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These models are also used for prediction: Predicting the possible outcome if you have new values on your independent variables and this is why independent variables are also called predictors. Multilevel models can accommodate such differences. Very roughly speaking, it is a repeatedmeasure version of linear models or GLMs. So, I won't go into detailed discussions about how we should consider these factors. Before jumping into examples of multilevel linear models, let's have a highlevel understanding of multilevel linear models.
Your measurement is performance time. In your experiment, 10 participants performed some tasks with both techniques; thus, the experiment is a withinsubject design. However, this analysis does not fully consider the experiment design you had: the differences between the participants. For example, some participants are more comfortable with using computers than the others, and thus, their overall performance might have been better.
Or the differences of the techniques might have caused different levels of the effects depending on the participants. Some participants had similar performance with both techniques, and some had much better performance with one technique.
The linear regression above tries to represent the data with one line, and unfortunately it aggressively aggregates such differences which may matter to your results in this case.
Multilevel regression, intuitively, allows us to have a model for each group represented in the withinsubject factors. Thus, in this example, instead of having one linear model, you will build 10 linear models, each of which is for each participant, and do analysis on whether the techniques caused differences or not.
In this way, we can also consider individual differences of the participants they will be described as differences of the models. What multilevel regression actually does is something like between completely ignoring the withinsubject factors sticking with one model and building a separate model for every single group making n separate models for n participants. But I think this exaggerated explanation well describes how multilevel regression is different from simple regression, and is easy to understand.
The previous section gave you a rough idea of what multilevel models are like. For the factors in which we want to take individual differences into account, we treat them as random effects and build each model for each level of these factors. But one question is still remaining. If we build a separate model for each participant, for example, analysis would be very timeconsuming. With the example we used above, we would have 10 models in total.
Some may have significant effects of Technique, and some may not. In that case, how can we generalize the results and say if Technique is really a significant factor? Multilevel models can remove this trouble. Instead of building completely different models, multilevel regression changes the coefficients of only some parameters in the model for each level of random effects.
Thus, the coefficients of the other factors remain the same, and model analysis becomes much easier. Roughly speaking, there are two strategies you can take for random effects: varyingintercept or varyingslope or do both. Varyingintercept means differences in random effects are described as differences in intercepts.
For example, in the previous example, we will have 10 different intercepts each for each participant , but the coefficient for Technique is constant. Varyingslope means vice versa: changing the coefficients of some factors. In many cases, factors, more precisely independent variables or predictors, are something you want to examine.
Thus, you want to generalize results for them. And the intercept is usually something you don't include in your analysis, so it can be very complicated.
Therefore, unless you have some clear reasons, varyingintercept models will work for you. They won't be computationally complicated and their results will be straightforward to interpret. In this page, I show an example of varyingintercept models. Then you won't get confused when you read other literature or try to use other statistical software. This is my interpretation of differences between fixed and random effects: In multilevel regression, you will build multiple models.
Random effects can be factors whose effects you are not interested in but whose variances you want to remove from your model. Generally, we are not interested in how different the performance of each participant is. But we do not want to let the individual differences of the participants affect the analysis. If you know a better way to understand the difference between fixed effects and random effects, please share it with us!
I prepare hypothetical data to try out multilevel linear regression. You can download it from here. We are going to use that file in the following example. Let me explain a hypothetical context of this hypothetical data :. We conducted an experiment with a touchscreen desktop computer. Our objective is to examine how mousebased and touchbased interactions affect performance time in different applications. In this system, participants could use either mouse click or direct touch to select an object or an item in a menu.
We just let them which way to interact with the system so that we could measure how people tend to use mousebased and touchbased interactions. Our design is also withinsubject across the three applications tested in this experiment. The file contains the results of this experiment. I think most of the columns are just guessable.
Time is the time sec for completing the task in each application indicated by Application. MouseClick , Touch , MouseWheel , and PinchZoom are the counts for mouse clicks, direct touch, zoom with the mouse wheel, and zoom with the pinch gesture. Of course, there are a number of models we can think of, but let's try something simple:. However, we want to take the effects of our experimental design into account. To do this, we make a tweak on the model above. What Random 1 Participant is trying to mean is that we are going to change the intercept for each participant.
Yes, we are making varyingintercept models. We assume that individual differences by participants can be explained by differences in the intercept. In this manner, we can remove some effects caused by the individual differences to the other factors.
There are a number of ways to do multilevel linear regression in R, but we are using the lme package. We also import the data. Again, 1 Participant is the part for the random effect. So this means we are changing the intercept for each participant.
To find the models, we use the restricted maximum likelihood REML. So you can see the estimated coefficient for each factor, but it is kinda unclear whether it is really significant or not. Let's try coefplot. Unfortunately coefplot in the arm package does not work with the lme object.
We thus use a fixed version of coefplot. A thick and thin line represent the 1SD and 2SD ranges. So it looks like that MouseClick has a significant effect because its 2SD does not overlap the zero. So far, so good. We successfully created a model and looks like we have something interesting there.
But we are not quite sure about which fixed effects are significant yet. However, it is not quite straightforward to run it because of random effects. In this case, we cannot really be sure about whether the test statistic is Fdistributed. There have been several attempts to address this and make an ANOVA test useful for multilevel regression, such as the KenwardRoger correction.
However, it is disputable if this correction is good enough so that we can assume the corrected test statistic is Fdistributed.
I will do it sometime later at a separate page. For now, let's simply think that MCMC tries to reestimate the coefficient for each factor based on the results we got with lmer so that we can have better estimation.
The parameter nsim is the number of the simulation to run. Here, I set , but you may need to tweak it to make sure that the estimation is converged. So we are going to use the results by MCMC. As you can see in the results, only MouseClick has a significant positive effect on increasing performance time. So the results imply that reducing the number of mouse clicks may decrease the overall task completion time in the applications tested here.
PinchZoom's effect 0. Thus, encouraging users to do pinching gestures for zoom operations might contribute to decrease in the overall task completion time. Unfortunately, there aren't many things to say from the results here, but I guess you have gotten the idea of how you interpret the results of multilevel linear models.
Lastly, let's make sure that we don't have multicollinearity problems. For lmer , we cannot use the vif function. Instead, we can use the function provided by Austin F. Copy the part of the vif. R , and paste them into your R console. And just use the vif. So, in this example, we are fine. If the value is higher than 5, you probably should remove that factor, and if it is higher than 2.
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These models are also used for prediction: Predicting the possible outcome if you have new values on your independent variables and this is why independent variables are also called predictors. Multilevel models can accommodate such differences. Very roughly speaking, it is a repeatedmeasure version of linear models or GLMs. So, I won't go into detailed discussions about how we should consider these factors. Before jumping into examples of multilevel linear models, let's have a highlevel understanding of multilevel linear models.
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Multilevel Linear Model
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