How does quantile regression compare to logistic regression with the variable split at the quantile?





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$begingroup$


I googled a bit but didn't find anything on this.



Suppose you do a quantile regression on the qth quantile of the dependent variable.



Then you split the DV at the qth quantile and label the result 0 and 1. Then you do logistic regression on the categorized DV.



I'm looking for any Monte-Carlo studies of this or reasons to prefer one over the other etc.










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  • 2




    $begingroup$
    Could you show us any reasonable way even to compare the results of the two regressions? After all, unless you have something a little less general in mind, the coefficients of the regressors in these two models have entirely different meanings and interpretations, so in what sense are we to understand what you mean by "prefer"?
    $endgroup$
    – whuber
    Apr 5 at 21:48


















8












$begingroup$


I googled a bit but didn't find anything on this.



Suppose you do a quantile regression on the qth quantile of the dependent variable.



Then you split the DV at the qth quantile and label the result 0 and 1. Then you do logistic regression on the categorized DV.



I'm looking for any Monte-Carlo studies of this or reasons to prefer one over the other etc.










share|cite|improve this question









$endgroup$








  • 2




    $begingroup$
    Could you show us any reasonable way even to compare the results of the two regressions? After all, unless you have something a little less general in mind, the coefficients of the regressors in these two models have entirely different meanings and interpretations, so in what sense are we to understand what you mean by "prefer"?
    $endgroup$
    – whuber
    Apr 5 at 21:48














8












8








8


1



$begingroup$


I googled a bit but didn't find anything on this.



Suppose you do a quantile regression on the qth quantile of the dependent variable.



Then you split the DV at the qth quantile and label the result 0 and 1. Then you do logistic regression on the categorized DV.



I'm looking for any Monte-Carlo studies of this or reasons to prefer one over the other etc.










share|cite|improve this question









$endgroup$




I googled a bit but didn't find anything on this.



Suppose you do a quantile regression on the qth quantile of the dependent variable.



Then you split the DV at the qth quantile and label the result 0 and 1. Then you do logistic regression on the categorized DV.



I'm looking for any Monte-Carlo studies of this or reasons to prefer one over the other etc.







logistic quantile-regression






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asked Apr 5 at 19:50









Peter FlomPeter Flom

77.3k12109217




77.3k12109217








  • 2




    $begingroup$
    Could you show us any reasonable way even to compare the results of the two regressions? After all, unless you have something a little less general in mind, the coefficients of the regressors in these two models have entirely different meanings and interpretations, so in what sense are we to understand what you mean by "prefer"?
    $endgroup$
    – whuber
    Apr 5 at 21:48














  • 2




    $begingroup$
    Could you show us any reasonable way even to compare the results of the two regressions? After all, unless you have something a little less general in mind, the coefficients of the regressors in these two models have entirely different meanings and interpretations, so in what sense are we to understand what you mean by "prefer"?
    $endgroup$
    – whuber
    Apr 5 at 21:48








2




2




$begingroup$
Could you show us any reasonable way even to compare the results of the two regressions? After all, unless you have something a little less general in mind, the coefficients of the regressors in these two models have entirely different meanings and interpretations, so in what sense are we to understand what you mean by "prefer"?
$endgroup$
– whuber
Apr 5 at 21:48




$begingroup$
Could you show us any reasonable way even to compare the results of the two regressions? After all, unless you have something a little less general in mind, the coefficients of the regressors in these two models have entirely different meanings and interpretations, so in what sense are we to understand what you mean by "prefer"?
$endgroup$
– whuber
Apr 5 at 21:48










3 Answers
3






active

oldest

votes


















7












$begingroup$

For simplicity, assume you have a continuous dependent variable Y and a continuous predictor variable X.



Logistic Regression



If I understand your post correctly, your logistic regression will categorize Y into 0 and 1 based on the quantile of the (unconditional) distribution of Y. Specifically, the q-th quantile of the distribution of observed Y values will be computed and Ycat will be defined as 0 if Y is strictly less than this quantile and 1 if Y is greater than or equal to this quantile.



If the above captures your intent, then the logistic regression will model the odds of Y exceeding or being equal to the (observed) q-th quantile of the (unconditional) Y distribution as a function of X.



Quantile Regression



On the other hand, if you are performing a quantile regression of Y on X, you are focusing on modelling how the q-th quantile of the conditional distribution of Y given X changes as a function of X.



Logistic Regression versus Quantile Regression



It seems to me that these two procedures have totally different aims, since the first procedure (i.e., logistic regression) focuses on the q-th quantile of the unconditional distribution of Y, whereas the second procedure (i.e., quantile regression) focuses on the the q-th quantile of the conditional distribution of Y.



The unconditional distribution of Y is the 
distribution of Y values (hence it ignores any
information about the X values).

The conditional distribution of Y given X is the
distribution of those Y values for which the values
of X are the same.


Illustrative Example



For illustration purposes, let's say Y = cholesterol and X = body weight.



Then logistic regression is modelling the odds of having a 'high' cholesterol value (i.e., greater than or equal to the q-th quantile of the observed cholesterol values) as a function of body weight, where the definition of 'high' has no relation to body weight. In other words, the marker for what constitutes a 'high' cholesterol value is independent of body weight. What changes with body weight in this model is the odds that a cholesterol value would exceed this marker.



On the other hand, quantile regression is looking at how the 'marker' cholesterol values for which q% of the subjects with the same body weight in the underlying population have a higher cholesterol value vary as a function of body weight. You can think of these cholesterol values as markers for identifying what cholesterol values are 'high' - but in this case, each marker depends on the corresponding body weight; furthermore, the markers are assumed to change in a predictable fashion as the value of X changes (e.g., the markers tend to increase as X increases).






share|cite|improve this answer











$endgroup$









  • 2




    $begingroup$
    I agree with all that. Yet, there does seem to be a similarity - that is, both look at the qth quantile as a function of the same independent variables.
    $endgroup$
    – Peter Flom
    Apr 5 at 22:16






  • 4




    $begingroup$
    Yes, but the difference is that one method looks at the unconditional quantile (i.e., logistic regression) while the other looks at the conditional quantile (i.e., quantile regression). Those two quantiles keep track of different things.
    $endgroup$
    – Isabella Ghement
    Apr 5 at 22:19



















2












$begingroup$

They won't be equal, and the reason is simple.



With quantile regression you want to model the quantile conditional of the independent variables. Your approach with logistic regression fits the marginal quantile.






share|cite|improve this answer









$endgroup$





















    1












    $begingroup$

    One asks "what is the effect on the nth quantile of the dependent variable's distribution?" The other one asks "what is the effect on the probability that the dependent variable falls into the nth quantile of its unconditional distribution?"



    I.e., the fact that they both have the word "quantile" in them let's them look more similar than they are.



    I guess if you first estimate a conditional quantile function, use this for the split and proceed from there, the two approaches would become more similar. But I don't see what you would stand to gain from such a detour.
    .






    share|cite|improve this answer









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      3 Answers
      3






      active

      oldest

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      3 Answers
      3






      active

      oldest

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      active

      oldest

      votes






      active

      oldest

      votes









      7












      $begingroup$

      For simplicity, assume you have a continuous dependent variable Y and a continuous predictor variable X.



      Logistic Regression



      If I understand your post correctly, your logistic regression will categorize Y into 0 and 1 based on the quantile of the (unconditional) distribution of Y. Specifically, the q-th quantile of the distribution of observed Y values will be computed and Ycat will be defined as 0 if Y is strictly less than this quantile and 1 if Y is greater than or equal to this quantile.



      If the above captures your intent, then the logistic regression will model the odds of Y exceeding or being equal to the (observed) q-th quantile of the (unconditional) Y distribution as a function of X.



      Quantile Regression



      On the other hand, if you are performing a quantile regression of Y on X, you are focusing on modelling how the q-th quantile of the conditional distribution of Y given X changes as a function of X.



      Logistic Regression versus Quantile Regression



      It seems to me that these two procedures have totally different aims, since the first procedure (i.e., logistic regression) focuses on the q-th quantile of the unconditional distribution of Y, whereas the second procedure (i.e., quantile regression) focuses on the the q-th quantile of the conditional distribution of Y.



      The unconditional distribution of Y is the 
      distribution of Y values (hence it ignores any
      information about the X values).

      The conditional distribution of Y given X is the
      distribution of those Y values for which the values
      of X are the same.


      Illustrative Example



      For illustration purposes, let's say Y = cholesterol and X = body weight.



      Then logistic regression is modelling the odds of having a 'high' cholesterol value (i.e., greater than or equal to the q-th quantile of the observed cholesterol values) as a function of body weight, where the definition of 'high' has no relation to body weight. In other words, the marker for what constitutes a 'high' cholesterol value is independent of body weight. What changes with body weight in this model is the odds that a cholesterol value would exceed this marker.



      On the other hand, quantile regression is looking at how the 'marker' cholesterol values for which q% of the subjects with the same body weight in the underlying population have a higher cholesterol value vary as a function of body weight. You can think of these cholesterol values as markers for identifying what cholesterol values are 'high' - but in this case, each marker depends on the corresponding body weight; furthermore, the markers are assumed to change in a predictable fashion as the value of X changes (e.g., the markers tend to increase as X increases).






      share|cite|improve this answer











      $endgroup$









      • 2




        $begingroup$
        I agree with all that. Yet, there does seem to be a similarity - that is, both look at the qth quantile as a function of the same independent variables.
        $endgroup$
        – Peter Flom
        Apr 5 at 22:16






      • 4




        $begingroup$
        Yes, but the difference is that one method looks at the unconditional quantile (i.e., logistic regression) while the other looks at the conditional quantile (i.e., quantile regression). Those two quantiles keep track of different things.
        $endgroup$
        – Isabella Ghement
        Apr 5 at 22:19
















      7












      $begingroup$

      For simplicity, assume you have a continuous dependent variable Y and a continuous predictor variable X.



      Logistic Regression



      If I understand your post correctly, your logistic regression will categorize Y into 0 and 1 based on the quantile of the (unconditional) distribution of Y. Specifically, the q-th quantile of the distribution of observed Y values will be computed and Ycat will be defined as 0 if Y is strictly less than this quantile and 1 if Y is greater than or equal to this quantile.



      If the above captures your intent, then the logistic regression will model the odds of Y exceeding or being equal to the (observed) q-th quantile of the (unconditional) Y distribution as a function of X.



      Quantile Regression



      On the other hand, if you are performing a quantile regression of Y on X, you are focusing on modelling how the q-th quantile of the conditional distribution of Y given X changes as a function of X.



      Logistic Regression versus Quantile Regression



      It seems to me that these two procedures have totally different aims, since the first procedure (i.e., logistic regression) focuses on the q-th quantile of the unconditional distribution of Y, whereas the second procedure (i.e., quantile regression) focuses on the the q-th quantile of the conditional distribution of Y.



      The unconditional distribution of Y is the 
      distribution of Y values (hence it ignores any
      information about the X values).

      The conditional distribution of Y given X is the
      distribution of those Y values for which the values
      of X are the same.


      Illustrative Example



      For illustration purposes, let's say Y = cholesterol and X = body weight.



      Then logistic regression is modelling the odds of having a 'high' cholesterol value (i.e., greater than or equal to the q-th quantile of the observed cholesterol values) as a function of body weight, where the definition of 'high' has no relation to body weight. In other words, the marker for what constitutes a 'high' cholesterol value is independent of body weight. What changes with body weight in this model is the odds that a cholesterol value would exceed this marker.



      On the other hand, quantile regression is looking at how the 'marker' cholesterol values for which q% of the subjects with the same body weight in the underlying population have a higher cholesterol value vary as a function of body weight. You can think of these cholesterol values as markers for identifying what cholesterol values are 'high' - but in this case, each marker depends on the corresponding body weight; furthermore, the markers are assumed to change in a predictable fashion as the value of X changes (e.g., the markers tend to increase as X increases).






      share|cite|improve this answer











      $endgroup$









      • 2




        $begingroup$
        I agree with all that. Yet, there does seem to be a similarity - that is, both look at the qth quantile as a function of the same independent variables.
        $endgroup$
        – Peter Flom
        Apr 5 at 22:16






      • 4




        $begingroup$
        Yes, but the difference is that one method looks at the unconditional quantile (i.e., logistic regression) while the other looks at the conditional quantile (i.e., quantile regression). Those two quantiles keep track of different things.
        $endgroup$
        – Isabella Ghement
        Apr 5 at 22:19














      7












      7








      7





      $begingroup$

      For simplicity, assume you have a continuous dependent variable Y and a continuous predictor variable X.



      Logistic Regression



      If I understand your post correctly, your logistic regression will categorize Y into 0 and 1 based on the quantile of the (unconditional) distribution of Y. Specifically, the q-th quantile of the distribution of observed Y values will be computed and Ycat will be defined as 0 if Y is strictly less than this quantile and 1 if Y is greater than or equal to this quantile.



      If the above captures your intent, then the logistic regression will model the odds of Y exceeding or being equal to the (observed) q-th quantile of the (unconditional) Y distribution as a function of X.



      Quantile Regression



      On the other hand, if you are performing a quantile regression of Y on X, you are focusing on modelling how the q-th quantile of the conditional distribution of Y given X changes as a function of X.



      Logistic Regression versus Quantile Regression



      It seems to me that these two procedures have totally different aims, since the first procedure (i.e., logistic regression) focuses on the q-th quantile of the unconditional distribution of Y, whereas the second procedure (i.e., quantile regression) focuses on the the q-th quantile of the conditional distribution of Y.



      The unconditional distribution of Y is the 
      distribution of Y values (hence it ignores any
      information about the X values).

      The conditional distribution of Y given X is the
      distribution of those Y values for which the values
      of X are the same.


      Illustrative Example



      For illustration purposes, let's say Y = cholesterol and X = body weight.



      Then logistic regression is modelling the odds of having a 'high' cholesterol value (i.e., greater than or equal to the q-th quantile of the observed cholesterol values) as a function of body weight, where the definition of 'high' has no relation to body weight. In other words, the marker for what constitutes a 'high' cholesterol value is independent of body weight. What changes with body weight in this model is the odds that a cholesterol value would exceed this marker.



      On the other hand, quantile regression is looking at how the 'marker' cholesterol values for which q% of the subjects with the same body weight in the underlying population have a higher cholesterol value vary as a function of body weight. You can think of these cholesterol values as markers for identifying what cholesterol values are 'high' - but in this case, each marker depends on the corresponding body weight; furthermore, the markers are assumed to change in a predictable fashion as the value of X changes (e.g., the markers tend to increase as X increases).






      share|cite|improve this answer











      $endgroup$



      For simplicity, assume you have a continuous dependent variable Y and a continuous predictor variable X.



      Logistic Regression



      If I understand your post correctly, your logistic regression will categorize Y into 0 and 1 based on the quantile of the (unconditional) distribution of Y. Specifically, the q-th quantile of the distribution of observed Y values will be computed and Ycat will be defined as 0 if Y is strictly less than this quantile and 1 if Y is greater than or equal to this quantile.



      If the above captures your intent, then the logistic regression will model the odds of Y exceeding or being equal to the (observed) q-th quantile of the (unconditional) Y distribution as a function of X.



      Quantile Regression



      On the other hand, if you are performing a quantile regression of Y on X, you are focusing on modelling how the q-th quantile of the conditional distribution of Y given X changes as a function of X.



      Logistic Regression versus Quantile Regression



      It seems to me that these two procedures have totally different aims, since the first procedure (i.e., logistic regression) focuses on the q-th quantile of the unconditional distribution of Y, whereas the second procedure (i.e., quantile regression) focuses on the the q-th quantile of the conditional distribution of Y.



      The unconditional distribution of Y is the 
      distribution of Y values (hence it ignores any
      information about the X values).

      The conditional distribution of Y given X is the
      distribution of those Y values for which the values
      of X are the same.


      Illustrative Example



      For illustration purposes, let's say Y = cholesterol and X = body weight.



      Then logistic regression is modelling the odds of having a 'high' cholesterol value (i.e., greater than or equal to the q-th quantile of the observed cholesterol values) as a function of body weight, where the definition of 'high' has no relation to body weight. In other words, the marker for what constitutes a 'high' cholesterol value is independent of body weight. What changes with body weight in this model is the odds that a cholesterol value would exceed this marker.



      On the other hand, quantile regression is looking at how the 'marker' cholesterol values for which q% of the subjects with the same body weight in the underlying population have a higher cholesterol value vary as a function of body weight. You can think of these cholesterol values as markers for identifying what cholesterol values are 'high' - but in this case, each marker depends on the corresponding body weight; furthermore, the markers are assumed to change in a predictable fashion as the value of X changes (e.g., the markers tend to increase as X increases).







      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited Apr 8 at 9:33









      Richard Hardy

      28.2k642130




      28.2k642130










      answered Apr 5 at 20:29









      Isabella GhementIsabella Ghement

      7,9661422




      7,9661422








      • 2




        $begingroup$
        I agree with all that. Yet, there does seem to be a similarity - that is, both look at the qth quantile as a function of the same independent variables.
        $endgroup$
        – Peter Flom
        Apr 5 at 22:16






      • 4




        $begingroup$
        Yes, but the difference is that one method looks at the unconditional quantile (i.e., logistic regression) while the other looks at the conditional quantile (i.e., quantile regression). Those two quantiles keep track of different things.
        $endgroup$
        – Isabella Ghement
        Apr 5 at 22:19














      • 2




        $begingroup$
        I agree with all that. Yet, there does seem to be a similarity - that is, both look at the qth quantile as a function of the same independent variables.
        $endgroup$
        – Peter Flom
        Apr 5 at 22:16






      • 4




        $begingroup$
        Yes, but the difference is that one method looks at the unconditional quantile (i.e., logistic regression) while the other looks at the conditional quantile (i.e., quantile regression). Those two quantiles keep track of different things.
        $endgroup$
        – Isabella Ghement
        Apr 5 at 22:19








      2




      2




      $begingroup$
      I agree with all that. Yet, there does seem to be a similarity - that is, both look at the qth quantile as a function of the same independent variables.
      $endgroup$
      – Peter Flom
      Apr 5 at 22:16




      $begingroup$
      I agree with all that. Yet, there does seem to be a similarity - that is, both look at the qth quantile as a function of the same independent variables.
      $endgroup$
      – Peter Flom
      Apr 5 at 22:16




      4




      4




      $begingroup$
      Yes, but the difference is that one method looks at the unconditional quantile (i.e., logistic regression) while the other looks at the conditional quantile (i.e., quantile regression). Those two quantiles keep track of different things.
      $endgroup$
      – Isabella Ghement
      Apr 5 at 22:19




      $begingroup$
      Yes, but the difference is that one method looks at the unconditional quantile (i.e., logistic regression) while the other looks at the conditional quantile (i.e., quantile regression). Those two quantiles keep track of different things.
      $endgroup$
      – Isabella Ghement
      Apr 5 at 22:19













      2












      $begingroup$

      They won't be equal, and the reason is simple.



      With quantile regression you want to model the quantile conditional of the independent variables. Your approach with logistic regression fits the marginal quantile.






      share|cite|improve this answer









      $endgroup$


















        2












        $begingroup$

        They won't be equal, and the reason is simple.



        With quantile regression you want to model the quantile conditional of the independent variables. Your approach with logistic regression fits the marginal quantile.






        share|cite|improve this answer









        $endgroup$
















          2












          2








          2





          $begingroup$

          They won't be equal, and the reason is simple.



          With quantile regression you want to model the quantile conditional of the independent variables. Your approach with logistic regression fits the marginal quantile.






          share|cite|improve this answer









          $endgroup$



          They won't be equal, and the reason is simple.



          With quantile regression you want to model the quantile conditional of the independent variables. Your approach with logistic regression fits the marginal quantile.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Apr 5 at 22:54









          FirebugFirebug

          7,76123280




          7,76123280























              1












              $begingroup$

              One asks "what is the effect on the nth quantile of the dependent variable's distribution?" The other one asks "what is the effect on the probability that the dependent variable falls into the nth quantile of its unconditional distribution?"



              I.e., the fact that they both have the word "quantile" in them let's them look more similar than they are.



              I guess if you first estimate a conditional quantile function, use this for the split and proceed from there, the two approaches would become more similar. But I don't see what you would stand to gain from such a detour.
              .






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                One asks "what is the effect on the nth quantile of the dependent variable's distribution?" The other one asks "what is the effect on the probability that the dependent variable falls into the nth quantile of its unconditional distribution?"



                I.e., the fact that they both have the word "quantile" in them let's them look more similar than they are.



                I guess if you first estimate a conditional quantile function, use this for the split and proceed from there, the two approaches would become more similar. But I don't see what you would stand to gain from such a detour.
                .






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  One asks "what is the effect on the nth quantile of the dependent variable's distribution?" The other one asks "what is the effect on the probability that the dependent variable falls into the nth quantile of its unconditional distribution?"



                  I.e., the fact that they both have the word "quantile" in them let's them look more similar than they are.



                  I guess if you first estimate a conditional quantile function, use this for the split and proceed from there, the two approaches would become more similar. But I don't see what you would stand to gain from such a detour.
                  .






                  share|cite|improve this answer









                  $endgroup$



                  One asks "what is the effect on the nth quantile of the dependent variable's distribution?" The other one asks "what is the effect on the probability that the dependent variable falls into the nth quantile of its unconditional distribution?"



                  I.e., the fact that they both have the word "quantile" in them let's them look more similar than they are.



                  I guess if you first estimate a conditional quantile function, use this for the split and proceed from there, the two approaches would become more similar. But I don't see what you would stand to gain from such a detour.
                  .







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Apr 6 at 8:06









                  sheßsheß

                  230419




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