 Let's take a look at how multi-level models with just random intercepts are commonly reported in journal articles. Our example paper is Holkom's article from 2010 in organizational research methods. This article serves as a tutorial for how to apply multi-level modeling for modeling trends over time in the context of strategy and entrepreneurship. The article contains an empirical example as well as explanation on what each of these analysis steps do as well as some equations that show what the model is like. So this is a pretty good simple paper for learning how to do different kinds of multi-level modeling and what kind of modeling strategies we could apply if we are interested in time trends. So the model that they estimate, the simplest model, the first year model is just a linear time trend. So this is, we say that there is a trend over time that is linear either increasing or decreasing and their interest is in estimating if this time trend here varies between different kinds of companies. For now we're just going to be looking at the first model, which is simply a random intercept model. So this is a multi-level model because they actually have three levels of data. They have the year observations, repeated observations of different companies that each belong to one industry. So they have different industries, each have multiple companies, each companies observe multiple times. So this is an estimate of a multi-level model. We have also, in the model, we can see that the intercept varies between firms and it varies between industries. The slope here is constant, so we simply say that the slope is gamma 100 and there is no variation between firms, no variation between industries. So these gammas are what we call fixed effects. So that is a specific value that we calculate that allows us to calculate predictions based on the model. And we have three random effects here. So we have the year observations here. So this is the, we would call it the error term or error variance or disturbance. So this is the each individual unit, individual observation specific variation that is random. Then we have the firm level variation and the industrial level variation. If we write this equation into the mixed format, we can see that we have the fixed part that contains the grand intercept, the slope and the year. And we have the random part which contains the random intercept for the industry level, from the firm level and then also the error term of the model. So we have five different things that we estimate and how do this kind of model typically get reported in a journal article. The article reports this kind of table. So quite often when we have these multi-level models, we are interested in both the fixed part and the random part. So these tables typically have one part for the fixed effect parameters, another part for the random effect parameters. So we have the fixed part here, the recursion line with random part here, various in our recursion line. And we report our estimates, standard errors, z or t values depending on the model and p values for all sets of coefficients. The t values are, for some reason, missing in this table, but they are present in all other tables. So that is probably a copy paste error or a layout error in the article. Then we also have the likelihood statistic which is pretty similar to the F-statistic in the sense that you can do model comparisons using the likelihood statistic, but it really can't be interpreted directly. And sometimes we have, if we have a variance components model, so we are really interested in what level the data varies, we have all kinds of explained variation statistics. So this just, this is basic, thought the variance explained by year effect is basically the same as r square. So how much of the variation of the data the fixed part explains and how much the random part explains. So this is a simple example of how to report these random interest, random effect coefficients. If you don't find the random effects coefficients interesting and you want to omit them from the table, then you probably shouldn't be using a multi-level model in the first place because you can get the fixed effects coefficients with a lot simpler techniques and still estimate it consistently.