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Say I want to do a logistic regression on whether it snows on a given month of the year and day of the week. I'd have 12 months and 7 days. My understanding is that this would translate into 17 dummies (11 for months and 6 for days).

First off is this correct? Second, how would I interpret regression results (i.e. assuming I dropped January, what's the p-value for January)?

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Short answer is yes; you can use multiple groups. Many researchers do it all the time.

What is the response here? Snows or doesn't or (snow if precipitation or rain if precipitation)?

What's correct here depends on your model of the data generating process. The prejudices here draw upon some experiences with environmental data, including climate data.

Day of the week as a control presumably depends on some idea that human-produced effects may mean more or less chance of snow on certain days. So, 7 days, 6 indicators (dummies if you will).

But in what sense does the climate system know (e.g.) when it changes from June to July? Or in what sense do the irregularities of the months have any possible bearing on the climate? Time of year is much more likely to be better modelled as one or more sine and cosine terms.

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Yes, as Nick Cox says, you can use dummy variables.

In answer to the second part of your question, the interpretation is a bit different from what you seem to be thinking.

For months, if you decide that January is the reference group and therefore don't include a dummy variable for January, then the interpretation of the p-values for all the other 11 months is "the probability that the observed difference in the probability of snow in month X compared to that in January would be observed if there was actually no difference (i.e. if the null were true)". So, there is no p-value for January, only for the other months, because the question you are asking is "is it more or less likely to snow in a given month than in January?". This tests the hypothesis: $\beta_X=0$, where $\beta_X$ is the parameter estimate for month X and represents the change in probability of snow in month X compared to January.

If you want to address the more general question: "does month have any effect on whether or not it snows?" you need to use the p-value from a test of the global null hypothesis. The global null hypothesis states that $\beta_1 = \beta_2 =...=\beta_{11} = 0$, that is, that all the parameter estimates for all the categories of your month variable (all the dummy variables for month) are equal to 0. Most statistics programs should have an option for this, but it may not be in the default output.

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  • $\begingroup$ For the global null hypothesis, that should be up to $beta_{12}$. Also, in this case you need to omit the intercept otherwise you will fall into dummy variable trap. $\endgroup$ – Metrics Jun 25 '13 at 16:00
  • $\begingroup$ There are only 11 dummy variables - there is no $\beta_{12}$, and I've never seen a requirement to omit the intercept for the global null. The global null is just for the month variables and doesn't depend on other covariates, etc (in so far as those variables have independent effects). $\endgroup$ – Ellie Jun 25 '13 at 17:25
  • $\begingroup$ My apology. The one I mentioned before is another way of testing the global null hypothesis if you also need the p-value of January (in which case you have to drop intercept but not January) $\endgroup$ – Metrics Jun 25 '13 at 17:55

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