# Regression model for a count proces

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 ? • Your data is not at all Poisson distributed so that this is not giving a good fit makes sense, I do not know a better approach however – Jan van der Vegt Jun 2 '16 at 11:28 • It seems like possibly the design of the analysis is out of whack in some sense. For a start, have you plotted the new users against either the days or against the cumulative number of promotions? I also note that apparently you expect a non-linear relationship (or at least a relationship that implies a square term - you want to find the 'sweet spot') but you don't appear to have attempted to fit a model that will account for that effect. – Robert de Graaf Jun 2 '16 at 11:39 • Yes I have plotted new users against promotion here: datascience.stackexchange.com/questions/11915/… . So by square you mean I should make a regression model where I square the independent variable ? – Ole Petersen Jun 2 '16 at 12:04 • Having a quick look at the other question I note that your plots suggest 'new users' has a relationship with promotion, but 'new good users' does not . Yes, squaring the promotion term is broadly what I meant, as a quadratic has a maximum, which you say is what you are trying to find, but again while the 'new users' vs 'promotion' plot looks like a maximum could plausibly found in those data, the plot of 'new good users' doesn't look all that likely to yield such a relationship. I say 'broadly what I meant' because it may really be non-linear, as said previously in the other question by XR SC. – Robert de Graaf Jun 3 '16 at 1:36 • Your plot is wrong. plotting point(model$fitted) will plot the points with 1:N on the X-axis, yet the fitted values are those for the corresponding values of promotion in the data. Try plotting points(data$promotion, m$fitted). it should look a bit better, but there's still no obvious linear trend in your data... – Spacedman Jun 4 '16 at 12:18