4
$\begingroup$

To study the quantile-quantile plot, I used the following codes (modified from here). The first group of pictures is derived from 100 data points while the second from 10000.

# Q-Q plots
par(mfrow=c(1,3), pty="s")

# create sample data
y <- rnorm(100)
x <- rt(100, df=3) # Heavy-Tail

# normal fit
qqnorm(y); qqline(y)
qqnorm(x); qqline(x)

# t(3Df) fit
qqplot(rt(100,df=3), x, main="t(3) Q-Q Plot", ylab="Sample Quantiles")
abline(0,1) 

100

When the sample size is small (100), the second and the third graph are similar and hence it is very difficult to make decision.

10000

However, when the sample size is large (10000) the second and the third graph are very different and hence it is very easy to make conclusion.

As the two groups of pictures show, sample size clearly affects one's judgement. In practice, the sample size is frequently given. Therefore, my question is as follows. What to do, when we are in situation one, where sample size is small. Is there any more effective diagnostic tool to use? Thank you!

$\endgroup$
4
$\begingroup$

I think there is less here than meets the eye. You need to recognize that the appearance of these plots will bounce around with different data. I modified your code with:

set.seed(2501)
par(mfrow=c(3,3), pty="s")

And then ran the rest of your code three times. Here is the resulting plot:

enter image description here

Sometimes the distinction between the left and center plots is clear and sometimes it isn't. That's the way it goes. Data are information. More data give you more information (all else being equal), and it is easier to see / figure out what you want to know.

One thing that may help you is to explore the qqPlot function in the car package, which will plot a 95% confidence band around the plot to help you see how much a dataset might vary from the ideal form to help you judge the deviations that you see in your observed data. Here it is with the last iteration of y:

enter image description here

Given the amount that 100 data can vary from the ideal, you just don't have enough information to reject the possibility of normality for these data (even though they were drawn from a $t$-distribution with 3 degrees of freedom).

$\endgroup$
  • $\begingroup$ Thank you for your answer. I think I get your point. My question is what to do, if I have to choose a distribution between normal and $t(3)$. Do I choose one arbitrarily, please? I am afraid not. $\endgroup$ – LaTeXFan Jan 12 '15 at 4:24
  • 1
    $\begingroup$ The 'check, then decide' recipe for statistical testing is frowned upon. If you are worried that your data may not be normal, it is best to simply use a more robust option. For more information, see: A principled method for choosing between t test or non-parametric e.g. Wilcoxon in small samples. $\endgroup$ – gung - Reinstate Monica Jan 12 '15 at 4:30
  • $\begingroup$ @20826 that depends on your data and what you know about your data $\endgroup$ – shadowtalker Jan 12 '15 at 4:39
  • $\begingroup$ I like the confidence band on the QQ plot :) $\endgroup$ – wolfies Jan 12 '15 at 17:36
1
$\begingroup$

I could think of at least two approaches to better diagnostics for a small sample size case:

  1. To use a different scale for Q-Q plots in order to visually emphasize deviation from the normal line;

  2. To augment visual diagnostics with analytical approach, as described, for example, here.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.