Let a sample ( x , y ) ∈ R 2 n be given, where y only attains the values 0 and 1. We can try to model this data set by either linear regression y i = α 0 + β 0 x i with the coefficients determined by the method of least squares or by logistic regression π i = exp ⁡ ( α 1 + β 1 x i ) 1 + exp ⁡ ( α 1 + β 1 x i ) ,where π i denotes the probability that y i = 1 under the given value x i and the coefficients are determined by the Maximum-Likelihood method. My question is whether the following statement holds true.Claim: If β 0 &gt; 0 ( β 0 &lt; 0), then β 1 &gt; 0 ( β 1 &gt; 0).I figure this could be due to the sign of the correlation coefficient.

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Let a sample   (  x  ,  y  )  ∈        R          2      n       be given, where   y only attains the values   0 and   1. We can try to model this data set by either linear regression      y    i    =      α    0    +      β    0        x    i  with the coefficients determined by the method of least squares or by logistic regression      π    i    =            exp      ⁡      (              α        1            +              β        1                    x        i            )              1      +      exp      ⁡      (              α        1            +              β        1                    x        i            )        ,where       π    i   denotes the probability that       y    i    =  1 under the given value       x    i   and the coefficients are determined by the Maximum-Likelihood method. My question is whether the following statement holds true.Claim: If       β    0    &amp;gt;  0 (      β    0    &amp;lt;  0), then       β    1    &amp;gt;  0 (      β    1    &amp;gt;  0).I figure this could be due to the sign of the correlation coefficient.

assumintdz · Accepted Answer

If       β    0    &amp;gt;  0, i.e.,       ∂          ∂      x        E  [  Y  |  X  =  x  ]  =      β    0    &amp;gt;  0, is suggests that increase in   x will increase the probability that   Y  =  1, since   E  [  Y  |  X  =  x  ]  =  P  (  Y  =  1  |  X  =  x  )  =  p. Now, the logistic model is equivalent to  ln  ⁡      (          p              1        −        p              )    =      α    1    +      β    1    x  .The right hand side can be viewed as linear approximation of   ln  ⁡  (  p  /  (  1  −  p  )  ). Given that the original linear model is ok, and       β    0    &amp;gt;  0, it translates into   ∂  /  ∂  x  ln  ⁡  (  p  /  (  1  −  p  )  )  =  (  p  (  1  −  p  )      )          −      1            β    0    &amp;gt;  0 for the   ln odds model, namely,       β    1   must be positive as well (as the slope of the linear approximation).

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