 Se on huomattu, että autot tuntuvat varioita, joilla on tuntuvat tuntuvat. Tämä videon olen tuntuvat, että se on oikein tarpeeksi. Ja mitä pitäisi tehdä tai ei. In HECNA's paper, autot tuntuvat, että tuntuvat varioita, joilla on tuntuvat multikollinarit. Se idea tuntuvat ja multikollinarit, on, että jos on X ja M, ja sitten tuntuvat formaprodukta X ja M, niin tuntuvat tuntuvat varioita on koko X ja M, koska tuntuvat varioita formaprodukta. Ja sen tuntuvat voimme tuntuvat tuntuvat varioita. Joten tuntuvat tuntuvat dataa. Tuntuvat 2 random numbers, X1 ja X2. Tuntuvat X1 ja X2 on tuntuvat 2, ja tuntuvat varioita X1 ja X2 on tuntuvat 0. Se idea tuntuvat on, että tehtäisit tuntuvat varioita, ja sitten tuntuvat tuntuvat tuntuvat varioita, ja se on tuntuvat varioita, jota on 0, ja me sanoit, että tuntuvat varioita on tuntuvat. Tuntuvat bar-symboli over the X tuntuvat tuntuvat varioita, tuntuvat tuntuvat varioita. Ja standardization on tuntuvat ja tuntuvat tuntuvat standardization. Me voisimme, että X1 ja X2 eivät ole hyvin koko kovin, joten se on tuntuvat tuntuvat tuntuvat tuntuvat, mutta kun me multiplaamme X1 ja X2 niin se on tuntuvat koko kovin X1 ja X2. Joten se on todella tyhjelmää. Kun me tuntuvat varioita, tuntuvat varioita on tuntuvat tuntuvat tuntuvat tuntuvat mutta me voisimme nähdä, että koko kovin X1 ja X2 ja tehtäisiin se on aika eri. On vielä todella tyhjelmää, joten X1 tai X2 on 0, joten ne on vain varioita ja dataa ja se on voittu suomalaisia, kun X1 ja X2 on todella tyhjelmää. Joten se on todella tyhjelmää, mutta se ei ole aslında kuciaa edellisestä. Joten mitä on se, yhdessä hallintain analytiottani tässä koko krikan, jossa on varioita rikroson analytiossa touched data, ja yhdessä hallintain analytiot kuten sen takaisin. Kun nähdään, että kuka on erityisessä, y- ja x1 ja x2, se on vain yleensä yleensä yleensä. Tämä yleensä yleensä on yksi, ja yleensä yleensä yleensä yleensä yleensä yleensä yleensä yleensä yleensä yleensä yleensä. Se on ihan yksi, koska kun senttiin, olemme yleensä yleensä yleensä yleensä yleensä x2, X2 ja se on yksi, koska sinä telekiin kertaa samaa numeroa every observation, se vain alkaa intersepti, koska se ei ole keskustelua X1, X2 ja co-variationsi between those two variables and why. Those are unaffected by centering. So centering will only affect means and in normal regression analysis it only affects the interest. What's the downside of centering is that once we calculate predictions, here the predictions for this model are on the original metric. So we will get predictions on whatever the y is. And if we calculate predictions using this model, then the predictions will be off by the amount that we centered. So for example if we're predicting a salary and let's say this model would give us 10,000 euros per year, then this model could give us minus 2,000 euros, which doesn't make sense unless we back convert or back translate that effect to the non-centered variables. So centering makes predictions and makes doing plots that apply predictions more difficult, and that's important for interactions for reasons that I'll explain in the last slide. When we take an interaction term, we can see now that there are some more differences. Importantly, the differences are only in the first three coefficients. So intercept again is different, which is expected, but now x1 and x2 coefficients are different, but the interaction of x1 and x2 is the exact same number. So the centering actually doesn't influence the interaction term at all. It influences only the first order coefficients. So is that something that you want to do or not? We have to consider, to answer that question, we have to consider what exactly the centering means and what exactly it means that we have this interaction term here. Let's take a look at a graph. So here the x1 and x2 effects are when x1 and x2 is zero and here the x1 and x2 effects are the mean effects. So when x1 and x2 are their means, then that's what the x1 and x2 effects are. What that means can be understood by looking at this graphically. So we have here a space and there is a plane in the space. Here we have x1 on this axis. We have x2 on this axis and then we have y here. So when we have two coefficients or two variables in the regression analysis as two independent variables, then the regression is a plane in three dimensional space. And we can see the plane here and because of the interaction the effect of x1 on y is the strength of that effect is contingent on the value of x2. So here when x2 is at zero, then x1 simply increases a little. So the effect is not that great. When x2 is at five, the effect is a lot greater. So we see a lot steeper slope here. So the idea is that the regression slope of x1 changes as a function of x2. Also the intercept changes so this line goes down here. So what centering does is that normally when we do an interaction term we take the effect of x1. So the regression with interaction gives you the effects of x1, effects of x2 and their product. When we don't center our data, the effect of x1 is this blue line here. So it's the effect of x1 when x2 is zero. Similarly, the effect of x2 is the effect of x2 when x1 is zero. When we center instead of taking the effect of x2 is at zero for x1, we take an effect of x1 when x2 is at its mean. So we take this green line in the middle. So the centering just influences which of these possible lines do we take it from here, from here, or perhaps all the way from the other end of the data. So it just changes at which part of the regression plane we are looking at. But the problem is that you have to look at multiple places. So you can't summarize this plane by saying that the effect of x1 is this line. You have to show multiple lines. So it doesn't really matter which of these lines you show in your regression table. And that's the problem. So you have to do these interaction plots. So you have to show multiple plots. So you show that the slope of x1 depends on the value of x2. And which of these lines we show in the regression table is arbitrary. So it doesn't really matter because we have to present these kind of plots anyway. So what we show here, whether we have the effect of x1 here to be the blue, green or red line, doesn't really make a difference. We have to show all the lines anyway. The problem with centering is that once we center our variables, then the interaction plot, the values of the predictive values of y will be incorrect by the amount that we center the data. So we can no longer do predictions usefully. We have to convert the predictions back to the non-centered metric for them to make sense. So centering is not useful because it doesn't do anything for the interpretation. You will have to interpret the results with this kind of plot anyway. And centering will be harmful for this plot because it makes forming these plots more difficult because you have to back convert your variables to the original metric to get the predictions correct. So because of these considerations, my recommendation is never center your data. It's not useful and it is harmful.