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Suppose we have a binary outcome variable $y$ and we plot an ROC curve with a predictor variable $x$. What is the interpretation of this? An ROC curve plots the FPR (false positive rate) and TPR (true positive rate) across all thresholds. Here, we don't have any probabilities/thresholds.

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closed as unclear what you're asking by AdamO, kjetil b halvorsen, Michael Chernick, Peter Flom Oct 5 '17 at 10:50

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  • $\begingroup$ The ROC is FPR/TPR. I don't understand the question. $\endgroup$ – AdamO Oct 4 '17 at 19:45
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I think it's unorthodox, yet I've seen it done, this approach is common in credit scoring. It's calculated this way:

  1. Sort your $X$ variable in ascending order
  2. At each point $x_i \in X$ (or every $n$ points if the set is too big) calculate:

    $b_i = (\#$ of $0$'s in $Y \, | \, X < x_i)/(\#$ of $0$'s in $Y) =$ propotion of negative observations when $X < x_i$ and all negative observations.

    $g_i = (\#$ of $1$'s in $Y \, | \, X < x_i)/(\#$ of $1$'s in $Y) =$ propotion of positive observations when $X < x_i$ and all positive observations.

  3. Plot each pair $(b_i, g_i)$ and you receive a curve similar to ROC.

Interpretation is quite analoguous to the ROC curve - area under the curve you just plotted will be a measure of discriminatory power of given variable. Line y = x will indicate that variable X won't be a good predictor. If the curve is convex, the variable is promising, you should check how it performs in a model.

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