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

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


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5












$begingroup$


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









$endgroup$








  • 1




    $begingroup$
    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
    2 hours ago


















5












$begingroup$


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









$endgroup$








  • 1




    $begingroup$
    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
    2 hours ago














5












5








5


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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




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asked 4 hours ago









Peter FlomPeter Flom

77k12109215




77k12109215








  • 1




    $begingroup$
    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
    2 hours ago














  • 1




    $begingroup$
    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
    2 hours ago








1




1




$begingroup$
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
2 hours ago




$begingroup$
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
2 hours ago










2 Answers
2






active

oldest

votes


















4












$begingroup$

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











$endgroup$









  • 1




    $begingroup$
    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
    2 hours ago






  • 2




    $begingroup$
    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
    2 hours ago



















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









$endgroup$














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    4












    $begingroup$

    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











    $endgroup$









    • 1




      $begingroup$
      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
      2 hours ago






    • 2




      $begingroup$
      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
      2 hours ago
















    4












    $begingroup$

    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











    $endgroup$









    • 1




      $begingroup$
      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
      2 hours ago






    • 2




      $begingroup$
      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
      2 hours ago














    4












    4








    4





    $begingroup$

    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 2 hours ago

























    answered 3 hours ago









    Isabella GhementIsabella Ghement

    7,821422




    7,821422








    • 1




      $begingroup$
      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
      2 hours ago






    • 2




      $begingroup$
      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
      2 hours ago














    • 1




      $begingroup$
      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
      2 hours ago






    • 2




      $begingroup$
      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
      2 hours ago








    1




    1




    $begingroup$
    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
    2 hours ago




    $begingroup$
    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
    2 hours ago




    2




    2




    $begingroup$
    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
    2 hours ago




    $begingroup$
    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
    2 hours ago













    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 1 hour ago









        FirebugFirebug

        7,74923280




        7,74923280






























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