Independence Tests: Videos & Practice Problems

Understanding the concept of independence between two variables is crucial in statistics, particularly when analyzing categorical data. Independence implies that the two variables do not influence each other. For instance, when examining students' heights in relation to their grade levels, we may want to determine if there is any relationship between these two variables. To assess this, we can utilize an independence test, which is conceptually similar to a goodness of fit test.

In an independence test, we start by formulating hypotheses. The null hypothesis (H₀) posits that the two variables are independent, meaning that one does not affect the other. Conversely, the alternative hypothesis (H_a) suggests that the variables are dependent. For example, we might state that students' heights are unaffected by their grade levels.

The test statistic used in an independence test is the chi-squared statistic, calculated using observed and expected frequencies. The expected frequencies (E) are determined by the formula:

E = (Row Total × Column Total) / Grand Total

This calculation is essential for establishing the expected distribution of data under the assumption of independence.

To compute the chi-squared statistic, we use the formula:

χ² = Σ((O - E)² / E)

where O represents the observed frequencies. After calculating the chi-squared statistic, we can determine the degrees of freedom (df) for the test, which is given by:

df = (Number of Rows - 1) × (Number of Columns - 1)

For example, if there are 2 rows and 3 columns, the degrees of freedom would be (2 - 1) × (3 - 1) = 2.

Once we have the chi-squared value and the degrees of freedom, we can find the corresponding p-value. This p-value is then compared to a predetermined significance level (α), often set at 0.05. If the p-value is greater than α, we fail to reject the null hypothesis, indicating insufficient evidence to claim that the variables are dependent. In our example, if the p-value is 0.19, we would conclude that there is not enough evidence to suggest a relationship between students' heights and their grade levels.

It is also important to ensure that certain conditions are met before conducting an independence test. These include having random samples, observed frequencies for all categories, and expected frequencies of at least 5 for each category. Meeting these criteria ensures the validity of the test results.

In summary, an independence test allows us to evaluate whether two categorical variables are related. By following the steps of hypothesis formulation, calculating the chi-squared statistic, determining the p-value, and checking the necessary conditions, we can draw meaningful conclusions about the relationship between the variables in question.

In this example, we explore the relationship between symptom improvement in ADHD patients and whether they received a placebo or not, using a chi-squared independence test. The goal is to determine if symptom improvement is independent of the treatment type. We begin by establishing our null hypothesis, which states that symptom improvement is independent of whether a patient received a placebo. Conversely, the alternative hypothesis posits that symptom improvement is dependent on the treatment type.

Before proceeding, we confirm that our data meets the necessary criteria for the test: we have random samples, observed frequencies for all categories, and we will later verify that expected frequencies exceed five for each category. The expected frequencies are calculated using the formula:

$\mathrm{Expected\; Frequency}=\frac{\mathrm{Row\; Total}\times \mathrm{Column\; Total}}{\mathrm{Grand\; Total}}$

For our observed frequencies, we have values such as 18, 37, 30, and 15, with corresponding expected frequencies of 26.4, 28.6, 21.6, and 23.4, respectively. We then calculate the chi-squared test statistic using the formula:

$\mathrm{\chi ²}=\frac{\sum \left(O-E\right)²}{E}$

Calculating each term, we find the chi-squared value to be 11.42. The degrees of freedom for our test, calculated as (rows - 1) × (columns - 1), results in 1. Using this value, we determine the p-value to be 0.0007, which is significantly lower than our significance level of α = 0.01.

Since the p-value is less than α, we reject the null hypothesis. This indicates that there is sufficient evidence to support the alternative hypothesis, suggesting that symptom improvement is indeed dependent on whether patients received a placebo. This outcome implies that the ADHD medication tested is effective in improving symptoms.