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]?
Regression – How to Interpret Negative Quantiles on a QQ Plot
qq-plotquantilesregression
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This QQ plot has the following salient features:
The stairstep pattern, in which only specific, separated heights ("sample quantiles") are attained, shows the data values are discrete. Almost all are whole numbers from $3$ through $21$. A few half-integers appear. Evidently some form of rounding has occurred.
Because the extreme "theoretical quantiles" are at $\pm 3.2$ (roughly), there must be around $1400$ data shown. This is because the extremes for this much Normally distributed data would have Z-scores about $\pm 3.2$. (This estimate of $1400$ is rough, but it's in the right ballpark.)
There is a large number of values at the minimum of $3$, far more than any other value. This is characteristic of left censoring, whereby any value less than a threshold ($3$) is replaced by an indicator that it is less than that threshold--and, for plotting purposes, all such values are plotted at the threshold. (For more on what censoring does to probability plots, see the analysis at https://stats.stackexchange.com/a/30749.)
Apart from this "spike" at $3$, the rest of the points come fairly close to following the diagonal reference line. This suggests the remaining data are not too far from Normally distributed.
A closer look, though, shows the remaining points are initially slightly lower than the reference line (for values between $5$ and $10$) and then slightly greater (for values between $13$ and $20$) before returning to the line at the end (value $21$). This "curvature" indicates a certain form of non-normality.
This particular kind of curvature is consistent with data that are starting to follow an extreme-value distribution. Specifically, consider the following data-generation mechanism:
Collect $k\ge 1$ independent, identically distributed Normal variates and retain just the largest of them.
Do that $n = 1400$ times.
Left-censor the data at a threshold of $3$.
Record their values to two or three decimal places.
Round the values to the nearest integer--but don't round any value that is exactly a half-integer (that is, ends in $.500$).
If we set $k=50$ or thereabouts and adjust the mean and standard deviation of those underlying Normal variates to be $\mu = -10$ and $\sigma = 7.5$, we can produce random versions of this QQ plot and most of them are practically indistinguishable from it. (This is an extremely rough estimate; $k$ could be anywhere between $8$ and $200$ or so, and different values of $k$ would have to be matched with different values of $\mu$ and $\sigma$.) Here are the first six such versions I produced:
What you do with this interpretation depends on your understanding of the data and what you want to learn from them. I make no claim that the data actually were created in such a way, but only that their distribution is remarkably like this one.
This is R code to reproduce the figure (and generate many more like it if you wish).
k <- 50
mu <- -10
sigma <- 7.5
threshold <- 3
n <- 1400
#
# Round most values to the nearest integer, occasionally
# to a half-integer.
#
rnd <- function(x, prec=300) {
y <- round(x * prec) / prec
ifelse(2*y == floor(2*y), y, round(y))
}
q <- c(0.25, 0.95) # Used to draw a reference line
par(mfcol=c(2,3))
set.seed(17)
invisible(replicate(6, {
# Generate data
z <- apply(matrix(rnorm(n*k), k), 2, max) # Max-normal distribution
y <- mu + sigma * z # Scale and recenter it
x <- rnd(pmax(y, threshold)) # Censor and round the values
# Plot them
qqnorm(x, cex=0.8)
m <- median(x)
s <- diff(quantile(x, q)) / diff(qnorm(q))
abline(c(m, s))
#hist(x) # Histogram of the data
#qqnorm(y) # QQ plot of the uncensored, unrounded data
}))
The "flatter" part of a QQ plot suggests that from corresponding normal scores on the X-axis where it is flat, you have more data than would be expected according to a normal probability model. These Z-scores are (low) to -2, -1 to 1, and 2 to (high). For instance, on a normal curve, you'd expect 66% of data to lie within 1 SD of the mean. However, in your residual distribution, you have far more than 66% in that interval. Projecting the curves value at X=-1 and X=1 seems to give a Y of about -.33 to 0.33. That means that the central $\pm$ 0.33 SD of the residual distribution holds 66% of the data, a much higher concentration than in a normal distribution.
Similarly, for the steeply sloped (greater than identity, or the 45 degree line) sections of the QQ-plot, you have fewer observations than would be expected by a normal probability model. That seems to match the residuals histogram you show. It looks like a mixture of platykurtic and leptokurtic distributions. As noted in the comments, a trimodal distribution seems to fit the ticket as well.
Best Answer
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.
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.