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Suppose I have an event $Y$ that can occur as a result of events $X_1$, $X_2$, $X_3$......$X_n$. Let us assume that I have already observed past historical data on the co-occurrence of $Y$ and the $X_i$s. As a result I have conditional probabilities on the occurrence of $Y$|$X_i$ and $Y|X_i, X_j$ ($i{\neq}j$), $Y|X_i, X_j, X_k$ ($i{\neq}j{\neq}k$) etc.

Now at a certain time, I observe that certain $X_i$s and $Y$ actually occurred. How do I attribute the occurrence of $Y$ to the individual $X_i$s? In other words, let's say at a certain point in time, $Y$ occurs together with $X_1$, $X_3$ and $X_{10}$. How do I attribute the occurrence of $Y$ to each of $X_1$, $X_3$ and $X_{10}$?

Here is a concrete example:

Let's say the variable of interest ($Y$) is whether a car is stolen or not. Let's say that I also observe 3 attributes of the car - it's color (red or not red), origin (domestic or not) and the type of car (SUV or not). I observe 10 cars and their statuses and the data is as follows:

| S No | Color   | Type    | Origin        | Stolen? |  
|------|---------|---------|---------------|---------|  
|  1   | Red     | Not SUV | Domestic      | Yes     |  
|  2   | Red     | Not SUV | Domestic      | No      |  
|  3   | Red     | Not SUV | Domestic      | Yes     |  
|  4   | Not Red | Not SUV | Domestic      | No      |  
|  5   | Not Red | Not SUV | Not Domestic  | Yes     |  
|  6   | Not Red | SUV     | Not Domestic  | No      |  
|  7   | Not Red | SUV     | Not Domestic  | Yes     |  
|  8   | Not Red | SUV     | Domestic      | No      |  
|  9   | Red     | SUV     | Not Domestic  | No      |  
| 10   | Red     | Not SUV | Not Domestic  | Yes     |  

Now let's say I observe a new car that is red, a SUV and domestic being stolen. How could I assign the extent to which each of these factors caused this PARTICULAR car to be stolen. Note that I'm not talking about the aggregate, but a scheme to attribute the observed factors to the status of this particular car.

Any pointers would be helpful, even if they are keywords that I can search for.

Edit: Bad example. Let's say a red SUV that is not domestic is identified as stolen, how would would you credit each of the attributes with a contribution for being stolen? The issue at hand here is that the logistic regression model would provide the probability of the car being stolen based on the factors. But we already know that the car is stolen. In other words, I would like a proportion for each of the factors Color=Red, Type=SUV and Origin=Not Domestic. Those proportions should sum to 1.

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  • $\begingroup$ I'm not sure what you mean by "attribute". Are you thinking of causation? That is, are you asking which of $X_1$, $X_3$, and $X_{10}$ caused $Y$? $\endgroup$ Oct 9, 2016 at 19:00
  • $\begingroup$ Yes, but not in the strict sense of the word. Basically, I have already made the assumption that $X_1$, $X_3$ and $X_{10}$ cause Y. I would like to know how much credit to give each of them. $\endgroup$ Oct 9, 2016 at 19:07

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Perhaps what you want is a logistic-regression model. The coefficient of each term (which could be a single $X_i$ or a function of several $X_i$s) tells you the effect on the probability of $Y$ occurring (as a log odds ratio) when the term is present compared to when it is absent.

Edit: Regarding your example, it looks like you don't have any information with which to explain specific observations. After all, if after observing one domestic red SUV being stolen, you then went on to see another domestic red SUV not being stolen, these two cases would have identical values on the $X_i$s, so the difference in $Y$ can't be explained by differences in the $X_i$s. It is a usual assumption in data analysis, especially in statistics, that there's a baseline unaccountable variability involved in each observation. In regression, this is represented by the model's error term.

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  • $\begingroup$ That's a good thought. So let's say I build a logistic regression model for predicting $Y$ from the $X_i$s. The coefficients would of course provide the log odds ratio. So let's say I have a new observation of $Y$ and $X_i$s. Remember that in this case, I observe both the $Y$ and $X_i$s. For example, let's say for this instance that $Y$ occurs with $X_1$ and $X_2$. How would I use the calculated coefficients to assign the credit to $X_1$ and $X_2$? $\endgroup$ Oct 9, 2016 at 21:26
  • $\begingroup$ @InfiniteExistence It is no more clear to me how "credit" relates to statistical notions than "attribution" does. Do the log odds ratios themselves not tell you what you want? $\endgroup$ Oct 9, 2016 at 21:32
  • $\begingroup$ @InfiniteExistence It seems that the closest thing to what you want that makes sense is to divide all the coefficients through by their sum. I think you may have some underlying confusions regarding regression and regarding statistical (as opposed to philosophical) notions of causation. For the latter, I found the chapter on causal models in Larry Wasserman's All of Statistics very helpful. $\endgroup$ Oct 10, 2016 at 3:10
  • $\begingroup$ Yup, seems like the best solution. Further research did lead me to a problem called Multi Touch Attribution for online advertising campaigns (link). The problem there is to attribute a given conversion to one or more advertising campaigns that the user was exposed to. $\endgroup$ Oct 10, 2016 at 18:21
  • $\begingroup$ @InfiniteExistence From a glance at that article, it looks like the goal is to understand the value (that is, causal effect on customer dollars spent) of each of several advertising campaigns. Doing this requires randomly assigning subjects to various combinations of the campaigns; you can't answer the question at all if all subjects were exposed to all campaigns. $\endgroup$ Oct 10, 2016 at 18:32

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