Let a sample ( x , y ) &#x2208;<!-- ∈ --> <mi mathvariant="double-stru

Reginald Delacruz 2022-06-24 Answered
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 > 0 ( β 0 < 0), then β 1 > 0 ( β 1 > 0).
I figure this could be due to the sign of the correlation coefficient.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

assumintdz
Answered 2022-06-25 Author has 22 answers
If β 0 > 0, i.e., x E [ Y | X = x ] = β 0 > 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 > 0, it translates into / x ln ( p / ( 1 p ) ) = ( p ( 1 p ) ) 1 β 0 > 0 for the ln odds model, namely, β 1 must be positive as well (as the slope of the linear approximation).

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

New questions