 Yksi ensimmäinen asia, johon olemme käyttäneet multivallista dataa, on se, johon lopulta yksinkertaisuissa. Olemme myös kokeilla entri-klaista korrelatiosta, jotta nämä variaat olemme käyttäneet. Lopulta katsotaan esim. yksinkertaisuista variaatioita ja yksinkertaisuista variaatioita. Olemme täällä yksinkertaisuista dataa. Olemme 5 yksinkertaisuissa, ja yksinkertaisuissa yksinkertaisuissa. Mutta ajattelisiin suunnastamme yksinkertaisuvuus on koko 14 illsipuolelainen. Yksinkertaisuissa avilaisuudestaan on hyvä vualia, se on myös yksinkertaisuista. Yksinkertaisuissa ei ole silloin täydellisi, niin on sitä fundauksia veristä. Mutta se ei ole oikein yksinkertaisuista. diezotuja, yksinkertaisuiden profeetavuus seurataan veritus bunta. Varmista meidät alueet avilaisuja. Varmista mumsistolta on company-level-variations. These red mounts each present one company, and they are all working within an industry. And this blue area here represents the variation of the performance of all companies within that industry. So we can see that different companies vary. Their performance vary within company, but there are also variations between companies, so that this company here is consistently less profitable than this company here. So we have two levels, we have the within company level, and we have the between company level, which is also the within industry level. We can also add more levels. There is no limit on how many levels we can do, but let's go for an industry level. So we have these blue, five different industries, and the industries are different in their profitability. Some industries are highly profitable, others are not so. And we can see that the individual variation of the data here is a function of these three sources of variation. The between industry level, the between company level, and the year-to-year variation within companies. To understand our data and understand the phenomenon that the data represents, we typically need to decompose the variance, to somehow come up with percentages or some other statistics that quantify how much of the variation is here, and how much of the variation is here in our data. If our data is this small, we typically start with a graphical analysis. So we can just upload the data. This is 25 observations, five observations for each company, for five companies within one industry. We can see that there are some patterns. For example, this company, there is not much variation in performance. This company is less profitable than that company and so on. This kind of analysis works well when you have a small set of observations. If we have a large number of observations, but still a fairly manageable number of clusters, let's say up to 30 companies or 30 industries or whatever is our level to unit, we can use box plots. Box plots are graphical representations of variation of individual variables, and we can do box plots by groups. The idea of a box plot is that we first calculate for a variable, calculate the median, and the median gets this thick line here, that marks the median. Then we calculate the first quartile and the third quartile of the data. So quartile means that below this line lies 25% of our observations and above the line 75%. Median is half and half, and third quartile is 75% and 25%. We draw a box between the first quartile and the third quartile, and half of our data is within this box. Then we have these whiskers that indicate the minimum and maximum, and sometimes we also have outliers that the box plot algorithm identifies as circles. So why is this box plot presentation useful and how can we analyze the box plots? We can, first of all, start to understand the between and within variance by looking at the box plots. We can compare these medians, or we can do box plot with means, and we can check how much variation there is between these means or medians, and that is our between variation. We can also take a look at how high the boxes are, and that quantifies the within variance. And comparing these two dimensions tell us if the variation in this variable is more due to the differences between firms or is it just random variation or some other variation within firms. So is it a within firm or between variation that expands the data? We can quantify the level of variation between two levels also numerically by calculating the within variance and between variance. This is our data, and we start by calculating group means. So we take each of these companies and we calculate the mean of this. So these are the group means or cluster means for these five firms. And we check how much these means vary. The variation is quantified here with this statistic. And then we calculate how much these individual observations vary from the group mean. In practice, we do group mean centering. So we take each of these observations, we subtract the group mean, and that gives us the group mean centred values. Then we calculate how much the group mean centred data varies. And this is our between variation. This is our within variation. And this is our total variation, which is the sum of the between variation and the within variation. So the variation or variance is a statistic that depends on the scale. It will be useful to have a scale free way to explain on which level the data varies. And this is where the intra class correlation comes to play. So intra class correlation is simply calculated as variance between groups divided by the total variance. So it answers the question how much of the various data is attributed to the groups and how much is attributed to the variation within the groups. This is called ICC1 for reason that there are many other kinds of intra class correlations. So intra class correlation generally refers to correlation between observations. And because there are many, this is called the ICC1. There are like a few others, but this is the most important one that you need to understand when you work with multilevel data. Other intra class correlations are mostly about reliability of multiple raters, but ICC1 is this simple equation that simply quantifies various. When ICC1 is zero, then that indicates that there is no variance between groups. So the box plots are all on the same level here. There are no difference between means, and in this case the medias are close as well. And all variation is simply because there is variation within these groups. Then when intra class correlation is one, then that means there is no variance within clusters at all. All observations equal the level two means. So this firms profitability is always here, this firms always here and so on. So there is no within unit variation. Why do people calculate intra class correlation and how it is typically reported? The role of intra class correlation, the first role is to make a decision whether something needs to be done for the cluster. If all the observations within clusters are the same, then you can just pick one observation for each cluster and use those in regression analysis. And it doesn't really matter that you have the remaining observations because they don't provide you any more data. If ICC1 is zero, then there is no clustering in that variable. And if all your ICC1s are very low, then there is no meaningful clustering in your data and it's possibly safe to go without a multi-level modeling. There are exceptions to the rule, but generally when ICC1 is close to zero or when ICC1 is close to one, then a multi-level modeling may not be needed. But if it's somewhere between, like it's 50%, then you typically need to take levels into account in your analysis somehow. Let's take a look at the example of how ICC1 has been reported in published research. So this comes from HouseNet's paper and this is a good example because they first explain what the statistic is. So quite often people just report a statistic or report a number without explaining what ICC1 is. And this study provides this concise description. ICC1 values can be interpreted as the total amount of variance in the dependent variable that is attributable to between unit rather than within unit differences over time. So that explains what the statistics interpretation is and also if the values are high, then regression analysis could be inappropriate. And then you would have to do something else or, for example, use cluster over standard errors or multi-level modeling. And then they go on and they explain what is the actual statistic, absenteeson 0.76 and then they explain what the statistic means. So giving this short introduction to your statistics is very useful for your readers because your readers may not be experts in using multi-level data. So make it easier for them.