Chi-square is a statistical test that is used to determine if there is a significant association
between two categorical variables. It is often used in research studies to analyze data and
determine if there is a relationship between variables. To perform a chi-square test, you need to
have a contingency table that shows the frequencies or counts of each category for the two
variables being studied. The contingency table is typically organized into rows and columns,
with each cell representing the count for a specific combination of categories. This table allows
you to compare the observed frequencies with the expected frequencies under the assumption
of no association.
For example, let's say we are studying the relationship between smoking habits and lung
cancer. We collect data from a sample of individuals and create a contingency table that shows
the number of smokers and non-smokers who have been diagnosed with lung cancer. The table
might look like this:
Lung Cancer
No Lung Cancer
Smokers
50
100
Non-smokers
30
120
In this example, the observed frequencies are the actual counts in each cell of the contingency
table. We can calculate the expected frequencies under the assumption of no association by
using the row and column totals. For instance, the expected frequency for the cell representing
smokers with lung cancer would be (row total for smokers * column total for lung cancer) / total
sample size.
Once we have the observed and expected frequencies, we can calculate the chi-square test
statistic. This statistic measures the difference between the observed and expected frequencies
and follows a chi-square distribution. The formula for calculating the chi-square test statistic is:
χ^2 = Σ((O - E)^2 / E)
where Σ represents the sum of all cells, O is the observed frequency, and E is the expected
frequency.
In our example, we would calculate the chi-square test statistic by summing the values of ((5040)^2 / 40), ((100-110)^2 / 110), ((30-40)^2 / 40), and ((120-110)^2 / 110).
The next step is to compare the calculated chi-square test statistic to a critical value from the
chi-square distribution. The critical value is determined based on the desired level of
significance and the degrees of freedom, which is calculated as (number of rows - 1) * (number
of columns - 1). If the calculated chi-square test statistic is greater than the critical value, we
reject the null hypothesis and conclude that there is a significant association between the
variables. On the other hand, if the calculated chi-square test statistic is less than the critical
value, we fail to reject the null hypothesis and conclude that there is not enough evidence to
suggest an association.
It is important to note that the chi-square test assumes certain conditions. One assumption is
that the categories being compared are independent of each other. This means that the
frequency of one category does not depend on the frequency of another category. Another
assumption is that the sample size is sufficiently large. If the sample size is too small, the chisquare test may not provide reliable results. Violation of these assumptions can affect the
validity of the test results. For example, if the categories are not independent, the chi-square
test may falsely detect an association. Similarly, if the sample size is too small, the test may fail
to detect a true association.
In summary, the chi-square test is a useful tool for analyzing categorical data and determining if
there is a significant association between variables. It helps researchers make informed
conclusions based on their data. However, it is important to carefully consider the assumptions
and limitations of the test before drawing any conclusions.