 Olemme nyt katsottaneet interpretation- ja regresson koeffisuus. Se, mitä tehtäisiin, on hieman hienoja, kuin kalkkulaiset tehtäisiin. Kun olet jälkeen regresson analysointi, regresson koeffisuus beta on hienoja, koska tehtäisiin, että tehtäisiin ei tiedä, mitä beta on, joten pitäisi sanoa. Ja tehtäisiin myös, että regresson analysointi on hienoja. Regresson analysointi näkyy, mitä on hieman hienoja, ja se, mitä hienoja on hienoja. Se, mitä hieman hienoja on, hienoja on hienoja ja se, mitä hienoja on hienoja. Se, mitä hienoja on hienoja, on hienoja, regresson koeffisuus on hienoja. Eli nää puna Pri titlesa asuu, niin hienoja on tarkasti... hienoja, mitä hienoa validate on, hienoja, j ambushate on clay. Usually they go into� metronomeulla kone metronomeutta, J Finnish realistic tells us how much the variables — the independent variables together explain the dependent variable – and it's an estimate of the quality of the model in some sense – sometimes it's referred to as goodness of fit of a recursion model, or as coefficient of determination. Most people just refer it to as an R-square. So the R-square varies between zero and one. Zero means that the independent variables don't explain the dependent variable. One means that the independent variables completely explain the dependent variable. One problem with R-square is that it always goes up when you add variables to the model. So when your number of variables starts to increase toward the number of observations, for example if you fit a model with 99 variables to 100 observations, the R-square will be exactly one. So you always increases and goes up and it's positively biased. Bias here means that if we calculate the regression analysis using sample data, the results can be expected to be larger than if we run the same regression analysis on the full population. Because the R-square is positively biased, we have introduced the adjusted R-square statistic, which penalizes complex models. So when your R-square goes up just because you have too many variables in the model, then adjusted R-square adjusts the R-square down to compensate for that bias. So it calculates an adjusted value and the adjustment is based on the number of observations and the sample size. When the sample size is large and you have a very small number of variables, for example if you have five independent variables and 500 observations, so you have 100 observations for each independent variable, the adjustment is very small. If you have let's say 25 observations and 100 units in your sample, then the adjustment is pretty large because you have only four observations for each independent variable. One problem is that the adjusted R-square is not unbiased either, but it can be expected to be less biased than the actual R-square. To actually get an unbiased estimate of the population R-square is quite difficult, so we don't normally do that. The R-square tells us whether the model explains the data at all. So when R-square is zero, then it's end of interpretation. The variables, the independent variables, don't explain the dependent variable at all. Then the question is how much is meaningful explanation. If you explain one person of a phenomenon in some context that is meaningful, in other context it's not meaningful. There are behavior of people and performance of organizations. It's very difficult to predict or explain because it depends on so many different things. And therefore in social sciences the R-squares typically vary in the 10, 20, 30 percent ball part. So if you have a 30 percent R-square then you have a pretty good explanation or you could also have a flawed study, but we'll talk about that a bit later. So you have to consider the context. In natural sciences R-square of 99 percent could be considered not large enough. So R-square is useful for the first check of whether the interpretation of the results further makes sense. If R-square is too small then we know that none of these variables in the model actually matter for the dependent variable. So interpreting the effects of each independent variable separately is a waste of time. Also the R-square offers us an intuitive way of explaining whether the results are large or not. If I can tell you that choosing the choice between three investment strategies for example explains 30 percent of the variation of your investment profits, then that's a big deal. We understand 30 percent. It's a big deal in that context. So because R-square can be understood as a percentage, it has a natural interpretation for most people. We'll take a look at how Hekman uses the R-square in his paper. So Hekman doesn't really interpret what the actual regression coefficients in their study mean, but they are basing their interpretation of the magnitude of the effects on the R-square. And they're saying that between their control variables only model and the variables, the model where there were this gender and race variables, the R-square increases between 15 to 20 percent. That can be interpreted to mean that the effects of race and gender are in the ballpark of 15 to 24 percent, assuming that there's no bias in R-square, which is not true. So they should really be looking at that justice R-square in this case. But everyone understands that if we say that the customer satisfaction scores variation, one fourth of that is explained by gender and race. Everyone understands that that's a big deal, everyone who understand percentages. So it provides us an easy way of saying whether the results are of any practical meaning. When you have looked at the R-square, the next thing that we want to know is which of the individual variables matters. And that's where we get to the interpretation of the regression coefficients. Let's take a look at the Talouselma 500 example. So we have a sample where the women-led companies are 4.7 percent points more profitable than men-led companies. And that's a big difference in our way. We want to know whether the difference is caused by a woman or whether it's caused by some third factor. So we have to present alternatively competing hypothesis. One competing hypothesis is that it is not an effect of sea and gender. Instead it's an effect, it's a spurious correlation caused by firm revenue, so that smaller companies are more likely to hire women and smaller companies are also more profitable. Another competing hypothesis or second competing hypothesis is that this is an industry difference. For example, manufacturing companies are less profitable in ROA metric, because ROA depends on assets and these companies tend to have more assets than service companies. And manufacturing companies are more likely to hire male CEOs than women CEOs. So we have the other variable here. Now Rigorsen analysis tells us what is the effect of CO-gender Ceteris paribus, which is an economics term for holding other variables constant. So when the CO-gender changes from zero indicating man to one indicating a woman, what is the expected increase in return on assets? Holding things constant means that you are comparing two cases that are exactly comparable on the other variables. So if we have two companies that are of the same size and same industry, then a woman-led company is on average beta one more profitable. So the Rigorsen coefficient directly tells us what is the profitability difference. If it's one percentage points, two percentage points or three percentage points, then it's up to us to interpret whether it's a big effect or not. We know that 4.7 percentage points is a big difference, one point probably not so big difference. Okay, so interpreting Rigorsen coefficient is relatively straightforward when these variables have a meaningful unit. So we know that ROA has a meaningful unit for managers. Everyone, if we say to a manager that my company's ROA is 20%, they know that it's pretty good for most industries. We also know that COS-female one is a woman, zero is a man, so it has some meaning for us. Sometimes we have units that don't really have any meanings and that complicates the interpretation. So let's take a look at this question. Does one unit increase of education, does it pay off? We have a statement, a Rigorsen result, that one unit increase in education leads to one unit increase in salary. Is it a big deal? We would need to know what is the unit of education, what is the unit in salary? Let's say that the unit is education in years and salary is euros per year. So we say one year increase in education leads to one euro increase in annual salary. Does it make a difference? I would think not for most people, pretty much every people. No one really wants to go to school if you just get one additional euro of income per year. So that way it's not meaningful. How about one year increase leads to one thousand year increase in annual salary? That's a more problematic question. If we consider Finland where our salaries annually are in tens of thousands of euros, maybe in the lower end, if you make like twenty thousand euros per year, maybe one thousand is worth one year of education. Maybe not, depends on, it's five percent. Depends on how much you like to go to school. On the other hand, if these data were from a developing country where the annual salaries are in the thousand, two thousand euro ballpark, then one euro increase in annual salary is a big deal. You can double your income basically in some cases if you go to one additional year of school. And that's a big thing for those people. So you have to think of what are the units, what's the unit of the independent variable, what's the unit of the dependent variable and what is the context that you're evaluating, the effecting. What if we say that one year increase leads to one bitcoin increase in annual salary? So we get one additional year of education and we get one bitcoin per year more. Well, that's more problematic because people don't have any intuitive understanding of what is the value of bitcoin. So obviously when you tell somebody that I'll give you a bitcoin, then the first question they'll ask, what's the value of bitcoin in euros? So in this case we could convert the value of bitcoin to euros so we can do a conversion and express the regression coefficient in a way that's more understandable. Let's say that one year increase leads to a three thousand increase in annual salary. I don't know what's the value of bitcoin now, but let's assume it's three thousand euros. So then we know that it's probably a big deal for some people. So sometimes we can convert the units to something that we can understand even if the original unit was something that we don't understand easily. What if we have a case of a unit that cannot be converted? So let's say that our result is that one year increase leads to one bukasoid increase in annual salary. Bukasoid is a fictional currency in a computer game and I don't think that anyone has ever developed an exchange rate from bukasoid to euros. So we can't convert this effect into euros. So what do we do? One way of dealing with this bukasoid issue is that we have to first understand what's the average salary in bukasoid in this fictional universe and also what is how much of the salary is dispersed. If we say that I'll give you ten bukasoids or I'll give you a million bukasoids, it doesn't really make sense unless we know whether what's the mean income. If we know that the mean income in that fictional world is ten bukasoids, if we tell somebody that you'll get a million bukasoids, then a million bukasoid is probably a lot. If we tell them that we give you a million bukasoids and the annual income is a billion bukasoids, then not a big deal as much. To understand how the variable varies, we have to look at its mean and standard deviations and it's useful in this case when we have these variables that don't have any units, any naturally interpretable units, look at how is it distributed. So we take a look at our mean and standard deviation. Let's assume that in our sample the income in bukasoids is distributed normally and normal distribution implies that one standard deviation, two standard deviations from the mean have a special interpretation. So in normal distribution 68% of observations are plus or minus one standard deviation about the mean. So if we say that our income is one standard deviation about the mean, then we know that we are solidly in the high income segment. So we are pretty good, pretty well about the average. If we say that our income is two standard deviations in bukasoids about the mean, then we know that we are in the top two and a half percent of the income distribution. We can also see that generally the effect of one standard deviation increase is pretty big. You are solidly here below mean, one standard deviation takes you to average. Then two standard deviations you are pretty rich. So you are in the top two and a half percent. So standard deviation units can be useful for interpreting rigorous analysis results. So if we say that one additional year of education increases your income by one standard deviation in the bukasoid units. Is it a large effect? Well then we would have to, for people it is, but we would have to think what is the life span of these aliens. If they only live on average one year, then one year investment in education is a huge deal for them. So we have to think about the context again. Let's take a look at an empirical example. So this is the deep houses paper, table two and model two from the regression results. And we'll be interpreting these purely through standard deviations. The dependent variable ROA has a meaningful unit, but we'll just ignore it for now. So we'll just be looking at standard deviations. Their regression coefficient was minus 0.02 for the effect of strategic deviation or relative return on assets. So is it a big effect? To understand that we would need to understand what is the unit of strategic deviation. That's a completely made up number by them, so it doesn't have a meaning. ROA has meaning, but we'll just ignore it for now. We need to know what are the standard deviation of these variables. So the standard deviation of ROA is 0.7 and standard deviation of strategic deviation is 2.9. That tells us that if the data are normally distributed then 95% of the observations of ROA are plus or minus 1.4 units. That's two standard deviations from the mean. The difference between top 2.5% and bottom 2.5% is then 2.8 units. So top 2.5, bottom 2.5, four standard deviations, it's 2.8 units. So what is the effect of strategic deviation? The strategic deviation, one standard deviation increase of strategic deviation is then 2.932 multiplied by minus 0.020, which equals minus 0.058, decreasing relative ROA. Then we compare is this minus 0.058, is it large than the 2.8 units? So the full scale is from the minus 2.5% to the worst 2.5% to the best 2.5% is 2.8 units. And if you increase your relative, your strategic deviation by one standard deviation you get minus 0.058 decrease in ROA. So it's a smallish effect. We can also understand the effects as interpretation and how it's reported while looking at this nice example about sauna. So when we ask whether the sauna is warm or not, sauna is a finished thing and a normal research paper would say that the temperature of the sauna statistically significantly different from normal room temperature. It tells us that maybe the sauna is heating, maybe it's ready for going in, maybe it's too hot. Maybe it was on a day before and it's still cooling. It doesn't really tell us anything about whether the sauna is warm or not. And that's equivalent of saying that the effect of statistical deviation on ROA is negatively and statistically significantly different from zero. So the statistical significance just tells that there is some effect. It doesn't tell us whether the effect is large or not. Then even better, a slightly better answer is that the temperature of sauna is currently 80 degrees and comparable that the effect of strategic deviation of ROA is minus 0.20. So that is useful for people who understand what 80 degrees means and what this minus 0.020 means. So for most people who go to sauna often know what 80 percent means but you can't assume that your readers of your research study will understand your units so you have to explain what it means. So a really good answer to whether the sauna is hot is to say that the temperature is currently 80 and then tell that most people who go to sauna regularly would say that the sauna is too hot but they could still do it. So that quantifies that the sauna is pretty hot. More so than just saying that it's 80 centigrade. The same thing, you can see that the effect of ROAs is minus 0.20 and the difference between ROAs of top 25 and bottom 25 percent for standard deviation is minus 12. So if you go from the least deviant to the most deviant is 0.12 and the same scale for the ROA is 2.8 so we can see that 0.12 is pretty small compared to 2.8 so the effect is quite small. There are other things that you can do to improve your profitability than to be more statistically deviant. Let's take a look at yet another example. So this is from Hekmans paper and Hekmans paper shows a regression table and these effects are number of patients in a panel so how many people go to see a doctor is minus 0.04 and the age of the doctor is minus 0.13, the regression coefficient. Are these large effects or not? We would have to look at the correlation table and standard deviations and means to understand whether these are large effects in a normal case. But this is actually not a normal case because these are standardized regression coefficients. They don't report it but you can see it by comparing if you start to interpret this effect of number of patients in the panel which is in the thousands and age which is in the tens. You can see that the effect sizes don't make any sense. Also all these effects are variable in 0, riddling plus or minus 1 which is the typical range for a standardized regression coefficient. They can be more or less but they are typically 0.0 or minus 0.0. So these are standardized coefficients which means that the data have been standardized so every variable has standard deviation of 1 and mean of 0 before regression estimation. In that case we estimate this directly as standard deviations. One unit increase in physician-proactivity associated with beta one increase in patient satisfaction so we say that these are one standard deviation increase in physician-proactivity as one standardized increase in satisfaction. So we interact directly as standard deviations. This looks like the way to do it always so it would simplify our life to always use standardized estimates but that's actually not the case. I recommend that you never standardize a variable that has a meaningful scale. So if you have euros or years or something that makes sense to people as a unit then don't standardize. The reason for that is that standardized estimates depend on the scale of the variables because the standard deviation is a sample standard deviation. So let's say that here the standard deviation of 8 is 6.58 and the mean is 50.34 so the doctors are quite old. What would happen if the doctors in this sample were actually newly graduated between 24 and 28 and standard deviation would be 1? What would happen is that the standardized regression coefficient for the same effect would be only minus 0.02 which has a very different interpretation from minus 0.14 so it's seven times as small. It's exact same effect. It's a scale differently. So the differential scaling means that these effects 0.02 and 0.40 are not comparable. So standardization doesn't make your results comparable so if you can interpret the results without standardization it is always better to do so. So rule of thumb use standardization only if your variables none of them have a natural scale. Otherwise interpret the standard deviation units only for those variables for which a natural scale does not exist.