Standard Deviation comparison

Question

Currently i am working on my master thesis. I performed a research of the weather on sales. Different weather types are included in the model such as sunshine duration, duration of rainfall and level of humidity. I performed the analysis 3 times, one time for all the sales data i have and 2 times for a specific product category (f.e. icecream and cars sales). I want to see if the effect from duration of sunshine for cars sales is significant different from the category icecream. I was told to perform a test based on their standard deviation and look if they significantly differ. But when i have a whole formula with 10 different parameters can I get the standard deviation per parameter? And how can i compare them?

Answer 1

Effects of sunshine, rain and humidity on the sales of various products correspond to some parameters of your statistical model, which you, then, each estimate by some function $f_i$ of the data in the sample. [of course, to be a good guess of the real value of parameter, the function $f_i$ has to be carefully chosen — that's what statistical theory is all about]

Since we view the data as a realization of some random variables, say $X_1,\ldots,X_n$ whose joint probability distribution is determined by the model, the parameter estimator $f_i(X_1,\ldots,X_n)$ is also (another) random variable with its own standard deviation, and we can further estimate this deviation by $\tilde{f_i}(X_1,\ldots,X_n)$ where $\tilde{f_i}$ is another appropriately chosen function.

In your question you didn't specify statistical model, but situation calls for Poisson regression, a special case of generalized linear model. In that case, one can show that appropriate parameter estimators are asymptotically normally distributed. You can use this result to test whether some of the parameters are equal to zero. For instance, if you add dummy variable Product which is 0 for a car sale, and 1 for an ice cream sale, and estimate parameters $\alpha_i$, $i=1,\ldots,4$ of the following generalized linear model: $$Y_i \sim Poiss(\mu_i)$$ $$ln \mu_i = \alpha_1 \text{Sunshine}_i+\alpha_2 \text{Rainfall}_i+\alpha_3 \text{Humidity}_i+\alpha_4 \text{Sunshine}_i \text{Product}_i$$

where $Y_i$ represent sales, then we can interpret parameter $\alpha_4$ as a difference in the effects of sunshine on the sales of ice cream and cars, and test if this difference significantly differs from zero. More generally, there is a theory that enables one to test whether parameters lie in arbitrary linear subspace, the so called linear hypothesis.

Details of all this go well beyond the scope of single math.stackexchange answer. I learned all this from these lecture notes by P. Altham, but they are targeted towards an advanced undergraduate student of mathematics and are not really suitable for someone without substantial mathematical background; I leave to others to recommend a better textbook for that kind of reader (maybe Linear Models with R and Extending the Linear Model with R by J. Faraway).

In conclusion, I would like to emphasize that, even though there are various software packages that perform these kinds of analysis at the press of a button, to know what you are doing, you really need an equivalent of a few courses in probability and mathematical statistics.