Chi-squared Test for Independence (DP IB Applications & Interpretation (AI): SL): Revision Note

Chi-Squared Test for Independence

What is a chi-squared test for independence?

• A chi-squared (${\chi }^2$) test for independence is a hypothesis test used to test whether two variables are independent of each other

  • This is sometimes called a ${\chi }^2$ two-way test

• This is an example of a goodness of fit test

  • We are testing whether the data fits the model that the variables are independent

• The chi-squared (${\chi }^2$) distribution is used for this test

• You will use a contingency table

  • This is a two-way table that shows the observed frequencies for the different combinations of the two variables

    • For example: if the two variables are hair colour and eye colour then the contingency table will show the frequencies of the different combinations

What are the degrees of freedom?

• There will be a minimum number of expected values you would need to know in order to be able to calculate all the expected values

• This minimum number is called the degrees of freedom and is often denoted by $\nu$

• For a test for independence with an m × n contingency table

  • $\nu =(m-1)\times (n-1)$ 

  • For example: If there are 5 rows and 3 columns then you only need to know 2 of the values in 4 of the rows as the rest can be calculated using the totals

What are the steps for a chi-squared test for independence?

• STEP 1
  Write the hypotheses

  • H₀ : Variable X is independent of variable Y

  • H₁ : Variable X is not independent of variable Y

    • Make sure you clearly write what the variables are and don’t just call them X and Y

• STEP 2
  Calculate the degrees of freedom for the test

  • For an m × n contingency table

    • the degrees of freedom is $\nu =(m-1)\times (n-1)$

• STEP 3
  Enter your observed frequencies into your GDC using the option for a 2-way test

  • Enter these as a matrix

  • Your GDC will give you a matrix of the expected values (assuming the variables are independent)

  • Your GDC will also give you the χ² statistic and its p-value

  • The χ² statistic is denoted as ${\chi }_{calc}^2$

• STEP 4
  Decide whether there is evidence to reject the null hypothesis

  • EITHER compare the χ² statistic with the given critical value

    • If χ² statistic > critical value then reject H₀

    • If χ² statistic < critical value then accept H₀

  • OR compare the p-value with the given significance level

    • If p-value < significance level then reject H₀

    • If p-value > significance level then accept H₀

• STEP 5
  Write your conclusion

  • If you reject H₀

    • There is sufficient evidence to suggest that variable X is not independent of variable Y

    • Therefore this suggests they are associated

  • If you accept H₀

    • There is insufficient evidence to suggest that variable X is not independent of variable Y

    • Therefore this suggests they are independent

How do I calculate the chi-squared statistic?

• You are expected to be able to use your GDC to calculate the χ² statistic by inputting the matrix of the observed frequencies

• Seeing how it is done by hand might deepen your understanding but you are not expected to use this method
  

• STEP 1:
  For each observed frequency O_i calculate its expected frequency E_i

  • Assuming the variables are independent

    • E_i = P(X = x) × P(Y = y) × Total

    • Which simplifies to $E_i=\frac{\mathrm{Row}\mathrm{ }\mathrm{Total}\mathrm{ }\times \mathrm{ }\mathrm{Column}\mathrm{ }\mathrm{Total}}{\mathrm{Overall}\mathrm{ }\mathrm{Total}}$

• STEP 2:
  Calculate the χ² statistic using the formula

  • ${\chi }_{calc}^2=\sum \frac{{(O_i-E_i)}^2}{E_i}$

  • You do not need to learn this formula as your GDC calculates it for you

• To calculate the p-value you would find the probability of a value being bigger than your χ² statistic using a χ² distribution with $\nu$ degrees of freedom