 In this video I will explain the basics of ordinal data models. Ordinal data is one level of measurement. So there are four levels of measurement typically introduced in research methods books such as Singleton and Straits. On the most primitive levels, the data are nominal, which means that we have numbers. The numbers refer to different categories, so its value is different, but we can't really compare the numbers. So Finland is one Sweden is two Norway is three. We can't say that Finland plus Sweden is Norway. We can't even say that Sweden is more than Finland or Norway is more than Sweden. These are just numbers that are labeled from some abstract categories. When we add a more refined variable we add order. That's the first thing that we add. We can say after adding order that one is less than two, two is less than three and one is less than three. But we can't say that the difference between one and two is the same as the difference between one and three. For example, we could have a rating scale one is disagree, two is not agree or disagree and three is agree. We can't say that the difference between agree and not agree or disagree is the same as difference between disagree and not agree or disagree. Also we can have ranks in sports competitions. We can't say that the difference between first and second is the same as the difference between second and third. When we add a constant interval between categories then we have interval measurement. We can say that two minus one is the same as three minus two and an example would be a Fahrenheit or Celsius temperature scale. The unit has a meaning so the difference between one and two is the same as two and three. But the difference between an interval scale and ratio scale is that there is no meaningful zero point. Meaningful zero point here means that we have a zero which means an absence of a quantity. So for example, length in meters, weight in kilograms and absence is a radio scale because zero means that the weight is non-existent. Zero meters non-existent are length and so on and we can say that our two kilos is twice as much as one kilo. We can't say that two Fahrenheit is twice as much as one Fahrenheit. Does it make sense? Rigorson analysis assumes interval measurement and also works with ratio measurement because all ratio measurement is also interval measurement. What if we have ordinal measurement? So we now know an order but we don't know what's the difference with the categories or different units that we order. There are tools that we apply for ordinal models are typical tools are ordered probitory rigorson and ordered logistic rigors. These are fairly straightforward extensions to the binary case where the dependent variable is one and zero. Let's take a look at the dependent variable receiving three values, one, two and three. One way to formulate the ordered model is to use the latent variable formulation which I cover in the probitory rigorson video. So we have a latent variable that is sum of these different rigorson coefficients multiplied by the data plus, and normally this is their term with the variance of one, know that there is no intercept here, so an intercept is not estimated in these models. Then we have two thresholds. If the latent variable y star is smaller than the first threshold, then y receives value of one. If the latent variable is between thresholds one and two, then y receives value of two, and if the threshold is less than or more than two, then y receives value of three. Graphically, we can look at the probabilities here. So let's say that the fitted value for one case is exactly zero, then we draw a normal distribution here, and then there are area here between threshold two and infinity is the probability of getting a three. The area between threshold one and two is the probability of getting two, and the area before threshold one is the probability of getting one. Of course, where exactly we draw this distribution depends on the fitted value, so for a case with a larger fitted value, we could have something like this, so observing two would be the most probable alternative for this case, and observing three and one would be equally, but less likely than observing two. So the idea is the same as in a probit model, but instead of splitting the normal distribution at zero, we split different thresholds that we estimate from the data. Another way of looking at this issue is the link function formulation. We can look at the link function and the idea is that we estimate two logistic curves or two probit curves that only differ in their intercept. The idea is that if the curve goes here, the first curve tells what is the probability of observing a value of at least one. Then the second curve tells us what is the effect of observing value that is at least two. So these curves differ only in their intercept. If the intercept is greater, then it means that the curve moves to the right, and then the probability increases. So what's the probability of getting one here? The probability of getting one is simply one minus the probability of getting more than one here. So this is the probability of getting more than one, then the probability of getting one is one minus that probability. The probability of getting two is the difference between the probability of getting more than one and the probability of getting more than two. So that's the probability of getting two, and then the probability of getting three is this final curve here. So that's the probability of getting a value greater than two from the data. So, when you look at books about ordinal models, ordinal recursion models, you quite often see these kind of distributions that are graphs that show the different probabilities, these red lines for the different response options. We can see here that when x has a small value, then all things being equal, the most probable answer is 1 or most probable observation. When x approaches 0, then the answer 2 is the most probable and then when we have a large value of x, then answer 3 is the most probable. So the probability for these different options changes as a function of this x value here. Okay, so what's the relationship between these? The link function formulation and the latent variable formulation. It turns out that the thresholds in this latent variable formulation are the negative of the intercept. So, if a threshold is large, then that means that the intercept must be negative. So, the practical implication of this difference is that when you interpret the regression results for ordered models from your statistical software, you have to understand whether the software presents you the intercepts or the thresholds when you calculate these curves to do the probabilities. Then the model makes some assumptions. The assumption here, the important assumption is the parallel lines or proportional odds assumption, which means that the linear prediction for all these observations is the same. So, the factors that explain the difference between the first category and the second category are the same as the factors that explain the difference between the second category and the third category. So if one variable has twice the effect of another variable when we differentiate between the first and second category, then it's assumed that the same two variables, one has twice the effect than the other when we differentiate between the second and third options. Also, the parallel lines assumption here means that these lines here, these curves are differ only in their interest. They differ on how much we shift them sideways and not in how steep they go up. This is an assumption that is empirically testable and whenever you use logistical or probate models and the ordered versions of these, you have to, you should test this assumption because it's fairly easy to do with your statistical software. Let's take a look at an empirical example and how these models are interpreted. The example is a paper by Antonakis and colleagues and they look at question of whether a charisma can be taught and we are looking at our rankings of leaders. So they did an intervention study and they applied leadership training to people and then they video those people before and after the training and then they had students or some people who were rating those videos and then they look at whether the ranking, the expected rank of a video was influenced by the training. So whether the ranks before and after the training were statistically significantly different and by how much. Here we can see that the thresholds are here, so they should be reported and there's no intercept because basically these thresholds are the same as the intercepts would be for the individual logistic regression curves. So we don't have any any common interest for these models. So these thresholds if you multiply them by minus one could be interpreted as intercepts for the regression models. So we don't normally interpret these directly. Instead we look at the predictive probabilities. So this is, they're looking at predictive probabilities based on the model and the independent variable is charismatic leadership tactics CLT and they look at whether the use of these tactics that they were teaching are the subjects whether they influenced the rating and when you have high charismatic leadership tactics then you're most likely to be rated a very good leader or very persuasive person and if you have low use of charismatic leadership tactic the most likely rating for you is going to be one the weakest out of four. So this isn't a way of interpreting the results of presenting them in the journal paper. Again you do graphically the interpretation looking at these regression coefficients really doesn't tell us much because they are their logistic regression or probit regression coefficients interpreting the indirect is hard interpreting the thresholds directly is hard but if you plot the predictive probabilities for different kinds of cases then you can look at okay what do the results actually tell us. Final question is when should these models be applied and the common question is that if we have an agreement scale like this strongly disagree one strongly agree five is this an interval scale or is this an ordinal scale normally if you have just one variable then you would consider it as an ordinal. So single item from one to five probably not you can't assume that it if strongly disagree is the same as disagree and do not agree or disagree. So probably not. If you have multiple items then maybe under certain scenarios. The problem with if you have multiple items from one to five scale is that if you want to take a mean out of those items then you must assume that they're ordinal because calculating a mean of you must assume that they're interval you can't calculate the mean from ordinal data because calculating the mean requires that the intervals are the same. When you go to more advanced modeling literature you can see that there's a debate on whether this should be a model as ordinal or nominal ordinal or interval when you have multiple measures in the latent variable literature and there is a this really good paper by Remtula and colleagues that demonstrates that if you have three or more items and the scaling is from one to five or more and the items are not extremely are not extreme so that you only get fives and fours instead of these one two three then in most cases assuming that this is a an interval scale doesn't make a difference. So a practical recommendation if you are a beginner then if you have one item it's better to treat that item as ordinal scale if you have two items or more take a mean and then treat it as an interval scale. Taking a mean and then treating it as an ordinal scale doesn't make any sense because you need to assume interval scales to take the mean.