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I have a series of non-negative integers $y=(y_1,y_2,..., y_n)$ and a design matrix $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_1 x_2$, where $x_0$ and $x_1$ are $0$ or $1$, $x_1x_2$ is the interaction, and $\beta_0 \ldots \beta_3$ are parameters we want to estimate. For example, the data look like

y    x1    x2    x1*x2
10   0     0     0
23   0     1     0
18   1     1     1
19   1     0     0
25   0     1     0
...

I want to estimate the $\beta_0$, $\beta_1$, $\beta_2$ and $\beta_3$ coefficients and perform a test to see if any coefficient is nonzero.

There are several different regression models that might be applied to this case:

  1. Simple linear regression: lm
  2. Poisson regression (when $y$ follows a Poisson distribution): glm with family = poisson
  3. Quasi-poisson regression (when $y$ is over-dispersed; that means $\text{sd}(y) \gt \text{mean}(y)$): glm with family = quasi-poisson
  4. Negative binomial regression (when $y$ is over-dispersed, $\text{sd}(y) \gt \text{mean}(y)$): glm.nb, in MASS package.

The questions I want to ask are:

  1. How should I select the model for this dataset? Is there any way to choose the right model based on some descriptive statistics of my dataset?
  2. How should I check and validate if the fitted selected model is right for my data?
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  • $\begingroup$ Could you specify what kind of outcome ($y$) you are studying? It might further help to decide among the various modeling strategies you proposed, or suggest other ones. $\endgroup$ – chl Oct 3 '11 at 11:57
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    $\begingroup$ It looks like you're overthinking this one. You have only four distinct categories, corresponding to $(x_1,x_2) \in \{(0,0),(0,1),(1,0),(1,1)\}$. Allowing four parameters is tantamount to fitting the $y$'s separately in each category. Therefore you really have four independent problems and each of them reduces to summarizing a univariate dataset. Where to go from here depends on the nature of the $y$'s and the purpose of the model, neither of which is apparent in the question. $\endgroup$ – whuber Oct 21 '11 at 20:24
  • $\begingroup$ Use a random forest? $\endgroup$ – Jase Feb 8 '14 at 7:32
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Your model is fully saturated, because you have indicators for every possible combination of categories. As such you have correctly specified the conditional expectation.

Any MLE estimate based on a distribution in the linear exponential family is consistent when the conditional expectation is correctly specified. Therefore you can use Poisson or a number of other distributions.

As whuber implies though, this problem reduces to estimating means and testing for their differences, which could conveniently be done in a regression framework.

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