I have revised this in the light of other answers (before and after) that in my view do not give the full picture.
There are some differences. The usual convention is the difference between a Bernoulli- which is when the observed response is binary 1 and 0, while the binomial is concerned with an observed proportion between 0 and 1. The latter has a variable denominator, the number of ‘trials ‘ , thus 3 out of 10 people in the village are unemployed. But fundamentally there is no difference in the underlying method of analysis. Indeed some software silently takes binary Bernoulli data and converts into binomial; same answer but potentially much more efficient. Take some data and analyse it both ways and you will see. The efficiency can be very important - thus millions of records on men and women could be reduced to two proportions. Two parameters based on processing millions of records or two records, but the same answers,; this matters! The binomial formulation is in effect an aggregation of the individual data. The aggregation depending on the unique values of the predictor variables. The binomial formulation is also useful for extra binomial variation ( under and over dispersion) and hence better estimated standard errors, and better residual analysis. I am old and the issue of efficiency used to loom very large and can still be important.
You can also perform a multinomial analysis with individual multi category outcomes or as a set of proportions that sum to 1;the latter is often more computationally stable. This makes it clear that a closed ratio will have negative covariances. Over dispersion is again possible.
Going off on one; apologies. Today there is often a gap in knowledge between textbooks and real analysis; and between theoretical and practical/ experiential understanding .
There is basically no difference between binary and binomial logistic regression. Actually we use the terminology multinomial logistic regression when the outcome variable has more than two categories. In that reference we use the terminology binomial logistic regression when outcome variable has two (binary)
categories.
A binomial logistic regression is simply referred as logistic regression. Logistic regression models the probability of outcome of a categorical dependent variable given all other independent variables. Generally, there may be two or more than two categories in the dependent variable. When there is only two categories (1 vs 0), we define this as binary logistic regression and for more than two categories it is multinomial logistic regression. To make the relationship linear here we use the logit link function and Binomial distribution as family of distribution. We can say, binary logistic regression is a special case of (binomial) logistic regression where the dependent variable has only two categories.
The binary logistic regression is a special case of the binomial logistic regression where the dependent variable has only two categories 1 and 0.
Ehab M. Almetwally the table is perfectly correct but your text is a little adrift. First distinction is binomial and multinomial. The former involves two categories, the latter more than two. The binomial comes in two flavors, one is when the response is proportion of an outcome that lies between 0 and 1. It is produced when the numerator is a subset of the denominator. The other flavor is sometimes given the name of the Bernoulli, it is a discrete 1 or 0 outcome. This binary is a special case of of the binomial when the denominator takes the value 1, so a 'Yes' is 1 out of 1 = 1, and a 'No' is 0 out of 1 = 0. The Bernoulli values are discrete; the binomial values are, in their ratio form, continuous. Both are forms of the binomial.
The multinomial also comes into two forms. A set of closed ratios that sum to 1 (e.g. the proportion of travelers that came by car, bus, or walked) or a dis- aggregated form in in which there is a set of 0/1 responses which sum across the choice set to 1.
For further discussion see
Book Percentages, Ratios and Inbuilt Relationships in Geographica...
and my responses to a similar question
https://www.researchgate.net/post/what_is_the_difference_between_linear_regression_and_logistics_regression
The binomial regression model is the case where the stochastic component in our generalized linear model (GLIM) is the binomial distribution. And as we know, any GLIM is composed of three main components. The link function is an important component of them. Then, if the logit (probit) link function is used in the binomial model, the binomial model is called the logistic (probit) regression.
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