NAME

Statistics::ChiSquare - How random is your data?

SYNOPSIS

  use Statistics::ChiSquare;

  

  print chisquare(@actual_occurrences);
  print chisquare_nonuniform([actual_occurrences], [expected_occurrences]);

The Statistics::ChiSquare module is available at a CPAN site near you.

DESCRIPTION

Suppose you flip a coin 100 times, and it turns up heads 70 times. Is the coin fair?

Suppose you roll a die 100 times, and it shows 30 sixes. Is the die loaded?

In statistics, the chi-square test calculates "how random" a series of numbers is. But it doesn't simply say "random" or "not random". Instead, it gives you a confidence interval, which sets upper and lower bounds on the likelihood that the variation in your data is due to chance. See the examples below.

If you've ever studied elementary genetics, you've probably heard about Gregor Mendel. He was a wacky Austrian botanist who discovered (in 1865) that traits could be inherited in a predictable fashion. He performed lots of experiments with cross-fertilizing peas: green peas, yellow peas, smooth peas, wrinkled peas. A veritable Brave New World of legumes.

How many fertilizations are needed to be sure that the variations in the results aren't due to chance? Well, you can never be entirely sure. But the chi-square test tells you how sure you should be.

(As it turns out, Mendel faked his data. A statistician by the name of R. A. Fisher used the chi-square test again, in a slightly more sophisticated way, to show that Mendel was either very very lucky or a little dishonest.)

There are two functions in this module: chisquare() and chisquare_nonuniform(). chisquare() expects an array of occurrences: if you flip a coin seven times, yielding three heads and four tails, that array is (3, 4). chisquare_nonuniform() is a bit trickier---more about it later.

Instead of returning the bounds on the confidence interval in a tidy little two-element array, these functions return an English string. This was a deliberate design choice---many people misinterpret chi-square results; the text helps clarify the meaning. Both chisquare() and chisquare_nonuniform() return UNDEF if the arguments aren't "proper".

Upon success, the string returned by chisquare() will always match one of these patterns:

  There's a >\d+% chance, and a <\d+% chance, that this data is random.

or

  There's a <\d+% chance that this data is random.

unless there's an error. Here's one error you should know about:

  (I can't handle \d+ choices without a better table.)

That deserves an explanation. The "modern" chi-square test uses a table of values (based on Pearson's approximation) to avoid expensive calculations. Thanks to the table, the chisquare() calculation is quite fast, but there are some collections of data it can't handle, including any collection with more than 21 slots. So you can't calculate the randomness of a 30-sided die.

chisquare_nonuniform() expects two arguments: a reference to an array of actual occurrences followed by a reference to an array of expected occurrences.

chisquare_nonuniform() is used when you expect a nonuniform distribution of your data; for instance, if you expect twice as many heads as tails and want to see if your coin lives up to that hypothesis. With such a coin, you'd expect 40 heads (and 20 tails) in 60 flips; if you actually observed 42 heads (and 18 tails), you'd call

  chisquare_nonuniform([42, 18], [40, 20])

The strings returned by chisquare_nonuniform() look like this:

  There's a >\d+% chance, and a <\d+% chance, 
       that this data is distributed as you expect.

EXAMPLES

Imagine a coin flipped 1000 times. The most likely outcome is 500 heads and 500 tails:

  @coin = (500, 500);
  print chisquare(@coin);

which prints

  There's a >99% chance, and a <100% chance, 
       that this data is evenly distributed.

Imagine a die rolled 60 times that shows sixes just a wee bit too often.

  @die1  = (9, 8, 10, 9, 9, 15);
  print chisquare(@die1);

which prints

  There's a >50% chance, and a <70% chance, 
       that this data is evenly distributed.

Imagine a die rolled 600 times that shows sixes way too often.

  @die2  = (80, 70, 90, 80, 80, 200);
  print chisquare(@die2);

which prints

  There's a <1% chance that this data is evenly distributed.

How random is rand()?

  srand(time ^ $$);
  @rands = ();
  for ($i = 0; $i < 60000; $i++) {
      $slot = int(rand(6));
      $rands[$slot]++;
  }
  print "@rands\n";
  print chisquare(@rands);

which prints (on my machine):

  9987 10111 10036 9975 9984 9907
  There's a >70% chance, and a <90% chance, 
       that this data is evenly distributed.

(So much for pseudorandom number generation.)

All the above examples assume that you're testing a uniform distribution---testing whether the coin is fair (i.e. a 1:1 distribution), or whether the die is fair (i.e. a 1:1:1:1:1:1 distribution). That's why chisquare() could be used instead of chisquare_nonuniform().

Suppose a mother with blood type AB, and a father with blood type Ai (that is, blood type A, but heterozygous) have one hundred children. You'd expect 50 kids to have blood type A, 25 to have blood type AB, and 25 to have blood type B. Plain old chisquare() isn't good enough when you expect a nonuniform distribution like 2:1:1.

Let's say that couple has 40 kids with blood type A, 30 with blood type AB, and 30 with blood type B. Here's how you'd settle any nagging questions of paternity:

    @data = (40, 30, 30);
    @dist = (50, 25, 25);
    print chisquare_nonuniform(\@data, \@dist);

which prints

  There's a >10% chance, and a <30% chance, 
       that this data is distributed as you expect.

AUTHOR

Jon Orwant

MIT Media Lab

orwant@media.mit.edu

cpanm Statistics::ChiSquare

perl -MCPAN -e shell
install Statistics::ChiSquare