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Discriminant analysis of purchasers of general aviation aircraft avionics
DISCRIMINANT
ANALYSIS
AVIATION
OF PURCHASERS
AIRCRAFT
OF GENERAL
AVIONICS
STEPHEN G. VAHOVICH~ Energy
Federal Aviation Administration, Office of Environment and Energy. Division (AEE-200). 800 Independence Avenue, S.W.. Washington. D.C.. U.S.A. (Received 25 October 1978; in revisedform
15 March 1979)
Abstract-Using data from the Federal Aviation Administration’s national sample of general aviation (GA) aircraft owners (1975). this study explores the factors that influence aircraft owners’ equipment purchase decisions for eight different types of communication and navigation instrumentation. The discriminant function is developed for noncompany owners of GA aircraft and is applied to data for company owners to generate the classification results. The results for almost all types of avionics suggest that the discriminant function-based on owner’s type of use of aircraft. the aircraft’s age. type of aircraft. and intensity of use-effectively discriminates between the equipped and nonequipped groups. The classification results, tested against both a proportional chance and pure chance criteria. are for the most part statistically significant at the 0.01 level. Thus the discriminant functions effectively identifies individuals most apt to purchase a particular type of avionics. Hence, it could be used by the avionics manufacturing industry to establish market profiles and by the FAA in estimating the expected demand on their ground facility services and equipment.
1. INTRODUCTION
fluenced by GA activity. More particularly, certain types of GA flying, especially instrument flight. require (Office of Federal Register. 1977) specified types of communication and navigational instrumentation and draws heavily on FAA ground facility manpower and equipment services. Currently GA accounts for almost one-third of all instrument flights handled at FAA control centers and well over onehalf of all FAA tower instrument operations (FAA 1978). Thus, in order to facilitate its planning for future manpower needs as well as for the ground aid system per se. knowledge of the factors that influence GA avionics equipage is critical to the FAA. As may be expected. factors influencing the purchase of communication and navigation instrumentation is also of intense interest to the avionics producing industry. GA’s commanding share of the civil aircraft fleet together with recent advances in microprocessors. which has made many types of avionics more practical (both in size and in cost), target GA as a prime market (Terry. 1977; FAA. 1977). Thus. identifying the characteristics of GA aircraft and their owners that are associated with the purchase of the various types of avionics forms a basis for defining market demand. planning future sales/marketing strategies and arranging production schedules. The GA specific data utilized in the analysis are from a stratified systematic random sample (for details see Vahovich. 1977) of the universe of 177,631 GA aircraft as of 1 January 1975 (aircraft owned bl persons not residing in the U.S.. nonengine-propelled craft, and government-owned alrcraft are excluded). The Bureau of the Census collected the data for the FAA. The primary method of data collection was tele-
Using data from the Federal Aviation Administration’s (FAA) national sample of general aviation (GA) aircraft owners, this study explores the factors that influence aircraft owners’ equipment purchase (equipage) decisions for eight different types of communication and navigation instrumentation (avionics). Because of the previous lack of representative cross-sectional data on the characteristics of the GA community and their aircraft avionics, this study marks the exploration of a virgin area of research. Briefly defined, GA includes all aircraft except certified air carriers. supplemental air carriers and military. Earlier analysis. utilizing these cross-sectional data. suggest that GA aircraft owners are a heterogeneous group (encompassing diverse types of aircraft. user groups. and utilization rates; Vahovich, 1977). and constitute an important segment of the national airspace system (GA accounts for 98% of all civil aircraft and over 80”,6 of all operations at FAA towered airports; FAA. 1978). They are unique as compared to the U.S. population (via GA owners’ higher income levels and less skewed geographical distribution: Vahovich, 1977). and are strongly committed to flying (consider hours flown as a necessity rather than a luxury good; Vahovich. 1978). This characterization of GA emphasizes the need to conduct and present analysis utilizing GA specific data. Given its large share of total aircraft operation, FAA manpower staffing at its facilities is strongly intThe author is an Industrial Economist wrth the Federal Aviation Administration. the views expressed in this paper are those of the author and do not necessarily reflect those of the FAA or an) other government agency. 75
STEPHEN G. VAHOVICH
26
phone interviews (with personal visits and mall followup for telephone non-respondents). The usable response rate was 94.?‘, (.V = 936). The analytical technique utilized to identify the factors intluencing GA avionics equipage and the extent of the influence is multiple discrimant analysis (e.g. Cooley and Lohnes. 1971). Eight types of avionics are considered for analysis. Company owners (i.e. the registered owner is some business entity such as a corporation. etc.) and noncompany owners of GA aircraft are considered separately.
2. THEORETICAL
FRAMEWORK
This section describes the model that is used to predict whether aircraft owners will (i.e. belong to the equipped group) or will not purchase (i.e. belong to the nonequipped group) a particular type of avionics. The fundamental assumption underlying the model is that GA avionics equipage is significantly affected by the primary use of the aircraft. the type of aircraft, the aircraft’s age, and the intensity of use. Besides the reasonableness of the economic hypothesis and statistical significance criteria. two important factors were taken into consideration in researching the structure of the discriminant function. First. a simple linear formulation was desirable because of its advantage of clarity of interpretation of the effect of each of the independent variables (Morrison, 1969). Second. the variables chosen as discriminators should be of the type for which information is easily available, so that this research-could be utilized by both the FAA and the avionics industry. The general form of the function utilizing the discriminating variables is described in eqn (1).
(1) The coefficients bj u = 1.. , m) are generated such that the function maximizes the differences between the respective means of the equipped and nonequipped groups. The estimated signed value of hj represents the direction (positive or negative) and extent (relative to the other coefficients) of the contribution that its associated. standardized ith variable makes to differentiating between the two groups. Si is the ith individual’s discriminate score. estimated by substituting the sample values (standardized) into the estimated function on the basis of which assignments to group membership are made. The discriminant variables xjio = 1, , m) are described below along with their expected direction of impact. The average cruising speed (in mph) of the aircraft is a surrogate variable representing the type of aircraft. Preliminary estimates. utilizing dummy variables to represent various aircraft types, indicated serious multicollinearity problems between these dummy variables and other independent variables. Since average cruising speed increases monotonically
from single-engine piston aircraft to the turbojets, this continuous variable was adopted. eliminating the estimation problem while simultaneously preserving the desired theoretical structure. Because turboprops and turbojets are structurally designed for longer distance and high altitude flights, requiring avionics for communications and navigation. and the smaller piston aircraft are not. cruising speed is expected to have a positive influence on aircraft avionics equipage. Four user group dummy variables (the business use category being constant) are utilized to reflect variations in avionics equipage due to type of use. Briefly. these dummy variables (i.e. equal to one when the sample respondent indicated that primary use, and zero otherwise) may be defined as follows: instructional use represents formal training with a flight instructor on board; personal use is pleasure flying andior maintenance of pilot proficiency; the composite air taxi and rental category is used by a lessee in return for payment to the owner; and the composite aerial application and industrial/special category represents specialized applications such as crop dusting, survey work, advertising, etc. Since business users are more likely than the other users*to fly in congested airspace (e.g. around large cities), where FAA avionics requirements are especially strmgent. the user group dummy variables are expected to have a negative influence on avionics equipage. For several reasons. aircraft manufactured prior to 1960 are expected to have lower avionics equipage rates than more recent vintage aircraft (the category held constant). Compared to the more recently built aircraft. pre-1960 vintage aircraft may be expected to have shorter trip lengths and generally lower utilization rates. The later is attributed to the deleterious effects of age/use and argues against avionics equipage. Further, the lower market value of such aircraft and their shorter expected useful life. relative to more recent vintage aircraft. argues against the installation of costly avionics. Thus the dummy variable (equal to one if the aircraft was manufactured prior to 1960) representing older aircraft is expected to have a negative parameter estimate. Itinerant hours is expected to vary directly with avionics equipage. This is because of the added pilot safety margin derives from avionics equipage, especially on trips where origin and destination differ. Two initial tests are used to determine whether the estimated discriminant function is a statistically significant discriminator between the avionics equipped and nonequipped groups. The computed chi-square statistic associated with Wilks’ lambda is used to test the null hypothesis that a statistically significant amount of discriminating information is not accounted for by the discriminant function. If the computed chi-square is greater than some critical value (e.g. x&95 or z;,~~. which are the critical values at the 0.05 and 0.01 significance levels. respectively), the null hypothesis is rejected at the corresponding level of significance. A further aid in judging the use-
Discriminant analysis of purchasers of aircraft avionics fulness of a discriminant function is its associated cannonical correlation-the square of which indicates the proportion of the variance in the discriminant function explained by the variables that define group membership. The above tests may be viewed as the preliminary screening tests to judge the effectiveness of the estimated discriminant function as a predictor of group membership (i.e. its predictive efficacy). The acid test of the effectiveness of the discriminant function is its ability to correctly identify/classify individuals not included in the analysis sample, given information on their discriminant variables only. This validation requirement is difficult to satisfy because it requires additional data. The results available from the analysis sample are usually presented in the form of a classification table, as shown in Fig. 1. N, is the total number of individuals actually in the ith group, Ck is the total number of individuals that were classified as being in the kth group. ni, is the number of individual of group i that were classified as group k. Then, Q/C& is the proportion of the predicted group i individuals that are correctly classified, and the sum of the main diagonal elements divided by Iv is the overall proportion of individuals correctly classified. While such results are obtained for the present study, it is noted that using the analysis sample to generate the discriminant function and then attempting to validate its predictive efficacy via a classification procedure based on the identical analysis sample is a biased test. Frank, Massy and Morrison (1965) point out that the primary cause of this bias is due to errors of sampling when estimating the means of the population, the basis for the discriminant COefficients. The direction of the bias is upward-the classification results show greater predictive power than actually exists among the true population. In addition, these authors note that bias may also be introduced when searching for variables that work best for a particular sample (which may have peculiar characteristics not exhibited by the population). To avoid the biases that can occur in classification tables constructed in this way. the present study uses only part of the available data. That is, only noncompany owners of GA aircraft are used to fit the discriminant function. then company owners of GA aircraft are classified into the equipped or nonequipped groups ulilizing the discriminant function generated from the analysis sample. Thus, the classification results (presented in Table 2) show what part of the observed proportion of correctly classified observations is due to true differences between the means of the discriminant variables in the avionics equipped and nonequipped groups. An alternative to this split sample approach is to use randomly generated data to form a synthetic validation sample. For purposes of the present study, and since original data was available for the validation sample. the split sample approach was preferred. The classification results obtained from the com-
27
pany owners are judged against a proportional chance criterion to determine if the function is an effective discriminator between the avionics equipped and nonequipped groups. If the latter is found true. the advantage of the company-noncompany owner dichotomy lies in the simplicity of application of one function in both of these avionics markets. The proportional chance criterion used is: P = 2
+ (1 -
2)’
(2)
where tl is the proportion of individuals in group 0 in group 1. Morrison and (1 - a) is the proportion (1969) points out that this proportional chance criterion is a more appropriate yardstick against which to judge the classification results than the pure chance criterion, because the former takes into account the fact that the discriminant function defies the odds by classifying individuals in the smaller group. The pure chance criterion would simply classify everyone into the group with the greatest membership. Since it is important for the FAA and manufacturers to identify correctly members of both groups, and since the group that is equipped with avionics is often the smaller of the two groups. P should be used. Thus. if the proportion of company owners (the validation sample) correctly classified [(nrr + nz,)!h’; as per Fig. l] is greater than P, the discriminant function (derived from the analysis sample) will be considered an effective discriminator between the avionics equipped and nonequipped groups. Finally, it should be noted that in deriving the classification results. a priori probabilities are not utilized. That is, individuals are assigned to the group for which they have the greatest probability of membership. For purposes of this study, the intent is to avoid incorporating a powerful set of prior probabilities which can lead to successful classification results even when the discriminant function is not performing effectively (Cooley and Lohnes. 1971). 3.
EMPIRICAL
RESULTS
Table 1 presents, the standardized discriminant function coefficients, the canonical correlation. and the computed chi-square associated with Wilks’ lambda for the analysis sample for each type of avionics. Without exception. the signs of the coefficients are as hypothesized in Section 3. The magnitude of the standardized coefficients of the discriminant variables suggest that for almost all types of
PREDICTED Group 0
Group I
Group
0
‘111
,112
Group
1
n21
1122
co
C,
.\ 0
ACTUAL
H4
Fig. 1.
.Y ,
.2
0.49 I 135.9”
648.0”
- 0.629 - 0.030
-G93
0.453 -0.439 0.179
0.42
-0.093
0.635 -0.342 0.264 -0.115 -0.338
OMNIdirectional receiver (N = 4153)
I. Standardized
.
function
1030.1”
0.47
-0.191 -0.081
0.690 -0.206 0.203 -0.193 -0.368
Distance measuring equipment (N = 4145)
discriminant
1342.6”
0.53
- 0.270 - 0.054
0.672 -0.241 0.266 -0.145 -0.272
owners)
180.6”
0.21
-
0.881 -0.175 0.191 - 0.096
Radar (N = 4146)
Variables
(Noncompany
Dependent Automatic direction finder (N = 4151)
coefficients
80.97”
0.14
-0.400 - 0.265
-0.359 -0.318
0.496 - 0.565
Area navigation equipment (N = 4144)
1277.1”
0.55
-0.220 -0.043
0.513 -0.456 0.325 - 0.073 -0.236
Transponder (N = 3596)
_
._
_
_.
Significance Icvels: a = 0.99 ~ Means that the variable did not enter because it did not add a statistically significant amount of centroid separation beyond that achieved by the previously (i.e. partial multivariate F ratio less than 1.0 was selected so that any variable with discriminatory power is chosen and retained for the analysis.) IV = Number of observations for the relevant avionics category
Canonical correlation Chi-square on Wilks’ lambda
Average cruising speed Pre-1960 vintage Itinerant hours Instructional use Personal use Aerial application & industrial/special Air taxi and Rental
Discriminant variables
Instrument landing system (N = 3351)
Table
entered
variables
844.9
0.54
-0.153 _
0.516 - 0.297 0.434 -0.155 -0.300
Altitude encoding transponder (N = 2636)
landing
nnvigution
with altitude
equipment
Total
Total
Total
Total
Total
Total
Total
Total
107 464 II73
5.53 1363 1916
2190 361 2551
1966 244 2210
811 1331 2208
1401 809 2210
26-l 1943 2210
735 1215 I950
*“Total” value is C max = max (x. 1 - x), where x is the proportion **For t value computation see eqn (3). ***For proportional chance computation see eqn (2).
Nonequipped Equipped
Transponder
Nonequipped Equipped
Transponder
Nonequipped Equipped
Aera
Nonequipped Equipped
Radar
encoder
equipment
receiver
system
Autontatic direction finder Nonequipped Equipped
Distance measuring Nonequipped Equipped
Nonequipped Equipped
OMNI-direction
Nonequipped Equipped
Instrument
Actual number
Company
or individuals
60.3 39.7 60.3
28.9 71.1 71.1
85.8 14.2 85.8
89.0
11.0
89.0
39.1 60.3 60.3
63.4 36.6 63.4
12.1 87.9 87.9
37.1 62.3 62.3
= 18.77)
in the nonequipped
group.
94.5 68.8 80.6 (t = 19.54)
82.7 70.2 80.0(t
93.8 18.7 46.2
99.3 27.9 72.1
54.8 64.0 62.2 (t = 9.50)
90.6 59.8 74.0(1 = 19.23)
52.1 95.7 88.6 (C = 11.37)
69.8 16.2 74.3 (C = 18.85)
owners Predicted correctly classified (%)**
results by type OTavionics
Pure chance criterion (%)*
Table 2. Classification
52.1
58.9
75.6
80.4
52.1
53.6
78.1
53.0
Proportional chance criterion (“A)***
2280 356 2636
1851 1745 3596
3802 342 4144
4086 60 4146
2451 1700 4151
3546 599 4145
726 3427 4153
1985 1366 3351
Actual number
Noncompany
95.6 46.9 85.2
78.8 77.9 78.3
94.6 11.7 57.0
99.3 5.7 85.2
80.3 73.5 77.6
95.0 40. I 80.2
43.8 94.4 78.7
75.8 65.8 71.8
owners Predicted correctly classified (70
30
STEPHEN G.VAHOVICH
avionics, type of aircraft (as represented by average crulsmg speed) is by far the most important factor in predicting avionics equipage. Age of aircraft is, in general, also relatively powerful in discriminating between the equipped and nonequipped groups. Itinerant hours flown is relatively unimportant except for the radar and altitude encoding transponder equipage decisions. Since among avionics devices, the latter two are particularly useful at higher altitude flying, generally associated with longer distance itinerant Rights. these results are not surprising. The relative contribution of the four type of use dummy variables in discriminating between the equipped and nonequipped groups varies widely with the types of avionics. The canonical correlation and the chisquare values suggest that each of the discriminant functions is a statistically significant discriminator between the groups. The chi-square value for each equation is greater than its corresponding critical value at the 0.01 level of significance. Thus the null hypothesis (that a statistically significant amount of discriminating information is not accounted for by the current function) is rejected. Given the cross-sectional nature of the data and the high degree of variability usually associated with this type of data, the canonical correlations are respectable. Further, excess emphasis should not be placed on their values; the true test of predictive efficacy is presented in Table 2. Table 2 presents results which may be used to test the effectiveness of the discriminant function as a predictor of group membership. This table presents the classification resuits for the validation sample (company owners) and the analysis sample (noncompany owners) for each type of avionics by group category. Column 1 shows the actual number of company owners, Column 2 gives the pure chance criterion, Column 3 shows the per cent of predicted that are correctly classified and Column 4 gives the computed proportional chance criterion [see eqn (2)]. The information to the right of the double line in Table 2 presents the classification results for noncompany owners. Since the discriminant function was generated from data on these individuals, Columns 5 and 6 are not considered in the validation test of the discriminant function. The results presented in Table 2 are very encouraging. The overall percentage of correctly classified (“total” rows in Column 3) range from 46.2% for area navigation equipment to 88.6% for OMNI-directional receiver. Except for radar and area navigation equipment, the per cent of predicted correctly classified exceeds both the proportional chance criterion (Column 4) and the pure chance criterion (Column 2) for each type of avionics. The minimum difference between the per cent correctly classified and their -orresponding proportional chance values is 109; and the maximum is 29%. To test further the hypothesis that the proportion of the correctly classified cases is significantly different from the proportion that would be expected if the proportional chance
criterion were used. the follofiing performed (results are presented Table 2. Column 3):
t=(Q-
computation was in parenthesis m
P) ___ 'VF '
(3)
where Q is the proportion of the validation sample observations correctly classified by the discriminant analysis, P is the proportion expected using the proportional chance criterion, and :V is the number of observations in the validation sample. This test is similar to that suggested by Frank. Massy and Morrison (1965). except that they use the pure chance criterion to define P. Comparing the computed t values against their corresponding critical values for each type of avionics suggests that, with the possible exception of radar and area navigation equipment, the proportion of correctly classified cases is significantly greater at the 0.01 level than that expected using the proportional chance criteria. In fact, if eqn (3) computations were performed using the more stringent pure chance criterion to define P,an identical statement is true for all types of avionics except the automatic direction finder (which is significant at the 0.10 level). These results are impressive and suggest that most of the discriminant functions given in Table I could be used effectively to identify individuals who would be most apt to purchase a particular type of avionics (i.e. to establish market profiles) and, more generally. to estimate increases or decreases in the derived demand on FAA ground facility services and equipment. Further, it may be noted (Table 2) that, without exception, the per cent of predicted correctly classified for each of the individual groups (nonequipped and equipped), within each avionics category, exceeds their corresponding pure chance expectations. Even for radar and area navigation equipment, the per cent correctly classified. when the equipped and nonequipped groups are taken se5arately, is greater than that expected under pure chance. Thus, the results for these two functions are not as bleak as first suggested.
4. CONCLUSlONS The findings presented in this paper suggest that the type of use. aircraft age, type of aircraft, and intensity of use can effectively discriminate between owners whose aircraft are equipped with various types of communication and navigation instrumentation and those not so equipped. The discriminant functions presented in this paper, validated by using additional sample data, appear to classify correctly a larger proportion of the aircraft owners (at the 0.01 level of statistical significance) than could be expected using either the pure chance or the proportional chance criterion. Thus. these discriminant functions could be used by the avionics manufacturing industry to establish market profiles and by the FAA in estimating the
Discriminant expected
demand
on their
ground
facility
analysis services
of purchasers and
Frank
of aircraft
avionics
31
R. E.. Mass\ W F. and Morrison D. G. (1965) Bias discriminant analysis. J. Marketiq Rrs. 15G-
m multiple 258.
equipment.
REFEREYCES Coole) W. and Lolmes P. R. (197 I ) Multituriate Atlal~ws. Wiley. New York.
Data
Federal Aviation Administration. Office of Aviation Policy (1978) Aaiattori Forrcasts Fiscal Ycwrs IYW IYYO. National Technical Information Service. Springfield. Virginia. Federal Aviation Administration, Office of Aviation Policy (1977) T/w Intpact qf Microcotnputrrs ON Aciatioti. National Technical Information Servtce. Springfield. Virginia.
Morrison D. G. (1969) On the Interpretation of discriminant analysis. J. hfarkctirtg Rcs. 6, I5fS-lh.3. Terry .I (1977) Microprocessors are key to new digital COM,‘NAV ‘IDENT designs. ICAO f3ttlletir1. 19-X. L’.S.. Code of Fedrral Rcgu/ariou Title 14-Aeronautics and Space. Part 9 1.33 (I 977). Office of the Federal Register. National Archives and Records Service. Washmgton. D.C. Vahovich S. G. (1977) Grrierol Ariatiort: Arrcr~~ff Owwr & L’rilicariori Characteristics. National Technical Information Service. Springfield. Virginia. Vahovich S. G. (i978) Income&d cost impact on general aviation hours flown by individual owners. Transprl Res. 12.315319.
ANALYSIS
AVIATION
OF PURCHASERS
AIRCRAFT
OF GENERAL
AVIONICS
STEPHEN G. VAHOVICH~ Energy
Federal Aviation Administration, Office of Environment and Energy. Division (AEE-200). 800 Independence Avenue, S.W.. Washington. D.C.. U.S.A. (Received 25 October 1978; in revisedform
15 March 1979)
Abstract-Using data from the Federal Aviation Administration’s national sample of general aviation (GA) aircraft owners (1975). this study explores the factors that influence aircraft owners’ equipment purchase decisions for eight different types of communication and navigation instrumentation. The discriminant function is developed for noncompany owners of GA aircraft and is applied to data for company owners to generate the classification results. The results for almost all types of avionics suggest that the discriminant function-based on owner’s type of use of aircraft. the aircraft’s age. type of aircraft. and intensity of use-effectively discriminates between the equipped and nonequipped groups. The classification results, tested against both a proportional chance and pure chance criteria. are for the most part statistically significant at the 0.01 level. Thus the discriminant functions effectively identifies individuals most apt to purchase a particular type of avionics. Hence, it could be used by the avionics manufacturing industry to establish market profiles and by the FAA in estimating the expected demand on their ground facility services and equipment.
1. INTRODUCTION
fluenced by GA activity. More particularly, certain types of GA flying, especially instrument flight. require (Office of Federal Register. 1977) specified types of communication and navigational instrumentation and draws heavily on FAA ground facility manpower and equipment services. Currently GA accounts for almost one-third of all instrument flights handled at FAA control centers and well over onehalf of all FAA tower instrument operations (FAA 1978). Thus, in order to facilitate its planning for future manpower needs as well as for the ground aid system per se. knowledge of the factors that influence GA avionics equipage is critical to the FAA. As may be expected. factors influencing the purchase of communication and navigation instrumentation is also of intense interest to the avionics producing industry. GA’s commanding share of the civil aircraft fleet together with recent advances in microprocessors. which has made many types of avionics more practical (both in size and in cost), target GA as a prime market (Terry. 1977; FAA. 1977). Thus. identifying the characteristics of GA aircraft and their owners that are associated with the purchase of the various types of avionics forms a basis for defining market demand. planning future sales/marketing strategies and arranging production schedules. The GA specific data utilized in the analysis are from a stratified systematic random sample (for details see Vahovich. 1977) of the universe of 177,631 GA aircraft as of 1 January 1975 (aircraft owned bl persons not residing in the U.S.. nonengine-propelled craft, and government-owned alrcraft are excluded). The Bureau of the Census collected the data for the FAA. The primary method of data collection was tele-
Using data from the Federal Aviation Administration’s (FAA) national sample of general aviation (GA) aircraft owners, this study explores the factors that influence aircraft owners’ equipment purchase (equipage) decisions for eight different types of communication and navigation instrumentation (avionics). Because of the previous lack of representative cross-sectional data on the characteristics of the GA community and their aircraft avionics, this study marks the exploration of a virgin area of research. Briefly defined, GA includes all aircraft except certified air carriers. supplemental air carriers and military. Earlier analysis. utilizing these cross-sectional data. suggest that GA aircraft owners are a heterogeneous group (encompassing diverse types of aircraft. user groups. and utilization rates; Vahovich, 1977). and constitute an important segment of the national airspace system (GA accounts for 98% of all civil aircraft and over 80”,6 of all operations at FAA towered airports; FAA. 1978). They are unique as compared to the U.S. population (via GA owners’ higher income levels and less skewed geographical distribution: Vahovich, 1977). and are strongly committed to flying (consider hours flown as a necessity rather than a luxury good; Vahovich. 1978). This characterization of GA emphasizes the need to conduct and present analysis utilizing GA specific data. Given its large share of total aircraft operation, FAA manpower staffing at its facilities is strongly intThe author is an Industrial Economist wrth the Federal Aviation Administration. the views expressed in this paper are those of the author and do not necessarily reflect those of the FAA or an) other government agency. 75
STEPHEN G. VAHOVICH
26
phone interviews (with personal visits and mall followup for telephone non-respondents). The usable response rate was 94.?‘, (.V = 936). The analytical technique utilized to identify the factors intluencing GA avionics equipage and the extent of the influence is multiple discrimant analysis (e.g. Cooley and Lohnes. 1971). Eight types of avionics are considered for analysis. Company owners (i.e. the registered owner is some business entity such as a corporation. etc.) and noncompany owners of GA aircraft are considered separately.
2. THEORETICAL
FRAMEWORK
This section describes the model that is used to predict whether aircraft owners will (i.e. belong to the equipped group) or will not purchase (i.e. belong to the nonequipped group) a particular type of avionics. The fundamental assumption underlying the model is that GA avionics equipage is significantly affected by the primary use of the aircraft. the type of aircraft, the aircraft’s age, and the intensity of use. Besides the reasonableness of the economic hypothesis and statistical significance criteria. two important factors were taken into consideration in researching the structure of the discriminant function. First. a simple linear formulation was desirable because of its advantage of clarity of interpretation of the effect of each of the independent variables (Morrison, 1969). Second. the variables chosen as discriminators should be of the type for which information is easily available, so that this research-could be utilized by both the FAA and the avionics industry. The general form of the function utilizing the discriminating variables is described in eqn (1).
(1) The coefficients bj u = 1.. , m) are generated such that the function maximizes the differences between the respective means of the equipped and nonequipped groups. The estimated signed value of hj represents the direction (positive or negative) and extent (relative to the other coefficients) of the contribution that its associated. standardized ith variable makes to differentiating between the two groups. Si is the ith individual’s discriminate score. estimated by substituting the sample values (standardized) into the estimated function on the basis of which assignments to group membership are made. The discriminant variables xjio = 1, , m) are described below along with their expected direction of impact. The average cruising speed (in mph) of the aircraft is a surrogate variable representing the type of aircraft. Preliminary estimates. utilizing dummy variables to represent various aircraft types, indicated serious multicollinearity problems between these dummy variables and other independent variables. Since average cruising speed increases monotonically
from single-engine piston aircraft to the turbojets, this continuous variable was adopted. eliminating the estimation problem while simultaneously preserving the desired theoretical structure. Because turboprops and turbojets are structurally designed for longer distance and high altitude flights, requiring avionics for communications and navigation. and the smaller piston aircraft are not. cruising speed is expected to have a positive influence on aircraft avionics equipage. Four user group dummy variables (the business use category being constant) are utilized to reflect variations in avionics equipage due to type of use. Briefly. these dummy variables (i.e. equal to one when the sample respondent indicated that primary use, and zero otherwise) may be defined as follows: instructional use represents formal training with a flight instructor on board; personal use is pleasure flying andior maintenance of pilot proficiency; the composite air taxi and rental category is used by a lessee in return for payment to the owner; and the composite aerial application and industrial/special category represents specialized applications such as crop dusting, survey work, advertising, etc. Since business users are more likely than the other users*to fly in congested airspace (e.g. around large cities), where FAA avionics requirements are especially strmgent. the user group dummy variables are expected to have a negative influence on avionics equipage. For several reasons. aircraft manufactured prior to 1960 are expected to have lower avionics equipage rates than more recent vintage aircraft (the category held constant). Compared to the more recently built aircraft. pre-1960 vintage aircraft may be expected to have shorter trip lengths and generally lower utilization rates. The later is attributed to the deleterious effects of age/use and argues against avionics equipage. Further, the lower market value of such aircraft and their shorter expected useful life. relative to more recent vintage aircraft. argues against the installation of costly avionics. Thus the dummy variable (equal to one if the aircraft was manufactured prior to 1960) representing older aircraft is expected to have a negative parameter estimate. Itinerant hours is expected to vary directly with avionics equipage. This is because of the added pilot safety margin derives from avionics equipage, especially on trips where origin and destination differ. Two initial tests are used to determine whether the estimated discriminant function is a statistically significant discriminator between the avionics equipped and nonequipped groups. The computed chi-square statistic associated with Wilks’ lambda is used to test the null hypothesis that a statistically significant amount of discriminating information is not accounted for by the discriminant function. If the computed chi-square is greater than some critical value (e.g. x&95 or z;,~~. which are the critical values at the 0.05 and 0.01 significance levels. respectively), the null hypothesis is rejected at the corresponding level of significance. A further aid in judging the use-
Discriminant analysis of purchasers of aircraft avionics fulness of a discriminant function is its associated cannonical correlation-the square of which indicates the proportion of the variance in the discriminant function explained by the variables that define group membership. The above tests may be viewed as the preliminary screening tests to judge the effectiveness of the estimated discriminant function as a predictor of group membership (i.e. its predictive efficacy). The acid test of the effectiveness of the discriminant function is its ability to correctly identify/classify individuals not included in the analysis sample, given information on their discriminant variables only. This validation requirement is difficult to satisfy because it requires additional data. The results available from the analysis sample are usually presented in the form of a classification table, as shown in Fig. 1. N, is the total number of individuals actually in the ith group, Ck is the total number of individuals that were classified as being in the kth group. ni, is the number of individual of group i that were classified as group k. Then, Q/C& is the proportion of the predicted group i individuals that are correctly classified, and the sum of the main diagonal elements divided by Iv is the overall proportion of individuals correctly classified. While such results are obtained for the present study, it is noted that using the analysis sample to generate the discriminant function and then attempting to validate its predictive efficacy via a classification procedure based on the identical analysis sample is a biased test. Frank, Massy and Morrison (1965) point out that the primary cause of this bias is due to errors of sampling when estimating the means of the population, the basis for the discriminant COefficients. The direction of the bias is upward-the classification results show greater predictive power than actually exists among the true population. In addition, these authors note that bias may also be introduced when searching for variables that work best for a particular sample (which may have peculiar characteristics not exhibited by the population). To avoid the biases that can occur in classification tables constructed in this way. the present study uses only part of the available data. That is, only noncompany owners of GA aircraft are used to fit the discriminant function. then company owners of GA aircraft are classified into the equipped or nonequipped groups ulilizing the discriminant function generated from the analysis sample. Thus, the classification results (presented in Table 2) show what part of the observed proportion of correctly classified observations is due to true differences between the means of the discriminant variables in the avionics equipped and nonequipped groups. An alternative to this split sample approach is to use randomly generated data to form a synthetic validation sample. For purposes of the present study, and since original data was available for the validation sample. the split sample approach was preferred. The classification results obtained from the com-
27
pany owners are judged against a proportional chance criterion to determine if the function is an effective discriminator between the avionics equipped and nonequipped groups. If the latter is found true. the advantage of the company-noncompany owner dichotomy lies in the simplicity of application of one function in both of these avionics markets. The proportional chance criterion used is: P = 2
+ (1 -
2)’
(2)
where tl is the proportion of individuals in group 0 in group 1. Morrison and (1 - a) is the proportion (1969) points out that this proportional chance criterion is a more appropriate yardstick against which to judge the classification results than the pure chance criterion, because the former takes into account the fact that the discriminant function defies the odds by classifying individuals in the smaller group. The pure chance criterion would simply classify everyone into the group with the greatest membership. Since it is important for the FAA and manufacturers to identify correctly members of both groups, and since the group that is equipped with avionics is often the smaller of the two groups. P should be used. Thus. if the proportion of company owners (the validation sample) correctly classified [(nrr + nz,)!h’; as per Fig. l] is greater than P, the discriminant function (derived from the analysis sample) will be considered an effective discriminator between the avionics equipped and nonequipped groups. Finally, it should be noted that in deriving the classification results. a priori probabilities are not utilized. That is, individuals are assigned to the group for which they have the greatest probability of membership. For purposes of this study, the intent is to avoid incorporating a powerful set of prior probabilities which can lead to successful classification results even when the discriminant function is not performing effectively (Cooley and Lohnes. 1971). 3.
EMPIRICAL
RESULTS
Table 1 presents, the standardized discriminant function coefficients, the canonical correlation. and the computed chi-square associated with Wilks’ lambda for the analysis sample for each type of avionics. Without exception. the signs of the coefficients are as hypothesized in Section 3. The magnitude of the standardized coefficients of the discriminant variables suggest that for almost all types of
PREDICTED Group 0
Group I
Group
0
‘111
,112
Group
1
n21
1122
co
C,
.\ 0
ACTUAL
H4
Fig. 1.
.Y ,
.2
0.49 I 135.9”
648.0”
- 0.629 - 0.030
-G93
0.453 -0.439 0.179
0.42
-0.093
0.635 -0.342 0.264 -0.115 -0.338
OMNIdirectional receiver (N = 4153)
I. Standardized
.
function
1030.1”
0.47
-0.191 -0.081
0.690 -0.206 0.203 -0.193 -0.368
Distance measuring equipment (N = 4145)
discriminant
1342.6”
0.53
- 0.270 - 0.054
0.672 -0.241 0.266 -0.145 -0.272
owners)
180.6”
0.21
-
0.881 -0.175 0.191 - 0.096
Radar (N = 4146)
Variables
(Noncompany
Dependent Automatic direction finder (N = 4151)
coefficients
80.97”
0.14
-0.400 - 0.265
-0.359 -0.318
0.496 - 0.565
Area navigation equipment (N = 4144)
1277.1”
0.55
-0.220 -0.043
0.513 -0.456 0.325 - 0.073 -0.236
Transponder (N = 3596)
_
._
_
_.
Significance Icvels: a = 0.99 ~ Means that the variable did not enter because it did not add a statistically significant amount of centroid separation beyond that achieved by the previously (i.e. partial multivariate F ratio less than 1.0 was selected so that any variable with discriminatory power is chosen and retained for the analysis.) IV = Number of observations for the relevant avionics category
Canonical correlation Chi-square on Wilks’ lambda
Average cruising speed Pre-1960 vintage Itinerant hours Instructional use Personal use Aerial application & industrial/special Air taxi and Rental
Discriminant variables
Instrument landing system (N = 3351)
Table
entered
variables
844.9
0.54
-0.153 _
0.516 - 0.297 0.434 -0.155 -0.300
Altitude encoding transponder (N = 2636)
landing
nnvigution
with altitude
equipment
Total
Total
Total
Total
Total
Total
Total
Total
107 464 II73
5.53 1363 1916
2190 361 2551
1966 244 2210
811 1331 2208
1401 809 2210
26-l 1943 2210
735 1215 I950
*“Total” value is C max = max (x. 1 - x), where x is the proportion **For t value computation see eqn (3). ***For proportional chance computation see eqn (2).
Nonequipped Equipped
Transponder
Nonequipped Equipped
Transponder
Nonequipped Equipped
Aera
Nonequipped Equipped
Radar
encoder
equipment
receiver
system
Autontatic direction finder Nonequipped Equipped
Distance measuring Nonequipped Equipped
Nonequipped Equipped
OMNI-direction
Nonequipped Equipped
Instrument
Actual number
Company
or individuals
60.3 39.7 60.3
28.9 71.1 71.1
85.8 14.2 85.8
89.0
11.0
89.0
39.1 60.3 60.3
63.4 36.6 63.4
12.1 87.9 87.9
37.1 62.3 62.3
= 18.77)
in the nonequipped
group.
94.5 68.8 80.6 (t = 19.54)
82.7 70.2 80.0(t
93.8 18.7 46.2
99.3 27.9 72.1
54.8 64.0 62.2 (t = 9.50)
90.6 59.8 74.0(1 = 19.23)
52.1 95.7 88.6 (C = 11.37)
69.8 16.2 74.3 (C = 18.85)
owners Predicted correctly classified (%)**
results by type OTavionics
Pure chance criterion (%)*
Table 2. Classification
52.1
58.9
75.6
80.4
52.1
53.6
78.1
53.0
Proportional chance criterion (“A)***
2280 356 2636
1851 1745 3596
3802 342 4144
4086 60 4146
2451 1700 4151
3546 599 4145
726 3427 4153
1985 1366 3351
Actual number
Noncompany
95.6 46.9 85.2
78.8 77.9 78.3
94.6 11.7 57.0
99.3 5.7 85.2
80.3 73.5 77.6
95.0 40. I 80.2
43.8 94.4 78.7
75.8 65.8 71.8
owners Predicted correctly classified (70
30
STEPHEN G.VAHOVICH
avionics, type of aircraft (as represented by average crulsmg speed) is by far the most important factor in predicting avionics equipage. Age of aircraft is, in general, also relatively powerful in discriminating between the equipped and nonequipped groups. Itinerant hours flown is relatively unimportant except for the radar and altitude encoding transponder equipage decisions. Since among avionics devices, the latter two are particularly useful at higher altitude flying, generally associated with longer distance itinerant Rights. these results are not surprising. The relative contribution of the four type of use dummy variables in discriminating between the equipped and nonequipped groups varies widely with the types of avionics. The canonical correlation and the chisquare values suggest that each of the discriminant functions is a statistically significant discriminator between the groups. The chi-square value for each equation is greater than its corresponding critical value at the 0.01 level of significance. Thus the null hypothesis (that a statistically significant amount of discriminating information is not accounted for by the current function) is rejected. Given the cross-sectional nature of the data and the high degree of variability usually associated with this type of data, the canonical correlations are respectable. Further, excess emphasis should not be placed on their values; the true test of predictive efficacy is presented in Table 2. Table 2 presents results which may be used to test the effectiveness of the discriminant function as a predictor of group membership. This table presents the classification resuits for the validation sample (company owners) and the analysis sample (noncompany owners) for each type of avionics by group category. Column 1 shows the actual number of company owners, Column 2 gives the pure chance criterion, Column 3 shows the per cent of predicted that are correctly classified and Column 4 gives the computed proportional chance criterion [see eqn (2)]. The information to the right of the double line in Table 2 presents the classification results for noncompany owners. Since the discriminant function was generated from data on these individuals, Columns 5 and 6 are not considered in the validation test of the discriminant function. The results presented in Table 2 are very encouraging. The overall percentage of correctly classified (“total” rows in Column 3) range from 46.2% for area navigation equipment to 88.6% for OMNI-directional receiver. Except for radar and area navigation equipment, the per cent of predicted correctly classified exceeds both the proportional chance criterion (Column 4) and the pure chance criterion (Column 2) for each type of avionics. The minimum difference between the per cent correctly classified and their -orresponding proportional chance values is 109; and the maximum is 29%. To test further the hypothesis that the proportion of the correctly classified cases is significantly different from the proportion that would be expected if the proportional chance
criterion were used. the follofiing performed (results are presented Table 2. Column 3):
t=(Q-
computation was in parenthesis m
P) ___ 'VF '
(3)
where Q is the proportion of the validation sample observations correctly classified by the discriminant analysis, P is the proportion expected using the proportional chance criterion, and :V is the number of observations in the validation sample. This test is similar to that suggested by Frank. Massy and Morrison (1965). except that they use the pure chance criterion to define P. Comparing the computed t values against their corresponding critical values for each type of avionics suggests that, with the possible exception of radar and area navigation equipment, the proportion of correctly classified cases is significantly greater at the 0.01 level than that expected using the proportional chance criteria. In fact, if eqn (3) computations were performed using the more stringent pure chance criterion to define P,an identical statement is true for all types of avionics except the automatic direction finder (which is significant at the 0.10 level). These results are impressive and suggest that most of the discriminant functions given in Table I could be used effectively to identify individuals who would be most apt to purchase a particular type of avionics (i.e. to establish market profiles) and, more generally. to estimate increases or decreases in the derived demand on FAA ground facility services and equipment. Further, it may be noted (Table 2) that, without exception, the per cent of predicted correctly classified for each of the individual groups (nonequipped and equipped), within each avionics category, exceeds their corresponding pure chance expectations. Even for radar and area navigation equipment, the per cent correctly classified. when the equipped and nonequipped groups are taken se5arately, is greater than that expected under pure chance. Thus, the results for these two functions are not as bleak as first suggested.
4. CONCLUSlONS The findings presented in this paper suggest that the type of use. aircraft age, type of aircraft, and intensity of use can effectively discriminate between owners whose aircraft are equipped with various types of communication and navigation instrumentation and those not so equipped. The discriminant functions presented in this paper, validated by using additional sample data, appear to classify correctly a larger proportion of the aircraft owners (at the 0.01 level of statistical significance) than could be expected using either the pure chance or the proportional chance criterion. Thus. these discriminant functions could be used by the avionics manufacturing industry to establish market profiles and by the FAA in estimating the
Discriminant expected
demand
on their
ground
facility
analysis services
of purchasers and
Frank
of aircraft
avionics
31
R. E.. Mass\ W F. and Morrison D. G. (1965) Bias discriminant analysis. J. Marketiq Rrs. 15G-
m multiple 258.
equipment.
REFEREYCES Coole) W. and Lolmes P. R. (197 I ) Multituriate Atlal~ws. Wiley. New York.
Data
Federal Aviation Administration. Office of Aviation Policy (1978) Aaiattori Forrcasts Fiscal Ycwrs IYW IYYO. National Technical Information Service. Springfield. Virginia. Federal Aviation Administration, Office of Aviation Policy (1977) T/w Intpact qf Microcotnputrrs ON Aciatioti. National Technical Information Servtce. Springfield. Virginia.
Morrison D. G. (1969) On the Interpretation of discriminant analysis. J. hfarkctirtg Rcs. 6, I5fS-lh.3. Terry .I (1977) Microprocessors are key to new digital COM,‘NAV ‘IDENT designs. ICAO f3ttlletir1. 19-X. L’.S.. Code of Fedrral Rcgu/ariou Title 14-Aeronautics and Space. Part 9 1.33 (I 977). Office of the Federal Register. National Archives and Records Service. Washmgton. D.C. Vahovich S. G. (1977) Grrierol Ariatiort: Arrcr~~ff Owwr & L’rilicariori Characteristics. National Technical Information Service. Springfield. Virginia. Vahovich S. G. (i978) Income&d cost impact on general aviation hours flown by individual owners. Transprl Res. 12.315319.