Logistic Regression -- Understanding Maximum Likelihood

[NKormanik]

Please see the following quote:

When Maximum Likelihood Exp(Est) is less than 1, increasing values of the variable correspond to decreasing odds of the event's occurrence.

Q1:  Just want to make sure....  So, decreasing values of the variable correspond to increasing odds of the event's occurrence.

Q2:  Additionally, the further from 1 the variable is (on either side), the greater the effect.

Example:

X1 MLE(Est) = 0.7

X2 MLE(Est) = 0.4

X2 will have a substantially greater effect on the odds of event's occurrence.

Thoughts appreciated.

Nicholas Kormanik

(p.s. -- If you feel there is a more appropriate forum for this question, please let me know.)

[pink_poodle]

It may be easier to think about it in percents. One unit increase in X1 associates with 30% decrease in the odds of the event. One unit increase in X2 associates with 60% decrease in the odds of the event. Therefore, you are correct, X2 associates with a greater negative effect on the odds of the event’s occurence.

[NKormanik]

@pink_poodle 

Super.  But the specific question I'm requesting confirmation of in Q1 is:

True or false: If MLE(Est) < 1, decreasing the values of that variable correspond to increasing odds of the event's occurrence.

[NKormanik]

A bit more clarification:

In the above example, let's look at X2 values.

X2 in our dataset ranges from 0 to 100.  Assume a fairly normal distribution.

Does it not matter where in the distribution the X2 value is?

I.e., X2 = 70 vs. X2 = 30.

If X2 MLE(Est) = 0.4, decreasing X2 in either of the above will amount to an increase in the odds of the event in question.

Please enlighten with your wisdom.

[pink_poodle · Accepted solution]

Assuming a linear relationship between x and outcome, it does not matter where x value is. Yes, decreasing x would associate with increased odds of outcome.

[Ksharp]

"True or false: If MLE(Est) < 1, decreasing the values of that variable correspond to increasing odds of the event's occurrence."
It is true I think. @Rick_SAS has wrote many bolgs about proc logistic . If you read his blog you would know the answer.

Basically, Yx+1 - Yx = beta(x+1 - x) => beta . Here Yx+1 - Yx = LOG( Px+1/(1-Px+1) ) - LOG( Px/(1-Px) ) = LOG( ODDS ).
Therefore ,Exp(Est)=Exp(beta)=Exp( LOG( ODDS ) ) = ODDS . if ODDS>1 they have the same direction, whereas vice verse .

[Rick_SAS]

In spite of the title and the notation in the OP's question, I don't this question is related to maximum likelihood estimations. It is merely a statement about interpreting the parameter estimate for a linear logistic model. How that estimate was obtained is irrelevant.

[NKormanik]

@Rick_SAS @Ksharp @pink_poodle 

Correct!  Doing a Google and Google Scholar search to answer the question brings up LOADS of the math.

[NKormanik]

@Rick_SAS @Ksharp @pink_poodle 

Thanks all.  Rick's article was helpful in pointing to a far better perspective viewing odds ratio:

plots=oddsratio(logbase=2 order=descending)

Wow!  What a difference!

[NKormanik]

@Rick_SAS @Ksharp @pink_poodle 

Thanks all.  Rick's article was helpful in pointing to a far better perspective viewing odds ratio:

plots=oddsratio(logbase=2 order=descending)

Wow!  What a difference!