Which statistical test should I use to see if the month of onset of a disease is random or clusters in the winter? I have data showing which month a disease started for 119 subjects. The months are represented by 1-12 (Jan - December). We hypothesize the disease more often onsets during winter months. What test should I use?
This is the data we have.
Month (line 1);  Number of people with disease onset in this month (line 2):
1,  2, 3,  4, 5, 6, 7, 8, 9, 10, 11, 12
11, 11, 9, 10, 7, 7, 5, 9, 9, 16, 11, 14
So far, I've attempted to use a Poisson test in SPSS and my interpretation of the results is that there's no significant difference from a random distribution (Omnibus test 0.155 significance).
However, I was hoping I could confirm with someone that this is the correct test and that I'm interpreting this accurately? Just eye-balling the data suggests there does seem to be less counts in the summer months, so I was a bit surprised there was no significance.
If there's a different test I should do in SPSS, please let me know!

You have a specific hypothesis tantamount to an annual cycle peaking around January (month 1) and a trough around July.  One way to test this would be to fit a simple family of cyclic functions with these properties. You can't get any simpler than a single such function $f$ (along with an intercept to model the average count), leading to a model of the form
$$X(t) \sim \mathcal{D}(\beta_0 + \beta_1f(t))$$
where $\mathcal D$ is an appropriate family of possible count distributions parameterized by their expectation.
Because these responses are counts, it is appropriate to choose the Poisson family, as you have done.
One of the simplest and nicest such functions would be the cosine; namely,
$$f(t) = \cos\left(\frac{2\pi(t-1)}{12}\right).$$
Any positive multiple of $f$ has a peak in January and a trough in July, varying smoothly between the two.
For your counts $X(t)$ ($t$ denotes the month) this gives the model
$$X(t) \sim \text{Poisson}\left(\beta_0 + \beta_1 \cos\left(\frac{2\pi(t-1)}{12}\right)\right).$$
The null hypothesis is that there is no such cyclic pattern: this is equivalent to $H_0:\beta_1 \le 0.$  The alternative is that there is such a pattern, $H_A:\beta_1 \gt 0.$ Notice that this is a one-sided test, unlike almost all tests of regression coefficients.
I have described a Generalized Linear Model.  Fit it in SPSS by creating a variable with the values $f(1), \ldots, f(12)$ corresponding to your $12$ counts.  In the Type of Model tab of the Analyze | Generalized Linear Models dialog select the Poisson loglinear model.
The output will include a p-value for the cosine, but this is the wrong p-value, because it conducts a two-sided test.  Convert it into the correct p-value as follows.  First, divide it by $2.$  Next, if the estimated cosine coefficient $\hat\beta_1$ is negative, subtract that from $1.$  If your p-value is sufficiently small, you have evidence against this null model in favor of the winter-peak, summer-trough cosine approximation.

To illustrate this approach, and to give you a feel for what data look like under the null hypothesis, I generated 20 more random datasets in several ways.  One way, shown here, randomly permutes the data points. Here are plots of each of those datasets against time along with the corresponding cosine curves that were fit:

(The points are colored according to their distances from the fitted curve, with blues being the closest.)
I find it interesting how often even random points seem to follow a cosine curve pretty well: the best cosine fit is rarely flat.  This should inspire us to be cautious.  Just because counts seem a little high in the winter and a little lower in the summer, you shouldn't necessarily conclude you have found a real pattern.
Any plot with a p-value of $0.05 = 1/20$ or less is shown in red.  Under the null hypothesis, we should expect one in every 20 such plots to be red on average.
There are $21$ datasets shown here, because your data are included.  They are second from the right in the second row, and it just so happens these are the only dataset shown with a p-value less than $0.05.$  Your p-value is $0.02$ (that is, $2\%$).
This low p-value is evidence that your data do exhibit a pattern.  It's not one likely to show up in many random plots, as shown above.
Note that if we had mistakenly accepted the usual two-sided p-value output by the software, the random dataset second from left in the top row would have a low p-value (around $0.01$).  However, as you can see from the fitted curve, it would be evidence of a trough in the winter, which is the opposite of what you are looking for.

It can be helpful to do a sensitivity analysis.  For instance, what if we were to use a different choice of $f$?  Maybe with a slightly different peak?  Maybe a more flexible model without prespecifying when the peak occurs?
Here is an example where the random datasets are drawn from the multinomial distribution summing to $119$ (the sum of your data) and a periodic sawtooth function (centered at the very beginning of the year) is used:

Although the p-value of your data changed a little, we should draw the same conclusions as before.  This time one of the random datasets has a low p-value ($0.0189$ in the top row second from right), but--as claimed--we expect to see about one p-value below $0.05$ in any $20$ random datasets.
Finally, I also fit models using a cosine and a sine term, which means the peak is not fixed in January: it is allowed to occur any time of the year.  This flexibility can be helpful in the analysis (since "winter" and "summer" are somewhat vague periods anyway), but at the cost of requiring another parameter in the model.  There isn't really any sense of "sidedness" to the hypothesis test, because now the alternative hypothesis is only that the data follow the sum of a sine and a cosine (plus a constant).  No adjustment to the software's overall p-value is needed.  Nevertheless, your data still have a p-value below $0.05,$ continuing to suggest you have observed a real pattern.

Again the random data are multinomial (the same as in the second example).  Notice how the peaks in the random data occur at random times of the year now.
