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The χ²-test
When a problem involves multiply categories, we use the χ²-test.
Introduction
• We know that when a problem only involves two categories, the z-test is
appropriate.
• For instance, in the ESP experiment, we only classified each guess being
correct or incorrect.
• Here, “correct” and “incorrect” are the two categories in this problem.
We modeled the 0-1 box, 1 represented “correct”, and 0 represented
“incorrect”. Then we could look at the sum of 1’s to count the frequency
of the guesses being correct.
• However, when a problem involves more than two categories, we have
to use the χ²-test.
• For instance, you might want to see if a die is fair, then there are 6
categories. The χ²-test will help to check whether these categories are
equally likely.
Example
• A gambler is accused of using a loaded die, but he pleads innocent.
• A record has been kept of the last 60 throws:
Example
• If we only focus on one line of the table, say, the number of 3’s.
• The SE for this number is 60 × 1/6 × 5/6 ≈ 2.9. Then the
observed number is about 2.4 SEs above the expected number.
• But different lines have different variance. This could happen even if
the die is fair.
• The idea is to combine all these differences into one overall measure
of the distance between the observed and expected values.
• The χ² is to square each difference, divide by the corresponding
expected frequency, and take the sum.
Example
• Here is the formula: χ² = ∑
(𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 − 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦)²
.
𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
• From the formula we see, when the observed frequency is far from the
expected frequency, the corresponding term in the sum is large; when
the two are close, this term is small.
• So the χ² measure the distances between the observed and expected
values.
• In our example, using the above formula, there is one term for each line
in the table. With the data, the χ²-statistic is:
•
(4−10)²
10
(6−10)²
+
10
(17−10)²
+
10
+
(16−10)²
10
+
(8−10)²
10
+
(9−10)²
10
=
142
10
= 14.2.
Example
• With the χ²-statistic, we need a curve to approximate the probability,
that is the P-value.
• The χ²-curve: is a bunch of curves. More precisely, there is one curve
for each number of degrees of freedom.
Example
• To figure out the degrees of freedom, we use the following formula:
• Degrees of freedom = number of terms in χ² − one.
• To figure out the P-value, we look at the area to the right of the χ²statistic under the χ²-curve: from the formula of χ², we see that the
values to the right of χ² represent the more extreme ones to the
observed value.
• In our example: there are 6 – 1 = 5 degrees of freedom.
Example
• To find the area, one needs to read the χ² table:
• For instance, look at the column for
1% and the row for 5 degrees.
• It reads 15.09, meaning that the area
to the right of 15.09 under the curve
for 5 degrees of freedom is about 1%.
• So in our example, the P-value is just a
bit more than 1%. (χ² = 14.2)
χ² approximation
• From probability theory, we
actually compute the exact
probability for each value of
the χ²-statistic.
• We see from the graph, the
probability histogram
indeed follows the χ²-curve:
Difference between z and χ²
• If it matters how many tickets of each kind are in the box, then we
use the χ²-test. (involve multiply categories and we know the content)
• For instance, fair die test, fair coin test, and etc.
• If it is about the average or the sum of the box, then we use the ztest. (including counting two categories and we only know the
average/sum)
• For instance, fair coin test, average of heights test, and etc.
Another Example
• The χ²-test can also be used to test for independence:
• Are handedness and sex independent?
• Take people age 25-34 in the U.S. The question is whether the
distribution of “handedness” (right-handed, left-handed,
ambidextrous) among the men in this population differs from the
distribution among the women.
• Here is a probability sample of 2,327 Americans 25-34:
Another Example
• In order to compare the distribution of handedness for men and
women, it is better to look at the data in percentage:
• To make a χ²-test of the null hypothesis, we have to compare
the observed frequencies with the expected frequencies (based
on the null).
Another Example
• Figuring out the expected frequencies takes some effort here:
• The null says handedness and sex are independent, then the difference in distribution is due to
chance.
• So the expected percentage of each type of handedness for each gender must be the same as the
percentage of each type of handedness for the whole population. This percentage can be estimated
from the sample.
2,004
• For instance, the percentage of right-handers is
× 100% ≈ 89.6%. Then the expected
2,237
frequency of right-handed men is 89.6% × 1,067 ≈ 956.
Another Example
• Therefore, we obtain the following table of data:
• There are 2×3 categories, and the χ²-statistic can be computed as:
•
(934−956)²
(1,070−1,048)²
(113−98)²
(92−107)²
(20−13)²
(8−15)²
+
+
+
+
+
956
1,048
98
107
13
15
≈ 12.
Another Example
• The formula for calculating the degrees of freedom has to be
changed:
• When testing independence in an m × n table:
• Degrees of freedom = (𝑚 − 1) × (𝑛 − 1).
• So in our example, it is only 3 − 1 × 2 − 1 = 2 degrees of
freedom.
• From the table, we see the P-value:
Another Example
• The P-value is only about 0.2% < 1%. It is highly significant.
• So the null hypothesis should be rejected. The observed difference in
the sample seems to reflect a real difference in the population, rather
than chance variation.
• Therefore, we have the conclusion:
• There is strong evidence to show that the distribution of handedness
among the men in the population is different from the distribution for
women.