A DECISION FRAMEWORK FOR DETERMINING
THE LEVEL OF MONETARY INCENTIVE IN MAIL SURVEYSDavid A. Gautschi
andDouglas L. MacLachlan

June 1995

Both authors are professors, Department of Marketing and International Business at the
University of Washington, School of Business, Box 353200, Seattle, WA 98195-3200 U.S.A. Email: gautschi@u.washington.edu and macl@u.washington.edu.

INTRODUCTION
Widely used in marketing research, mail surveys appeal to marketing researchers for anumber of reasons: they are low in cost, geographically flexible, and able to reach a widelydispersed sample simultaneously (Kanuk and Berenson, 1975). However, mail surveys sufferfrom a number of disadvantages, principal among these is the problem of low response rates.
Acknowledging this problem of mail surveys, marketing researchers have invented a number ofmeasures to increase response rates (Heberlein and Baumgartner, 1978). One ploy which has
been used with some success is to include or promise a monetary reward for completing andreturning the questionnaire (Wotruba, 1966; Armstrong, 1975; Cox, 1976; Gunn and Rhodes,
1981; Schewe and Cournoyer, 1976).
In this paper we investigate the relationship between the size of monetary inducement
and the probability of response. One should expect the response probability to be a
monotonically increasing function of the level of monetary inducement. In this context, the basic
question of interest to the manager is how much of monetary inducement is needed to generate a
response of a desired rate (or amount). The trade-off which the manager faces typically is either
to vary the monetary incentive to achieve some target rate of response or to vary the size of thesample to assure a responding sample of desired size. Both of these options involve costs.
Obviously, the cost of administering each questionnaire rises directly with the value of themonetary inducement. Similarly, as the sample size increases, additional costs of administration
are incurred.
In the next section of the paper, we discuss the principal features of the response model
called NORMIT. We then discuss an empirical application illustrating the calibration of the
response model. Because the response model is similar to the well-known quantal responsemodels, logit and probit, we compare the results of normit, probit and logit analyses. In thesubsequent section of the paper, we consider the issue of integrating the response model into apractical methodology for the research manager.
1

THE RESPONSE MODEL: NORMIT
During the fifties, Berkson conducted pioneering research in discrete probability models
and their estimation. Although his area of application was biometry and medical statistics, his
contributions (in particular, logit analysis) have been adapted to problems in other fields such as
economics, psychology, and marketing. We describe here one of the discrete probability (quantal
response) models which is attributed to Berkson (1955, 1957), called normit analysis. The normit
model has two principal attractions:1.

Its estimator, which is asymptotically equivalent to the maximum likelihood
estimator, is in the class R.B.A.N. (Regular Best Asymptotically Normal)(Neyman, 1949) with a variance less than that of the maximum likelihood
estimator in most cases, and2.

the estimation of the model is extremely simple, not requiring a computer.

MODEL
Let Pi denote the probability of response to some stimulus x at level i. The response
function for Pi is given as:
Pi = (equation needs to be inserted)(1)

This is the well-known normal frequency function describing an ogive. The normit of Pi
is derived from(2)(xi - μ) /σ

by defining
α = -μ/σ and β = 1/σso that the normit of Pi is(3)

Yi = α + βxi
The observed relative frequency (maximum likelihood of estimate of Pi) corresponding to

xi is denoted fi, and the observed normit of fi is denoted Yi. The estimation method is based on2the realization that Yi plotted against xi should fall along the straight line defined by (3). The
estimates of α and β are obtained by minimizing(4)

M = (equation needs to be inserted)

Where Yi = α + βxi is the estimate of (3), ni is the number of individuals exposed to xi, and(5)wi = (equation needs to be inserted)and summation is over i.
Berkson (1954) has shown that M is asymptotically distributed as X2. The estimates of αand β are obtained by a weighted least squares on (3) where α and β are given directly as
follows:
ΣniwiYixi2 - ΣniwiYi Σniwixi / Σniwi
Σniwixi2 - (Σniwixi)2 / Σniwi

(6)

β=

(7)

a = (equation needs to be inserted)

The estimate of the inflection point of the ogive (where E(Pi) = .5) is given as:(8)x50 = -α/βFormulae for standard errors of the estimates are given in the Technical Appendix. For

values of ni > 2, Tables are provided in Berkson (1957) for calculating wi and wi Yi.
In the next section we describe an empirical application, comparing the normit modelwith the probit and logit models.
AN EMPIRICAL EXAMPLE
DATA
To evaluate the proposed response model, we developed a questionnaire pertaining tofood shopping behavior of French consumers. Wording of the questionnaire and assignment oftreatment levels were refined by means of a pilot test conducted on ten French housewives. The
3

treatment of interest for this study was level of monetary inducement. Ten treatment levels were
established on an approximately logarithmic scale as follows:
0 francs.10 franc.20 franc.50 franc
l franc

2 francs
5 francs
10 francs
20 francs
50 francs

In the interest of maintaining the anonymity of respondents, we decided to affix the monetary
incentive to the covering letter enclosed with the questionnaire rather than to promise a monetary
reward for returning the questionnaire. The questionnaire was six pages in length, asking anassortment of background questions and involving a sorting task associated with a conjoint
analysis exercise. Fifty respondents were included in each treatment cell so that the total sample
size was 500.
As it is illegal to send coin and currency through the French postal system, we were
obliged to deliver the questionnaires to households. We selected the Fontainebleau-Avon area, acommunity of approximately 50,000 residents sixty kilometers south of Paris.
The 500 questionnaires were distributed according to a systematic random sampling
scheme in which all treatments were randomized to avoid any geographic, income or otherbiases. The questionnaires were placed in mailboxes during a single day and addresses werenoted for eventual follow-up mailings. After two weeks time (the exact date being designatedbeforehand) returns within each treatment were tabulated as shown in Table 1. A follow-up
postcard was mailed, but questionnaires returned later are not considered further in this paper.
A question of some interest to us was whether inclusion of a very small monetary
incentive would actually be worse than no incentive. Ten centimes (.10 FF) is worth very little(approximately 1.5 cents in 1989 U.S. terms). However, we found no evidence that people were
insulted or "turned-off" by the small values. The hypothesized monotonicity of responses was4observed over practically the entire range of results. Therefore, our quantal response modelingapproach was deemed appropriate.

5TABLE 1
Responses by Treatment Group

Treatment
Group

Treatment
Group

Cell
Size

Responses

1

.00

50

8

.16

2

.10

50

9

.18

3

.20

50

11

.22

4

.50

50

12

.24

5

1.00

50

16

.32

6

2.00

50

15

.30

7

5.00

50

20

.40

8

10.00

50

21

.42

9

20.00

50

25

.50

10

50.00

50

31

.62

500

168

33.6

Totals6

Observed Response
Frequency

TABLE 2
Estimates Obtained from Alternative
Response Models

Estimatedparameters*
α

β

x50

NormitProbit

Logit

-.491

-.493

-.816

(-8.07)(-8.05)(-7.82)

.157

.158

.167

( 6.29)( 6.23)( 6.08)

23.00

22.59

21.19

*The t-statistics are in parentheses.

7TABLE 3
Predictive Ability of Alternative
Response Models
Stimulus
Value(Francs)

Observed
Relative
Frequency

Predicted Response Probability
NormitProbit

Logit

.00

.16

.113

.112

.115

.10

.18

.197

.196

.193

.20

.22

.229

.226

.224

.50

.24

.274

.274

.269

1.00

.32

.312

.311

.307

2.00

.30

.351

.351

.347

5.00

.40

.406

.405

404

10.00

.42

.448

.448

.449

20.00

.50

.491

.492

.496

50.00

.62

.549

549

557

.0281

.0276

.0251

M.A.D.**Mean Absolute Deviation between observed relative frequencies and predicted response
probabilities.

8

ANALYSISThe inducement levels for the experiment were selected purposely so as to be spaced
approximately according to a logarithmic scale, because of the Weber-Fechner Law of justnoticeable differences.l The latter suggests that responses to sensory stimuli are proportional to
the logarithms of the stimuli. Therefore, our estimations of the response functions employed thenatural logs of the inducement levels.2 We estimated the functions in original inducement level
units as well, but obtained considerably poorer fit and we decided to omit them from the paper in
the interest of space.3
Values of wi and wiYi corresponding to the treatment responses were obtained fromBerkson's (1957) tables and inserted in equations (6), (7), and (8) to yield the normit function
estimates shown in the first column of Table 2. (Note that the ni dropped out of the equations,because they are equal for all treatments.) For comparison purposes we include also in Table 2
the results of probit and logit estimations obtained with an available computer program( A Q D ). All coefficients are highly statistically significant. Very similar coefficient estimates
were obtained using all three quantal response models, except for a slightly steeper slope andlower intercept for the logit model. The inflection point of the function occurs at x50. Each of themodels estimated this value to be slightly higher than 20FF.
Table 3 shows the relative predictive accuracy of the three estimated models. By thiscriterion as well, there is very little difference in the models, although the mean absolutedeviation measure is slightly better for the logit model.
One can never guarantee that the "right" model of any particular phenomenon is selected.
There is no theoretical rationale, for example, for preferring normit over one of the others. We
introduced normit because it is not known in the marketing literature and because of theadvantages mentioned earlier. Since each model yields approximately the same predictions forresponse probability, the normit approach can be recommended if one has no access to acomputer or to the probit/logit programs.

9In order to employ the decision models to follow, the research manager must select one
response model that allows prediction of response rate as a function of incentive level. For theabove reasons, we will use the normit function in the ensuing discussion.
DECISION MODELS
Once a response function has been estimated, various approaches can be suggested for
deciding the level of inducement to use in a particular study. Two basic approaches are
diagrammed in Figure 1. It is understood that prior to the inducement level decision various steps
have been taken such as determination of the target population, development of questionnaire,and pilot test of the questionnaire with experimentally manipulated incentive levels such as wasdone in this paper. Response to the inducement levels will depend, of course, on the nature of thetarget population and the questionnaire itself. The decision models are described in greater detailand illustrated with numerical examples in the following.
Model 1 - Sampling Budget Fixed
Given a sampling budget B (which excludes fixed costs not depending on sample size orlevel of incentive) and costs per mailed questionnaire c (including, for example, printing, twoway postage, and labor for stuffing and sealing envelopes), it is desired to obtain the optimallevel of incentive x* which yields the largest expected return sample size nr. That is, we want toselect the x that will maximize nr subject to B and c being given constants. We know bydefinition that:(9)nr = n p

where p is the response probability for a given x.
Also, it is clear that(10)

B = n (c + x)

combining (9) and (10) we have(11)

nr = pB/(c + x)
10

FIGURE 1(David and Mac, Figure 1 needs some work. I would be happy to do it if you like. Eva)
Two Approaches to Use of Response del
1. Sampling Budget Fixed
Target Population
Questionnaire Design
Pilot Test
Response Model Estimation

Sampling Budget

Optimal Incentive
Decision Model

Cost/questionnaire

Number to be Mailed
Expected Sample Size
ContinueResearch
Process

2. Minimum Sample Fixed
Target Population
Questionnaire Design
Pilot Test
Response Model Estimation

Desired Sample Size

Optimal Incentive
DecisionModel

Cost/questionnaire

Number to be Mailed11Budget Required
ContinueResearch
Process12

By the earlier discussion, p is a function of x, the latter appearing in the upper limit of theintegral of a density function. Standard calculus methods for determining the optimum are
intractable, given the complex nature of the objective function. Fortunately, however, there are
only a finite number of incentive levels which need to be explored for optimality (it would besilly, for example, to include an incentive with the questionnaire of 37FF or any other odd-sizedamount). Therefore, a simple numerical solution is possible in a given practical situation.
As an example, suppose we are given a sampling budget of 10,000FF (French francs) andknow the per questionnaire costs are 20FF. For each level of incentive under consideration wecan compute n = 100000/ (20 + x) and an expected response probability (from our selected
response model). The calculations for a range of incentive levels are shown in Table 4 (using thenormit function estimated earlier). We see that the optimal incentive level is 5FF, since it yields
the highest expected returned sample size, 162. As secondary outputs, we expect this level of
incentive to require a mailed sample of 400 and to achieve a response rate of 40.6%.
Model 2 - Sample Size Fixed
Given a minimum returned sample size nr and sampling costs per questionnaire c, one's
task would be to determine the optimal incentive level x* which would yield the lowest budget
B. That is, we want to determine the x* that will minimize B subject to nr and c being given
constants.
We know, by combining (9) and (10) that(12)

B = (nr/p)(c + x)

Again, standard calculus methods are intractable, but we can easily obtain a simple numericalsolution in any particular case. For example, suppose we desire a returned sample of 350
questionnaires and have a cost per questionnaire of 15FF. Calculations for a range of incentive
levels are shown in Table 5. In this situation, the optimal incentive is 2FF, since it requires thelowest budget, 16,949FF. As secondary outputs, we determine that the mailed sample size shouldbe 997 and will achieve an anticipated response rate of 35.1%.
13

TABLE 4
Calculations for Obtaining Optimal Incentive Given
Budget 10,000FF and Sampling Costs 20FF
Incentive
Level

p

n

nr

.00

.113

500

57

.10

.197

497

98

.20

.229

495

113

.50

.274

488

134

1.00

.312

476

149

2.00

.351

455

160

5.00

.406

400

162

10.00

.448

333

149

20.00

.491

250

123

50.00

.549

143

79

14

TABLE 5
Calculations for Obtaining Optimal Incentive
Given Sample Size 350 and Sampling
Costs 15FF
Incentive
Level

p

n

B

.00

.113

3097

46,455

.10

.197

1777

26,833

.20

.229

1528

23,226

.50

.274

1277

19,794

1.00

.312

1122

17,952

2.00

.351

997

16,949

5.00

.406

862

17,240

10.00

.448

781

19,525

20.00

.491

713

24.955

50.00

.549

638

41,47015

DISCUSSION
Limitations. This study was performed on a specific target population, so the substantive
findings cannot be generalized beyond that population. However, the major purpose of the study
was to illustrate a proposed methodology which should be applicable to any study where an
S-shaped response to monetary incentive level can be safely assumed.
An important issue that was ignored in this study is the extent to which monetary
incentives can influence the nature of responses to mail surveys. That is, monetary incentives
have been postulated to cause volunteer biases (i.e., the tendency for the incentive to induce
some groups to respond over others) and response bias (i.e., the tendency for the incentive toaffect some questionnaire item responses). Both types of biases have been observed (Gelb, 1975;
Rush et al., 1978; Nederhof, 1983). As examples of response bias, a study of elites by C,odwin(1979) demonstrated that large monetary incentives substantially improved questionnairecompletion levels and caused respondents to give more correct information than when theyreceived lesser incentives.
Potential biases caused by different levels of incentive deserve further investigation. The
research manager will need to weigh the likelihood that such biases are significant against theimportance of achieving a large (or optimal) returned sample of questionnaires.
Some Statistical Issues. It turned out in this study that we only used one incentive level
above the estimated inflection point of the response function. Although it is never known until
after the data have been collected whether sufficiently high incentive levels have been included,
it is helpful to employ a large range of values so that one would avoid ever having to makeout-of-range predictions of response probabilities. This was not a problem for our examples,
which yielded optima well within the empirical range of incentives, but we could haveconstructed examples for which our probability estimates would be considerably less secure.
The quantal response functions have a theoretical range of from negative to positive
infinity. However, of course there is no way to give a negative monetary incentive (unless wewere to ask respondents to pay to respond). Furthermore, it is possible to postulate that there is16

an absolute maximum response probability below 1.0 (Sudman, 1982). By adding two moreparameters to the response functions, we could estimate both the probability of response at zeromonetary incentive and an upper limit on the response probability. However, the degrees of
freedom would be reduced, requiring a larger number of experimental treatments to achieve thesame level of statistical precision. We do not believe the additional information provided bythese parameters would justify the expense or the additional complexity of estimation.
Another way to extend the model would be to add other independent variables thansimply monetary incentive to the response functions. It is possible to experiment with a variety
of stimuli during the pretest. Although it would take some effort to extend the normit model toaccommodate multiple independent variables, the multivariate logit model would be directly
applicable.
CONCLUSION
Low response rates are the critical problem of mail surveys. It has been shown that
response rates are monotonically related to level of monetary incentive at least in someinstances. We have demonstrated in this paper how one can use that information to decide on thelevel of monetary incentive to employ in a particular survey. Although we determined that any ofseveral quantal response functions could adequately model the relationship under investigation,
we introduced the normit model to the marketing literature as a model with some attractive
features, perhaps the most important of which is estimation simplicity.
We believe we have described a practical procedure that might be employed in any mailsurvey setting. Of course the primary requirement is that one experimentally manipulate the levelof monetary incentive during a pretest of the survey. The cost and time required to do this mightcause the method to have its greatest appeal to those companies who conduct multiple surveys
using essentially the same questionnaire on similar populations over time. On the other hand, werecommend use of the method in any mail survey where the research manager desires anappropriate balance between response and survey budget.
17

FOOTNOTES

1.

Readers interested in the Weber-Fechner Law are referred to Mueller (1965) and Corso(1967). An important marketing reference is Monroe (1971).

2.

To avoid taking the logarithm of zero, the value 0.01 was substituted for zero francs inestimating the functions.

3.

Although it was not a purpose central to our research, the fact that a better fit wasachieved with the incentive level measured in log units provides a partial validation of
the Weber-Fechner Law for our data.

18

TECHNICAL APPENDIX

Standard errors for normit parameters*(Equation needs to be inserted)(1)(Equation needs to be inserted)(2)(Equation needs to be inserted)(3)

where s2 (Equation needs to be inserted)*Derived in Berkson (1955).

19

REFERENCES
Armstrong, J. Scott (1975), "Monetary Incentives in Mail Surveys," Public Opinion Quarterly,
39, 111-16.
Berkson, Joseph (1955), "Estimate of the Integrated Normal Curve by Minimum Normit
Chi-Square with Particular Reference to Bio-Assay," Journal of the American Statistical
Association, 50, 529-49.
(1957), "Tables for Use in Estimating the Normal Distribution Function by Normit Analysis,"
Biometrika, 44, 411-35.
Corso, John F. (1967), The Experimental Psychology of Sensorv Behavior New York: Holt,
Rinehart & Winston, Inc.
Cox, Eli (1976), "A Cost Benefit View of Prepaid Monetary Incentives in Mail Questionnaires,"
Public Opinion Quarterly, 40, 101-4.
Gelb, Betsy (1975), "Incentives to Increase Response: Social Class Considerations," Journal ofMarketing Research, 12 (February), 109-9.
Godwin, R. Kenneth (1979), "The Consequences of Large Monetary Incentives in Mail Surveys
of Elites," Public Opinion Quarterly, 43, 378-87.
Gunn, Walter J. and Isabelle N. Rhodes (1981), "Physician Response Rates to a Telephone
Survey: Effects of Monetary Incentive Level," Public Opinion Quarterly, 45, 109-15.
Heberlein, Thomas A. and Robert Baumgartner (1978), "Factors Affecting Response Rates to
Mailed Questionnaires: A Quantitative Analysis of the Published Literature," American
Sociological Review, 43, 447-62.
Kanuk, Leslie and Conrad Berenson (1975), "Mail Surveys and Response Rates: A Literature
Review," Journal of Marketing Research, 12 (November), 440-53.
Monroe, Kent B. (1971), "The Information Content of Prices: A Preliminary Model for
Estimating Buyer Response," Management Science, 17, (April), B-519-32.
Mueller, Conrad (1965), Sensory Psychology, Englewood Cliffs, New Jersey: Prentice-Hall, Inc.
Nederhof, Anton J. (1983), "The Effects of Material Incentives in Mail Surveys: Two Studies,"
Public Opinion Quarterly, 47, 103-11.
Neyman, Jerzy (1949), "Contribution to the Theory of the Chi-Square Test," in J. Neyman, ed.,Proceedings of the Berkeley Symposium on Mathematical Statistics and Probabilities,
Berkeley and Los Angeles: University of California Press, 23973.
20

Rush, M.C., J.S. Phillips, and P.E. Panek (1978), "Subject Recruitment Bias: The Paid Volunteer
Subject," Perceptual and Motor Skills, 47, 443-49.
Schewe, Charles D. and Norman G. Cournoyer (1976), "Prepaid vs. Promised Monetary
Incentives to Questionnaire Response: Further Evidence," Public Opinion Quarterly, 40,
105-7.
Sudman, Seymour (1982), "Estimating Response to Follow-Ups in Mail Surveys," Public
Opinion Quarterly, 46, 583-4.
Wotruba, Thomas R. (1966), "Monetary Inducements and Mail Questionnaire Response,"
Journal of Marketing Research, 3 (November), 398-400.

21