Hi everyone. In the video, my first multilevel model, we've discussed how the familiar multiple regression model could be extended to multilevel data. We also saw that the random parameters, are parameters whose value is allowed to vary across higher level units, whereas fixed parameters have one single value that applies to everyone. Finally, we looked at the simplest of multilevel models. A model would just run random effect, or random intercept. But what about the slopes? Can they be random too? In this video, we will see that the answer is yes. At least for predictors that aren't on the highest level. In short, this video will introduce you to the random effects model, that is a multilevel model with random slopes. Let's return to the basic multilevel model with random intercepts and the school achievement example. To remember the conditions of this example, we allowed b_0 to vary across schools, making the dependent variable in its own analysis on level 2. Now what about slopes? Can we simply make the slope for IQ and school size dependent variables on level 2 as well and make them random effects that way? The answer is, we can do that for IQ, but not for school size. The reason for this is that school size is a predictor at the highest level. There are no units above schools across which the effect of schools size can differ. The effect of IQ, on the other hand, can differ between schools. Now before seeing how to incorporate random slopes, let's consider what random slopes represent. What does it mean to have a b_1 value that has a different value for each higher level unit. In the school achievement example, school achievement is our dependent variable, and IQ is our level one predictor and this regression coefficient for this predictor that we were allowed to vary across schools in this video. Allowing different regression coefficient for IQ means that we think that the relation between IQ and school achievement is not the same for each school. Mathematically, this is represented by giving b_1 a subscript j, just like we did to b_0 in the video, my first multilevel model. Every school now has their own b_1 value, and just as was the case would b_0, b_1 will now be made the dependent variable in its own level 2 regression equation. To clean up the notation a bit, the level 2 intercept and the slope for school size, it will be renamed Gamma_00, and Gamma_01 respectively. In addition, u_j will be renamed u_0j. The level 2 equation for the slope of IQ would look like this, assuming for now that we don't have any predictors to explain differences between schools in the slope of IQ, b_1 as the dependent variable, an intercept Gamma1_0, that represents the average effect of IQ on school achievement across schools and an error term u_1j, that captures each school's deviation from this average IQ effects with positive values implying a stronger than average effect of IQ and school achievements and negative values implying a smaller than average effect. Graphically, it would look like this. Assume we have three schools, each with their own effect of IQ. In school 1, the effect is the strongest, so the line for this school is the steepest. In school 2, the effect is the weakest, so IQ influences school achievement the least in school 2. This can be seen from the regression line for school 2 which is the flatness of the bunch. School 3 in the meantime, it's somewhere between school 1 and school 2. If we take the average of these three lines, the thick black one in the figure, the slope of this line is equal to Gamma0_1. The slope of school 1 will be Gamma_01 plus u_11. The slope of school 2 will be Gamma_01 plus u_12 and for school 3, it'll be Gamma_01 plus u_13, giving each school their own slope. Now as we saw in the previous video, each school can also have its own intercept. So the line might also differ on that respect. With the averages of the intercepts being Gamma_00, and each school being allowed to differ from the average intercept. That's it for now, we saw that slopes can differ across level 2 units, just like the intercept and we saw that this makes the slopes dependent variables in the separate level 2 regression model. Next time, we'll look at including predictors for the random slopes.