Now in this video we will be discussing about the concept of odds and odds ratio now what is odds also the ratio of probability of an event divided by probability of a known event mathematically it can be represented as odds equal to probability by equals to one divided by probability equals to zero, where probability y equals to one denotes probability of an event and probability y equals to zero denotes probability of unknown event. So, this is the standard form of registration model that is the equation the standard equation policy regression model where probability y equals to r is equal to one by one plus exponential means e to the power minus beta naught plus beta naught or V naught plus b one x one plus EI and probability y equals to zero is one minus y equals to one which is equal to one by one plus exponential z to the Part B naught B naught plus b one x one plus e Here we have considered that there is one independent variable that is x one P naught are the intercept that is B not as intercept and B one is the regression coefficient and EA are the error terms.

Now, standard equation for logistic regression model if we calculate the odds, then we get the equation for Arthur they are the expression for Odyssey to the Part D naught plus b when expand if we do not consider their terms and if we want to consider their atoms it will be equal to the Part D non plus b when expand plus AI here we have not considered the error terms while calculating odds. So, therefore, the expression for x is equal to e to the V naught plus b one x one e to the V naught plus b one x one now, if you take the natural logarithm on both the sides that is ln ln of odds gives you B naught plus b one x one that is it's a linear model. So, this is the assumption of logistic regression which we had discussed before that that there is a linearity between the independent variables and log of odds in logistic regression models.

That assumption caught clarified in this derivation that if we take natural log on both the sides, we get a linear model, therefore, ln of x is equal to P naught plus b one x. Now, let's understand the concept of change in odds. If we change x by one unit that the change in ops is given by T or x is equal to e to the bar b naught plus b one x, if we increase the value of x by one and makes it and we the value of x becomes x plus one, then the expression for us becomes about V naught plus d one into x plus one. Now if we find that change of odds, then we'll find the new odds will be e to the power b one times the original rounds. So if the value of my x increases by n times and it becomes x becomes x plus n, then my new odds will change by e to the power n b one times.

So this is a very important derivation you can understand that when we increase the value of x by a particular term, say it is n r In our independent variable becomes x plus n from x that is our independent variable changes from x to x plus n then our new odds will be e to the power n b one times the original logs now let's come to the concept of odds ratio odds ratio is basically the ratio of two odds in order to explain you all the concept of odds ratio let me take an example here we have assumed a case where the probability of y equals to event is that accident will occur that is a person is going to get hit by a car and probability of no event is a distributed device goes to zero his accident will not occur that is the person will not get hit by a car now I have divided the individuals into two parts one is poor vision individuals and other is good vision individuals for poor vision individuals probability y equals to one that is the accident will occur is going to be equal to point seven.

Let's assume we have assumed that probability by equals to one for a poor vision individual is point seven therefore gravity equals to zero for a poor Which individual will be point three that is one minus point seven because contracted by equals to zero is one minus probability by equals to one similarly for good vision individual probability by equals to one that is collaborative I will show you even which denotes that accent will occur is equal to point four that we have assumed then collaborative idols to zero will be one minus probability by equals to one for good vision person which will be equal to point six so see for a poor vision person my priority by equals to one a probability of an event is point seven or good vision person probability by equals to one is point four so, for a poor vision percent probability by equals to zero will be one minus point seven because some of the gravities are equal to one one minute concept is equal to point three and probability by equals to zero for good vision percent will be one minus point four which is equal to point six so, odds for a poor vision person will be what probability equals to one by gravity by equals to zero, so our supervision person is point seven by point three net is seven by three which is approximately equal to 2.33.

And also a good vision person will Again, point four by point six, that is probability y equals two, y equals to zero, which is fine for vacancies, which is equal to two by three, which is equal to 0.67. Now also the ratio is ratio of two odds. So either we can do poor vision by good vision, or we can do good vision by position here I have done coefficient a good vision, which is equal to 2.33 by 0.67, which is equal to 3.48. Approximately, we can take it as a parasite. So the interpretation of odds ratio is, if I'm moving from a good vision person to a poor vision person, then the odds of getting hit by a car increases by 3.5 times and if we take the reciprocal of it, that is, if we do good by output, then the interpretation will be when I'm moving from poor vision person to a good vision person, the odds of getting hit by a car will increase by point three tenths.

So now let's understand the interpretations of odds ratio offs ratio is basically the ratio of two odds odds ratio helps us to understand the effect of a predictor the interpretation of the odds ratio depends on whether predictor is categorical or continuous when the predictors are categorical that is when my independent variables are categorical for categorical predictors, the odds ratio compares the odds of the event occurring at two different levels of the of the predictors of ratios that are greater and greater than one for safe there are two different levels the Level A and level B. So odds ratio data greater than one for Level A indicates that the event is more likely to happen at level eight and odds ratio that is less than a one for Level A indicates that the event is less likely to happen for level eight. Similarly for level D, and if we want to understand the concept of odds ratio for continuous predictors odds ratio that is greater than one indicates that the event is more likely to occur as the character increases and odds ratio that are less than one indicates that the event is less likely to occur as a percentage increases.

So in this video we will be learning to hear for now let's end this video over here. Good bank. See you all for the next video.