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Say I am running a store and I have the daily customer count for the last 5 years.

  1. I note that the average customer count on Black Fridays is three times higher than the average for the rest of the year.

  2. I build a table with two columns:

    IsDayBlackFriday (0 or 1), CustomerCount (integer)

    and run linear regression where the independent variable is customer count and the independent variable is whether it is a Black Friday or not. As expected, the p-value for the Black Friday variable is extremely low, but the R-squared val for the model is also quite low. I note that I am not bothered by it since I am aware that there are many other factors that determine the demand, and I am really looking for some statistical validation of the correlation between the customer count and one particular day in a year, and not trying to build a predictive model for customer count.

My questions are:

  1. Can I conclude that on Black Fridays, I should hire 3 times more people as compared to the rest of the year to cover the demand, assuming the staffing requirement increases linearly with the customer count? For example, if on normal days, I have a staff count of 5, will hiring 15 people be a "statistically informed" decision?

  2. If not, what can I really infer from the facts given above that I can translate to an informed decision? Should the staffing question be approached differently?


Thank you for your answers Chris and ERT.

I think what I am interested in right now is getting a more solid understanding about the relation between 1) the p-value of the Black Friday binary variable in the regression results and 2) the percentage increase in the customer count that we can compute separately. More particularly, if the percentage increase is 200% (3 times higher value) and the p-value is low, can we say that for the next Black Friday, we can expect the customer count to be three times as high as the average we observed for the rest of the year? Or is that an overreach and the most we can say is that the variables are strongly correlated but we cannot predict anything since our R-squared value is low.

The hiring staff part is a detail in which I am not interested in at the moment (sorry if that wasn't clear from the original post) and was just used as an example of where the insight could be applied.

Chris, you mentioned that regression can help us quantify how busier it is on a holiday as opposed to the rest of the year. Can you elaborate on that?

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  • $\begingroup$ Please register &/or merge your accounts (you can find information on how to do this in the My Account section of our help center), then you will be able to edit & comment on your own question. $\endgroup$ – gung Aug 10 '18 at 13:11
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I don't think you needed a regression to know that Black Friday was busier. The regression may be useful in helping you to quantify how much busier it is, if you are controlling for other factors too. It would be more informative if "Black Friday" was one example of an event that occurs multiple times, perhaps whenever there are other large promotions.

Why do you want to hire more people? To prevent wait times from increasing? To ensure that you have a ratio of x:1 staff to customers to help with questions? You could model one of these outcomes as your dependent variable (though I imagine that data is not available, perhaps there is some proxy).

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  • $\begingroup$ If you built a model with more than one independent variable (day of the week, promotion activity, weather, whatever) then that coefficient on your BlackFriday dummy will be put into more context. Let's say that the weather was really good, but next year the weather is really bad. The coefficient on weather (which might be a series of dummies) shows how adverse weather affects customers. Then for next year, you can use the coefficients for weather and use each unique combination of possible weather outcomes to say, eg: if it rains, only 50% more customers. If cold, only 2x as many customers. $\endgroup$ – Chris Umphlett Aug 10 '18 at 13:06
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A slightly more rigorous approach to solving this problem would be to use queueing. Queueing theory is the broad term for the study of waiting in queues (lines). This sort of analysis could give you more insight you are looking for, that regression likely will not.

For example, if you are looking to predict the number of employees (agents) you should have on-hand for the day, you would do something similar to the following:

  1. Project the rate at which people enter your store (system) and the amount of time a customer would normally spend in the system shopping. Both of these can be modeled using relatively simple functions.
  2. Estimate the amount of time it takes for a customer to "check out," or be serviced by an agent. This is also easy to model with a simple function.
  3. Now that you know the rate at which people (i) enter the store, and (ii) enter the queue, and (iii) are serviced, you can estimate the time a customer will spend in the queue. Ideally, you are looking to keep the probability a single customer waits for longer than $m$ minutes less than some threshold value $p$, by hiring multiple agents $a$.

This is the general approach to this type of problem. Big retailers (Macys, Costco, etc.) do this type of analysis before any sort of large event to determine the minimum number of employees to retain to keep waiting time below some pre-determined maximum. You would need to account for many factors (not achieving steady-state, jockeying, and many other situation-specifics). You also rely heavily on estimations and assumptions, like assuming customers arrive according to a random process $\sim \lambda$, or estimating how long it will take an average customer to be serviced.

In your case, since you are not dealing with this rigorous approach, you may as well rely on intuition. Black Friday is absolute mayhem, regardless of the exact number of employees you retain on the day.

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Can I conclude that on Black Fridays, I should hire 3 times more people as compared to the rest of the year to cover the demand, assuming the staffing requirement increases linearly with the customer count? For example, if on normal days, I have a staff count of 5, will hiring 15 people be a "statistically informed" decision?

No. Staffing is not part of your model, nor is any other aspect of the business such as inventory sales and so forth. All you can do is predict people entering the store based on your comments. It also would not be an informed decision because you should have a fixed number of cash registers and a roughly fixed floor space. Without some form of optimization package being involved, no you cannot. You lack the required variables.

If not, what can I really infer from the facts given above that I can translate to an informed decision? Should the staffing question be approached differently?

You can infer customer count rates of change ($\hat{\beta}$). Yes, it should be an element of the problem itself.

I think what I am interested in right now is getting a more solid understanding about the relation between 1) the p-value of the Black Friday binary variable in the regression results and 2) the percentage increase in the customer count that we can compute separately. More particularly, if the percentage increase is 200% (3 times higher value) and the p-value is low, can we say that for the next Black Friday, we can expect the customer count to be three times as high as the average we observed for the rest of the year? Or is that an overreach and the most we can say is that the variables are strongly correlated but we cannot predict anything since our R-squared value is low.

There are two issues here. The first is that the p-value is both irrelevant and uninformative. Second, the form of your regression precludes you from forming an association between background data and Black Friday data.

First, the p-value. The p-value has no informative content here. You should be ignoring it. The p-value is intended as a device of rhetoric. If P is true then Q is true, Q is not true, therefore P is not true. It is intended as a statistical form of modus tollens. If the null is true then the data should appear in a particular way, the data does not appear in that way, therefore the null is false (to some degree of confidence). Of course, the default null is the no effect model. This would imply that no customer ever enters your store, EVER. This is trivially false. Furthermore, even if the p-value was p<.5 you would still have to act on staffing. This is still the only data you have, so you still must act.

Now as to your Black Friday data. By creating a binary variable, you have forced independence between the background data and the Black Friday data. There is no interaction effect between the background data and the Black Friday data. From an OLS perspective, this is no different than cutting out the Black Friday data and running two regressions. One regression is on regular data. The other only includes historical Black Fridays. You have not created any interaction between the time period surrounding Black Fridays and Black Friday.

Or is that an overreach and the most we can say is that the variables are strongly correlated but we cannot predict anything since our R-squared value is low.

You said you were not interested in prediction. As mentioned above, staffing is an overreach. If you are interested in prediction, then you should be using the predictive interval as your tool and not the coefficients or the confidence intervals. See Duke Lecture on Regression Prediction as an example.

The R square doesn't matter either if you have no further data. If you have no other way to make a prediction, all the R-square is telling you is that your prediction quality will be weak and so you should allow for a high level of natural variability. It is a warning that your model is fragile for the intended purpose, but again, if you have no further data, then this is all you have to work with. If you chose to alter your model and it improved $r^2$ you should first check whether or not it is an improvement through some model testing process such as the AIC or BIC. Preferentially, you would do formal validation of the model if you had enough data.

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