

Aa A

AA

A A

A

AA SAA

AA

380 38)A .0 .40 -60 480 030 $20 5.0 560 Sao o20 63 t-0 .

?IEDICTED VAIVE



101

AA

" 0 . .........................................................................................................................

700 75 710 7.5 72 715 730 735 70 ;S 750 755 760 765 770 775 780 785 790 795 800 05 9 .3 815 820

P9UDSCTED VALUt

Figure 65. Data Set 540.Dependency Status '0' and Paygrade '3,.


102

F .-r C CST-7 7? :--;E. I :'3. 2 OS. E-:

SCO

A.A

A AA A

-DC. BA AIB AAA A A

2^ C AA A *

000 5 700 ;2O ------ 00 H600 .-00 .900 2100 2300 3500 3700 :?30TOT?



103

I SH]P. 1 P, 2

!70

S-80

-AC A AAB

A

AAAA-20

6 .A AAA

310

AA

3¢2

270

ooo00 O0 ;.400 2600 s.m00 z~O0 0200 2*,00 2800 o.a00 5000 3200 "*00

TO"t

Figure 67. Data Set 540.Dependency Status '1' and Paygrade '2 .


104

350

TOTP

Figure 68. Data Set 540.Dependency Status '1' and Paygrade '3,.


105

00 AAA

A A A

-020. A

260 270 280 290 300 300 300 330 040 330 360 370 380 290 - 0 .10 420 430 440 430 *60 470 480 A90 500

?IEOCTZD VAt.E



106

A~H

A A

AA

.. . ......

.0 *5 .0 *5 .0 .5 .0 .8 45 1 *0 .61 . .....5 *80 .PIIDICTID VALUE

Figure 70. Data Set 540.Dependency Status I1l and Paygrade '2'.


107

*:30. A

.00 '60 470 .80 '90 000 520 520 030 40 000 080 070 060 090 600 820 620 60 640 600 660 670 6S0 890

PXLD0COE VALIU



108

D. USING DATA SET 540 AS AN EXAMPLE, STEM AND LEAF, NORMAL PLOTS,AND RESIDUAL PLOTS FOR TEE ANCOVA MODEL.

CSO

SAC4. . 50ASiAlACoC C C •i *0CsAI LA A l A A A A

-9S*AACC KEIK PA"$ tA: AACAA 9. C

ASI Kwi iC1PO 3 AAA IO . A C1 . . . . A:

CCA AlA :I z :A~E.~CCtA III .. A a C... st i

--------- .. --- ----------------ACCVAC-SA..ACC ---- - A--------- ---- ---- ----

ISO C3C??? ! I'S Is. .lo $,-:a o:1 . so . C


109

HISTOGRAM

'B I

-700-- 2700. - 0

..... .. .... 9

.. 20

- 700--I 6

I.

-2 -1 *l

Figure 73. Data Set 540. Stem and Leaf and Normal Plots.

110

APPENDIX B. SAS PROGRAM EXAMPLE

/I XT4 JOB (1668 9999) 'WILL MIS',CLASS-S//'-MAIN SYSTEMS2,LIN =(99) CARD~S=(500)// EXEC SAS//WORK DD SPACE=(CYL (202)//DArAIN DD DISP=SHR D N-r1~4W DPDVHA.EDITSR.CCG45.M540//DATAOUT DP~ DISP=(OLD,KEEP),DSN=MSS. 51668. EXT/ISY SIN DD-fATA DATA54O.

INFILE DATAI1INPUT PG 18-10 NSHR 20-21 HT 22-23 BR 24-25 RO 26-27 COST 30-33El 34 E2 35;

VWP=269;BW2=269;BI-3=282;BW4=3 04;BW5r 349;BW6= 388;BW7= 420;BW8=452;BW9=49 1

BW 1 431;BW]2=469,BWl 3 511;BW14=428,BW15=463;BVI16=513;B~s'7=365;Bt-18=408;BWl9-4'8;BI-2O 578;Bl-21=655;BW22=680;BlW23=755;BWO1=150;BVIO2=169;BVO 3 208;BW04 =2 12;BW05=244;BWO 6 =26 4;BWO17=292;PWO =342;BWO9 372-BVIOlO 28J;BWO 1=338;BWO 2=31BWO14=318;BVIO15=370,BWO16=434;BWOl7=2b9;13W0]8=319;BV701 9=402;BV1020=502,BU021=542;1M422=562;BlW023=6 13;TP1' 1054;TP2=1178;TP3= 1238;TP4=139E.TP5= 1631;TP6= 1914;T = 2 2 3 8;T8=2590;

TP9=3072-TP10=20Ui;TP11=2ei 12;T P12=2811;IP13-332];TP14=2281;TP15=2759

TPl6=3343;T 17=1815,TP18=239 4;TP19=2966;TP2O=3628;TP21=4321,TP22:5 179;TP23=6517IF El EQ OR E2 EQ 2 THEN DELETE;IF El E U 7 OR E2 EQ 7 THEN DELETE;IF El GE 8 OR E2 RE 8 THEN DELETE;IF NSIIR CT 2 THEN DELET;IF NSHR EQ 2 AND PC GT4 ;THEN DELETE;IF RO EQZ THEN DELETE;IF COST LT 1 THEN COST = 1;ICOST:1/COST;DATA DATA5 0O

SET DATA54O-ARRAY BW(?3j BW1-BW23-ARRAY BWUI2 ) BWOl-BwC)23;ARRAY TP(23) TPl-TF23;DO I =I TO 23-

IF PG EQ I )ANP NSHR EQ 0 THEN DO;BAQ BW(I)PAY Tf()TIP TP( Ij - BAQ;TOTP TTP * BAQ;ITOTP 1/TOT?;

END'ELSt-

IF~~ P E N SHR NE 0 THEN DO;BA :WQCIPA: TF(I)TTP PAYN BW(I)TOTP BAQ I TP;ITOTP 1 (TOT?;

EN*END;DATA DATA 4O-

SET DAfA540-PROC SORT DATA = hATA540i

BY PG NSHR HT BA COST ICOST ITOIP TOT?;DATA DATAOUT. DATA54O;

SET DATA540WKEEP PG NSHA HT BR COST ICOST ITOTP TOT?;

PROC UNIVARIATE DATA=DATA54O NOPRINT;VAR COST ICOSTBY PG NSHR HI iR ITOTP TOT?;OUTPUT OUT=DATA54lMEDIAN=MCOSTMEDIAN: IMCOSTN: NUMB;

DATA DATAOUT. DATA541;SET DATA541bKEEP PG NSH~ HT BR. MCOST IMOOST ITOTP TOT? NUMB;

PROC PLOT DATA-PATA541;PLOT MCOST-TOTP-PLOT IMCOST- IIofP T 1PO OMLPROC UNIVARIATE DATA=DAT5IPO OMLVAR MCOSI;

PROC UNIVARIATE DATA=DATA541 PLOT NORMAL;VAP IMCOST

PROC REG DATA DAtA541 SIMPLE;MODEL MCOST: TOT?.OUTPUT OUT=DATA546

PMC STHTR=RESID,

MODEL IMCO ST=ItOIP;OUTFUT OUT=DATA5 47

P= IMCSTHTR=RESID;

PROC PLOT DATA=DATA546;

112

PLOT RESID'*TOTP/IVREF=O;PLOT RESID"-'MCST HT/VREFl-O;

PROC PLOT DATA=PATA547-PLOT RESIDI:ITOTP/I ~REF=O;PLOT RESID*'IMCSTIITIVREF=O.

PROC LNIVARlATE DATA=DATA546 PLOT NORMAL;VAR RESID;

PROC UNIVARIATE DATA=DATA547 PLOT NORMAL;VAR RESID;

PROC SOPT DATA = DATA541 OUT=DATA541A;BY TOTP.,

DATA DATAOUT. oArA IA;SET DATA541A;KEEP PG NSHR MT BR MCOST IMCOST ITOTP TOTP;

PROC RSREG DATA=DATA541A,MODEL MCOST=TOT P4 LACY.FIT-

PROC SORT DATA =DATA5 1 OUT=DATA541B;BY.I TOTP

DATA D AT AOU. .1)A T.Ai4 1B;SET DATA54~1B:KEEP PG NSHR HT DP MCOST IMCOST ITOTP TOTP NUMB;

PROC RSREG DATA=DATA5 41BMODEL IMCOST=ITOTP LACKFIT;

DATA DATA541C;SEr DArA54'IF NSHRP GT 1. THEN NSHR' 1:DATA DATAOUr. DAIA541(;

SET DATA541C:VEEP PG NSHR TIT BR MCOST IMCOS T ITOTP TOTP NUMB;

PROC SORT DATA =DATA541C OUTrDATA54 1D;BY NSHR'

DATA DATADU!T. DATi541D;,SET DATA54LD:KEEP PG NSHR HT PR MCOST IMCOST ITOTP TOTP NUMB;

PROC PLOT DATAMQATA541D);['LOT IICOST-TOTP;BY NSHP*

PROC PLOT DAfA=DATA541D;PLOT IMCOSTITOTP;BlY UsHR

PROC UNIVARIATE DATA=DATA541D PLOT NORMAL;VAR MCOST;ply NSHR.

PROC U:;IVARIAiE DATA=DATA541D PLOT NORMAL;VAR I!ICOST;BY NSHR;

PROC REG DATA=DATA541D SIMPrLE;MODEL MCOST=TyTF.OUTPUT OUT=DATA546D

P=MC S IH-TR=RESID;

BY NSHR'PROC REG DAtA=DATA541D SIMPLE;MODEL IMCOSlIIITI'OUTPUT OUTVDAIA547

P=1MiCSTHTR=RESID;

BY NSIR-PROC PLOT LATA~pATA546D'

PLOT RESID ITr/VRftFO;BY NSHRP

PROC PLOT DATA- PATA546DPLOT RESID MCS11T/4REF=O;

BY NSHP-PROC PLOT DATA'PATA547D

PLOT PESIl'ITOIP,'VAEF=O;BY NSHP-

PROC PLOT DATArDATA547DPLOT RESID<-IflCSTHT)VREFO;

BY NSHR-PROC UNIVARIATE DATAzDATAS46D PLOT NGRIAL,

VAR RESID;

1 13

BY NSHRPROC IJNIVARiATE DATArDATA547D PLOT NORMAL;

VAR RESID;BY NSHR.

PROC SORT DATA =DATA541D OUT=DATA541E;BY NSHR TQTP;

DATA DATAOUT. DAIA5 4IE;SET DATA541E;KEEP PG NSHR H-T BR MCOST IMCOST ITOIP TOTP NUMB;

PROC RSREG DATA=DATA541F-MODEL MCOST=TOTP/LACKFIT;

BY NS"R.PROC SORT DATA =DATA541D OUT=DATA54 IF;

BY NSHR ITOTP;DATA DATADUT. DATA54IF;

SET DATA541F;VEEP PG NSHR HT BR MCOST IMCOST ITOTP TOTP NUMB;

PROC PSREG DATA=DATA54FMODEL IMCOST=J.TOTP)LACKFIT;BY NS"R;

DATA DATA541G,SET DATA541IF PG GE I AND PG LE 9 THEN PG~l;IF FG GE 10 AND PG LE 19 THEN PG'2*IF PG GE 20 AND PG LE 23 THEN PG=3;DATA DATAOUT. DATA541G;

SET DATA54IG;KEEP PC NSIIR HT BR MCOST IMCOST ITOTP TOTP NUMB;

PROC SORT DATA = DATA54IG OUT DATA5 41H;BY PG-

DATA DATAOUT. DATA54 iN;SET DATA541H;KEEP PG NSHR HT BR. MCOST IMCOST ITOTP TOTP NUMB;

PROC PLOT DATA=PATA541H;PLOT MCOSTTOTP;BY PG&ADTA4H

PROC PLOT ~T=AA4HPLOT IMCOST"'lTOTP.

PRO U "1VAIATE DATA=DATA541H PLOT NORMAL;VAR MCOST;BY P

PROC UNIV1RMATE DATA=DATA54IF{ PLOT NORMAL;VAR IMCOST;BY PG.

PROC REG DAtA=DATA541H SIMPLE;MODEL MCOST=TOIPOUTPUT OUT=DATA546H

P MC ST HTBRRESID;BPG&ADT51

PROC REG f~T=AA4HSIMPLE;MODEL IMCOST=ITOTP:OUTPUT OUT=DATA547H

P=IMCSTHrYR=RESID;BYPG.

PROC PLOT~ DATA=QATA54 6H-PLOT RESIDT-:OTP/VRtF=O;

BY PG;PROC PLOT DATA=DATA546H

PLOT RESID1MCSTHT/ 'REF=O;BY PG;

PROC PLOT DATA=IPATA547HPLOT RESID%:ITOTP/ViEF=O;

BY PG;PROC PLOT DATA=QATA547H

PLOT RESID"<IMCSTHT)VREF=0;BY PG AEDT=DT56 LO OML

PROC UNIVAAT DTDTA4HPONRMLVAR RESID;

BY PC;

114

PROC UNIVARIATE DATA=DATA547H PLOT NORMAL;VAR RESID;

BY PG;PROC SORT DATA =DATA541H OUT=DATA54II;

BY PG TOTPjIDATA DATAOUT. DATA54 i

SET DATA54].1;KEEP PG NSHR HTT BR MCOST IMCOST ITOTP TOTP NUMB;

PROC RSREG DATA=DATA541I,MODEL MCOST.=TOTP/1.ACKFIT;

BY PG-DATA DATA 41J3

SET DAT 541HPPROC SORT DATAz DATA541H1;

BY PG ITOTP;DATA DATAOUT. DATA .4 ii

SET DATA541J;KEEP PG NSHR HT BR MCOST IMCOST ITOTP TOTP NUMB;

PROC RSREG DATAzDATA541J-MODEL I1COST~ITOTP2LACKFIT;BY UPK

DATA DATA5 4KSET DATA54DIF NSHR CT i THEN NSIIR~1:IF PG GE 1 AND PG LE 9 IIhEN PG=1;IF PG GE 10 AND PG LE 19 1HEN FG=2;IF PG GE 20 AND PG L.E 23 THEN PG=3;DATA DATAOUT. DATA541K;

SET DATA541K;KEEP PG NSHR HTT BR MCOST IMCOS T ITOIP TOTP;

PROC SORT DATA =DATA541K OUT=DATA5 4 L;BY NSHR PG-

DATA DATADUT. DATA54 iL;SET DATA541L,;KEEP PG NSHR ITIT BR MCOST IMCOST ITOTP TOTP;

PROC PLOT DATA=PATA541L;PLOT MCOSTTOTP;BY NSHR PG'

PROC PLOT DATA~?IATA541L;PLOT IMCOST"ITOTP;BY NSHR PG-,

PROC UNIVARIATE DATA=DATA541L PLOT NORMAL;VAR ?COSTBY NSHR P :

PROC UNIVARIATE DATA=DATA541L PLOT NORMAL;VAR lI-.' fBY NSF, FG!

PROC REG DATA D~tA541L SIMPLE;MODEL MCOST=TOTi'OUTPUT O1'T=DATA546L

P MC STHTR=RESID;

BY NSHR PG:PROC REG DATA=DAlA541L SIMPLE;MODEL. IM-COST=ITOTPOUTPUT OUT=DATA547L

P= IMCSIHTrR=RESID;

BY NSH. PG;PROC PLOT DATA=PATA546L

PLOT RES1D--TOTP/VR F=O;BY NSHR PG;

PROC PLOT DATA=QATA546L;PLOT RESID-MCSTHT/VREF=O;

BY NSHR P;;PROC PLOT DATA=PATA547L

PLOT RESID,:ITOTP/ViEF=O;BY NSHRz PGC;

PROC PLOT DATAIPATA547LPLOT RESIO"eIMCSTlIT7VREF=O;

BY NSHR PG*PROC UNIVARIATt DATA=DATA546L PLOT NORMAL;

VAR RESID;BY NSHR NPG ADT57LPO OML

PROC UNIVARIA1~DT~AA4LPO ONLVAR RESID;

BY NSHR PG;PROC SORT DATA =DATA541L OUT=DATA54IM;

BY NSHR PG TOTP;DATA DATAOUT. DATA541I;

SET DATA41M;KEEP PG NSHR H-T BR !ICOST IMCOST ITOTP TOT? NUMB;

PROC RSREG DATA=DATA541MODEL MCOST=TOTP/LCKFIT;

BY NSHR PG;DATA DATA541N'

SET DATA54lL'PROC SORT DATA =DATA541L,;

BY NSHR PQ ITOTP;DATA DATAOUT. DATA54 IN;

SET DATA541N;K~EEP PG NSHR MT R MCOST IMCOST ITOTP TOT? NUMB;

PROC RSF.EG DATA= DATA5 IN;MODEL lllCOST~lTQTP/LACKFIT;BY NSHR PG;

OKTIONS LINESIZE=80

116

APPENDIX C

TABLES 1 - 14

117

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LIST OF REFERENCES

1. American Management Systems, VHA Current ProcessDescriptions, pp. 2-1 to 2-64, February 21, 1989.

2. Draper, N. R., and Smith, H., Applied Regression Analysis,pp. 11-147, John Wiley & Sons, 1981.

3. McGill, R., Tukey, J. W., and Larsen, W. A., The AmericanStatistician, Variations of Box Plots, Vol. 32, No. 1, p.16, February 1978.

4. McCalla, P., and Nelder, J., Generalized Linear Models,Chapman & Hill Monograph on Statistics and Probability,1983.

298

INITIAL DISTRIBUTION LIST

1. Defense Technical Information Center 2Cameron StationAlexandria, Virginia 22304-6145

2. Library, Code 0142 2Naval Postgraduate SchoolMonterey, California 93943-5002

3. Defense Manpower Data Center 299-100 Pacific St.Suite 155AMonterey, California 93940

4. Laura D. JohnsonCode 55joNaval Postgraduate SchoolMonterey, California 93943-5000

5. Donald P. GaverCode 55GvNaval Postgraduate SchoolMonterey, California 93943-5000

6. Michele Williams6185 Wild Valley Ct.Alexandria, Virginia 22310

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