# How to determine if the log likelihood of logistic regression is too large or not?

I am running a logistic regression on STATA with binary response variable, and 2 predictor variable that are discrete, as such one is in % (but takes only 2 values strictly i.e., 5% or 10%) and another takes value strictly from 1 to 10. My goal is to identify the relationship between these predictor variables and success of binary variable (occurrence of event, where y=1). I also have control variables I would like to add which are either discrete, continuous or categorical. Since I am new to running a logistic regression, I have some queries.

I understand that the loglikelihood is always negative (the statistic provided when you run a regression on stata), and that larger values i.e. farther from zero (-100 as opposed to -50) is an indication of poor fit of the model. " Since the log likelihood is negative, −2LL is positive, and larger values indicate worse prediction of the dependent variable". My question is how to determine whether log likelihood is too large? Is it dependent on number of observations.

I have over 300k observations, and I am trying to ascertain if the primary predictor variable (in %)decrease the probability y =1 . In this case, how large of log likelihood is considered as too large, as in a poor fit of the model. My estimated log likelihood from only running regression with predictor variables (excluding control variables) is -207079.88. Is this too large for my dataset? My LR chi square is significant and so are the coefficients of predictor variables.

Default | Coefficient  Std. err.      z    P>|z|
x1 |   4.941906   .1605616    30.78   0.000
x2 |  -.1427967   .0059592   -23.96   0.000
_cons |   .0465623   .0433171     1.07   0.282

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Logistic regression Number of obs = 312,748 LR chi2(2) = 1768.93 Prob > chi2 = 0.0000 Log likelihood = -207079.88 Pseudo R2 = 0.0043

I understand the Pseudo r2 is too low, and preferred Mc fadden r2 minimum range is 0.2. However, I assume it will be better when I add control variables

Could someone advise when the value of log likelihood matters? Is a significant LR chi sq and significant coefficient sufficient to conclude the relationship between dependent variable and independent variable?

• I'm curious where you read that the log-likelihood is an absolute measure of model fit and can have values that are too large. I have never heard of this. Where did you read that the R2 needs to be high?
– Noah
Commented Sep 30, 2022 at 6:46
• @Noah "Close parallels to F and R2 exist for the logistic regression model. Just as the sum of squared errors is the criterion for selecting parameters in the linear regression model, the log likelihood is the criterion for selecting parameters in the logistic regression model. Some software presents information on the log likelihood itself; however, other software presents not the log likelihood itself, but the log likelihood multiplied by −2, which forms the basis of calculating χ2 statistics based on the likelihood ratio" - continued in the next comment
– LKho
Commented Sep 30, 2022 at 7:36
• @Noah"For convenience, the log-likelihood multiplied by −2 will be abbreviated as −2LL in general, with subscripts to identify specific −2LL statistics. Since the log likelihood is negative, −2LL is positive, and larger values indicate worse prediction of the dependent variable". Source: Chapter 3: Quantitative Approaches to Model Fit and Explained Variation in Logistic Regression: From Introductory to Advanced Concepts and Applications by Scott Menard.
– LKho
Commented Sep 30, 2022 at 7:38
• @Noah Since he mentioned log likelihood is a close parallel to R2 - which is a measure of goodness of fit model, i thought he implied the same for log likelihood
– LKho
Commented Sep 30, 2022 at 7:39
• @Noah Regarding your second comment about R2, McFadden states in his book Behvioural Travel Modelling "while the R2 index is a more familiar concept to planner who are experienced in OLS, it is not as well behaved as the ρ2 measure, for ML estimation. Those unfamiliar with ρ2 should be forewarned that its values tend to be considerably lower than those of the R2 index...For example, *values of 0.2 to 0.4 for ρ2 represent EXCELLENT fit." Since, my R2 square value was only 0.0043, it was not a good fit.
– LKho
Commented Sep 30, 2022 at 7:45