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Is the assumption of constant variance violated in the following?

I am measuring point in a 3D coordinate system (X,Y,Z) using 10 different methods. One method is the golden standard (reference). The measurements are done at the same time for 40 different subjects repeated 2-3 times per subject. I end up with 105 measurements for each of the 10 methods. I am then interested in the length of the error compared to the reference (|Xerror| = |Xref-Xmethod|) in the X, Y and Z direction and the total error as the Euclidian distance (=sqrt(X^2+Y^2+Z^2)).

I want to see which method gives the smallest error in each of the directions (X,Y,Z) and also the lowest Eucledian distance. I have been using linear regression for this. With a model in for example the X direction as: |X| ~ 1+ method, where method is a categorical variable representing the 9 different methods tested (not including the reference). I have 9 x 105 = 945 observations.

Some different diagnostic plots can be seen below. I have used square-root transformation of the data to get a decent QQ-plot. My biggest concern is on the assumption of constant variance? But is there anything else which should concern me?

Additional info: the measurements are discrete (can only be integer numbers).

enter image description here

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  • $\begingroup$ Are the estimates also restricted to be integers? Note that even if X, Y and Z are integers usually the Euclidean distance will not be. $\endgroup$ Commented Oct 21, 2017 at 21:56
  • $\begingroup$ Thank you for interrest. The coefficients in the model are not restricted to be integers. The observations are integers. The eucledian distance is continues. $\endgroup$
    – SupAnne
    Commented Oct 22, 2017 at 5:26

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You should look at the lower plot in the left column, titled Plot of residuals vs. fitted values. Does it show some tendency for the residuals do be more spread out with increasing fitted values? I do not think so, so no indication of non-constant variance.

The qq-plot is close to linear, except the steep gradient at the lower left (corresponding to a hump in the corresponding histogram.) That indicates some lower bound on the residuals, which is maybe natural, since there is a lower bound (zero) on the observations, distances. The leverages seems close to constant, which is good. So nothing in particular to worry about.

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