RESEARCH AND PRACTICE

Effective Segmentation of University Alumni: MiningContribution Data with Finite-Mixture Models

Elizabeth J. Durango-Cohen • Siva K. Balasubramanian

Received: 11 January 2013� Springer Science+Business Media New York 2014

Abstract Having an effective segmentation strategy is key to the viability of any orga-

nization. This is particularly true for colleges, universities, and other nonprofit organiza-

tions—who have seen sharp declines in private contributions, endowment income, and

government grants in the past few years, and face fierce competition for donor dollars

(Wall Str J p. R1, 2011). In this paper, we present a finite-mixture model framework to

segment the alumni population of a university in the Midwestern United States based on

the monetary value of annual contributions. A salient feature of the model is that it exploits

longitudinal data, i.e., contribution sequences. Another important feature of the model is

that it supports the identification of unobserved heterogeneities in the population’s giving

behavior. Our empirical study presents substantive insights gained through the processing

of the full contribution sequences, and establishes the presence of seven distinct segments

of alumni in the population. Results provide a basis to support the design of segment-

tailored solicitations, and guide the allocation of resources (e.g., telemarketing dollars) to

fundraising activities.

Keywords Alumni giving � Fundraising � Customer segmentation � Finite mixture

models � Data mining � Customer-based analysis

Introduction

Alumni contributions have become a major source of funding for American colleges and

universities. In 2011, 44 % ($13.45 billion) of funds raised by higher education

E. J. Durango-Cohen (&) � S. K. BalasubramanianStuart School of Business, Illinois Institute of Technology, 565 W. Adams Street, Chicago, IL 60661,USAe-mail: durango-cohen@iit.edu

S. K. Balasubramaniane-mail: sivakbalas@stuart.iit.edu

123

Res High EducDOI 10.1007/s11162-014-9339-6

institutions, to support academic and athletic programs, came from contributions by

individuals (Council for Aid to Education 2012). Increases in alumni contributions are in

response to, and a driver of, the considerable effort of fundraisers to identify and target

sub-groups with appropriately tailored fund-raising solicitations. As a consequence, the

success of fundraising strategies is predicated on effectively segmenting the alumni

population.

In this paper, we analyze the giving behavior of the alumni population at a private,

Ph.D.-granting university in the Midwestern United States. A multivariate normal mixture

model is used to segment individuals based on the monetary value of their annual con-

tributions. Finite mixture models provide a flexible framework that allows for the for-

mulation and estimation of descriptive and predictive models to segment alumni. In

general, the aim of descriptive models is to capture how individuals differ in their realized

(contribution) behavior (e.g., using K-means clustering), whereas predictive models

attempt to explain how different factors, such as demographics and affinity variables, drive

contribution behavior (e.g., via econometric/regression models). Our focus in this paper is

to develop a descriptive model of alumni giving that can be used by development offices to

support fundraising activities.

The work discussed herein is an important application of mixture models in the context

of fundrasing, particularly given universities’ increasing reliance on contributions for

operational support. It is noteworthy, however, that this framework can be applied to other

areas of higher education research, including for the analysis of student course taking

patterns and enrollment management. Lastly, the proposed segmentation model has

applicability to other settings, including those where purchase behaviors of individuals

generate large numbers of transactions, i.e., panel data sets with recorded longitudinal data,

and where it is of interest to focus on/allocate resources based on a small number of

segments instead of a large number of individuals, such as catalog mailing and customer

relationship management (Ha et al. 2002).

Data

Alumni records for 75,922 ‘active’ alumni, who donated to the university from fiscal year

2000 through 2010, were assigned to distinct classes based on the monetary value of

contributions. Active donors were defined as alumni who contributed at least $10, but no

more than $25,000 over any consecutive three years1. Our analysis is based on sequences

of annual contributions, meaning that for each individual in the data set, we summed the

monetary value of all transactions in a given year (a total of 342,847 transactions were

present in the original dataset), which, in turn, resulted in a total of 282,888 aggregate

annual receipts.

Table 1 shows data summary statistics by fiscal year, including the number of active

alumni, mean and standard deviation of the annual contributions. Figure 1 shows the

distribution of annual contributions for fiscal year 2000 (other years have similar distri-

butions). We observe that the distribution of donations is skewed toward smaller values;

1 Alumni who contribute at higher levels are not targeted via direct-mailings or phon-a-thons by thedevelopment office. Major donors and major donor prospects are generally engaged via face-to-facemeetings.

Res High Educ

123

that is, few alumni donors contributed large sums of money, while the majority contributed

smaller amounts—this is also characteristic of other data sets in the literature.

Because the university in our study kept little information about how solicitations were

targeted to the alumni population, we are unable to discern whether the individuals rep-

resented in our database were recipients of different solicitation strategies. In general, most

alumni for which the university had a valid address, or active phone number, were con-

tacted annually. From our perspective, a more important piece of information would be to

know the ‘‘ask’’ amounts in the solicitations, given that we are segmenting alumni based on

the monetary value of their contributions. This, however, was never stored in the database.

The university also provided additional information on the alumni, including: age, class

year(s), degree(s) obtained from the university, major(s), marital status, and several (yes/

no) indicator variables on whether the spouse is also an alumnus/a, a child is also an

alumnus/a, the home phone is listed in the database, Greek affiliation, participation in the

alumni travel program, and participation in any alumni reunion event.

Table 1 Data summary

Fiscal year No. active donors Annual contribution

Mean SD

2000 22,550 $270 $740

2001 23,536 $278 $729

2002 21,952 $264 $668

2003 23,059 $251 $622

2004 22,860 $272 $719

2005 26,026 $262 $654

2006 26,655 $281 $685

2007 27,567 $311 $785

2008 29,356 $325 $821

2009 28,928 $305 $800

2010 30,399 $340 $912

Fig. 1 Distribution of alumnicontributions in fiscal year 2000

Res High Educ

123

Related Literature

In this section, we contrast the approach implied by the proposed segmentation model to

others appearing in the fundraising and segmentation literatures. For broader reviews of

fundraising and segmentation studies readers are referred to Bekkers and Wiepking (2011a,

b) and Wedel and Kamakura (2000).

Based on the taxonomy detailed in Wedel and Kamakura (2000), we characterize

models of alumni giving, most of which appear in economics, education, sociology, and

psychology journals, by whether they utilize a priori or post-hoc segmentation methods. In

a priori segmentation, the number and types of segments are determined in advance by the

researcher, whereas in post-hoc segmentation the type and number of segments is deter-

mined as a result of the data analysis (e.g., K-Means Clustering). The conditions used to

assign individuals to segments are referred to as segmentation bases. Segmentation bases

are classified as either observed, i.e., relying on observed/measured trait such as demo-

graphic or socioeconomic characteristics; response or institutional variables such as size,

private versus public institutions, or unobserved, frequently attributed to latent (psycho-

graphic) variables.2 Finally, with respect to the capability to explain responses, models are

classified as predictive or descriptive. Predictive models relate explanatory variables to the

outcomes of a set of dependent variables. In contrast, descriptive models represent the

(joint) distribution of the variables without distinction between outcome and explanatory

variables (Green 1977). Following this taxonomy, the proposed model can be categorized

as a post-hoc segmentation procedure where the underlying statistical models are

descriptive, and where the segmentation basis is unobserved, which is consistent with the

assumption that cross-sectional heterogeneity can be, at least partly, attributed to unob-

served, and potentially unobservable factors.

Most papers in this literature rely on a priori techniques (e.g., cross-tabulation, dis-

criminant analysis), and rely on observed bases comprised of variables related to demo-

graphic (e.g., donors versus nondonors, Sun et al. 2007), donor patronage (e.g., major

contributors vs. regular donors, Lindahl and Winship 1992), and/or contribution frequency

(e.g., consistent donors versus occasional donors, Wunnava and Lauze 2001) and estimate

predictive models of giving. In most studies, predictive statistical models are estimated for

each segment to explain the segment’s underlying behavior. A common specification is

often used for all segments to make qualitative judgments about the significance of dif-

ferent variables in explaining the behavior of the different segments. Baade and Sundberg

(1996) is an example of a study where individuals in a large, multi-institution database of

alumni donors are classified based on the type of institution they attended for their

undergraduate studies: public university, private university, or liberal arts college. The

authors then estimate differential effects of variables, including gender and age, on the

monetary value of annual contributions for the three segments. In addition, some papers

employ econometric models to study a particular subset (one segment) of the alumni

population, such as alumni who were financial aid recipients and graduated between 1988

and 1990 (Marr et al. 2005), alumni from a Carnegie-classified anonymous research

institution (Okunade et al. 1994), or graduate degreed alumni (Okunade 1996).

Papers generally assume that differences in donation behavior within each segment can

be explained by personal characteristics (marital status, gender, age, race), socio-economic

variables (income and education, past giving, sector of employment, type of financial aid

received), behavioral factors (membership in Greek fraternities, volunteering for the

2 We note that segmentation bases can be unobserved due to missing data.

Res High Educ

123

college, membership in alumni chapters, reunion attendance), and institutional character-

istics such as size, type of institution, or endowment value (Ehrenberg and Smith 2003).

Lindahl and Winship (1992) is a seminal example in the context of university fundraising.

They present logit models for both major gifts and annual fund prospects, using both trait/

socio-economic and response/contribution data. In their analysis, past giving turns out to

be the strongest single factor in predicting future giving in both segments, but other factors

are also statistically significant. Other papers in this literature aim to learn about what

motivates alumni in different segments to contribute. Reasons commonly cited are:

awareness financial support need (Pearson 1999; Taylor and Martin 1995; Weerts and

Ronca 2008), satisfaction with undergraduate experience (Clotfelter 2001, 2003; Gaier

2005; Monks 2003), institutional academic/athletic reputation (Holmes 2009; Rhoads and

Gerking 2000; Tucker 2004), tax deductions for charitable contributions (Beranek et al.

2010; Feldstein 1975; Holmes 2009), availability of personal resources (income and/or

wealth) (Bristol 1990; Bruggink and Siddiqui 1995; Olsen et al. 1989), influence of

reunions (Grant and Lindauer 1986), altruism (Andreoni 1989; Becker 1974; Kim et al.

2011; Meer and Rosen 2009a), and prestige/recognition (Yoo and Harrison 1989).

The mixture model presented herein differs from the models in the aforementioned

studies in that these models assume that sources of heterogeneity in the population are

observed, i.e., can be explained by observed traits. As such, they do not account for

unobserved heterogeneity; we refer to unobserved heterogeneity as systematic, but

unobserved differences between individuals. These differences are unobserved either

because they are not measurable/observable, e.g., personality traits that might explain an

individual’s propensity to donate in response to solicitations, or because it is too costly to

collect the data, or because data are (inadvertently) omitted. One of the consequences of

unobserved heterogeneity is that individuals sharing the similar traits, do not respond in a

homogeneous fashion as expected and, in turn, this can lead to the implementation of

inappropriate or suboptimal marketing strategies (Wedel et al. 1999). To adjust for

unobserved individual heterogeneity that might be correlated with giving behavior, some

models include a random effects specification (for example, Holmes 2009). It is important

to note that instead of explaining heterogeneity at the individual level as is done in fixed or

random effects specifications of predictive models (cf. Clotfelter 2003; Meer and Rosen

2009b; Gottfried 2010; Meer 2011), in the proposed model heterogeneity is captured at the

segment level. We present evidence of unobserved heterogeneity in the data set used in the

empirical study, and discuss how this information can be used by fundraisers to discern

patterns in sub-populations that can guide fundraising actions. For example, the ability to

identify alumni in different age groups that contribute to high value segments gives fun-

draisers the ability to focus development efforts (e.g., by developing targeted call lists).

Post-hoc segmentation procedures are less common in the alumni-giving literature.

Weerts and Ronca (2009), for example, employ the classification and regression tree

(CART) methodology to distinguish between donors and non-donors, and then to predict

characteristics tied to alumni giving. As described in Wedel and Kamakura (2000), CART

and other tree search procedures are used to set thresholds that are used to assign indi-

viduals to segments (when there are multiple dependent variables of interest). Blanc and

Rucks (2009), on the other hand, utilize cluster analysis to segment an alumni population.

An algorithm based on the nearest centroid sorting method, where an alumnus is assigned

to the cluster for which the distance between the alumnus’ average gift and the center of

the cluster is smallest, is used. We add to this literature by formulating segmentation

models under the assumption that the traits that cause heterogeneity in the data are not

Res High Educ

123

known in advance. Thus, segments and individual alumni membership must be inferred

post-hoc from the data—the contribution sequences associated with each individual.

Segmentation and Finite Mixture Models

Prior research suggests that market segmentation has a positive impact on business per-

formance outcomes (Peterson 1991; Verhoef et al. 2002). Simply stated, market seg-

mentation involves the decomposition of a large heterogeneous market into a number of

segments where elements are similar within a given segment and dissimilar across seg-

ments. A number of methods are available to implement segmentation, ranging from

traditional approaches such as cluster analysis and discriminant analysis, to applications

that involve optimization, finite mixtures, or Bayesian methods using effective segmen-

tation criteria (Kim et al. 2012).

This study employs finite-mixture modeling as a framework to analyze alumni contri-

butions. Within the social, behavioral, and health sciences, this approach is characterized

as latent class analysis (LCA). Our model follows in a long line of works estimating finite

mixture models (or latent class models) using the EM Algorithm. Examples of works in

this vein can be found in McLachlan and Peel (2000) (and the references therein), Collins

and Lanza (2010), and McLachlan and Thriyambakam (2001), dating back to the seminal

works of Goodman (1974a, b) and Dempster et al. (1977).

Finite-mixture models allow for the probabilistic description of sub-populations within

an overall population, and are used to make statistical inferences about the sub-popula-

tions. The framework’s capability to process contribution sequences, i.e., longitudinal data,

provides fundamental new insights into the dynamics of donor contribution behavior. The

methodology also provides a rigorous mechanism to infer and segment the population

based on unobserved heterogeneities (as well as based on other observed characteristics).

While there is ample empirical evidence that such differences can be (statistically) sig-

nificant, existing models to support direct marketing activities are only capable of

explaining these differences as random variations within the population.

More specifically, the underlying multivariate normal mixture model framework used in

this study assumes that the population is comprised of a finite set of (latent) classes in

unknown proportions, where each segment is characterized by a continuous probability

distribution function that describes the annual monetary value of individual contributions.

The assumption of a population comprised of latent classes, and the development of a post-

hoc segmentation methodology is appealing to describe populations with unobserved

heterogeneity. Under prevailing approaches, segmentation bases rely on observed variables

and subjective judgements, e.g., monetary value thresholds to identify major donors. While

intuitive and simple, it is clear that such approaches can introduce bias, particularly

in situations where the cross-sectional heterogeneity is caused/influenced by unobserved,

possibly unobservable, factors.

As stated earlier, finite mixture models provide an exceedingly flexible modeling

framework; one that allows for the formulation of different families of segmentation

models. As examples, Durango-Cohen et al. (2013a) and Durango-Cohen et al. (2013b)

model different aspects of contribution-behavior heterogeneity. Durango-Cohen et al.

(2013a) analyzes Markov mixture models to segment alumni based on contribution-

behavior patterns. Using a dataset from a different university, the authors identify the

presence of three distinct classes of donors in the population: a class of ‘‘loyal alumni’’

who tend to contribute at the same level from year-to-year; a class of ‘‘transient’’ alumni

Res High Educ

123

who donate once or twice, and then stop contributing; and a class of ‘‘high-variance’’

alumni whose contributions shows significant fluctuations from year-to-year, possibly in

response to appeals from the university. In Durango-Cohen et al. (2013b), the authors

formulate a Bernoulli–Gaussian mixture model that jointly describes the likelihood and

monetary value of donations. The paper’s focus is on identifying segments that differ not

only in terms of contribution amounts, but in terms of ‘‘loyalty’’ or contribution frequency.

The work presented herein focuses instead on identifying donor potential, as well as on the

managerial implications tied to capturing variability within contribution sequences as a

way, for example, to devise solicitation ‘Ask Amounts.’

As with other post-hoc segmentation methods, the number of classes/segments in finite-

mixture models is established based on goodness-of-fit criteria. In our analysis, we con-

sidered segmenting the alumni population in up to eight classes. In the numerical study, we

present the estimated parameter values for each of the models considered, but focus the

discussion on the managerial insights resulting from the seven-segment mixture, which

provided the best fit-to-data based on the consistent Akaike information criterion (CAIC)

and Log-likelihood (LL) values.

In addition to providing estimates that describe the distribution of contribution amounts

in each of the segments, multivariate mixture models also generate estimates of posterior

probabilities of segment membership—the probability that an individual belongs to each of

the segments based on his/her sequence of contributions. The identification of alumni

belonging to each of the segments is of particular importance in this setting, as direct-

marketing activities are developed based on the given segments, but are deployed in the

form of mailings to individual alumni. Specifically, over the entire dataset, the proposed

approach provides a powerful tool to (i) aggregate alumni that belong to the same segment

(e.g., for mailing distributions), and (ii) guide the allocation of scarce resources (e.g.,

telemarketing volunteers) to target alumni in different sub-populations that belong to more

higher contribution segments.

Methodology

The notation and underlying assumptions needed to formulate finite-mixture models are as

follows: The sequence of contributions for member m is denoted by ym ¼fy1

m; y2m; . . .; yT

mg ¼ fytmg

Tt¼1, where yt

m is the contribution of member m in period (year) t,

m ¼ 1; . . .;M, t ¼ 1; . . .; T . M represents the total number of individuals in the population,

and T corresponds to the total number of periods in the sample. In terms of specifying the

contribution sequences (for the data used in the empirical analysis), no entries exist in the

periods preceding the first contribution, and thus the sequences can be unbalanced. We use

y to represent the set of sequences, i.e., y � fymgMm¼1. We assume the population is

comprised of S segments in proportions k1; k2; . . .; kS, corresponding to the fraction of the

population belonging to each of the segments. k represents the set of proportions, i.e.,

k ¼ fksgSs¼1. The mixture proportions correspond to the probability mass function

describing an individual’s segment membership. That is, ks corresponds to the (a priori)

probability that a randomly selected individual from the population belongs to segment s.

Mathematically, letting zms ¼ 1 if individual m belongs to segment s, and zms ¼ 0 other-

wise, and assuming every individual belongs to exactly one segment, i.e.,PS

s¼1 zms ¼1; �m ¼1; ; . . .; , we have Pðzms ¼ 1Þ ¼ ks; �s ¼1; ; . . .;S. Thus, ks� 0; �s ¼1;

. . .; S, andPS

s¼1 ks ¼ 1.

Res High Educ

123

We further assume that each segment is characterized by a stochastic model, fsðymjhsÞ,i.e., a probability mass/density function, representing the probability that an individual

belonging to s contributes sequence ym. hs represents a set of parameters that define the

function fsð�Þ, e.g., hs ¼ ðls; rsÞ for a Normal distribution. The total probability of

observing sequence ym is thus:

f ðymjhÞ ¼XS

s¼1

ksfs ymjhsð Þ ð1Þ

where h � fhsgSs¼1. The total probability is a weighted sum, i.e., a mixture, of the prob-

abilities associated with each segment. Equation (1) is referred to as a finite-mixture

model.

In addition to describing the distribution of sequences across the population, the

specification of a finite-mixture model provides a segmentation framework based on

updating each individual’s membership probabilities in response to her contribution

sequence. That is, given ym, Bayes Law can be applied to update the probability that

individual m belongs to segment s, pms, as follows:

pms � Pðzms ¼ 1jymÞ ¼ksfsðymjhsÞ

PSr¼1 krfrðymjhrÞ

ð2Þ

The segmentation is stochastic in that (2) yields conditional membership probabilities, as

opposed to deterministic assignments of individuals to segments.

For different number of classes/segments, s ¼ 1; . . .; S, each mixture model estimates

the proportion (or mixture) of alumni in each class (ks) and estimates a distinct mean (ls)

and standard deviation (rs) for each class/segment, under the assumption that contributions

follow a Normal distribution. In addition to Normal distribution, we also considered dif-

ferent families of distributions in our analysis, including the Exponential and Log-Normal

distributions. In spite of their appeal, because they are defined over the positive domain,

the fit-to-data of the respective mixture models was poor in our application, and so the

results are omitted.

A rigourous treatment of the formulation and estimation of (multivariate normal)

mixture models can be found in McLachlan and Peel (2000). As background, we present

the formulation of the maximum likelihood estimation (MLE) problem for multivariate

normal mixture models in the Appendix. A more general and descriptive presentation on

mixture models in market segmentation, including the maximum likelihood algorithms for

their estimation, is available in Wedel and Kamakura (2000).

Various commercial software programs are available to fit mixture models (SAS’s

ProcFMM, Stata’s fmm plug-in developed by Partha Deb, R’s mixtools package, among

others). While flexible in terms of distributional assumptions, these programs can only be

used to fit mixture models for pure cross sectional data (e.g., mixture regressions), or for

univariate data. Because our aim is to segment alumni based on their contribution

sequences, the aforementioned programs cannot be used. Accordingly, we have developed

a MatLab (MathWorks, Inc., 2013) implementation of EM-algorithm to estimate Normal

mixture models for longitudinal data (i.e., the individual contribution sequences). This

program is freely available from the authors upon request.

Res High Educ

123

Segmentation Results

One way to assessing the goodness-of-fit of different mixture models is to consider the

CAIC, a widely-used statistic that trades off goodness-of-fit and overfitting, and LL values.

We find that the CAIC improves at a decreasing rate, as a function of the number of

segments in the model (details appear in ‘‘Goodness-of-Fit Measures and Segment Esti-

mates’’ Appendix, along with the full set of parameter estimates for model with S ¼1; . . .; 8 segments3). The parameter estimates (in Table 2) suggest that improvements from

increasing the number of segments arise from fitting a smaller percentage of donors at the

higher contribution levels. For instance, as the number of segments increases from S ¼ 6 to

S ¼ 7, the algorithm refines the distribution of the population across the last three segments

(when S ¼ 6) into four segments with different segment-mean specifications and, perhaps

more appealingly, reduced segment-standard deviations. We also note that, like the CAIC,

the scaled LL values also improve at a decreasing rate with the number of segments in the

model. Based on the last two observations, we selected the seven-segment model as the

preferred segmentation model.

The majority of alumni in the seven-segment model are assigned to segments defined by

lower mean annual contribution values. Segments 1, 2 and 3 comprise approximately 80 % of

the population, and about 28 % of the funds raised. Alumni assigned to segments 4, 5, 6, and

7, comprise about 20 % of the population, contributed approximately 72 % of the funds

raised—segment seven with about 2 % of donors contributed 29 % of the total funds raised.

These results are consistent with findings in the literature. Not unexpectedly, the proportion of

the population assigned to segments decrease with the mean segment-contribution level.

Assignment of Alumni to Segments

As stated earlier, an output of the estimation procedure for mixture models is the proba-

bility that each alumnus belongs to the estimated segments. Thus, for each member m of

the alumni population, m ¼ 1; . . .;M ¼ 75; 922, the probability that the member belongs to

segment s, s ¼ 1; . . .; S ¼ 7 is given by pms. Each pms represents the conditional probability

of observing the sequence of contributions by member m, given the estimated distribution

of alumni across the segment (or mixture proportions), k1; :::; kS, and the segment

parameter estimates, ðl1; r1Þ; . . .; ðlS; rSÞ.In Table 3, we first present the contribution sequences, along with the average contri-

bution value, followed by the estimated membership probabilities for a sample of alumni.

The sample was selected to highlight the managerial implications of capturing the vari-

ability in contribution sequences. We note that alumni contribution sequences are of dif-

ferent lengths, as the first contribution period for individuals may differ, and alumni may

not contribute in every period. For example, alumnus 1’s first contribution period is 2003

and includes a total six contributions, while alumnus 4’s first and only contribution occurs

in 2009. We make the following observations based on the results presented in Table 3:

• The sequences and probabilities for alumni 1 and 2 highlight an appealing feature of

the model. In these two cases, the average contributions for the two individuals are

identical, $110. Based on the mean of contributions, it is unclear to which of the

segment (segment 2 with a mean of l2 ¼ $78:61 vs. segment 3 with a mean of

3 The parameter estimates and mixture proportions for Normal mixture models with one to eight segments,as well as the scaled LL, the LL divided by the total number of observations, and CAIC associated with eachmodel is presented in Table 10, in ‘‘Goodness-of-Fit Measures and Segment Estimates’’ Appendix.

Res High Educ

123

l3 ¼ $149:13) the alumni are most likely to belong. We observe, however, that the

estimated probabilities that each belongs to segment 2, are vastly different (80 vs.

24 %). These results are driven by the variation within the contribution sequences, and

illustrate the potential bias introduced by aggregate summary statistics.

• To further highlight the benefits of processing the full contribution sequences, consider

the contributions of Alumnus 3, with a mean contribution of $300. He is assigned

probabilities of 84.1, 15.9, and 0.1 % that he belongs to segments 4 (l4 ¼ $279), 5

(l5 ¼ $601), and 6 (l4 ¼ $1; 277), respectively. While his mean contribution amount is

in line with segment 4, the posterior probabilities may seem somewhat surprising given

that two of his contributions (for $50) fall between the mean contribution levels of

segments 1 and 2, two (for $650 and $500) are near the mean of segment 5, and one (for

$250) falls near segment 4’s mean. The results are less surprising when one interprets

each membership probability as the likelihood that the entire contribution sequence was

generated by the Normal distribution function defining the given segment.

• The contribution sequences and posterior membership probabilities for alumni 4 and 5

highlight the relationship between data availability, i.e., length of the contribution

sequences, and the confidence with which individuals are assigned to segments.

Alumnus 4’s data sequence is shortest among the alumni in the example. The ensuing

membership probabilities, therefore, reflect high uncertainty, possibly due to insuffi-

cient data.

Alumnus 5’s contributions, with a mean of $241, is assigned to segment 4 (with

l4 ¼ $278:55) with probability 100 %. Qualitatively, this observation is opposite that

for alumnus 3—alumnus 4 is assigned to segment 4 with certainty despite appreciable

within-sequence variation.

• For alumni 6 and 7, the contributions are split between sets of adjacent segments. In

case of alumnus 6, it appears that she transitions between segments. It is important,

however, to emphasize that contributions are assumed to be serially independent and so

their order does not matter. Again, the correct interpretation of the membership

Table 2 Estimated parameters for normal mixture models with six to eight segments

Segment

1 2 3 4 5 6 7 8

S ¼ 6 segments, LLscaled ¼ �5:963, CAIC ¼ 3:374E þ 06

ls $32.10 $79.86 $159.37 $326.51 $843.82 $2,615.10

rs $15.19 $34.98 $82.62 $209.59 $557.36 $2,557.90

ks ð%Þ 30.4 28.6 19.2 11.6 7.6 2.6

S ¼ 7 segments, LLscaled ¼ �5:9501, CAIC ¼ 3:367Eþ 06

l~s$31.99 $78.61 $149.13 $278.55 $601.37 $1,277.30 $3,137.50

r~s$15.12 $34.09 $75.98 $163.57 $385.96 $877.48 $2,960.30

^k~s ð%Þ 30.3 27.8 17.9 11.1 7.9 3.3 1.7

S ¼ 8 segments, LLscaled ¼ �5:9413, CAIC ¼ 3:362E þ 06

ls $31.79 $76.70 $135.38 $235.09 $416.64 $862.00 $1,420.30 $3,484.00

rs $15.00 $32.85 $68.18 $128.70 $285.42 $417.87 $1,091.30 $3,168.10

ks ð%Þ 30.0 26.5 16.2 11.1 7.7 4.3 2.8 1.4

The model that provides the best fit is indicated in bold

Res High Educ

123

probabilities (and resulting uncertainty) is that they apply to the event that the entire

sequence is generated by one of the segment probability functions.

Practical Assignment Considerations

In general, the assignment of alumni to segments is overlapping, i.e., rather than deter-

minist assignments, the model estimates posterior probabilities of segment membership, as

discussed above. From a practical perspective, it is important to fundraisers to be able to

construct exact assignments to segments so the individuals can then be targeted with

specific and appropriate solicitations. In other words, fundraisers prefer to work with non-

overlapping assignments. The most common approach to address this issue in practice is to

assign individuals to the segment with the largest posterior membership probability.

To illustrate the segmentation results when alumni are assigned to the segment with the

highest posterior probability, we plot the mean and standard deviation associated with the

contribution sequence of each alumnus assigned to segments 1–4 (as shown in Fig. 2).4 For

the alumni in our sample, alumni 1 and 2 would be assigned to segments 2 and 3,

respectively, alumnus 3 would be assigned to segment 4, while alumnus 7 would be

assigned to segment 1. Thus, we note that the mean and variance are sufficient statistics to

assign individuals to segments for Normal mixture models.

Table 3 Contribution sequences for seven donor sample, along with posterior probabilities that individualm belongs to segment s, pms, for seven-segment model

m Member contribution sequence ($)

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Average

1 $50 $100 $150 $100 $125 $100 $150 $110

2 $100 $75 $150 $50 $200 $100 $100 $110

3 $50 $50 $650 $500 $250 $300

4 $150 $150

5 $150 $150 $150 $150 $150 $405 $165 $355 $380 $215 $385 $241

6 $100 $100 $200 $500 $2,000 $1,000 $650

7 $35 $35 $35 $75 $75 $35 $48

Posterior probabilities (%)

m pm1 ð%Þ pm2 ð%Þ pm3 ð%Þ pm4 ð%Þ pm5 ð%Þ pm6 ð%Þ pm7 ð%Þ

1 0 80 20 0 0 0 0

2 0 24 76 0 0 0 0

3 0 0 0 84.1 15.9 0.1 0

4 0 24 60 13 3 0 0

5 0 0 0 100 0 0 0

6 0 0 0 0 43 57 0

7 53 47 0 0 0 0 0

4 Alumni assigned to segments 5 through 7 are not shown because those segments require a different(larger) scale.

Res High Educ

123

Moreover, we observe from the shape of the segments that alumni described by higher

mean and lower variance are considered to have a similar giving potential as alumni with

lower mean, but higher variance in their contributions.

Across the population, we find that over 99.78 % of alumni have an associated segment-

specific posterior membership probability that exceeds 50 % (i.e., for some seg-

ment s; pms [ 0:5, for all m), while 72.12 and 47.23 % of the population, respectively, are

assigned to specific segments with at least 75 and 90 % confidence. From a managerial

perspective and using the 75 % confidence level as a threshold, the results suggest that the

segmentation supports segment-tailored solicitations for approximately three-quarters of

the population, while the remaining quarter can be approached with exploratory solicita-

tions partially aimed at refining the posterior membership probabilities. Hence, the pos-

terior probabilities serve not only to support targeted fundraising efforts, but can also be

used to drive informed data acquisition. The latter can be accomplished through field

experiments, where alumni with diffused (across two or more segments) posterior prob-

abilities can be targeted with different solicitations strategies.

Use of Segmentation Results in Devising ‘‘Ask Amounts’’

A key decision for university fundraisers is the ‘‘Ask Amount.’’ If fundraisers ask a

particular alumnus for too little, the university forgoes some surplus funding, and if too

much is asked, alumni often choose not to donate at all. Generally, organizations use

‘secret’ formulas tied to some donor contribution metric (e.g., last, maximum, or average

contribution amount) to calculate a target ask amount, and then send an appeal with three

ask amounts, centered around the target ask amount, along with a spot for ‘‘Other’’—for

those cases where they’ve completely missed the mark.

The segmentation results from our analysis can be used in a prescriptive manner to aid

fundraisers in selecting ask amounts. To illustrate this, we reexamine the contributions of

alumni 1 and 2, from Table 3. Their contributions, average contribution amount, and

assignment probabilities are reproduced in Table 4. The average contributions for the two

individuals are identical at $110; both are active donors, and have the same donation

frequency. Based on traditional approaches for setting ask amounts, both alumni would

likely receive identical appeals.

The probabilities that each belongs to segments 2 and 3 are vastly different, however.

These results, along with the segment estimates, can be used to set ask amounts that are

consistent with foot-in-the-door/door-in-the-face (FitD/DitF) fundraising techniques

(Rodafinos et al. 2005).5 In particular, the target ask amount can be set ‘‘close’’ to mean of

the segment with the highest membership probability, with the low(high) ask amount being

set at one standard deviation below(above) the segment mean. Thus, for alumni 1 with an

80 % probability of belonging to segment 2 (l2 ¼ $78:61; r2 ¼ 34:09), the target ask

amount should be close to the segment mean of $78. We set the target to $75. The low and

high ask amounts should be set approximately at $44 (l2 � r2) and $112(l2 þ r2),

respectively. Hence, the appeal for alumnus 1 may state:

I would like to make a gift of: $50:00 $75:00 $100:00 Other $Using a similar rationale, because alumnus 2 has a 76 % probability of belonging to

segment 3 (l3 ¼ $149:13; r3 ¼ 75:98), his target ask amount should be set close to seg-

ment 3’s mean of $149. The low and high ask amounts should be set at approximately $73

5 In this context, the organization asks for a low amount (FitD level), a high amount (DitF level), and theamount it hopes to get (the target level).

Res High Educ

123

and $225, respectively, based on being one standard deviation below/above the mean.

Consequently, the appeal for alumnus 2 can state:

I would like to make a gift of: $75:00 $150:00 $225:00 Other $While the specifics can be fine-tuned by university fundraisers, this approach makes use

of estimated membership probabilities and segment characteristics to create individually-

targeted appeals that are based on each individual’s sequence of contributions—something

that could not be accomplished using average contributions alone.

Practical Implications—Contribution Metrics

The segmentation results presented thus far have important practical implications for

fundraisers. Most importantly, fundraisers must expand the metrics used to summarize

contribution histories, in order to better identify systematic differences in contribution

behavior. In the proposed model, the segment assignment probabilities convey such sys-

tematic differences/similarities across alumni—they inform one’s expectations about

future contributions from individual alumni. And, knowing how much individual alumni

are likely to contribute is valuable because it allows fundraisers to engage donors differ-

ently. More to the point, the assignment probabilities can be exploited, as shown above, to

improve fundraising outcomes via ask amounts.

In the absence of segment specifications and assignment probabilities, fundraisers

should consider more than average contribution behavior when setting gift-string asks, and

incorporate a measure of dispersion (i.e., standard deviation) in contribution amounts.

When we consider the contributions for alumni 1 and 2, as shown in Fig. 3, we observe that

alumnus 2’s contributions exhibit much higher variation than those of alumnus 1, even

though their average donation amounts are identical. This higher variance is reflected in the

Fig. 2 Distribution of alumni across segments 1 to 4 based on the mean and standard deviation ofcontributions

Res High Educ

123

larger probability that alumnus 2 belongs to segment 3, and arguably indicates higher

donation potential.

Convergence of Assignment Probabilities

We now turn our attention to how the posterior probabilities are updated in response to

additional contribution data. To investigate this issue, we first fit the model using the first

seven years of data, using alumni donations from 2000 to 2006. We then study the

robustness of assignments, by estimating changes in posterior probabilities when we

consider contributions from 2007 to 2010, a year at a time.

Table 5 shows the parameter estimates based on the first seven years of data. We note

that the parameter estimates are very similar to those using the full data set. This points to

the robustness of the model and is also, in part, due to the fact that by year 7 (2006) the

sample size is sufficiently large, with 55,867 (or 73.6 % of the population) alumni having

contributed at least once.

Table 4 Contribution amounts, average contribution level, along with posterior probabilities

m Contributions Average pm2 ð%Þ pm3 ð%Þ

1 $50 $100 $150 $100 $125 $100 $150 $110 80 20

2 $100 $75 $150 $50 $200 $100 $100 $110 24 76

Fig. 3 Sequence of contributionsfor alumni 1 and 2

Table 5 Parameter estimates: seven-segment mixture model using first seven years of data

Segment

1 2 3 4 5 6 7

S ¼ 7 segments, LLscaled ¼ �5:834

ls $33.42 $83.88 $162.53 $315.45 $700.38 $1,487.00 $3,423.30

rs $15.16 $31.56 $74.50 $170.34 $394.91 $851.14 $2,998.90

ks ð%Þ 32.8 28.6 17.9 10.0 6.8 2.6 1.3

Res High Educ

123

Table 6 shows cumulative number of contributing alumni by year, broken down by the

number of contributions. For example, of the 55,867 alumni who contributed between 2000

and 2006, 19,281 alumni contributed only once, 10,092 contributed twice, and so on. The

table also shows the percentage of the active population who have contributed by each

year. By 2010, all alumni in the study have contributed at least once.

To better understand the convergence of assignments in response to newly available

data, we consider a sample of alumni, shown in Table 7. This sample, and the associated

posterior estimates shown in Table 11, are intended to be representative of the population

as a whole.

We make the following observations based on the results presented in Table 11 of

‘‘Goodness-of-Fit Measures and Segment Estimates’’ Appendix.

• Alumnus 1’s first contribution occurs in 2007, and so when the model is estimated

using the first seven years of data (2000–2006), no information is available to update

the posterior probabilities, and the alumnus is assigned the priors, as expected. In 2007,

a contribution of $80 leads to a significant increase (to 83.5 from 28.6 %) in the

posterior probability that the alumnus belongs to segment 2 (l2 ¼ $83:88), and drops

in the assignment probabilities to segments 1 and 3. Contributions of $20, $25, and $20

in the subsequent three years leads to the incremental increase (as each year’s

contribution data becomes available) that the alumnus is more likely to belong to

Table 6 Cumulative number of alumni contributions by year, broken down by number of contributions

No. of contributions 2006 2007 2008 2009 2010

1 19,281 20,275 21,801 23,181 24,659

2 10,092 10,735 11,364 11,985 12,715

3 7,262 7,419 7,892 8,267 8,628

4 5,283 5,511 5,609 5,987 6,337

5 4,515 4,333 4,461 4,503 4,758

6 4,358 3,921 3,763 3,843 3,851

7 5,076 3,856 3,454 3,265 3,278

8 4,497 3,425 3,027 2,877

9 4,051 3,105 2,757

10 3,600 2,786

11 3,276

No. of alumni 55,867 60,547 65,820 70,763 75,922

Pop. (%) 73.58 79.75 86.69 93.20 100.00

Table 7 Contribution sequences for sample alumni

Alumnus 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

1 $80 $20 $25 $20

2 $100 $2,000 $200

3 $1,500 $1,875 $1,000 $500 $1,000 $1,000 $1,000 $1,000 $750 $750

4 $50 $65 $50 $130 $65 $70 $75

5 $533 $541 $471 $483 $495 $545 $615 $200 $325 $660 $250

Res High Educ

123

segment 1 (l1 ¼ $33:42). Specifically, the pm1 increases from 17.8 to 69.7 to 96.2% in

response to the new contribution data.

• For alumnus 2, the contributions of $100 and $2,000 in 2005 and 2006 leads to an

85.9 % posterior probability that alumnus 2 belongs to segment 6 (l6 ¼ $1; 487), and

6.5 and 7.6 % probabilities that he belongs to segments 5 and 7, respectively. The $200

contribution made in 2007 leads to a reduction in the posterior probabilities that the

alumnus belongs to segments 6 and 7, and an increase in the probability that he belongs

to segment 4 (l4 ¼ $315:45). We note that because the alumnus makes no other

contributions between 2008 and 2010, the posterior probabilities are not updated in

those years. This is reflective of the model’s assignment of alumni to segments based

on actual contributions, i.e., zero contributions are ignored. A future research direction

is to formulate a model in which one accounts for zero contributions in the sequence.

• The initial posterior probabilities assigned to alumnus 3 in 2006 capture the fact that

alumnus 3 initially contributed at levels closely tied to segment 6 (with contributions

of $1,500 and $1,875 in 2000 and 2001), but then contributed at (lower) levels more

closely tied to segment 5. Because she did not contribute in 2007, we see no update in

the posterior probabilities in that year. Like alumnus 1, the contributions of $1,000,

$750 and $750 in years 2008, 2009 and 2010, respectively, lead to incremental

increases in the assigned posterior probabilities that alumnus 3 belongs to segment 5.

• The inclusion of alumnus 4 highlights the fact that despite the variability in the

contribution sequence, the availability of data, i.e., length of contribution sequence,

allows for high confidence in the assignment of alumni into segments.

• Finally, with alumnus 5, we observe that after the first seven years the assignment

probabilities exhibit a significant level of uncertainty (with posterior probabilities of

42.21, 56.05, 1.65, 0.09 % that alumnus 5 belongs to segments 4, 5, 6, and 7,

respectively). A contribution of $200 in 2007 leads an increase in the posterior

probability that alumnus 5 belongs to segment 4 (75.40 %). With a $325 contribution

in 2008, the assigned probability to segment 4 vaults to 91.82 %. This, however, is

reversed in 2009 in light of a $660 contribution, dropping pm4 to 77.22 % and

increasing pm5 to 22.76 from 8.15 %. The responsive in the posterior assignments is

further highlighted when a contribution of $250 in 2010 leads to an increase in pm4 to

93.33 % in the posterior probability assignment of alumnus 5 to segment 4.

As discussed in the previous section, higher levels of confidence in the assignment

probabilities are desirable. Table 8 shows the relationship between data availability and the

confidence with which individuals are assigned to segments; that is, the table shows the

proportion of the population whose posterior probabilities exceed the given confidence

thresholds as contribution data becomes available, on a yearly basis. As an example, the

table shows that 73.5 % of individuals in the population have an associated segment-

Table 8 Relationship between data availability and assignment probabilities

Assignment confidence (%) Year

2006 (%) 2007 (%) 2008 (%) 2009 (%) 2010 (%)

50 73.5 79.7 86.6 93.1 99.9

75 60.5 65.9 72.0 77.8 83.9

90 37.1 41.2 45.3 49.3 53.2

95 29.7 33.3 36.8 40.2 43.9

Res High Educ

123

specific posterior membership probability that exceeds 50 % (i.e., for some seg-

ment s; pms [ 0:5, for all m) based on the data available from 2000 to 2006, and that

this proportion increases to 79.7 % when data from 2007 are used to update the posterior

probabilities.

Practical Implications—Data Collection

An important observation from this analysis is that as data availability increases (data from

subsequent years becomes available), uncertainty decreases. Thus, as more data is col-

lected, fundraisers are better able to target their solicitations.

While assertions that ‘‘having more data is better’’ often come across as trite, here it is

an important practical implication of our results. To reproduce the efficiency and effec-

tiveness gains achieved by many for-profit organizations using data analytics, universities

will need collect more and better data. Put another way, unlocking the value of data-driven

fundraising will require that universities track every interaction with its ‘customers.’ For

fundraisers, this should include tracking what solicitations are being sent (i.e., general-fund

appeals, capital campaign, college fund solicitations, etc.), the ask amounts, as well as

response rates (to capture the success of specific appeals/mailings), in addition to alumni

contribution amounts.

Analysis of Behavior in Population Subsets

University fundraisers often target specific of subsets of the population with tailored

solicitation, such as that of alumni with different degree types, alumni with Greek affili-

ations, etc. Having estimated distributions for the population allows us to compute con-

ditional distributions for sub-populations. As an example, in Fig. 4, we show the

distribution of alumni across the segments based on the type of degree(s) earned at the

Fig. 4 Distribution of alumni based on degree type across seven segments

Res High Educ

123

university, i.e., undergraduate only, graduate/professional only, and both undergraduate

and graduate/professional degrees.

We note that the majority of alumni in all three categories belong to lower value

segments—82.7, 70.4, and 67.9 % of undergraduates, graduates, and alumni with both

degree types, respectively, are assigned to segments 1, 2, and 3.

There are, however, appreciable differences between degree types. In particular, while

we see that each of the alumnus types are represented in all segments, the segment

proportions are far from uniform, only 14 % of undergraduates contribute at the moderate-

level contribution value segments (4 and 5), whereas 24 % of graduates and 23 % of both

degree type alumni do so. For the higher contribution value segments (6 and 7), the

proportion of undergraduates and graduates contributing is 4 and 6 %, respectively.

Moreover, 9 % of alumni with both degree types contribute at the high contribution value

segments. Put another way, undergraduates make up just over a third of the alumni in the

high-value segments, 6 and 7. At the same time, graduate degreed alumni make up close to

50 % of high-value segments, with the remaining 15 % being both degree type alumni.

In the population as whole, 47 % the alumni received undergraduate degrees, 44 %

received graduate/professional degrees, and the remaining 9 % received both degree types.

In light of these facts, the results above illustrate the need for segment-and-degree type

specific appeals. In other words, there’s significant heterogeneity in alumni behavior—

alumni with the same degree type do not all behave similarly, and the make-up of segments

varies significantly with respect to the proportion of alumni with the different degree types.

Performing similar analyses to the above, we are able to consider other alumni char-

acteristics available to us, and observe the distribution of the population conditional on the

characteristic of interest. This allows us to determine whether a characteristic has any

influence in the monetary value of a donor’s contributions. For example, we find that

alumni whose spouses are also alumni (13 % of the 75,922 alumni in the population, or

about 9,870 individuals) are more likely to give at higher contribution segments (33 %

versus 23 % for alumni not married to another alumnus). Alumni with children who are

Fig. 5 Distribution of alumni across segments based on reunion attendance; proportion of segmentscomprised of reunion attendees

Res High Educ

123

also alumni (3 % of the populations, or about 2,277 individuals), are almost twice as likely

to belong to the segments defined by higher mean contributions; 77 % of alumni with non-

alumni children donate in segments 1, 2 and 3, whereas only about half of alumni with

alumni children do so).

When it comes to alumni who attend reunions (19 % of the population, or about 14,570

individuals), 39 % donate at the four higher value segments, whereas only about 20 % of

non-reunion attendees do so. Moreover, reunion attendees make up only 11 % of segment

1 (89 % of segment 1 alumni are non-reunion attendees) with the fraction of reunion

attendees increasing to 40 % in segment 7, as shown in Fig. 5.

Lastly, we consider the giving behavior of alumni as a function of age. Figure 6 shows

the proportion of the population in different age groups, as well as the proportion of total

funds raised by alumni in each of the age groups.

Younger alumni make up a small proportion on the population and funds raised. Alumni

in the 35–45 age group account for approximately 21 % of the population, and contribute

12 % of the funds raised. Alumni in the 45–55 and 55–65 age groups make up 40 % of the

population, and each contribute about a quarter of the funds raised. The proportion of the

alumni population belonging to older age groups is smaller, as is the proportion of funds

raised. These results are consistent with other studies in the literature, which reflect the

facts that: young alumni are less engaged and have less disposable income, middle aged

alumni have more disposable income and are more involved and responsive to alumni

giving campaigns, and that older/retired alumni tend to withdraw from the giving popu-

lation and/or reduce their giving.

More interesting, however, is the distribution of alumni across segments based on

contribution levels, as shown in Table 9. Specifically, the table shows the proportion of

alumni assigned to low contribution value segments (1, 2, and 3), medium contribution

level segments (4 and 5), and high contribution value segments (6 and 7).

These results show that, in general, as alumni age the proportion in medium and high

value segments increases. The ability to identify alumni in the medium and high value

segments in all age groups is very important from a fundraising perspective—it gives

Fig. 6 Proportion of alumni population in different age groups, and proportion of total funds raised by agegroup

Res High Educ

123

university fundraisers the ability to focus their alumni development efforts (e.g., by cre-

ating targeted appeals) on the top 25 % of alumni who give at the higher levels. In all, the

results presented in this section expose the presence of significant, systematic heteroge-

neity in the population in excess of that explained by the available trait data.

Conclusions

In this paper, we make use of a finite mixture model framework to analyze alumni contri-

bution behavior at a university in the Midwestern United States. The model segments the

population based on the monetary value of donor contributions. A salient feature of the model

is its ability to exploit longitudinal data, i.e., alumni contribution sequences. Another

important feature of the model is that it supports the identification of unobserved heteroge-

neities in individual giving behavior, in excess of that explained by trait characteristics.

The results arising from the analysis reveal key implications/benefits of the proposed

approach for practitioners. First, the model yields a segmentation that is sensitive to the

variations in the contribution amounts. This provides a richer characterization of alumni

giving potential. In particular, the results illustrate that alumni described by higher mean

and lower variance are considered to have a similar giving potential as alumni with lower

mean, but higher variance in their contributions. Prescriptively, this richer characterization

can be used, along with the estimated alumni segment-membership probabilities, to

improve the selection of appeal ‘‘Ask Amounts.’’ Second, the proposed approach provides

a powerful tool to aggregate alumni that belong to the same segment, based on contribution

behavior. This, in turn, allows for tailored, segment-specific solicitations to be designed

and deployed to support fundraising efforts, e.g., for mailing distributions. Second, the

proposed model allows fundraisers to discern patterns in sub-populations that can guide

fundraising actions. For example, the ability to identify alumni in different age groups that

contribute to high value segments gives fundraisers the ability to focus development efforts

(e.g., by developing targeted call lists).

We also note that while the proposed model, in general, results in overlapping

assignments (rather than determinist assignments of individuals to segments), this issue can

be easily circumvented in practice by assigning alumni to the segment with the largest

posterior membership probability. Across the population, we find that 72.12 and 47.23 %

of alumni, respectively, are assigned to specific segments with at least 75 and 90 %

Table 9 Proportion of alumni population in different age groups, and proportion of total funds raised byage group

Age group Proportion in segments with

Low value (%) Medium value (%) High value (%)

20–25 92.2 7.2 0.6

25–35 88.3 10.7 1.1

35–45 75.0 21.7 3.3

45–55 72.3 21.5 6.2

55–65 72.8 20.5 6.7

65–75 74.4 19.2 6.4

75–85 74.3 18.8 6.9

85? 79.1 15.2 5.8

Res High Educ

123

confidence. As stated earlier, from a managerial perspective and using the 75 % confidence

level as a threshold, the results suggest that the segmentation supports segment-tailored

solicitations for approximately three-quarters of the population, while the remaining

quarter can be approached with exploratory solicitations partially aimed at refining the

posterior membership probabilities. Thus, the posterior probabilities can be used to drive

informed data acquisition efforts. The results also highlight the increased convergence of

assignments in response to newly available data. Managerially, this allows fundraisers to

better target their solicitations, as more data is collected.

In terms of practical implications, the results suggest that fundraisers should look

beyond the average contribution behavior of donors, and incorporate a measure of dis-

persion (in contribution amounts) when setting ‘Ask amounts.’ They also imply the need

for more and better data collection by institutional fundraisers, including data on the types

of appeals sent and ask amounts (both of which directly affect the donation behavior of

alumni), as well as data on response rates and alumni contribution amounts.

Lastly, although the model is applied in the context of university fundraising, it has

applicability in other (for-profit) settings, e.g. catalog mailing, where individuals pur-

chasing behavior result in large numbers of transactions, and where it is of interest to

allocate resources based on a small number of segments rather than a large number of

individuals.

Appendix

1. EM Algorithm Implementation for Normal Mixture Models

In this section, we discuss the parameter estimation problem for Normal mixture models.

Given the alumni contribution sequences, and having specified the finite mixture model via

(1), we define the data likelihood for k and h as

Lðy;k; hÞ ¼YM

m¼1

Pðymjk; hÞ ð3Þ

The objective then is to find parameters k; h that maximize (3). To solve this problem, as is

commonly done in the estimation of finite mixture models, we rely on the EM Algorithm.

The EM Algorithm, formalized by Dempster et al. (1977), is a numerical method to solve

MLE problems in cases were data are missing. As applied in the estimation of finite

mixture models, the EM algorithm relies on the fact that if individual memberships, zm; 8m,

were known, the ensuing estimation problem would be simplified. To this end, we can

write the complete data likelihood for k and h as

Lc y; z; k; hð Þ ¼YM

m¼1

P ym; zmjk; hð Þ ¼YM

m¼1

P ym; k; hð ÞP zmjk; hð Þ

where z � zmf gMm¼1. The associated complete data LL is

ln Lc y; z; k; hð Þ ¼XM

m¼1

XS

s¼1

zms ln fsðymjzms; hsÞ þ zms ln ks ð4Þ

The EM Algorithm is a numerical method to maximize the complete data LL function (4).

The algorithm consists of two steps: An expectation step, E-Step, where we evaluate the

Res High Educ

123

expectation of (4) over the possible realizations of z, given the observed data, y, and

estimates of k and h, denoted k and h; and a maximization step, M-Step, where we update k

and h with arguments that maximize the expectation of ln Lcð�Þ. The EM algorithm

alternates between the E and M steps (until a convergence criterion is met). Mathemati-

cally, the EM algorithm can be written as follows:

Step 0: Initialization Choose randomly generated values for the parameters that define

the segments, h0, and the mixture proportions k0. The values for the mixture proportions

are randomly generated from a Dirichlet prior distribution. Set the solution index k 0.

Step 1: Expectation Step (E-Step) Calculate the expected value of the complete-data

LL function, shown in (4), with respect to the conditional distribution of the

(unobserved) data on segment membership, z, given the observed contribution

sequences, y, and the estimated parameters, hk, and kk at iteration k. We calculate the

expected value of the LL function, Q, as follows:

Q kk; hk� �

¼XM

m¼1

XS

s¼1

pms ln fs ymjpms; hks

� �� �þ ln kk

s

� �h ið5Þ

where, pms is the probability of member m belonging to segment s, or the expected value of

zms, E½zmsjym�.Step 2: Maximization Step (M-Step) Calculate the mixture proportions, kkþ1, and the

parameters defining the segments, hkþ1, that maximize the expected value of the

complete-data LL calculated in Step 1, assuming that the missing data, zm, are known—

this is done replacing each zms by its expected value, pms, at iteration k. We calculate the

mixture proportions, kkþ1, for s ¼ 1; . . .; S, as follows:

kkþ1s ¼ 1

M

XM

m¼1

pms ð6Þ

To calculate the optimizing parameters that define the normal distributions describing the

distribution of annual contributions in each of the segments, hkþ1s � lkþ1

s ; rkþ1s

� �, we do,

for s ¼ 1; . . .; S, as follows:

lkþ1s ¼ 1

Mkkþ1s

XM

m¼1

pms

1

T

XT

t¼1

ytm ð7Þ

rkþ1s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

Mkkþ1s

XM

m¼1

pms

1

T

XT

t¼1

ytm � lkþ1

s

� �2

vuut ð8Þ

Set the solution index k k þ 1.

Step 3: Check Stop Criteria While k\K and

Qðkkþ1; hkþ1Þ � Qðkk; hkÞ� �

=Qðkk; hkÞ���

���[ �

continue to iterate steps 1 and 2. The stopping criteria were set to � ¼ 1� 10�5, or a

maximum of K ¼ 100 iterations.

Relevant properties of the EM Algorithm, such as those related to its convergence, are

discussed in references such as McLachlan and Thriyambakam (2001).

Res High Educ

123

2. Derivation of Posterior Membership Probabilities

In this section, we derive expression presented in (2), which is used to update the mem-

bership probabilities in response to a contribution sequence. In particular, given a proba-

bility density function of k, f ð�Þ, and a contribution sequence, ym, the probability that

individual m is assigned to segment s, pms, is updated as follows:

pms ¼ E zmsjym½ � ¼ EkP zms ¼ 1jymð Þ

¼Z

k2SP zms ¼ 1jym; kð Þf ðkjymÞok

¼Z

k2S

P zms ¼ 1; ymjkð ÞPðymjkÞ

f ðkjymÞok

ð9Þ

where Bayes’ Law yields:

f ðkjymÞ ¼PðymjkÞf ðkÞR

k2S PðymjkÞf ðkÞokð10Þ

Substituting (10) into Eq. (9), we have:

E zmsjym½ � ¼Z

k2S

P zms ¼ 1; ymjkð ÞR

k2S PðymjkÞf ðkÞokf ðkÞok

¼R

k2S Pðymjzms ¼ 1; kÞPðzms ¼ 1jkÞf ðkÞokR

k2SPS

r¼1 Pðymjzmr ¼ 1; kÞPðzmr ¼ 1jkÞf ðkÞok

¼R

k2S Pðymjzms ¼ 1; kÞksf ðkÞokR

k2SPS

r¼1 Pðymjzmr ¼ 1; kÞkrf ðkÞok

¼ Pðymjzms ¼ 1; kÞksPS

r¼1 Pðymjzmr ¼ 1; kÞkr

¼ fs ymð ÞksPS

r¼1 fr ymð Þkr

Fig. 7 Consistent Akaike information criterion for S = 1 to 8

Res High Educ

123

3. Goodness-of-Fit Measures and Segment Estimates

The CAIC is given as �2 � LLþ ðp � Sþ S� 11Þ lnðnÞ þ 1ð Þ, where p, S and n represent the

number of parameters per segment (2 per segment in our analysis, ls; rs, for each segment

s), the number of segments in the model and the number of observations, respectively. The

CAIC renders AIC consistent such that Akaikes principle of minimizing the Kullback–

Leibler information quantity is not violated (Bozdogan 1987).

Table 10 Estimated parameters for multivariate normal mixture models with up to eight segments

Segment

1 2 3 4 5 6 7 8

S ¼ 1 segment, LLscaled ¼ �8:041, CAIC ¼ 4:549E þ 06

ls $289.99

rs $751.50

ks100%

S ¼ 2 segments, LLscaled ¼ �6:5683, CAIC ¼ 3:716E þ 06

ls $89.73 $876.86

rs $73.34 $1,319.80

ks ð%Þ 80.0 20.0

S ¼ 3 segments, LLscaled ¼ �6:2401, CAIC ¼ 3:531E þ 06

ls $59.38 $219.33 $1,299.80

rs $37.32 $153.35 $1,627.40

ks ð%Þ 60.1 29.2 10.7

S ¼ 4 segments, LLscaled ¼ �6:0943, CAIC ¼ 3:448E þ 06

ls $33.83 $93.96 $270.10 $1,433.50

rs $16.49 $47.22 $189.29 $1,712.80

ks ð%Þ 32.4 35.5 23.0 9.2

S ¼ 5 segments, LLscaled ¼ �5:982, CAIC ¼ 3:385E þ 06

ls $32.57 $84.19 $191.76 $589.50 $2,241.00

rs $15.51 $38.35 $111.99 $416.55 $2,289.90

ks ð%Þ 31.0 31.1 21.8 12.5 3.6

S ¼ 6 segments, LLscaled ¼ �5:963, CAIC ¼ 3:374E þ 06

ls $32.10 $79.86 $159.37 $326.51 $843.82 $2,615.10

rs $15.19 $34.98 $82.62 $209.59 $557.36 $2,557.90

ks ð%Þ 30.4 28.6 19.2 11.6 7.6 2.6

S ¼ 7 segments, LLscaled ¼ �5:9501, CAIC ¼ 3:367Eþ 06

ls $31.99 $78.61 $149.13 $278.55 $601.37 $1,277.30 $3,137.50

rs $15.12 $34.09 $75.98 $163.57 $385.96 $877.48 $2,960.30

ks ð%Þ 30.3 27.8 17.9 11.1 7.9 3.3 1.7

S ¼ 8 segments, LLscaled ¼ �5:9413, CAIC ¼ 3:362E þ 06

ls $31.79 $76.70 $135.38 $235.09 $416.64 $862.00 $1,420.30 $3,484.00

rs $15.00 $32.85 $68.18 $128.70 $285.42 $417.87 $1,091.30 $3,168.10

ks ð%Þ 30.0 26.5 16.2 11.1 7.7 4.3 2.8 1.4

The model that provides the best fit is indicated in bold

Res High Educ

123

The parameter estimates and mixture proportions for Normal mixture models with one

to eight segments, as well as the scaled LL, the LL divided by the total number of

observations, and CAIC (as shown in Fig. 7) associated with each model is presented in

Table 10. We observe that the one-segment model estimates for the mean and standard

deviation value of annual contributions equal the overall population mean and standard

deviation values, as expected. In other words, these (maximum likelihood) estimates are

analogous to modelling a homogeneous alumni population.

4. The Effect of Added Contribution Data on Convergence Probabilities

See Table 11

Table 11 Posterior probability updates in years 2007–2010, given initial estimates based on data upto 2006

Alumnus Update year pm1 ð%Þ pm2 ð%Þ pm3 ð%Þ pm4 ð%Þ pm5 ð%Þ pm6 ð%Þ pm7 ð%Þ

1 2006 32.8 28.6 17.9 10.0 6.8 2.6 1.3

2007 1.8 83.5 12.1 2.1 0.5 0.1 0.0

2008 17.8 75.8 5.8 0.6 0.1 0.0 0.0

2009 69.7 29.2 1.0 0.1 0.0 0.0 0.0

2010 96.2 3.7 0.1 0.0 0.0 0.0 0.0

2 2006 0.0 0.0 0.0 0.0 6.5 85.9 7.6

2007 0.0 0.0 0.0 0.0 15.4 80.6 4.1

2008 0.0 0.0 0.0 0.0 15.4 80.6 4.1

2009 0.0 0.0 0.0 0.0 15.4 80.6 4.1

2010 0.0 0.0 0.0 0.0 15.4 80.6 4.1

3 2006 0.0 0.0 0.0 0.0 51.1 48.9 0.0

2007 0.0 0.0 0.0 0.0 51.1 48.9 0.0

2008 0.0 0.0 0.0 0.0 66.7 33.3 0.0

2009 0.0 0.0 0.0 0.0 86.2 13.8 0.0

2010 0.0 0.0 0.0 0.0 95.1 4.9 0.0

4 2006 0.0 99 1 0.0 0.0 0.0 0.0

2007 0.0 100 0.0 0.0 0.0 0.0 0.0

2008 0.0 100 0.0 0.0 0.0 0.0 0.0

2009 0.0 100 0.0 0.0 0.0 0.0 0.0

2010 0.0 100 0.0 0.0 0.0 0.0 0.0

5 2006 0.00 0.00 0.00 42.21 56.05 1.65 0.09

2007 0.00 0.00 0.00 75.40 24.35 0.24 0.01

2008 0.00 0.00 0.00 91.82 8.15 0.02 0.00

2009 0.00 0.00 0.00 77.22 22.76 0.02 0.00

2010 0.00 0.00 0.00 93.33 6.67 0.00 0.00

ls $33.42 $83.88 $162.53 $315.45 $700.38 $1,487.00 $3,423.30

ks ð%Þ 32.8 28.6 17.9 10.0 6.8 2.6 1.3

Estimated segment means and population mixture proportions also shown

Res High Educ

123

References

Andreoni, J. (1989). Giving with impure altruism: Applications to charity and ricardian equivalence.Economics of Education Review, 97(6), 1447–1458.

Baade, R., & Sundberg, J. (1996). What determines alumni generosity. Economics of Education Review,15(1), 75–81.

Becker, G. S. (1974). The theory of social interactions. Journal of Political Economy, 82(6), 1063–1093.Bekkers, R., & Wiepking, P. (2011a). A literature review of empirical studies of philanthropy: Eight

mechanisms that drive charitable giving. Nonprofit and Voluntary Sector Quarterly, 40(5), 924–973.Bekkers, R., & Wiepking, P. (2011b). Who gives? A literature review of predictors of charitable giving. Part

I: Religion, education, age, and socialization. Voluntary Sector. Review, 2(3), 337–365.Beranek, W., Kamerschen, D. R., & Timberlake, R. H. (2010). Charitable donations and the estate tax: A

tale of two hypotheses. American Journal of Economics and Sociology, 69(3), 1054–1078.Bozdogan, H. (1987). Model selection and Akaike’s information criterion (AIC): The general theory and its

analytical extensions. Psychometrika, 52(3), 345–370.Bristol, R, Jr. (1990). The life cycle of alumni donations. Review of Higher Education, 13(4), 503–518.Bronfman, C., Solomon, J., & Edwards, M. (2011). Should philantropies operate like businesses? The Wall

Street Journal, page R1.Bruggink, T. H., & Siddiqui, K. (1995). An econometric model of alumni giving: A case study for a liberal

arts college. The American Economist, 39(2), 53–60.Clotfelter, C. T. (2001). Who are the alumni donors? giving by two generations of alumni from selective

colleges. Nonprofit Management and Leadership, 12(2), 119–138.Clotfelter, C. T. (2003). Alumni giving to elite private colleges and universities. Economics of Education

Review, 22(2), 109–120.Collins, L. M., & Lanza, S. T. (2010). Latent class and latent transition analysis: With applications in the

social, behavioral, and health sciences. Hoboken: Wiley.Council for Aid to Education (2012). Voluntary support of education report.Dempster, A., Lair, N., & Rubin, D. (1977). Maximum likelihood from incomplete data via the em algo-

rithm (with discussion). Journal of the Royal Statistical Society B, 39(1), 1–38.Durango-Cohen, E., Torres, R., & Durango-Cohen, P. (2013a). Donor segmentation: When summary sta-

tistics don’t tell the whole story. Journal of Interactive Marketing, 27(3), 172–184.Durango-Cohen, P., Durango-Cohen, E., & Torres, R. (2013b). A Bernoulli–Gaussian mixture model of

donation likelihood and monetary value: An application to alumni segmentation in a university setting.Computers & Industrial Engineering, 66(4), 1085–1095.

Ehrenberg, R., & Smith, C. (2003). The sources and uses of annual giving at selective private researchuniversities and liberal arts colleges. Economics of Education Review, 22(3), 223–235.

Feldstein, M. (1975). The income tax and charitable contributions: Part II the impacton religious, educa-tional, and other organizations. National Tax Journal, 28, 209–226.

Gaier, S. (2005). Alumni satisfaction with their undergraduate academic experience and the impact ofalumni giving and participation. International Journal of Educational Advancement, 5(4), 279–288.

Goodman, L. A. (1974a). The analysis of systems of qualitative variables when some of the variables areunobservable. Part I - a modified latent structure approach. American Journal of Sociology, 79,1179–1259.

Goodman, L. A. (1974b). Exploratory latent structure analysis using both identifiable and unidentifiablemodels. Biometrika, 61, 215–231.

Gottfried, M. A. (2010). School urbanicity and financial generosity: Can neighborhood context predictdonative behavior in spite of the economy? International Journal of Education Advancement, 9(4),220–233.

Grant, J. H., & Lindauer, D. L. (1986). The economics of charity life-cycle patterns of alumnae contribu-tions. Eastern Economic Journal, 12(2), 129–141.

Green, P. (1977). A new approach to market segmentation. Business Horizons, 20, 61–73.Ha, S., Bae, S., & Park, S. (2002). Customers time-variant purchase behavior and corresponding marketing

strategies: An online retailers case. Computers and Industrial Engineering, 43, 801–820.Holmes, J. (2009). Prestige, charitable deductions and other determinants of alumni giving: Evidence from a

highly selective liberal arts college. Economics of Education Review, 28, 18–28.Kim, M., Gibson, H., & Ko, J. Y. (2011). Understanding donors to university performing arts programs:

Who are they and why do they contribute? Managing Leisure, 16, 17–35.Kim, S., Fong, D. K., & Desarbo, W. S. (2012). Model-based segmentation featuring simultaneous segment-

level variable selection. Journal of Marketing Research, 49(5), 725–736.

Res High Educ

123

Le Blanc, L., & Rucks, C. (2009). Data mining of university philanthropic giving: Cluster-discriminantanalysis and pareto effects. International Journal of Education Advancement, 9(2), 64–82.

Lindahl, W., & Winship, C. (1992). Predictive models for annual fundraising and major gift fundraising.Nonprofit Management and Leadership, 3(1), 43–63.

Marr, K. A., Mullin, C. H., & Siegfried, J. J. (2005). Undergraduate financial aid and subsequent alumnigiving behavior. The Quarterly Review of Economics and Finance, 45, 123–143.

McLachlan, G., & Thriyambakam, K. (2001). The EM algorithm and extensions. New York: Wiley.McLachlan, G. J., & Peel, D. (2000). Finite mixture models. New York: Wiley.Meer, J. (2011). Brother, can you spare a dime? peer pressure in charitable solicitation. Journal of Public

Economics, 95, 926–941.Meer, J., & Rosen, H. S. (2009a). Altruism and the child cycle of alumni donations. American Economic

Journal, 1(1), 258–286.Meer, J., & Rosen, H. S. (2009b). The impact of athletic performance on alumni giving: An analysis of

microdata. Economics of Education Review, 28, 287–294.Monks, J. (2003). Patterns of giving to one’s alma mater among young graduates from selective institutions.

Economics of Education Review, 22, 121–130.Okunade, A. (1996). Graduate school alumni donations to academic funds: Micro-data evidence. American

Journal of Economics and Sociology, 55(2), 213–229.Okunade, A. A., Wunnava, P. V., & Walsh, R. J. (1994). Charitable giving of alumni: Micro-data evidence

from a large public university. The American Journal of Economics and Sociology, 53(1), 73–84.Olsen, K., Smith, A. L., & Wunnava, P. V. (1989). An empirical study of the life-cycle hypothesis with

respect to alumni donations. The American Economist, 33(2), 60–63.Pearson, J. (1999). Comprehensive research on alumni relationships: Four years of market research at

stanford university. New Directions for Institutional Research, 101, 5–21.Peterson, R. (1991). Small business usage of target marketing. Journal of Small Business Management,

29(4), 79–85.Rhoads, T., & Gerking, S. (2000). Educational contributions, academic quality, and athletic success.

Contemporary Economic Policy, 18(2), 248–259.Rodafinos, A. S., Vucevic, A., & Ideridis, G. D. (2005). The effectiveness of compliance techniques: Foot in

the door versus door in the face. Journal of Social Psychology, 145, 237–239.Sun, X., Hoffman, S., & Grady, M. (2007). A multivariate causal model of alumni giving: Implications for

alumni fundraisers. International Journal of Educational Advancement, 7(4), 307–332.Taylor, A., & Martin, J. (1995). Characteristics of alumni donors and nondonors at a research I, public

university. Research in Higher Education, 36(3), 283–302.Tucker, I. B. (2004). A reexamination of the effect of big-time football and basketball success on graduation

rates and alumni giving rates. Economics of Education Review, 23, 665–661.Verhoef, P., Spring, P., Hoekstra, J., & Leeflang, P. (2002). The commercial use of segmentation and

predictive modeling techniques for database marketing in the Netherlands. Decision Support Systems,34(4), 471–481.

Wedel, M., & Kamakura, W. (2000). Market segmentation: Conceptual and methodological foundations.Norwell: Kluwer Academic Publishers.

Wedel, M., Kamakura, W., Arora, N., Bemmaor, A., Chiang, J., Elrod, T., et al. (1999). Discrete andcontinuous representations of unobserved heterogeneity in choice modeling. Marketing Letters, 10(3),219–232.

Weerts, D., & Ronca, J. (2008). Characteristics of alumni donors who volunteer at their alma mater.Research in Higher Education, 49, 274–292.

Weerts, D. J., & Ronca, J. M. (2009). Using classification trees to predict alumni giving for higher edu-cation. Education Economics, 17(1), 95–122.

Wunnava, P., & Lauze, M. (2001). Alumni giving at a small liberal arts college: Evidence from consistentand occasional donors. Economics of Education Review, 20, 533–543.

Yoo, Y., & Harrison, W. (1989). Altruism in the market for giving and receiving: A case of highereducation. Education of Economics Review, 8, 367–379.

Res High Educ

123