 Lopulta logistic regression analysis results differs a bit from normal regression analysis interpretation. Let's take a look at the results from logistic regression analysis using the Menard's dataset. The RGLM command gives us these results. We'll just focus on the actual coefficients for now and leave these other things for another video. We have the estimate, which is an estimate. Then we have standard error, which quantifies how much the estimate is likely to change from one sample to another. If we repeat the study over, we have a z value, which is the ratio of the estimate divided by standard error. The z value is the same as the value in regression analysis. It is called a z-statistic instead of t-statistic because the maximum likelihood estimates are based on large sample theory. Instead of comparing against a t-distribution, we compare this against the normal distribution. Under the null hypothesis that this estimate is zero in the population, and if the sample size is large enough, the z value follows standard normal distribution. It allows us to calculate the p-values. Whether an age has an effect or not can be interpreted from these p-values, we can see that age has a very statistically significant result. We can confidently say that age has some kind of effect on the probability of having had Menard's. What is the magnitude of that effect? It's a bit more complicated question to answer. We really can't say that the probability of having had Menard's increases by 1.6 when the girl gets one year older. The one reason is that a 1.6 increase gets us beyond the range of the data. So if the probability is zero, initially you increase age by one, the predicted probability would be 1.62. So it doesn't work that way. The reason why we can't interpret this directly is that these are the effects before we apply the logistic link function. So these are effects on the linear predictor and not on the actual dependent variable. So it's the same thing like when you do a log transfer from dependent variable. The interpretation is the coefficient tells you what is the effect in log scale. And you want to know what's the effect in the original scale. This coefficient here tells you what is the effect in the scale of the linear predictor. But you are not really interested in that. You are interested in what is the effect on the observed variable scale. So we don't interpret these directly. Instead we interpret them as odds ratios. So the odds ratio is a concept that is useful for regression analysis. And for some other, for logistic regression analysis and for some other models as well. The idea is that odds are the ratio of two outcomes. So here we have the outcome of girl having had menards and not having had menards. So it's if one in 100 girls have had menards, then odds for having had menards is 1 to 99. Because one out of 100 has had it and the remaining 99 hasn't had it. You can think of one common use of odds is engambling. So if you have a team, two soccer teams, one has won two matches in the past. Another one has won five matches in the past. Then you say that based on that data the odds for the first team winning is 2 to 5. So that's the idea of odds. And more formally, if the probability of an outcome is P, then the odds is defined as P against 1 minus P. So it's the probability of one outcome divided by probability of another outcome. If you have only two possible outcomes. And the exponential, if you exponentiate the logistic regression coefficients. Those can be interpreted as odds ratios. And the idea is that when you exponentiate the coefficients. Then the coefficients tell you that one unit increase in independent variable causes the odds to change proportionally to the regression coefficient. I'll show you an example. Let's take a look at the idea of odds ratio and why we can interpret these coefficients as odds ratios. So example odds for the data. And this is some guess of the results. We have the linear prediction. We have the fitted like probability. We have fitted odds, which is there are the probability against the other probability. And we calculate the value. So the odds for this first girl having had menarties 74 to 26, which is 2.79. The odds for the second girl is 8 to 92, which is 0.09 and so on. So these are the odds. And when we calculate marginal prediction. So in regression analysis, we are interested in the marginal effect. What is the increase of one independent variable? The effect of increasing one independent variable by one unit holding everything else constant. So we are interested in marginal effects. And let's calculate marginal effects now for girls at different ages. So instead of using this actual data, we have a hypothetical girl at the age of 9, 10, 11, 12 and so on. We calculate the fitted probabilities using our model. And we calculate odds. We calculate the value of the odds. And when we compare two odds here, the ratio of these two zeros, they are actually not exactly zero, is 4.6. So every time we go and we increase the girls age by one, then the odds increase by 4.6. So every additional year increases the odds by 4.6 units. So that's the odds ratio implementation. So these always increase by 4.6. And how we use that in regression analysis? Well, we calculate the odds ratios and this is for the actual data. So we calculate the odds ratio, which is about five. And the interpretation is that each additional year of age increases the odds of having had menards by five-fold. So that kind of quantifies how large is the effect of age. We know that if something is increased five-fold, then it's pretty large effect. The problem still is that with odds ratios, we can't really say how much does the actual probability increase. Because odds and probability are not the same thing. And quite often we want to know how much does the probability of having had menards depend on the age and what does the effect look like. To do that, we would need to plot the marginal predictions from the model.