This question concerns the likelihood for logistic regression. Suppose your data

Question

This question concerns the likelihood for logistic regression. Suppose your data consist of (xi,yi),i=1,…,188, with values (2,0), (10,1), (5,0), (8,1), (9,0), (1,0), ...

Part a) Consider the likelihood for logistic regression. Which are the following are true? There might be more than one correct answer.

A. The likelihood is greater than or equal to 1

B. The residual deviance is greater than or equal to 0

C. The log-likelihood is greater than or equal to 0

D. The likelihood is less than or equal to 0

E. The likelihood is less than or equal to 1

F. The log-likelihood is less than or equal to 0

G. The residual deviance is less than or equal to 0

H. The likelihood is greater than or equal to 0

I. Under I have all data values, I cannot determine any of the above

Part b) The reason for the answer in (a) is because for the likelihood of discrete random variables, the likelihood is a ________ . [Fill in one suitable word.]

Explanation / Answer

Question-(a)

Logistic regression can be binomial, ordinal or multinomial. Binomial or binary logistic regression deals with situations in which the observed outcome for a dependent variable can have only two possible types, "0" and "1" (which may represent, for example, "dead" vs. "alive" or "win" vs. "loss").

The regression coefficients are usually estimated using maximum likelihood estimation.Unlike linear regression with normally distributed residuals, it is not possible to find a closed-form expression for the coefficient values that maximize the likelihood function, so that an iterative process must be used instead.

So inthis question concerns the likelihood for logistic regression.

and the data consist of (xi,yi) ; i=1,…,188, with values (2,0), (10,1), (5,0), (8,1), (9,0), (1,0), ..........

First we know that in log-likelihood Regression,the odds are defined as the probability that a particular outcome is a case divided by the probability that it is a noncase. Like other forms of regression analysis, logistic regression makes use of one or more predictor variables that may be either continuous or categorical. So

(A)

  The likelihood is greater than or equal to 1 is TRUE

   Because the Likelihood is the estimate of parameter that is why it must be greater than or equal to 1 from the data (xi,yi) ; i=1,…,188.

(B)

The residual deviance is greater than or equal to 0 is TRUE

In logistic regression, we calculate the probability of getting a residual deviance higher than the one and we got on a Chi-Square distribution with degrees of freedom equal to the degrees of freedom of the model.

(C)

   The log-likelihood is greater than or equal to 0 is TRUE

   Likelihood must be at least 0, and can be greater than 1.

For example, likelihood for three observations from a uniform on (0,0.1); when non-zero, the density is 10, so the product of the densities would be 1000.

Consequently log-likelihood may be negative, but it may also be positive.

(D)

The likelihood is less than or equal to 0 is False. (From the Part-(A))

(E)

   The likelihood is less than or equal to 1 is also False. (From the Part-(A))

(F)

     The log-likelihood is less than or equal to 0 is False. (from the part-(C) )

(G)

      The residual deviance is less than or equal to 0 is False. (from the part-(B))

(H)

   The likelihood is greater than or equal to 0 is False. (from the part-(A))

(I)

      Under I have all data values, I cannot determine any of the above is FALSE.

   Because we have data values so we can determine any one of the above parts.

Question-(b)

   The reason for the answer in (a) is because for the likelihood of discrete random variables, the likelihood is a MAXIMUM LIKELIHOOD FUNCTION.

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