qqnorm {stats}

R Documentation

Quantile-Quantile Plots

Description

qqnorm is a generic function the default method of which produces a normal QQ plot of the values in y. qqline adds a line to a “theoretical”, by default normal, quantile-quantile plot which passes through the probs quantiles, by default the first and third quartiles.

qqplot produces a QQ plot of two datasets. If conf.level is given, a confidence band for a function transforming the distribution of x into the distribution of y is plotted based on Switzer (1976). The QQ plot can be understood as an estimate of such a treatment function. If exact = NULL (the default), an exact confidence band is computed if the product of the sample sizes is less than 10000, with or without ties. Otherwise, asymptotic distributions are used whose approximations may be inaccurate in small samples. Monte-Carlo approximations based on B random permutations are computed when simulate = TRUE. Confidence bands are in agreement with Smirnov's test, that is, the bisecting line is covered by the band iff the null of both samples coming from the same distribution cannot be rejected at the same level.

Graphical parameters may be given as arguments to qqnorm, qqplot and qqline.

Usage

qqnorm(y, ...)
## Default S3 method:
qqnorm(y, ylim, main = "Normal Q-Q Plot",
       xlab = "Theoretical Quantiles", ylab = "Sample Quantiles",
       plot.it = TRUE, datax = FALSE, ...)

qqline(y, datax = FALSE, distribution = qnorm,
       probs = c(0.25, 0.75), qtype = 7, ...)

qqplot(x, y, plot.it = TRUE,
       xlab = deparse1(substitute(x)),
       ylab = deparse1(substitute(y)), ...,
       conf.level = NULL, 
       conf.args = list(exact = NULL, simulate.p.value = FALSE,
                        B = 2000, col = NA, border = NULL))

Arguments

x	The first sample for qqplot.
y	The second or only data sample.
xlab, ylab, main	plot labels. The xlab and ylab refer to the y and x axes respectively if datax = TRUE.
plot.it	logical. Should the result be plotted?
datax	logical. Should data values be on the x-axis?
distribution	quantile function for reference theoretical distribution.
probs	numeric vector of length two, representing probabilities. Corresponding quantile pairs define the line drawn.
qtype	the type of quantile computation used in quantile.
ylim, ...	graphical parameters.
conf.level	confidence level of the band. The default, NULL, does not lead to the computation of a confidence band.
conf.args	list of arguments defining confidence band computation and visualisation: exact is NULL (see details) or a logical indicating whether an exact p-value should be computed, simulate.p.value is a logical indicating whether to compute p-values by Monte Carlo simulation, B defines the number of replicates used in the Monte Carlo test, col and border define the color for filling and border of the confidence band (the default, NA and NULL, is to leave the band unfilled with black borders.

Value

For qqnorm and qqplot, a list with components

x	The x coordinates of the points that were/would be plotted
y	The original y vector, i.e., the corresponding y coordinates including NAs. If conf.level was specified to qqplot, the list contains additional components lwr and upr defining the confidence band.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988). The New S Language. Wadsworth & Brooks/Cole.

Switzer, P. (1976). Confidence procedures for two-sample problems. Biometrika, 63(1), 13–25. doi:10.1093/biomet/63.1.13.

See Also

ppoints, used by qqnorm to generate approximations to expected order statistics for a normal distribution.

Examples

require(graphics)

y <- rt(200, df = 5)
qqnorm(y); qqline(y, col = 2)
qqplot(y, rt(300, df = 5))

qqnorm(precip, ylab = "Precipitation [in/yr] for 70 US cities")

## "QQ-Chisquare" : --------------------------
y <- rchisq(500, df = 3)
## Q-Q plot for Chi^2 data against true theoretical distribution:
qqplot(qchisq(ppoints(500), df = 3), y,
       main = expression("Q-Q plot for" ~~ {chi^2}[nu == 3]))
qqline(y, distribution = function(p) qchisq(p, df = 3),
       probs = c(0.1, 0.6), col = 2)
mtext("qqline(*, dist = qchisq(., df=3), prob = c(0.1, 0.6))")
## (Note that the above uses ppoints() with a = 1/2, giving the
## probability points for quantile type 5: so theoretically, using
## qqline(qtype = 5) might be preferable.) 

## Figure 1 in Switzer (1976), knee angle data
switzer <- data.frame(
    angle = c(-31, -30, -25, -25, -23, -23, -22, -20, -20, -18,
              -18, -18, -16, -15, -15, -14, -13, -11, -10, - 9,
              - 8, - 7, - 7, - 7, - 6, - 6, - 4, - 4, - 3, - 2,
              - 2, - 1,   1,   1,   4,   5,  11,  12,  16,  34,
              -31, -20, -18, -16, -16, -16, -15, -14, -14, -14,
              -14, -13, -13, -11, -11, -10, - 9, - 9, - 8, - 7,
              - 7, - 6, - 6,  -5, - 5, - 5, - 4, - 2, - 2, - 2,
                0,   0,   1,   1,   2,   4,   5,   5,   6,  17),
    sex = gl(2, 40, labels = c("Female", "Male")))

ks.test(angle ~ sex, data = switzer)
d <- with(switzer, split(angle, sex))
with(d, qqplot(Female, Male, pch = 19, xlim = c(-31, 31), ylim = c(-31, 31),
               conf.level = 0.945, 
               conf.args = list(col = "lightgrey", exact = TRUE))
)
abline(a = 0, b = 1)

## agreement with ks.test
set.seed(1)
x <- rnorm(50)
y <- rnorm(50, mean = .5, sd = .95)
ex <- TRUE
### p = 0.112
(pval <- ks.test(x, y, exact = ex)$p.value)
## 88.8% confidence band with bisecting line
## touching the lower bound
qqplot(x, y, pch = 19, conf.level = 1 - pval, 
       conf.args = list(exact = ex, col = "lightgrey"))
abline(a = 0, b = 1)

---