Objectives

Example 1

The contingency table below shows the distribution of survival and different classes of passengers on the Titanic.

Goals:

1. To analyze whether the observed frequencies of class differ significantly from the expected frequencies of survived.
2. To test whether class and survived are independent by comparing observed and expected frequencies in a two-way table.

The Chi-Squared Test

The Chi-Square statistic is used in hypothesis testing to determine whether observed data differs significantly from expected data. Commonly used for categorical data analysis.

\[\chi^2 = \sum \frac{(O-E)^2}{E}\]

• \(O\) is the observed frequency
• \(E\) is the expected frequency

The degrees of freedom (df) in a chi-square test depend on the type of test being conducted:

• For a One-Way Table: \(df = r - 1\) where \(r\) is the number of categories.
• For a Two-Way Table: \(df = (r - 1) \times (c - 1)\) where \(r\) is the number of rows and \(c\) is the number of columns.

Example 1

1st Class Distribution

\[ \begin{aligned} H_0: & \text{The observed distribution of survived among 1st class passengers follows the expectation.} \\ H_A: & \text{The observed distribution of survived among 1st class passengers differs from the expectation.} \end{aligned} \]

df <- 1 # define degrees of freedom
chisq <- 18.78 # set chi-square statistic
1-pchisq(18.78,df) # compute the p-value

## [1] 1.466974e-05

Example 1: The Chi-Square Sampling Distribution

Example 2: Inference for One-Way Table

2nd Class Distribution

\[ \begin{aligned} H_0: & \text{The observed distribution of survived among 2nd class passengers follows the expectation.} \\ H_A: & \text{The observed distribution of survived among 2nd class passengers differs from the expectation.} \end{aligned} \]

Example 3

\[ \begin{aligned} H_0: & \text{There is no association between class and survived. The variables are independent.} \\ H_A: & \text{There is an association between class and survived. The variables are dependent.} \end{aligned} \]

Example 3

Consider the following problem description.

Students in grades 4-6 were asked whether good grades, athletic ability, or popularity was most important to them. A two-way table separating the students by grade and by choice of most important factor is shown below. Do these data provide evidence to suggest that goals vary by grade?

Source: Popular Kids Dataset. This is from a 1992 study and was revisited 30 years later.

Example 3: The Chi-Squared Test for Independence (1/2)

• The null and alternative Hypothesis \[H_0: \text{Grade and goals are independent. Goals do not vary by grade.}\] \[H_A: \text{Grade and goals are dependent. Goals vary by grade}\]

Example 3: The Chi-Squared Test for Independence (2/2)

• The Chi-Squared test statistic \[\chi^2_{k} = \sum_{i=1}^n \frac{(O_i - E_i)^2}{E_i}\]
  • \(O_i\) is the number of observations of type \(i\)
  • \(E_i\) is the expected frequency of type \(i\)
  • \(n\) is the number of cells in the table
  • \(k = (R-1)(C-1)\) is the degrees of freedom where \(R\) is the number of rows and C is the number of columns.

Example 3: Computing the \(\chi^2\) statistic - Expected Frequency (1/3)

• Start with the expected frequency of type \(i\)

Example 3: Computing the \(\chi^2\) statistic - Expected Frequency (2/3)

• Question - What is the expected count for the highlighted cell?

• \[\color{green}{E_{5th,Popular} = \frac{(176)(141)}{478} = 52}\]

Example 3: Computing the \(\chi^2\) statistic - Expected Frequency (3/3)

• The expected frequency for each \(\color{blue}{[cell]}\).

Example 3: Computing the \(\chi^2\) statistic

• The \(\chi^2\) statistic. \[\chi^2_{k} = \frac{(63-61)^2}{61} + \frac{(31-35)^2}{35} + \cdots + \frac{(32-34)^2}{34} = 0.967\]

• Degrees of freedom. \[k = (3-1) \times (3-1) = 2(2) = 4\]

Example 3: Computing the p-value

• \(\chi^2_{k} = 1.3121\) and \(k = 4\)

• We can use the pchisq function in R.

df <- 4
1-pchisq(0.967,df)

## [1] 0.9147579

The p-value of 0.9148.

• Note that in a chi-squared analysis, the p-value is the probability of obtaining a chi-square as large or larger than that in the current experiment.

Example 3: Conclusion

• Do these data provide evidence to suggest that goals vary by grade? \[H_0: \text{Grade and goals are independent. Goals do not vary by grade.}\] \[H_A: \text{Grade and goals are dependent. Goals vary by grade}\]

• Since the p-value is large, we fail to reject \(H_0\). The data do not provide convincing evidence that grade and goals are dependent. It doesn’t appear that goals vary by grade.

Summary: Steps for \(\chi^2\) Tests

• Compute the expected values.
• Set the significance value \(\alpha\).
• Compute the \(\chi^2\) test statistic and the degrees of freedom \(df\).
• Determine the p-value using the \(\chi^2\) test statistic.
• Make a conclusion.