 Yksi tärkeintä statistiikkia normaali- ja regresson analysointi on R-squarella, jossa on kysynyt, miten se maailma on esiintynyt ympäristövalvella. Yksi tärkeintä linear- ja ympäristövalvella ja ympäristövalvella on ympäristövalvella, jossa R-squarella ei ole esiintynyt, tai jos se voi olla kalkoitu, niin se on ympäristövalvella, joka ei ole sama kuin normaali- ja regresson analysointi. Me katsomme nämä alternativista statistiikkia as Pseud R-squarella. Tämä video on tämän videossa näkyy ympäristövalvella Pseud R-squarella ja regresson analysointi. Tämä ympäristövalvella asaitaa vielä ympäristövalvella, jotta pitää olla ympäristövalvella. Joten, että sillä on ympäristövalvella, jotka coincidivista regresson analysointit ovat ympäristövalvella, joilla ne ovat ympäristövalvella ja regresson analysointit ovat ympäristövalvella. Let's take a look at first what R-square quantifies so that we can understand what so R-squares tell us and what they don't tell us. So this is the equation for R-square. The R-square is calculated, this is one of the ways, but it's calculated based on the residuals. So we have the residual variance here and we take the residuals. It's the sum of it's a difference between the predicted value, predicted value and observed value. We take squares and then we take sum of squares. Then we have a prediction calculated by the sample mean only. So this is a prediction from a model with intercept only. So this is the prediction from new model, how far they are from the observed values and this is the predictions from the actual model. So we compare for example how much better the estimated model is over a model that shouldn't really predict the variance in the dependent variable at all. So R-square has a couple of different roles. It can quantify the amount of variance explained in the data. If R-square is zero, then the model doesn't explain any variance in the data. If R-square is one, then the model explains all the variance in the data. R-square also quantifies model fit. So R-square of one means that it's a perfect model, fits perfectly. R-square zero means that it's equivalent to the new model that doesn't explain the data at all. And then we have a third interpretation which is how well the predictions from the model correlate with the actual observed values. We also have a couple of different two other features in R-square. One is the estimation criterion in normal regression analysis is maximizing the R-square. So you maximize the variance explained and then that will give you the OLS estimates. Also R-square can be used in estimable comparisons using DevTest. These last two criteria are not they're useful for pseudo R-square. So pseudo R-square only focuses on these first three aspects of R-square. Because pseudo R-square are typically calculated based on the maximum likelihood estimates and there is not always a straight relationship between the estimation criterion and the pseudo R-square. And also pseudo R-square can be used for model comparisons or model comparison, nesting model testing the same way that R-square can be used. You can use the likelihood, log likelihood statistic directly as the maximization criteria and for model testing. But for interpretation purposes the pseudo R-square tell us some information about these three characteristics. So they can tell us how much the model explains of the data, how much better the data is than the null model and how much better the model is than the null model and how well the model predicts. So we have three how much it explains the improvement of a null model and prediction. So these three characteristics and generally we can have only at most two in these in any pseudo R-square. So pseudo R-square are if it's meant to quantify prediction then it's not necessarily good for quantifying how much the model improves over the null model or it doesn't tell us how much the model explains the data. The reason why we have so many different pseudo R-squares is that these three objectives, different objectives are important and sometimes we want to emphasize one objective over another. So this is a list of some of the pseudo R-square values of pseudo R-square statistics for logistic recursion analysis. There are maybe a dozen more but these are perhaps the most common. So this is the normal R-square. We're just calculating R-square and using the normal equation. So we take the predicted values and we calculate the residual using predicted values. Then we calculate the difference from the mean value and that gives us the R-square that's attributed to F1. So that's but this is not very very useful because while it's the same equation than the normal R-square it doesn't really tell us anything that normal R-square does. Then we have the ones that are more commonly used. So McFadden R-square compares the likelihood, log likelihood of the model against the null model. The idea is that the log likelihood of a perfectly fitting model is zero and this will go to one and the log likelihood of the worst possible model that doesn't explain the variation of the dependent variable at all is the null model. So this ratio goes to one and the McFadden R-square goes to zero. So that quantifies the improvement over the null model. Another commonly used is coaxon-snale. The idea of this equation is that it's basically the same as this equation but calculated using the deviance residuals. So this is if you calculate max normal linear regression analysis using maximum likelihood and you apply this equation then you will get the normal R-square. The problem with this equation is that the maximum is less than one. It can be substantially less than one and another pseudo R-square that takes that weakness into account if you consider the weakness is the Nagelkerkes R-square which is basically just a scaled version of the coaxon-snale R-square so that the maximum is one. So it's coaxon-snale divided by one minus low likelihood null to the power of two divided by n which is the maximum of this equation here. So those are the ones based on the likelihood. Then we have McHelveny and Chavoina's pseudo R-square. This is an R-squaring in the sense that it's a how much variation is explained. So there is variation explained plus variation explained plus variation of the error term. So this is if you apply this to normal regression analysis you will get the normal R-square. The difference is that we are not looking at the predictions in the observed variable instead we are using the latent variable formulation of the logistic regression model and we are looking at how well the model explains the latent response variable. The latent variable that underlies the response variable instead of the response variable itself. I have another video that explains the latent variable interpretation. Then we have two simple statistics that quantify actually how well the model predicts. We have the count R-square and the count R-square simply classifies each observation into the number that it's more likely. So if I'm predicting probability for an observation is 51 percent then the predictive value in the count R-square for that observation is 1. If the predicted probability for observation is 49 percent then the predicted value is zero for the count R-square then you just calculate how many predictions you got correctly and you divide by sample size. If you predict randomly the count R-square will be 50 percent. So basically this R-square will always be really high. It's almost never below 50 percent and it could go up to close to 100 percent. Then we have a coefficient of determination or discrimination which is a bit more recent introduced by Ture in 2009 and this is also called Ture observed R-square. The idea is that we focus on the probability of the prediction. So we calculate the mean probability of those cases that had a value of one. The mean of predicted probability of cases that had a value of zero and then we calculate the difference between the means. So this quantifies whether the model predicts or successes and failures differently. If there's a large difference in how successes and failures are predicted then it implies that the model is good for prediction and it has a high coefficient of discrimination on high pseudo R-square. Let's take a look at examples because these statistics are quite different. So this is some data sets and we have the blue line is linear recursion model and the red line is logistic recursion analysis. We have a couple of scenarios. In the first case the model doesn't explain the data at all. So the R-square in all is zero and we can see that all of these R-squares are close to zero except our adjusted version of McFadden's R-square which is negative. So we have this adjusted R-square here as well. So if we had adjusted R-square for the OLS that would probably be negative as well. Importantly the count R-square is 52 percent. So model doesn't explain data at all but still we get half of the predictions correctly because if we predict randomly then we will get one or zero variable correctly with a 50 percent probability. Then we have a scenario here where the logistic recursion curve and the recursion line overlap. So this is the scenario where it wouldn't really matter whether you use logistic recursion analysis or normal recursion analysis because the predictive probabilities are between 20 and 80 percent. Ideally in this case an ideal pseudo R-square would probably agree with the OLS R-square which we can see here that the Coxson-Snell R-square does and the two R-square does but and also the difference because that's calculated the same way as the normal R-square. But for example the negative R-square is pretty high. McFadden's is too low and count 74 percent so that really tells us something very different than what the normal R-square does. So if you want to say that the model doesn't explain the data at all then you can pick the McFadden adjusted R-square if you want to say that the model predicts well then you pick the count R-square. So it's kind of like pick your own result scenario. Of course you should justify your choice and if you want to answer a question how much various data explains then the count R-square will be inappropriate for that. Then we have a model where the S-curve and the line start to depart. So the S-curve clearly bends down here and the line gives us invalid predictions. Predictions exceeding one predictions are below zero and this is a scenario where we really should not be using a linear model. Again we see that the values are quite different. So some of them are close to R-square from or less some or not and then we have the final which is a scenario which is interesting. This is the perfect prediction scenario. So the model predicts perfectly when the x is below a cutoff then y is always zero when the x is above cutoff y is always positive. The count and U-R-square interestingly both show that it's perfect predictions. This scenario demonstrates that the Cox's and Snell even if the model predicts perfectly the Cox's and Snell will not be one. So all these provide you a different value. So the question now is which one should you use and of course this is a balanced model where we have ones and zeros that are equally on with equal probabilities. We can have a model where the ones are three times as prevalent as zeros and then these R-squares will give us again a slightly different values. The problem is that people don't realize that these quantify different things. So when you take a look at our published examples, published papers how they apply pseudo R-squares quite often you see pseudo R-square reported in a regression table. In an ideal case they are two or three and then they are just there they are not reported. Sometimes they are not interpreted. Sometimes they are interpreted and typically researchers interpret them as if they were the same as OLSR-square. So there for example you are comparing that there is a 11.9 percent difference and that would be the difference between 22 and 35 here. It's not clear whether you can actually interpret the difference in the algorithm R-square and how you would do that. But nevertheless people think that you interpret these the same way as R-squares. So what's the R? How should these be used then? There is quite a lot of disagreements on the value of these pseudo R-squares. I have my opinions others have their opinions. There's an article in strategy management journal that goes over the use of logistic and profit models in this journal and in the end they have recommendations table and they're basically saying that a model fit you have to assess it somehow and there is a disagreement on whether you should have a pseudo R-square or not. If you have one you should really pay attention to it and report that it's not a normal R-square and it should not be interpreted as such and then they give some other advice. So what's my take? It's that I don't include pseudo R-square. That's my first recommendation. If you don't know what to do just leave it out. It doesn't really hurt anyone that you leave out the statistic that your readers don't know how to interpret properly and if you don't interpret it yourself. There are exceptions to this rule. If you have a good reason to include one for example if you want to know how well the model classifies. If you have a problem where you are interested in classifying cases then the count or the cure R-square could be very useful. But you have to have a specific reason for a specific coefficient instead of just going for the coxons in the lanagelkaker which are the two most common perhaps. Interpret the effects as graphically instead. So show other readers how does the logistic regression could go. That's much more useful than giving them a pseudo R-square that they don't know how to interpret. Then if you're asked to include pseudo R-square sometimes you write the paper and then you send it to review and the reviewer is used to looking at recursion results and they always expect to have the R-square for the recursion results but they have your logistic recursion analysis results without any R-square statistic. The reviewer thinks that there should be one. So what do you do about it? Or your co-author thinks that there should be one. In one of my articles a co-author of mine wanted to have pseudo R-square and I gave him a pseudo R-square of 20 and a pseudo R-square of 50 and I told him which one would you like to have. And that was the end of the discussion. We didn't include pseudo R-square because there was no reason to. But it's very common that people want to have an R-square because normal recursion analysis has one. I wouldn't argue against the reviewer. So I guess the co-author it's okay but I would just include a pseudo R-square of few if a reviewer asks me to. It doesn't really make that much of a difference. In that case I would use multiple. So I would have at least the McFadden, Cox and Snail and Tures. I wouldn't use the Nagelkärkä because it tends to overestimate the OLS R-square and you don't want to use statistics that make your model look more impressive than it actually is. So that's one rule. These will be conservative in what you present. And then if you are asked to interpret the R-square, I would interpret the Ture R-square. So that's my point, my take. This is the take from a person who wrote a paper to the Society of Management Journal. So what do the experts say? Let's take a look at the book by Hosmer and Lemerson or Sturlivan. This is the best book in logistic recursion analysis. It's a bit technical but it's a very, very good explanation and for example many of the diagnostics that we use for logistic recursion analysis are based on the ideas presented in this book. They basically say that pseudo R-squares are not that useful. They don't allow you to test whether the model is mis-specified or not. There are better tools for that and also people who are used to reading logistic normal recursion analysis results and they see that there are R-squares from logistic analysis results, they will be confused. So you have a statistic that is not very useful for model diagnostics or analyzing whether the model is any good and a statistic that is likely to mislead your readers, then the clear recommendation is that you probably shouldn't include those. But understanding the pseudo R-squares is important because oftentimes reviewers want you to include one and quite often when you read a paper it does include one.