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{"sheet":"Solve for k","index":0,"n_rows":13,"n_cols":2,"rows_kept":13,"cut":false,"hidden":false,"merged":[]}
["Estimate k in NBD model","(Enable macros for this worksheet to function)"]
["Solve for k "]
["Mean per person per period (mu)",4.25]
["Proportion of non-buyers per period P(0)",0.3]
["k (to solve)",0.56020000000000003]
[]
["Model P(0) from k, mu",{"$f":"(k/(k+mu))^k","v":0.29982926454801928}]
["Error = Model P(0) - Observed P(0)",{"$f":"B7 - P_0","v":-1.7073545198070805E-4}]
[]
["Penetration (1 - Observed P(0))",{"$f":"1 - P_0","v":0.7}]
["Model penetration (1 - Model P(0))",{"$f":"1 - B7","v":0.70017073545198072}]
[]
["Predicted variance of purchase counts",{"$f":"mu * (1 + mu /k)","v":36.492948946804709}]

{"sheet":"Explanations","index":1,"n_rows":82,"n_cols":4,"rows_kept":82,"cut":false,"hidden":false,"merged":[]}
[]
[null,"1. The Negative Binomial as a “Poisson–Gamma mixture”"]
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[null,"The Negative Binomial Distribution (NBD) is often described as a Poisson distribution whose mean varies across people according to a Gamma distribution."]
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[null,"Each individual consumer has their own average purchase rate (say, “expected number of purchases per period”)."]
[null,"Across all consumers, these individual rates are not identical — some people buy a lot, some a little, many not at all."]
[null,"The Gamma distribution describes that heterogeneity in purchase rates."]
[null,"When you combine (or mix) the Gamma with a Poisson (for random event counts), you get the Negative Binomial Distribution."]
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[null,"2. The role of k — the “Size” or “Dispersion” parameter"]
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[null,"The parameter k (sometimes called r or shape) comes from that Gamma distribution that underlies the NBD."]
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[null,"It controls how much variation there is between people’s purchasing rates."]
[null,"So k is essentially a heterogeneity or dispersion parameter."]
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[null,"Here’s what it means in practice:"]
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[null,"Situation","k value","Interpretation"]
[null,"Large k (say >10)","Very little variation between people","Everyone buys at about the same rate → distribution looks almost Poisson"]
[null,"Small k (say <1)","Huge variation between people","A few heavy buyers, many light/non-buyers — a “long-tailed” pattern"]
[null,"Moderate k (~1–3)","Typical marketing scenario","Some moderate skew, common in brand purchase and repeat data"]
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[null,"Mathematically, the variance of the NBD is"]
[]
[null,"Var(X)=μ(1+μk)\\text{Var}(X) = \\mu \\left( 1 + \\frac{\\mu}{k} \\right)Var(X)=μ(1+kμ​)"]
[]
[null,"So when k is small, the variance is much bigger than the mean — the data are overdispersed."]
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[null,"3. Intuitive analogy"]
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[null,"Think of k as a measure of how similar or different your customers are:"]
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[null,"k → ∞ → everyone behaves the same → pure Poisson world"]
[null,"(e.g., every person buys 1.2 times per year on average)"]
[null,"k → 0 → extreme diversity → most people never buy, a few buy often"]
[]
[null,"So small k = more inequality in buying frequency, which is what we typically observe in real markets."]
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[]
[]
[null," 4. In the Ehrenberg–Bass / Byron Sharp context"]
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[null,"In How Brands Grow and related work:"]
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[null,"μ represents the average category purchase frequency (for the brand or the category)."]
[null,"k captures buyer heterogeneity — differences in propensity to buy."]
[]
[null,"Together, they let you model:"]
[]
[null,"the proportion of non-buyers (P(0))"]
[null,"penetration (1−P(0))"]
[null,"repeat buying rates"]
[null,"and patterns like duplication of purchase (when extended to the NBD–Dirichlet model)."]
[]
[null,"Typical empirical values for k are 0.5–1.5, which correspond to highly skewed distributions — many light buyers, few heavy ones."]
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[]
[]
[null,"What “Observed zero-class probability (P(0))” means"]
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[null,"In the Negative Binomial model, P(0) represents the proportion of consumers who make zero purchases during a given time period."]
[]
[null,"In other words:"]
[null,"P(0) = probability that a randomly chosen person makes no purchases at all in the period."]
[]
[null,"So in real data, your observed P(0) is simply the proportion of people who did not buy (the “non-buyers”)."]
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[null,"Example"]
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[null,"Suppose you have a customer panel of 10,000 people, and:"]
[null,"3,000 bought at least once,"]
[null,"7,000 bought nothing in that period."]
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[null,"Then:"]
[]
[null,"P(0)=700010000=0.7P(0) = \\frac{7000}{10000} = 0.7P(0)=100007000​=0.7"]
[]
[null,"and"]
[]
[null,"Penetration=1−P(0)=0.3\\text{Penetration} = 1 - P(0) = 0.3Penetration=1−P(0)=0.3"]