Posted: April 25th, 2025

Stat 4

Dear Students,

Please read module 4, Hypothesis Testing for Categorical and Ordinal Outcomes. Also, read your textbook chapter 7, section 7.4. You have to read carefully to understand all the information in the section. Look at the examples. Once you feel sure you understood the examples and the procedures of hypothesis testing, then do the following assignment.

If you will have any questions, please let me know in advance.

Thank you.

Assignment: 

Obesity is defined as having body mass index (BMI) of 30 or higher. There are 3 categories of obesity; Category 1 = BMI 30 to less than 35; Category 2 = BMI 35 to less than 40; and Category 3 = BMI 30 or more.

The National Center for Health Statistics reports that the prevalence of different categories of obesity are: Category 1: 60%, category 2: 30%, and category 3: 10%. A health education and nutrition program was launched for 5 years to mitigate the issue of obesity. A survey was conducted to see the impact of the program. The survey reports the following data:

Table: Distribution of obesity by category 

Obesity Category 1Obesity Category 2Obesity Category 3Total sample4503981521000

Based on the sample data, is there a evidence of a shift in the distribution of undernutrition after the health and nutrition mitigation program? Test the hypothesis.

Instructions:

Test the hypothesis following all the steps:

1. Set up hypothesis and determine the level of significance at α = 0.05;

2. Select the appropriate test statistic;

3. Set up the decision rule;

4. Compute the test statistic; and

5. Conclusion

Hypothesis Testing for Categorical and Ordinal Outcomes

Hypothesis Testing for Categorical and Ordinal Outcomes

Let us remember the 
categorical and ordinal variables. Variables that take on more than two distinct responses or categories are called categorical. When responses/categories are unordered, we call it categorical and when they are ordered, we call it ordinal variables. Let us give examples: Race may be of different categorical like, White, Black, Hispanic, American Indian, Alaskan Native, Asian, Pacific Islander. We can not order them, and hence they are categorical variables. On the other hand there are different categories of income group in the society, like low income, medium income, high income. They are categorical, but they can be ordered. Hence, these variables are ordinal. 

As we discussed in module 1, first we select a sample and compute descriptive statistics on the sample data. We compute sample size (n), and the proportions of participants in each response category, (p1^, p2^,…….. pk^), where k represents the number of response categories. 

In the hypothesis testing for categorical and ordinal variables/outcomes, we use the same steps in our previous module with continuous and dichotomous outcomes. Let us recall the steps again,

Step 1: Select the null and the alternative hypothesis;

Step 2: Select the appropriate test statics;

Step 3: Set up the decision rule;

Step 4: Compute the test statistic; and

Step 5: Conclusion.

For categorical and ordinal variables, we use χ2 (Chi-square) tests. This is simply an assessment of how far responses/outcomes in the sample fits a specific population distribution. Say it differently, we want to see whether the computed sample statistics fit the population parameter. This test (χ2) is used for categorical and ordinal data. Do not confuse that ordinal variables are also categorical variables. The test is used to determine whether our data are significantly different from what we expected. Usually two types of χ2 tests are used:

·
The chi-square goodness-of-fit test is used to test whether the frequency distribution of a categorical variable is different from our expectations; and

·
The chi-square test of independence used to test whether two categorical variables are related to each other.

In higher statistics, there are other chi-square tests, like chi-square test of homogeneity and McNemar’s tests that are not our topics of interests.

The chi-square formula: χ2=∑((O−E)2E); Where χ2= Chi-square statistic; ∑ = Summation;

O = Observed number; and E= Expected number.

Look at the formula and see the larger the difference between the observations and expectations, the bigger the chi-square value. To decide whether the difference is big enough to be statistically significant, we compare chi-square value to the critical value given 
table 3, Critical Values of the χ2
 Distribution at the end of the textbook.

Let’s see and practice an example.

The   National Survey reported that the prevalence of undernutrition among under 5 children in “X” country is as follows:

Wasting

Stunting

Underweight

30%

20%

50%

The Health Department of “X” county ran a child nutrition program for 3 years. A survey was conducted to see the impact of the program. The sample size (n) was

320

. The results of the survey are as follow: 

Wasting

Stunting

Underweight

Our task is to show, 
is there an evidence of a shift in the distribution of undernutrition after the the child nutrition program was conducted? 

Step 1: Set up hypothesis and determine the level of significance

           Ho : p1 = 0.30; p2 = 0.20; and p3 = 0.50. 

           H1 : Ho is false

          α = 0.05

 
Step 2: Select the appropriate test statistic

The formula for the test statistic is chi-square:   χ2 =∑((O−E)2E)

We must first assess whether our sample size is adequate. The minimum sample must be checked by min(np10 …….npk0) ≥5. The sample size here is 320 and the proportion specified in the null hypothesis are: 0.30, 0.20, and 0.50. Let us calculate the sample size:

min[320(0.30), 320(0.20), 320(0.50)] = min(96, 64, 160) = 64 is greater than 5. So, sample size is more than adequate, so the test statistic can be used. 

Step 3: Set up the decision rule

The decision rule for χ2 test is similar to the decision rules for z and t tests. Please recall that decision rule depends on the level of significance (α ) and the degree of freedom (df). The df is defined as df = k-1, where k is the number of response/outcome categories. Note that in χ2 tests, there are no upper or lower-tailed versions of the test. If we fail to reject the null, i.e., the null is true, the observed and expected frequencies are close in value and the χ2 is close to 0. If the null hypothesis is false, meaning we reject the null, then the χ2 statistic is large. The rejection region for the χ2 test is always in the upper tail.

If the calculated χ2 value is greater than the critical (tabulated) value we reject the null.

Reject H0 if χ2 ≥ tabulated value. 

How to read the χ2
 tabulated value?

Go to the table 3 in the appendix, Critical Values of the χ2 Distribution, the columns are the significant levels ( α ), like column .10 is 90% significant level, .05 is 95% significant level, .025 is 97.75%, .01 is 99%, and .005 is 99.99%. 

Let’s assume, our significant level is 95% (α=0.05). 

Now, in the row, see the degree of freedom (df = k-1), in our example, we have 3 categories (Wasting, stunting, and underweight). Therefore, df =3-1 = 2. Now see the column 0.05 and row df 2. See the χ2 value is 5.99. 

Our decision rule: Reject H0  if χ2 ≥ 5.99

Step 4: Compute the test statistic

χ2 = ∑((O−E)2E)= (80−96)296+(62−64)264+(178−160)2160= −16296+−2264+ 182160= 25696+464+324160 = 4.75

Step 5: Conclusion

The calculated χ2 = 4.75 < tabulated χ2 =5.99. Therefore, we fail to reject the H0 i.e., we accept the null. 

Interpretation: We have statistically significant evidence at α = 0.05 to show that H0 is true, or that the distribution of undernutrition is 0.30, 0.20, and 0.50. 

We can also approximate the p-value. We need to look at critical values for smaller levels of significance with df =2. Using table 3 in the appendix of the textbook, the p-value is p<0.025.