 Multi-level data allows us to estimate four different effects. The within effect, the between effect, the contextual effect and the population average effect. In this video I will show with an example what these effects actually are, and hopefully the video clarifies the meaning of these different effects. So let's take a look at the example. We have two teams here, team one and team two. And they both have five team members that all have intelligence of 100 and performance of 50%. So there is no within team variation in performance or in intelligence. So the average performance 50% and average intelligence is 100% in both teams. Now what will happen when we add one smart person to team number two and take one normal person away from that team? This happens. We can see that the smart person works better. His or her performance is better than the average people but also the performance of the average people increased when the smart person came to the team. For example, this kind of effect would happen if the smart person can provide leadership for the team and that leadership will make these other people in the team work better. So we have these two effects now. The within effect in this case is the individual's intelligence effect on the individual's performance. So this person is performing better because he or she is more intelligent than the other people. But there's also the contextual effect. So the contextual effect is the smart individual's impact on the performance by others for example by enabling better coordination or providing leadership. So when we add a smart person to the team then the performance of everyone else increases as well. And this is the contextual effect. So it is the effect of one's intelligence on everyone else in the team and of course everyone else includes the smart person. So if we were to change some of these people to normal people then the smart person's performance would go down. So the individual's performance is the sum of the within effect and the contextual effect and the contextual effect also affects other members in the team. The between effect is simply whatever is the difference in average performance on a team level as a function of average intelligence on a team level like so. So how much did the team performance increase when the average intelligence of a team increase. So that is the between effect. So within effect and contextual effect basically correspond to two causal mechanisms. How much better I am because I'm intelligence of some other attribute and how much my attribute has an effect on the others or everyone in the team. And then the between effect is simply sum of these two causal effects. So to show some numbers we saw that the intelligence difference for the person number one is normal person 100, smart person 120. So it's 20 points and plus four points to the average because we have five members in the team. Changing performance is the contextual effect that is 10% of its points. So this everyone became 10% of its points better because we introduced a smart person to the team and then one unit of average intelligence increases average performance by 2.5% of its points. So that's the actual contextual effect. The within effect for this person is changing performance within so this person was replaced with a person who is 20 points more intelligent and there is that much difference. So 20 points increase resulted in 20 points increase in intelligence here so we got one person this points difference per point of intelligence and if we take a look at the changing performance the between effect the change was 14% so 64 compared to 50% and the change in explanatory variable intelligence was 4 points on average so 14 divided by 4 is 3.5% which is the sum of the within and between effect. So that's the between effect. When you know the within effect and you know the between effect you can calculate the contextual effect if you know contextual effect you know between effect you can calculate the within effect. So always when you know two of these the truth can be calculated because they are governed by this equation here. Now this shows the difference between the between effect, the within effect and the contextual effect. How about the population average effect? So population average effect tells us basically what is the expected performance of an individual given his or her intelligence and we get the population average effect by running a regression analysis through the data. So we have intelligence varies between 100 that's 9 people and 121 person and then performance varies and the regression gives us 1.27 percentage points and what does that quantify? So that's not the within effect, that's not the between effect and that's not the contextual effect. So the increase is 1.278 and the within effect was 1 percentage points between effect was 3.5, contextual effect was 2.5 and this is none of them. So how do we actually get that? Well we can start thinking about this through different scenarios. So what if there was no between group varies on the intelligence? So all the groups are equal intelligence on average. Then the contextual effect really wouldn't have any effect. So the contextual effect on everybody would be the same and then how well the individual is performing that we pick randomly only depends on that person's intelligence but not on which team that person is because all teams are equally well off with regard to intelligence and if there is no variation between groups population average effect equals the within effect. In the other case if there is no within group variation so every member in every team has the same intelligence but the intelligence levels between the teams can vary. So in that case we would say that the intro class correlation which quantifies the between group variation is 1 and in that case if we know your group average then we will know your performance. So the between effect is equal to the population average effect. How about in between? Well it can be proven and for Snire's chapter he approves that the population average effect is the within effect multiplied by intro class correlation of the explanatory variable and the regression coefficient for the contextual effect. So when the intro class correlation is 0 then we simply have the within effect when the intro class correlation is 1 then we have the between effect and we can verify the calculation just by calculating the intro class correlation it is sum of squares between divided by sum of squares total so how much variation of intelligence can be attributed to the group membership that's what ICD tells us and it is about 11 percent and we multiply the regression coefficient or plug into the formula for ICCR for population average effect and we get 1.278 which is what we got from regression analysis. So this demonstrates the population average effect and we are really interested in understanding the effect of these variation between groups instead we want to have regression coefficients that generalize accuracy conditions and do not depend on how the explanatory variables vary and for that reason the population average effect is normally not the effect that we are after and some sources say that this is uninterpretable as a causal effect so if you want to estimate a causal effect of x and y you basically need to be looking at the within effect the contextual effect or the between effect depending on what level of variation you want to study so if you want to study the effect of average intelligence of a team on average performance or are you looking at more what happens inside the team