Regression model for a count proces

Question

In R I have data where head(data) gives

day   count   promotion
1        33        20.8
2        23        17.1  
3        19         1.6  
4        37        20.8    

Now day is simply the day (and is in order). promotion is the promotion-value for the day. It is simply the number of times an advertisement has been on television. count is the number of new users we got that day.

I want to investigate the impact the promotion-value has on new users (count). Since we have a count process I thought it would be best to make a poisson regression model.

model=glm(formula= data$count ~ data$promotion, data=data)

When we type summary(model) we get

Coefficients:
           (Intercept)              good_users$promotion  
              13.40216                   0.24342

Degrees of Freedom: 793 Total (i.e. Null);  792 Residual
Null Deviance:      9484 
Residual Deviance: 9325     AIC: 12680

Here is a plot of the data.

But when I plot the fitted values for the model

points(model$promotion, model$fitted, col="blue")

we get this

Here is another plot that shows the same but where days with 0 promotion are removed.

How should I chose my regression model (should I use lm instead of glm) or is the another better approach to solve this? Because the data is not pretty but more random like this what should one do ?

Updated

Finding the sweet spot

I have done the following for finding a sweet spot. I divide data into 10 groups. group1 is simply a subset where the promotion-value is within 1:10. group2 is data where the promotion-value is between 11:20, and so on for the other groups. So in R we have

group1 <- subset(data, data$promotion %in% 1:10)
group2 <- subset(data, data$promotion %in% 11:20)
group3 <- subset(data, data$promotion %in% 21:30)
...
group10 <- subset(data, data$promotion %in% 91:100)

Now I can use wilcox.test to test if there is a significantly difference between the groups by typing

wilcox.test(group2, group1, alternative="greater")

which gives a low p-value, ie group2 has significant higher new_good_users than group1. The same goes for

wilcox.test(group3, group2, alternative="greater")

but for wilcox.test(group4, group3, alternative="greater") I get a p-value at 0.20, ie there is no significant difference in new_good_users between group4 and group3. And the same goes for the rest of the group-pairs up to 10.

So this must mean that if we increase promotion in the first groups we have an increase in new_good_users but in the last groups we do not have that increase. This means that we have a sweet spot at group3 where the promotion-value is 21:30. Is this not correct ?

Answer 1

I have to quote Tukey, perhaps the grandfather of data science:

The combination of some data and an aching desire for an answer does not ensure that a reasonable answer can be extracted from a given body of data.

I see nothing wrong with your Poisson model. In fact its a pretty good fit to the data. The data is noisy. There is nothing you can do about it. Perhaps the noise if due to whatever else is on TV at the time, or the weather, or the phase of the moon. Whatever it is, its not in your data.

If you reasonably think the weather might be affecting your data, get the weather data and add it. If it decreases the log-likelihood enough for each degree of freedom then it's doing a good job and you leave it in. This is regression modelling 101.

Of course there's a zillion other things you can do. Scale the data by any old transformation you want. Fit a quadratic. A quartic. A quintic. A spline. You could include the date and possible temporal correlation effects. But always bear in mind what Tukey was saying - if your data is noisy, you won't get anything much out of it. So it goes.