Transcript Probability Models in Marketing

Probability Models in Marketing
• Marketing models attempt to describe or predict
behaviour
– Usually include a random element to allow for
imperfect knowledge
• We will develop probability models that specify a
random model for individual behaviour
– Sum this across individuals to get a model of
aggregate measures
– May need to incorporate differences between
individuals into the model
Uses of Probability Models
• Understand and profile individual behaviour
• Understand market-level patterns, and their
origin in individual behaviours
• Provide norms or benchmarks for comparison
– Ehrenburg: Understanding Buyer Behaviour; and
Repeat-Buying (1988)
– Latter book available free online at
http://www.empgens.com/ehrenberg.html#repeat
• Prediction or forecasting of:
– Aggregate results beyond current observation period
– Individual behaviour, given knowledge of past actions
Product Trial Example
• Have a newly launched product
– Multi-pack juice drink, aimed at children
– Launched in test market
– May be rolled out nationally if successful
• Measure trial over time
– Based on household scanner panel data, e.g.
ACNielsen’s HomeScan
• Have data from first 13 weeks
• Want to predict trial 13 weeks later
Cumulative Trial Penetration
Week
Cum. % Hhlds
Tried Product
(n=1499)
1
0.6%
2
1.1%
3
1.2%
4
2.5%
5
3.1%
6
3.6%
7
3.8%
8
4.0%
9
4.4%
10
4.6%
11
5.0%
12
5.1%
13
5.2%
Cumulative Trial
Cumulative % of Households Who Have Tried Product
10.0%
9.0%
8.0%
7.0%
6.0%
5.0%
4.0%
3.0%
2.0%
1.0%
0.0%
1
2
3
4
5
6
7
Week
8
9
10
11
12
13
Develop Probability Model
• Variable of interest (for individual households)
– When did they first try the product?
• Treat time of first purchase T as a random
variable
– Assume this has an exponential distribution, with trial
rate λ
– Probability of trial by time t for each household is
F t   PT  t   1  e
t
– Averaging this across all households would give the
same result, but this would not be realistic – why?
Market Level Model
• Assume there are two groups of consumers
– One group may try product (λ>0)
– Other group will never try product (λ~0)
– In proportions p and 1-p respectively
• “Exponential with never-triers” model:
PT  t   pFt |      1  p Ft |   0 
 p1  et 


– Note: technically this is not a cdf as it does not =1 as t
approaches infinity, but as we are only dealing in relatively
small values of t this approximation is valid.
Estimate Parameters
• Model has parameters p and λ
• Estimate these parameters using maximum
likelihood
– The likelihood function is the probability that this
dataset would be observed
• Viewed as a function of the parameters
• Assumes the model holds
• L(parameters) = P(this data observed|parameters)
– The maximum likelihood estimates (MLEs) of the
parameters are the values that maximise L(.), for the
given dataset
– Can equivalently maximise l(.), the log-likelihood
Implementing MLE
• The maximum likelihood method can be
implemented relatively easily in many software
environments
– E.g. R, SAS, Excel
– It may already be implemented if the model is
commonly used
• R code for exponential w. never-triers model:
• trial<-c(8,14,16,32,40,47,50,52,57,60,65,67,68)
• Trial1 <- trial – c(0,trial[1:12])
• F <- function(t,p,lambda) {
p*(1-exp(-lambda*t))
}
R Code (continued)
l <- function(p,lambda,data) {
week <- 1:13
if ((p>=0) && (p<=1)) {
sum(data*log(F(week,p,lambda) - F(week-1,p,lambda))) +
(1499-sum(data))*log(1-F(13,p,lambda))
} else {NaN}
}
optim(c(.2,.2),function(param) {-l(param[1],param[2],Trial1)})
• Result: maximum value of log-likelihood is -445.84,
which is achieved at p=0.060 and λ=0.109
• Complications due to sample design and weighting have
been ignored
Forecasting
• Can use fitted model to forecast trial
• Let N(t) be a random variable, being the
number of households in the panel
purchasing the product by time t
• Forecast trial as:
E N t   nFˆ t 

 npˆ 1  e
 ˆt

20
40
60
80
100
Cum. No. of Households Trying Product
Cumulative Trial Forecast
0
5
10
15
Week
20
25
Model Extensions
• Current model assumes same trial rate for
all households, except never triers
– May be overly simplistic
• Can allow for multiple segments of
households, each with different underlying
trial rate
F t    pk F t k ,
K
k 1
1  0,
K
p
k 1
k
1
Model Extensions
• Finite mixture models can
be hard to fit
– Local minima are common
• Another alternative that
allows for consumer
heterogeneity is a
continuous mixture model
– Assume trial rates are
distributed with pdf g(λ)
– The discrete mixture model
can be thought of as an
approximation to the
underlying continuous
distribution of trial rates
Gamma Trial Rate Distribution
• Assume trial rates are distributed
according to a gamma distribution
   1  
g   
 e ,  0
 
where α is a shape parameter and β is an
inverse scale parameter
• The gamma distribution is a flexible,
unimodal, mathematically tractable
distribution
Market-Level Model
• The resulting cumulative distribution of first
trial times, at an overall market level, is
F t   PT  t 
  PT  t  g  d

0
  

 1  
 t

– This is called an exponential-gamma model
Estimating Parameters
• R Code for finding MLEs:
Fg <- function(t,alpha,beta) {
1 - (beta/(beta+t))^alpha
}
lg <- function(alpha,beta,data) {
week <- 1:13
sum(data*log(Fg(week,alpha,beta) - Fg(week-1,alpha,beta))) +
(1499-sum(data))*log(1-Fg(13,alpha,beta))
}
optim(c(1,1),function(param) {-lg(param[1],param[2],trial1)})
• Result: maximum value of log-likelihood is -446.64,
which is achieved at α=0.0416 and β=6.32
Further Extensions
• Could add a “never try” component into
the exponential-gamma model
• Could incorporate the effects of marketing
covariates
– E.g. advertising weight over time
• Could incorporate the effects of household
covariates
– E.g. presence of children
Building a Probability Model:
General Approach
1. Determine the marketing problem or
information needed
2. Identify the behaviour of interest at the
individual level
– Make sure this is observable; denote by x
3. Choose an appropriate probability distribution
f(x|θ)
– The parameters θ of this distribution can be thought
of as latent traits of each individual
•
Latent or underlying traits; not observed directly but affect x
General Approach (continued)
4. Specify a distribution for the latent traits
across the population
– Denote this by g(θ)
•
•
Called the mixing distribution
Can be discrete, continuous or a combination
5. Obtain the resulting aggregate marketlevel distribution (if this is observed or of
interest) by integrating with respect to θ
General Approach (continued)
6. Estimate the parameters of the mixing
distribution
– Usually done using maximum likelihood
– Check model fit, graphically if possible
7. Use the fitted model to solve the
marketing problem or to obtain the
required information
Outdoor Advertising Example
• Advertisers can buy a “monthly showing” on a
set of specific billboards
• Effectiveness of the showing is primarily
evaluated through three measures
• Reach, frequency and gross ratings points (GRPs)
• Measures derived from daily travel maps filled in
by a sample of people
– An “exposure” is counted when a respondent goes
past one of the billboards, while facing the billboard
• Have data from each person for one week
• Want to project from this data to get measures
for the relevant month (or four weeks)
Measures of Advertising Exposure
• Three measures are commonly used
– Reach is the proportion of people exposed to the
advertising at least once during the month
– Frequency is the number of times each person is
exposed to the advertising message
• Usually summarised as the average frequency, which is the
average number of exposures experienced among those who
were exposed
– Gross rating points (GRPs) is the mean number of
exposures per 100 people
• This is just the product of the reach (expressed as a
percentage) with the average frequency
Distribution of
Billboard Exposures
(during one week)
# of Exposures
# of People
# of Exposures
# of People
0
48
12
5
1
37
13
3
2
30
14
3
3
24
15
2
4
20
16
2
5
16
17
2
6
13
18
1
7
11
19
1
8
9
20
2
9
7
21
1
10
6
22
1
11
5
23
1
Model: Aim and Approach
• Goal: Develop a model that uses one
week data to provide an estimate of the
monthly performance measures
• Approach
– Model the weekly exposure distribution
– Derive the monthly exposure distribution
under this model, and estimate summary
statistics for the month
Probability Model
• Let X denote the number of billboard
exposures during one week
• For each person, X is assumed to have a
Poisson distribution with
rate
parameter
λ
x 
P X  x   
e
x!
• We assume that the exposure rates λ
have a gamma distribution
   1  
g   
 e ,  0
 
Probability Model
• Aggregating across the population (i.e.
integrating with respect to λ) gives

P X  x    PX  x  g  d
0
  x    



 x!    1 

 1 


1  
x
• This Poisson-Gamma distribution is also
known as the negative binomial
distribution, or NBD
– It has mean α/β and variance α(β+1)/β2
Estimating Model Parameters
• R Code:
expodist <c(48,37,30,24,20,16,13,11,9,7,6,5,5,3,3,2,2,2,1,1,2,1,1,1)
lnbd <- function(alpha,beta,data) {
expos <- 0:23
prob <- beta/(beta+1)
sum(data*log(dnbinom(expos,alpha,prob)))
}
optim(c(1,1),function(param) {-lnbd(param[1],param[2],expodist)})
• Result: maximum value of log-likelihood is -649.7, which
is achieved at α=0.969 and β=0.218
Exposure Distributions
0
10
20
30
40
Observed exposure distribution
Exposure distribution from fitted model
0
2
4
6
8
10
12
14
16
18
20
22
NBD For More Than 1 Week
• Let X(t) denote the number of exposures
experienced by a person over t weeks
• Suppose that over one week, the
exposure distribution for that person is
Poisson(λ)
• Then X(t) is also Poisson, with rate
parameter λt
NBD For More Than 1 Week
• The market-level exposure distribution is

P X t   x    P X t   x  g  d
0
  x    



 x!    t 

 t

t 



• This has mean EX t   t  .
x
0
10
20
30
40
50
60
Exposure Distributions: 1 week vs 4 weeks
0
2
4
6
8
10
12
14
16
18
20
22
Performance of Monthly Showing
• For t=4:
– P(X(t)=0) = 0.056
– E[X(t)] = 17.82
• So:
– Reach = 1 - P(X(t)=0) = 94.4%
– Average Frequency = E[X(t)] / (1 - P(X(t)=0)
= 18.9
– GRPs = 100* E[X(t)] =1782
Log-Likelihood Calculation
• If data available as counts (for discrete or
discretised data), can use
– Sum of (count times log probability)
• E.g. sum(data*log(dnbinom(expos,alpha,prob)))
– Sum of (count times (increase in distribution
function))
• E.g.
sum(data*log(F(week,p,lambda) - F(week-1,p,lambda)))
+ (1499-sum(data))*log(1-F(13,p,lambda))
Direct Marketing Example
• Have customer database containing data
on past purchases
– 126 segments defined based on purchase
histories
• We’ll cover segmentation methods later
• Believe that some customers are more
likely to respond to mail-out than others
– Send test mail-out to 3% sample of customers
– Analyse response by segment to identify most
profitable groups to target
Target Segments
• Profitable to send mail-out if it costs less than the profit
on resulting sales
– i.e. if the expected rate of purchase response (PRR) is above the
following cut-off:
PRR > cost per letter of mail-out / unit margin
– Mail-out cost is 33.43 cents per letter
– Unit margin is $161.50
– Cut-off rate is 0.21%
• Standard approach
– Conduct full mail-out to all segments with test PRR above this
cut-off value – 51 segments in this case
• There is a problem with this rule – what is it?
• Manager chose to mail-out to 47 of these segments, plus
another 24 segments
Purchase Response Rates
Test vs Full Mail-out (47 Segments)
8%
7%
6%
Full PRR
5%
4%
3%
2%
1%
0%
0%
2%
4%
Test PRR
6%
8%
Develop Probability Model
• Objective is to enable better decisions
based on the test mail-out dataset
• Outcome variable is the number of
responses for a specified number of letters
mailed, by segment
• Suggests a binomial distribution
Model Development
• Notation:
– Ns = size of segment s (for s = 1, 2, …, S)
– ms = number of test letters sent to members of
segment s
– Xs = number of purchases due to responses from
segment s
• Assume that all members of segment s have the
same probability of purchase response ps, and
they respond/purchase independently
• Then Xs is a binomial random variable
Applying the Model
• What is our best estimate of ps given a
response of xs to a test mail-out of size
ms?
• Intuitively we might expect a weighted
average of the population mean response
and the response in that segment, i.e.:

xs
E  ps xs , ms   
 1   

ms
Bayes Theorem
• The prior distribution g(p) describes the distribution p is
believed to follow, before any data is collected
• The posterior distribution g(p|x) reflects the distribution p
is believed to follow, taking the observed data x into
account
• According to Bayes theorem,
g  p x 
f x p g  p 
 f x p g  p dp
i.e. the posterior is proportional to the prior times the
likelihood
Empirical Bayes Approach
• In a true Bayesian analysis, the prior
distribution is specified before looking at
the data
• For an empirical Bayes analysis, a “prior”
distribution is calculated from the data
• The posterior distribution is then
calculated using Bayes theorem, as on the
previous slide
Model-based Decision Rule
• Roll-out to segments with
0.3343
W  ps X s  xs , ms  
 0.0021
161 .5
• 66 segments qualify under this criterion
• To test this approach, compare its
performance with the manager’s approach
(and the standard rule)
Results
Standard
Manager
Model
# Segments
51
Actual # Seg.
47
71
55
682,392
858,728
732,675
4,463
4,804
4,582
$492,651
$488,773
$495,060
Contacts
Purchases
Profit
66
• Model is over $6,000 more profitable than the manager’s
selection
• The model is evaluated on the 55 segments for which there is data
Concepts Introduced
• Binary choice processes
• Beta-Binomial model
• Regression to the mean
– How to use models to allow for this effect
• Bayes theorem
– Empirical Bayes methods
• Application of EB to direct marketing campaigns
Types of Observed Variables
• Have introduced three types of
behavioural outcomes
– Timing – “when?”
– Counting – “how many?”
– Choice – “whether/which?”
• These are widely encountered in a range
of situations
Applications of Timing Models
• Product trial
• Repeat purchasing
• Response times
– Direct mail
– Mail or e-mail survey responses
• Customer retention or attrition
• Other durations
– Time spent on a web site
– Job tenure for salespersons
Applications of Counting Models
•
•
•
•
Number of advertising exposures
Number of pages viewed per web session
Salesperson productivity
Sales concentration among customers
– E.g. 80/20 rule
• Number of each item bought, or number of
distinct items, per shopping occasion
• Number of trips
– Shopping, bus or plane travel, park visits, fishing
Applications of Choice Models
• Brand choice, e.g.
– choice modeling questionnaire (exclusive choice)
– scanner panel data (non-exclusive choice)
• Media exposure
• Binary variables
– Response
• Direct mail
• Click-through for Web banner advertisements
• Survey non-response (non-contacts, refusals)
– Brand usage, awareness, image/associations
Combined Models
• Two outcome variables
– Counting + counting
• Purchase - # of shopping trips & # of units bought per trip
• Web site traffic - # of visits & # of pages viewed per visit
– Counting and timing
• Purchases – spacing of trips & # of items bought/trip
• Web site - # of visits & duration of each visit
– Counting and choice
• # of visits & whether trip involved purchase
• choice of brand & # of units purchased
Generalisations
• If there are problems with model fit, we can use
a different distribution or relax the usual
assumptions
– Non-exponential distribution for purchasing intervals
• E.g. gamma distribution (Exp=Gamma(1,β))
• Implies non-Poisson distribution of counts
– Non-gamma or non-beta heterogeneity
• E.g. never try/buy group
– Non-stationarity – latent traits may change over time
• However the usual models appear quite robust
to departures from the standard distributions and
assumptions
Other Extensions
• Introducing covariates
• Finite mixture/latent class models
• Hierarchical Bayes methods
– These account properly for uncertainty at the
population/market level