What is Poisson Regression?

Poisson regression is a type of generalized linear model used to model count data and contingency tables. It assumes that the response variable YYY follows a Poisson distribution, which means it models the rate at which events happen. This is particularly useful in marketing analytics where you often want to predict the number of occurrences of an event within a fixed period.

For example, Poisson regression can be used to model the number of transactions per customer, the number of clicks on a webpage, or the number of emails opened in a campaign.

The Poisson distribution formula is:

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where:

• P(Y=y) is the probability of y events occurring in a fixed interval.
• λ is the average rate of occurrence (mean of the distribution).
• e is the base of the natural logarithm (approximately equal to 2.71828).
• y is the actual number of occurrences (a non-negative integer).

For example, Poisson regression can be used to model the number of transactions per customer, the number of clicks on a webpage, or the number of emails opened in a campaign.

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When Not to Use Poisson Regression

While Poisson regression is powerful, there are scenarios where it may not be the best choice:

1. Zero-Inflated Poisson (ZIP) Models: If your data has an excess of zero counts (more zeros than expected in a Poisson distribution), a Zero-Inflated Poisson model might be more appropriate. This is common in datasets where some individuals have no chance of experiencing the event (e.g., non-buyers in transaction data).
2. Overdispersion: Poisson regression assumes that the mean and variance of the count data are equal. If the variance is significantly higher than the mean (overdispersion), Poisson regression might not fit the data well. In such cases, a Negative Binomial regression can be a better alternative.

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When to Use Ordinary Least Squares (OLS) Regression Again

If you are dealing with continuous outcome variables rather than count data, or if the assumptions of Poisson regression are violated (e.g., severe overdispersion or zero inflation), you might consider using OLS regression. OLS is suitable for modeling relationships where the dependent variable is continuous and normally distributed.

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Example: Predicting Number of Transactions

Let's consider an example where we want to predict the number of transactions per customer based on several predictors: net price, number of direct mails, number of SMS, number of emails, and household income.

Here is a summary of the results from a Poisson regression analysis:

| Parameter | Estimate | Standard Error | t value | Pr > |t| |

|------------|----------|----------------|--------|--------|

| Intercept | 0.05 | 0.02 | 2.50 | 0.012 |

| Net Price | -0.01 | 0.01 | -1.00 | 0.320 |

| Direct Mails | 0.10 | 0.03 | 3.33 | 0.001 |

| SMS | 0.15 | 0.02 | 7.50 | <0.001|

| Emails | 0.12 | 0.04 | 3.00 | 0.003 |

| HH Income | 0.02 | 0.01 | 2.00 | 0.045 |

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From this table, we can interpret the following:

• Intercept: The baseline level of transactions when all predictors are zero.
• Net Price: The estimate is negative, suggesting that as net price increases, the number of transactions slightly decreases, though this effect is not statistically significant (Pr > |t| = 0.320).
• Direct Mails: A positive estimate indicates that more direct mails are associated with an increase in transactions.
• SMS: Similarly, SMS have a strong positive association with the number of transactions.
• Emails: Emails also positively influence transactions.
• Household Income: Higher household income is associated with a slight increase in transactions.

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Conclusion

Poisson regression is a robust tool for modeling count data, particularly in marketing analytics where it can help predict the number of occurrences of an event. However, it is important to check for overdispersion and zero inflation before choosing Poisson regression. If these issues are present, consider alternative models such as Negative Binomial regression or Zero-Inflated Poisson models. For continuous outcome variables, OLS regression remains a valid choice.

By understanding when and how to apply Poisson regression, marketers can more accurately model and predict customer behaviors, ultimately leading to more effective and targeted marketing strategies.

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