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

Abstract

Having an effective segmentation strategy is key to the viability of any organization. This is particularly true for colleges, universities, and other nonprofit organizations—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.

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 $\varvec{\lambda }$ and $\varvec{\theta }$ as

$$\begin{aligned} L(\mathbf{y}{;} \varvec{\lambda }, \varvec{\theta }) = \prod _{m=1}^{M} P({\mathbf{y} _m}|\varvec{\lambda }, \varvec{\theta }) \end{aligned}$$

(3)

The objective then is to find parameters $\varvec{\lambda }, \varvec{\theta }$ 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, $\varvec{z}_m, \forall m$, were known, the ensuing estimation problem would be simplified. To this end, we can write the complete data likelihood for $\varvec{\lambda }$ and $\varvec{\theta }$ as

$$\begin{aligned} L_c\left( \varvec{y}, \varvec{z};\varvec{\lambda }, \varvec{\theta }\right)&= \prod _{m=1}^M P\left( \varvec{y}_m, \varvec{z}_m | \varvec{\lambda }, \varvec{\theta }\right) = \prod _{m=1}^M P\left( \varvec{y}_m; \varvec{\lambda }, \varvec{\theta }\right) P\left( \varvec{z}_m | \varvec{\lambda }, \varvec{\theta }\right) \nonumber \end{aligned}$$

where $\varvec{z}\equiv \left\{ \varvec{z}_m \right\} _{m=1}^M$. The associated complete data LL is

$$\begin{aligned} \ln L_c\left( \varvec{y}, \varvec{z};\varvec{\lambda }, \varvec{\theta }\right) = \sum _{m=1}^M \sum _{s=1}^S z_{ms} \ln f_s(\varvec{y}_m|z_{ms}, \varvec{\theta }_s) + z_{ms} \ln \lambda _s \end{aligned}$$

(4)

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 expectation of (4) over the possible realizations of $\varvec{z}$, given the observed data, $\varvec{y}$, and estimates of $\varvec{\lambda }$ and $\varvec{\theta }$, denoted $\hat{\varvec{\lambda }}$ and $\hat{\varvec{\theta }}$; and a maximization step, M-Step, where we update $\hat{\varvec{\lambda }}$ and $\hat{\varvec{\theta }}$ with arguments that maximize the expectation of $\ln L_c(\cdot )$. The EM algorithm alternates between the E and M steps (until a convergence criterion is met). Mathematically, the EM algorithm can be written as follows:

• Step 0: Initialization Choose randomly generated values for the parameters that define the segments, $\hat{\varvec{\theta }}^0$, and the mixture proportions $\hat{\varvec{\lambda }}^0$. The values for the mixture proportions are randomly generated from a Dirichlet prior distribution. Set the solution index $k$ $\leftarrow 0$.

• 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, $\varvec{z}$, given the observed contribution sequences, $\varvec{y}$, and the estimated parameters, $\hat{\varvec{\theta }}^k$, and $\hat{\varvec{\lambda }}^k$ at iteration $k$. We calculate the expected value of the LL function, $\mathcal {Q}$, as follows:

  $$\begin{aligned} \mathcal {Q} \left( \hat{\varvec{\lambda }}^k, \hat{\varvec{\theta }}^k \right) = \sum _{m=1}^M \sum _{s=1}^S \hat{p}_{ms} \left[ \ln \left( f_s\left( \varvec{y}_m|\hat{p}_{ms}, \hat{\varvec{\theta }}_s^k\right) \right) + \ln \left( \hat{\lambda }_s^k\right) \right] \end{aligned}$$

  (5)

  where, $\hat{p}_{ms}$ is the probability of member $m$ belonging to segment $s$, or the expected value of $z_{ms}$, $\mathbb {E}[z_{ms}|\varvec{y}_m]$.

• Step 2: Maximization Step (M-Step) Calculate the mixture proportions, $\hat{\varvec{\lambda }}^{k+1}$, and the parameters defining the segments, $\hat{\varvec{\theta }}^{k+1}$, that maximize the expected value of the complete-data LL calculated in Step 1, assuming that the missing data, $\varvec{z}_m$, are known—this is done replacing each $z_{ms}$ by its expected value, $\hat{p}_{ms}$, at iteration $k$. We calculate the mixture proportions, $\hat{\varvec{\lambda }}^{k+1}$, for $s = 1, \ldots , S$, as follows:

  $$\begin{aligned} \hat{\lambda }^{k+1}_s = \frac{1}{M}\sum _{m=1}^M \hat{p}_{ms} \end{aligned}$$

  (6)

  To calculate the optimizing parameters that define the normal distributions describing the distribution of annual contributions in each of the segments, $\hat{\varvec{\theta }}_s^{k+1} \equiv \left( \hat{\mu }_s^{k+1}, \hat{\sigma }_s^{k+1}\right)$, we do, for $s = 1, \ldots , S$, as follows:

  $$\begin{aligned} \hat{\mu }_s^{k+1}&= \frac{1}{M \hat{\lambda }_s^{k+1}} \sum _{m=1}^M \hat{p}_{ms} \frac{1}{T} \sum _{t=1}^T y_m^t \end{aligned}$$

  (7)
  $$\begin{aligned} \hat{\sigma }_s^{k+1}&= \sqrt{\frac{1}{M \hat{\lambda }_s^{k+1}} \sum _{m=1}^M \hat{p}_{ms} \frac{1}{T} \sum _{t=1}^T \left( y_m^t - \hat{\mu }_s^{k+1}\right) ^2} \end{aligned}$$

  (8)

  Set the solution index $k \leftarrow k+1$.

• Step 3: Check Stop Criteria While $k<K$ and

  $$\begin{aligned} \left| \left( \mathcal {Q}(\hat{\varvec{\lambda }}^{k+1}, \hat{\varvec{\theta }}^{k+1}) -\mathcal {Q}(\hat{\varvec{\lambda }}^k, \hat{\varvec{\theta }}^k)\right) /\mathcal {Q}(\hat{\varvec{\lambda }}^{k}, \hat{\varvec{\theta }}^{k})\right| > \epsilon \end{aligned}$$

  continue to iterate steps 1 and 2. The stopping criteria were set to $\epsilon = 1 \times 10^{-5}$, or a maximum of $K = 100$ iterations.

Relevant properties of the EM Algorithm, such as those related to its convergence, are discussed in references such as McLachlan and Thriyambakam (2001).

2. Derivation of Posterior Membership Probabilities

In this section, we derive expression presented in (2), which is used to update the membership probabilities in response to a contribution sequence. In particular, given a probability density function of $\varvec{\lambda }$, $f(\cdot )$, and a contribution sequence, $\varvec{y}_m$, the probability that individual $m$ is assigned to segment $s$, $p_{ms}$, is updated as follows:

$$\begin{aligned} p_{ms} = \mathbb {E} \left[ z_{ms}| \varvec{y}_m \right]&= \mathbb {E}_{\varvec{\lambda }} P\left( z_{ms}=1| \varvec{y}_m \right) \nonumber \\&= \int _{\varvec{\lambda }\in \mathbb {S}} P\left( z_{ms}=1| \varvec{y}_m, \varvec{\lambda }\right) f(\varvec{\lambda }| \varvec{y}_m) \partial \varvec{\lambda }\nonumber \\&= \int _{\varvec{\lambda }\in \mathbb {S}} \frac{P\left( z_{ms}=1, \varvec{y}_m| \varvec{\lambda }\right) }{P(\varvec{y}_m| \varvec{\lambda })} f(\varvec{\lambda }|\varvec{y}_m) \partial \varvec{\lambda } \end{aligned}$$

(9)

where Bayes’ Law yields:

$$\begin{aligned} f(\varvec{\lambda }|\varvec{y}_m) = \frac{ P(\varvec{y}_m|\varvec{\lambda }) f(\varvec{\lambda })}{\int _{\varvec{\lambda }\in \mathbb {S}} P(\varvec{y}_m|\varvec{\lambda }) f (\varvec{\lambda }) \partial \varvec{\lambda }} \end{aligned}$$

(10)

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

$$\begin{aligned} \mathbb {E}\left[ z_{ms}| \varvec{y}_m \right]&= \int _{\varvec{\lambda }\in \mathbb {S}} \frac{ P\left( z_{ms}=1, \varvec{y}_m | \varvec{\lambda }\right) }{\int _{\varvec{\lambda }\in \mathbb {S}} P(\varvec{y}_m| \varvec{\lambda }) f(\varvec{\lambda }) \partial \varvec{\lambda }} f(\varvec{\lambda }) \partial \varvec{\lambda }\\&= \frac{\int _{\varvec{\lambda }\in \mathbb {S}} P(\varvec{y}_m|z_{ms}=1, \varvec{\lambda }) P(z_{ms}=1|\varvec{\lambda }) f(\varvec{\lambda }) \partial \varvec{\lambda }}{\int _{\varvec{\lambda }\in \mathbb {S}} \sum _{r=1}^S P(\varvec{y}_m|z_{mr}=1, \varvec{\lambda }) P(z_{mr}=1|\varvec{\lambda }) f(\varvec{\lambda }) \partial \varvec{\lambda }} \\&= \frac{\int _{\varvec{\lambda }\in \mathbb {S}} P(\varvec{y}_m|z_{ms}=1, \varvec{\lambda }) \lambda _s f(\varvec{\lambda }) \partial \varvec{\lambda }}{\int _{\varvec{\lambda }\in \mathbb {S}} \sum _{r=1}^S P(\varvec{y}_m|z_{mr}=1, \varvec{\lambda }) \lambda _r f(\varvec{\lambda }) \partial \varvec{\lambda }} \\&= \frac{P(\varvec{y}_m|z_{ms}=1, \varvec{\lambda })\lambda _s}{\sum _{r=1}^{S}P(\varvec{y}_m|z_{mr}=1, \varvec{\lambda })\lambda _r} \\&= \frac{f_s\left( \varvec{y}_m\right) \lambda _s}{\sum _{r=1}^S f_r \left( \varvec{y}_m \right) \lambda _r} \end{aligned}$$

3. Goodness-of-Fit Measures and Segment Estimates

The CAIC is given as $-2\cdot LL + (p\cdot S + S-1 1)\left( \ln (n)+1\right)$, where $p$, $S$ and $n$ represent the number of parameters per segment (2 per segment in our analysis, $\mu _s, \sigma _s$, 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).

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.

Fig. 7

Consistent Akaike information criterion for $S$ = 1 to 8

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

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