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I am having troubles interpreting the qq plot below. The histogram is residuals from linear regression. How to interpret negative theoretical and sample quantiles? Shouldn't it be [0,1]?

qq_plot

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2 Answers 2

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The actual data values (in your case, the residuals) are plotted along the y-axis. The meaning of the x-axis is probably best illustrated through a simple example. There are five data values in the first column of the table below and they are sorted in increasing order. The remaining columns are rank order of the data values, percentile values (as proportions) associated with the ranks (see the NOTE below), and Z-values associated with those percentile values for a standard normal distribution.

NOTE: For n<=10, the percentile value (as a proportion) is computed as (rank-0.375)/(n+0.25). For n>10, we use (rank-0.5)/n. See http://en.wikipedia.org/wiki/Normal_probability_plot.

   Sorted        
    Data    Rank     Percentile        Z
      8       1        0.119        -1.180
     10       2        0.310        -0.497
     15       3        0.500         0.000
     20       4        0.690         0.497
     22       5        0.881         1.180

The normal Q-Q plot for this data set is shown below and you’ll see that it plots the Sorted Data values on the y-axis and the Z-values on the x-axis.

enter image description here

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    $\begingroup$ "For n<=10, the percentile value (as a proportion) is computed as (rank-0.375)/(n+0.25). For n>10, we use (rank-0.5)/n" This isn't true of all software implementations. The OP is using R, but many other languages and packages implement these plots. On CV there is no default software, even though some software is more popular than others. $\endgroup$
    – Nick Cox
    Nov 14, 2017 at 9:07
  • $\begingroup$ So "Theoretica quantiles" axis is just a z-score? $\endgroup$
    – YKY
    Nov 14, 2017 at 14:56
  • $\begingroup$ @YKY Yes, see the wiki here en.wikipedia.org/wiki/Normal_probability_plot $\endgroup$
    – James LT
    Feb 6, 2018 at 4:32
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Recall what a q-q-plot is: it plots theoretical against observed quantiles.

When you ask whether it shouldn't be values in [0,1], you are thinking of the corresponding confidence levels. These are not plotted. (You could think of the confidence levels as parameterizing the sequence of dots in your q-q-plot: the dots in the lower left correspond to low confidence levels, the ones in the upper right to high ones.)

When you create a q-q-plot for a linear regression model, you assume normally distributed errors, so the theoretical quantiles on the horizontal axis will be distributed around zero, and since a linear regression always forces the average residual to be zero, the observed residual quantiles on the vertical axis will also be distributed around zero. The vertical axis will be scaled proportionally to your estimated residual standard deviation $\hat{\sigma}$, and if your dots lie on a straight diagonal line (again, with slope equal to $\hat{\sigma}$), you are good. In particular, neither the theoretical nor the observed quantiles will necessarily lie in some specific interval, since the normal distribution takes values on the entire real axis.

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    $\begingroup$ Expecting [0,1] I suggest means rather that the OP is expecting a (cumulative) probability scale. After all, Q-Q plots are often still called probability plots, along with P-P plots. I don't see that confidence levels are expected. $\endgroup$
    – Nick Cox
    Nov 14, 2017 at 9:04
  • $\begingroup$ @NickCox: you are probably right. I honestly don't know what you call the-thing-in-[0,1]-that-you-plug-into-a-quantile-function, and I have seen this referred to as "confidence levels", although it doesn't feel right. Is this called the "probability scale"? $\endgroup$ Nov 14, 2017 at 9:12
  • $\begingroup$ Cumulative probability is the name I used. It is true that the more you use it, the less you feel the need to name it. Being obliged to teach it helps there! $\endgroup$
    – Nick Cox
    Nov 14, 2017 at 9:28

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