I have the following dataset:

> df1
     Location SalesQty product
1  location 3        1   prod1
2  location 3        0   prod1
3  location 3        3   prod1
4  location 5        3   prod1
5  location 5        0   prod1
6  location 5        4   prod1
7  location 3        2   prod2
8  location 3        5   prod2
9  location 5        2   prod2
10 location 5        1   prod2

I want to perform a poisson regression to predict/estimate the SalesQty of prod1 on location 3 and 5 and prod 2 on location 3 and 5 (I know there are not enough datapoints such that a predictor will be significant). The SalesQties can be visualised as: enter image description here

If you run:

Reg <- glm(SalesQty ~ Location, family = "poisson", data = df1)

The predictions on each location is just the average per location. This is due to the least squares error per location.

If you run:

Reg <- glm(SalesQty ~ product + Location, family = "poisson", data = df1)

I am figuring out how the coefficients of each categorized predictor play a role in the formula for predicting the SalesQty of a product on a location.

Only considering the location, the formula will be: ln(SalesQty) = $\beta_0$ + $\beta_5 I_5$ with $\beta_0$ beiing the intercept corresponding to location 3. Now $exp(\beta_0)$ is the average of the SalesQties of location 3 and $exp(\beta_0$ + $\beta_5)$ is the average of the SalesQties of location 5. But when considering 2 categorical predictors, the location ánd the product, I don't see how to interpret the coeficients and the modelling formula.

I hope someone can send me in the right direction

Suppose your model is $$\log(\lambda) = \beta_0 + \beta_1 X_1 + \beta_2 X_2$$ and $Y\sim \text{Poisson} (\lambda)$. $Y =$ SalesQty. $X_1 = 1$ for prod2, = 0, otherwise. $X_2=1$ for location 5, = 0 otherwise.

Then

$e^{\beta_0}$ is mean for prod1 and location 3

$e^{\beta_0+\beta_1}$ is mean for prod2 and location 3

$e^{\beta_0+\beta_2}$ is mean for prod1 and location 5

$e^{\beta_0+\beta_1+\beta_2}$ is mean for prod2 and location 5

  • Thanks for your answer. But if I run the regression and calculate $e^{\beta_0} = 1.920635$. However, the mean of weekly sales of product 1 on location 3 is $\frac{4}{3}$. I actually thought that 1.920635 is obtained by combining the mean of weekly sales of product 1 ($\frac{11}{6}$) and the mean of the weekly sales on location 3 ($\frac{11}{5}$) in some way. – Jorden Nov 9 at 7:56
  • If you add the interaction between location and production into model, you will get the estimates close to each observed mean. – a_statistician Nov 9 at 14:32

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