 Regression analysis is a basic tool for making causal claims from observational data. To understand regression analysis you need to understand what the technique does, when it is useful, and also what kind of results it produces and how those results are interpreted. In this video I'll take a look at the results in the presentation. Regression analysis results in a research article are typically presented in a table like this. These tables follow certain conventions, and understanding the conventions makes it easier to understand what the table is about. So let's start from the top and go to the bottom. On the top there's the explanation of what the table is about, and particularly what is the dependent variable in the regression model. So that is in the table title, typically says the purpose of the table, and then at least the dependent variable. It might say some of the independent variables as well. Here we have age, gender and objective performance. Then when we move down we can see that this table contains four models. They are labeled model 1, model 2, model 3, model 4. That's fairly typical. Sometimes they have more descriptive labels, or if we run the same model using different subsamples, we might have the subsample description as a table, as a model heading. Then come the regression coefficients. So these are the values of the betas. If we want to do a quick and dirty interpretation of the regression coefficients, we can look at these stars that are typically till the significance levels as shown in the table of footer, and check which ones are significant. If a coefficient is significant, then we conclude that we have enough evidence to conclude that there is an effect of that variable, and then we look at the sign to determine the direction of the effect. If there are no significant results, then we conclude that we don't have enough evidence to say that there is an effect of, for example, our productivity. Non-significant result does not mean that there is no effect. It just means that we don't have enough evidence to say that there is an effect. Then we have model indices. So these model indices quantify something about the models as a whole. They are not about individual variables, but they are about how well the model explains the data. If we have different subsamples or different sample sizes for different models, then we would have the sample size information here. Finally, we have table footer here, or table note, that tells some useful additional information about the table. For example, here it tells a sample size, and that these are 100 white person and 13 non-whites. For a paper that talks about gender bias, it is relevant to know that there are only 13 minority race people. And there is a legend for p-values. The one star means significant at 0.05 level. Two stars means significant at 0.01 level. So this is the table. Let's take a look at now. Can we interpret these actual numbers? The first thing that we look at the table is the model quality indices. So we look at this, and in this table the indices show us R-square and adjusted R-square statistics. The R-square tells us how much the independent variables together explain the dependent variable. 0 means that independent variables are linearly independent from the dependent variable, so they don't explain the dependent variable at all. Value of 1 means that the independent variables completely determine the value of the dependent variable. So if we think of regression as a line, then all observations are on that line. The R-square has a problem that it tends to be too large in small samples. So there are some correlations that occur by chance only, and then R-square capitalizes on those correlations. To address that issue, there is an adjusted R-square statistic. So adjusted R-square statistic is a bit more realistic of the amount of variance explained in small samples, in large samples, adjusted R-square, and the R-square will be the same. Then we also have something that does not really quantify an individual model, but that is used for model comparisons. So we have this delta R-square from model 1. So how much the second model explains data better than the first model, and how much the third model explains the data better than the first model, and so on. These stars here are significant tests for the delta R-square to be exactly 0. So we have statistically significantly different results from 0, which means that model 2, model 3, and model 4 explains the data statistically significantly better than their first model. So what's the point of having four models then? The point of having four models is that we want to have a baseline. So the first model is always a baseline that only includes the control variables and not the interesting variables. Then in the subsequent models, we add more interesting variables to the model, and if a model with interesting variables and the controls explain the data a lot better than the control variables on the baseline, then we can conclude that the interesting variables have an effect. So the objective with this kind of R-square comparison is to address the question, how much the independent variables explain the data beyond the controls only. R-square is also useful for quantifying the size of the effects. And this strategy is used in the Heckmann's paper that I use as an example here. So in Heckmann's paper, in the conclusions section, you can find this part in yellow that states that across the three studies that they conducted, our example comes from study one. There are interesting variables explained 15-24% of the performance variation in customer satisfaction scores. Where does the 15% come from? Well, it comes from this table here. It is the delta R-square value of the most comprehensive model and the baseline model. So when they add these interesting variables all together in the model, the model explains the data 15% more than the baseline only. So that's the 15%, 24% comes from another source. So this is where you get the 15% to 24%. Is it a large difference or not? So is 15% a lot or a little? There are two ways to answer the question. One is that we can use our understanding of the phenomenon without any other reference to any other statistics. I would say that 15% is pretty large effect of bias. So if something is 15% biased, that's a really big deal. Another strategy that people commonly apply, which is a bit more mechanistic, is that they look for benchmarks. And one commonly used benchmark is given by Cohen's 1980 book. And you can also find that in 1992 article where this psychology is just determined that certain levels of R-square correspond to weak, medium and large effects. But that's a pretty context-free benchmark. You can find more relevant benchmarks for management research, for example in this Bosco's article where these authors review existing research, how large R-square values they produced, and then make judgments on whether effects are considered large or not. But ideally you would look at this number 15 and tell what does the number 15 mean in your context. So that is the R-square interpretation. Various between 0 and 100, the more, the bigger effect. Then we also want to interpret the coefficients. I also talked about the quick and dirty interpretation by looking at what is significant and to what direction. But we also want to understand what the coefficients mean. So let's take a look at this regression model. So our model contains COS female, firm revenue, and manufacturing firm. The interesting variable is the COS female. We want to understand the gender effect. We use firm revenue and whether the company is a manufacturing firm or not as controls. So what is the interpretation of beta one? We get some estimate, let's say, for, and then the interpretation would be, say, terisparibus or everything else held constant. One unit increase in COS female is associated with beta one increase in ROA in this example. And what that means is that if two companies are of the same size and operate in the same industry, the one led by the woman would be four percentage points, more profitable than the one led by the men, led by a man if the regression coefficient was four. So the coefficient tells us one unit change. And this is easy to interpret, because we have ROA has a meaningful unit and the independent variable is one or zero. So we can just compare one is female, one is woman, zero is man, and the coefficient gives us the difference between man and woman led companies. Quite often our independent variables are continuous. So let's take a look at another example. Does education pay off? Let's say that our regression coefficient is one. The interpretation would be that one unit of education increases your salary by one unit. So is that a large difference or not? To understand it, you need to understand the units. So for example, if we have one year increase in education leads to one euro increase in annual salary, probably there are no context in which this would be important finding. We would say that the effect is trivial for any context that we could think of. Things become more interesting if we say that well, the unit of income is not actually one euro, but it's thousands of euros. So one additional year of education leads to one thousand euro increase in annual salary. Would education pay off? That's a more interesting question, because it depends on the context. If you make few tens of thousands of euros per year, then maybe not. Maybe you don't want to go to school to get an incremental increase in your income. But if this was the context of a developing country, where the salaries on annual level are in the thousands, instead of tens of thousands, this would be a huge effect. If you make two thousand euros, and someone says that you'll get another thousand euros for one year of schooling, I would go. Probably many other people would go as well. So when we understand the units, it's easy to understand if the effect is large or not when you consider the context. What if we have one year increase leads to one bitcoin increase in annual salary? The problem here is that bitcoin is not something that we have any idea of a value of a bitcoin. So if someone tells me that I'll pay you a bitcoin, I don't really know what to make of it. I would have to ask him how much is bitcoin in euros. And then I convert it to euros, and then it makes sense to me. Someone tells me that bitcoin is three thousand euros. I'll say, okay, now it makes sense. So if the effects are in a unit that you don't understand, sometimes the units can be converted to ones that they do understand. Someone tells me that the temperature outside is 80 Fahrenheit. I'll have to convert it to centigrade to understand if it's hot or not. The same thing of course works the other way around. If I tell an American person that is 20 centigrade outdoors, they will have to convert it to the Fahrenheit that they know. So conversions can make things easier to understand. What if the increase is one bugasoid? Bugasoid is a fictional currency, and I don't know of any exchange rate between bugasoid and euro. So we need to deal with a unit that we really don't understand at all what it means. This would be, for example, if we have some kind of scale, rating scale between 1 to 7, we don't really understand what the one point difference between 4 and 5, for example, means, or if we have some kind of performance rating that is, let's say 300. We don't know what is the unit. When we don't know what the unit is, we need to consider the standard deviations. So this is our incoming bugasoid in a hypothetical population. So we have a normally distributed incoming bugasoid in this fictional universe, and what we want to know is how, if we go from mean, how far do we have to go until we are in the top third or two and a half percent, or top one in 1000 or something like that? Standard deviation tells us that. So we know that one standard deviation gets us from the mean to about the top one third. Another standard deviation gets us to about top two and a half percent, and then another standard deviation gets us into the top 0.1 percent approximately. So we know that one standard deviation is a pretty big difference. Two standard deviations takes you from the bottom to the top. So one standard deviation is a lot. If you consider human height, the standard deviation of average of male height is 8 centimeters. If a person is 8 centimeters taller than another one, we would consider that other person, the taller person to be considerably taller than the shorter person. Two standard deviations, 16 centimeters, that would be a very big difference in height. So one standard deviation is a lot. When we don't know anything about the units, like bugasoid does not have a meaningful loss, meaningful meaning for us, then we can think about the phenomenon as standard deviation. Let's go and let's get back to the Hekman's paper. So now we have to make a call whether these are large effects or not. We already did that using R-square, and I would say that the effects are pretty substantial. So how do we interpret the individual coefficients? For example, our physics and productivity is also the beta one increase in patient satisfaction. We need to understand the units. But the productivity unit doesn't really have any meaning for us. The quality and accessibility units don't really have any meaning for us, and neither really does the customer satisfaction scores. So in this case, it might make sense to convert these to standard deviation units instead. And this is what we do when we use standard as regression coefficients, which this paper applies. So how we interpret these, that one standard deviation increase in physics and productivity is associated by 0.9 standard deviation increase in customer satisfaction. It seems quite small effect to me, and it is non-significant so we can't really claim that there is an effect at all. And this is a useful way because it makes the interpretation easier. So you can, with some serious limitations that I will not get into in this video, we can basically compare if, whether 10-year or age is more important. So age is a negative effect of minus 13, 10-year has a positive effect of 0.34. We could say that the effect of 10-year is roughly three times as large as the effect of age, and it's in the opposite direction. This interpretation has some serious reservations that I will not go into. But it's a starting point for understanding with regression analysis and how they are interpreted. One rule of thumb is that if your units have naturally meaningful scale, then you probably should not standardize. For example, age, we know age in years, has, it has meaning for us. So now, because these coefficients are standardized, we don't know what is the effect of increasing age by one unit. And that would probably be useful. If I'm behind the best person or the best doctor in customer satisfaction by 10 points, I would like to know how much of that is due to me being five years older than the best doctor. With these coefficients that are standardized, we lose the natural interpretation. So that's the downside. So whenever possible, using the raw coefficients, thinking about the raw units instead of standardization would be better. But in some cases, using these standardized coefficients is easier because it, in a way, allows comparisons with some reservations. So summary of regression analysis interface. What do we do in a regression analysis? We look at two different things. We look at the R-square. How much does the model, all the variables together in the model, how much do they explain the dependent variable? The R-square tells how much variation it is. It's explained between 0% and 100% or 0 and 1. Adjusted R-square is better because in small samples it's more realistic and often we want to compare models in their explanation. So we have the control models only, control variables model only, the baseline, and then we add the interesting variables and we compare how much R-square increases. This is done in the HECMA spectrum. Then we also look at the individual coefficients. A quick and dirty co-interpretation would be check which are significant and to what direction. Then we conclude that effects exist either in positive or negative direction, but we can't conclude what is the magnitude of the effect just by looking at the significance. So these coefficients tell us about the individual effect if all the other variables stay at the same values. Important to understand what is the metric because beta 1 quantifies one unit difference. So one unit change if the effect of one year of education on income is one, then it makes a big difference whether the income is measured as euros or as thousands of euros. So ideally when you start looking at Richardson coefficient, Richardson analysis results, you go in three stages. So you first look at the R-square. Does the model explain the independent variables at all? If it doesn't, then no further interpretation is needed. Then you look at which coefficients are significant to what direction and then you evaluate the magnitudes of the interesting coefficients taking the units into account.