Linear mixed model of Proportional variance

Analysis with linear mixed models

Looking at a Linear Mixed Model, you can create a model taking into account that "groups have different slopes".

For a linear mixture model, equal variance is assumed, so It is not a model that does not fit the characteristic that "the larger the X, the greater the variation in the Y direction". If you only want to focus on differences in slope, you can use a linear mixed model, but you can't analyze variability.

Analysis with a histogram

If we make Y the histogram only, we can see that the spread in the Y direction is different.

You can analyze the difference in variation. "The larger the X, the greater the variation in the Y direction" is no longer visible.

Analyzed as a Proportional variance

It seems abrupt, but I created a variable called Y/X and made it a histogram.

Then, the difference in slope of A, B, and C can be expressed by the difference in the center of distribution, and the variation of B is the largest, and the state where C is the smallest can be expressed. This is an application of Regression analysis of Proportional variance.

If there is a Proportional variance behind the mechanism of the data, then the histogram of Y has It includes both the property that "the larger the X, the greater the variation in the Y direction" and the property of "the greater the slope". Therefore, the slope is the largest, and the variation of A is the largest.

By looking at the variable Y/X, you can separate the difference in slope from the difference in variation. Therefore, in terms of Proportional variance variation, we can see that B variation is the largest.

For many groups

If the horizontal axis is the mean value of Y/X and the vertical axis is the standard deviation of Y/X, the information that can be understood from the histogram can be expressed in a simple graph. This method is useful if you have many groups.