Problem 2 - An Introduction to Generalized Linear Models

Question

The weights, in kilograms, of twenty men before and after participation in a "waist loss" program are shown in Table 2.8 (Egger et al. 1999 ). We want to know if, on average, they retain a weight loss twelve months after the program. Let \(Y_{j k}\) denote the weight of the \(k\) th man at the \(j\) th time, where \(j=1\) before the program and \(j=2\) twelve months later. Assume the \(Y_{j k}\) 's are independent random variables with \(Y_{j k} \sim \mathrm{N}\left(\mu_{j}, \sigma^{2}\right)\) for \(j=1,2\) and \(k=1, \ldots, 20\) (a) Use an unpaired t-test to test the hypothesis \\[\mathrm{H}_{0}: \mu_{1}=\mu_{2} \quad \text { versus } \quad \mathrm{H}_{1}: \mu_{1} \neq \mu_{2}.\\] (b) \(\operatorname{Let} D_{k}=Y_{1 k}-Y_{2 k},\) for \(k=1, \ldots, 20 .\) Formulate models for testing \(\mathrm{H}_{0}\) against \(\mathrm{H}_{1}\) using the \(D_{k}\) 's. Using analogous methods to Exercise 2.1 above, assuming \(\sigma^{2}\) is a known constant, test \(\mathrm{H}_{0}\) against \(\mathrm{H}_{1}\) (c) The analysis in (b) is a paired t-test which uses the natural relationship between weights of the same person before and after the program. Are the conclusions the same from (a) and (b)? (d) List the assumptions made for (a) and (b). Which analysis is more appropriate for these data?

Step-by-Step Solution

Answer

Perform both unpaired and paired t-tests; the paired t-test is more suitable due to its focus on individual differences.

Step 1: Identifying the Statistical Methods

In part (a) we use an unpaired t-test to compare the means of two independent groups, assuming each group follows a normal distribution. In part (b), we will perform a paired t-test using differences in weights, which accounts for the correlation between pre- and post-program weights for each individual.

Step 2: Performing the Unpaired T-test

For the unpaired t-test, calculate the sample means and standard deviations of both groups (before and after the program). Use these to find the test statistic \( t \): \[ t = \frac{\bar{Y}_1 - \bar{Y}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]where \( \bar{Y}_1 \) and \( \bar{Y}_2 \) are sample means, \( s_1^2 \) and \( s_2^2 \) are sample variances, and \( n_1 = n_2 = 20 \). Find the p-value using a t-distribution with \( n_1 + n_2 - 2 \) degrees of freedom.

Step 3: Performing the Paired T-test

Calculate the differences \( D_k = Y_{1k} - Y_{2k} \) for each individual. Compute the mean \( \bar{D} \) and standard deviation \( s_D \) of these differences. The test statistic \( t \) is given by: \[ t = \frac{\bar{D}}{s_D/\sqrt{n}} \]where \( n = 20 \), the number of differences. Use a t-distribution with \( n - 1 \) degrees of freedom to find the p-value.

Step 4: Comparing Conclusions

The conclusions from both tests will be the same if the p-values from both tests lead to the rejection or acceptance of the null hypothesis. If \( p \text{-value} < 0.05 \), we reject \( H_0 \), suggesting a significant weight change after the program.

Step 5: Listing Assumptions

For (a), assume independence between groups, normal distribution of weights, and equality of variances. For (b), assume normal distribution and independence of difference data, as well as the relationship between pre- and post-measurements of the same individual. The paired t-test (b) is more appropriate for these data as it accounts for within-person correlation.

Key Concepts

Unpaired t-test

The unpaired t-test, also known as the independent t-test, is used to compare the means of two separate groups to determine if they are statistically different from each other. This test is particularly useful when you have data from two independent and unrelated groups.
For example, suppose you have data on the weight of participants before and after a weight loss program, treated as separate groups. To perform an unpaired t-test, you will need to calculate the means and standard deviations for both groups, then use the t-test formula to find the test statistic:

• The test statistic is calculated as \( t = \frac{\bar{Y}_1 - \bar{Y}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \)
• Here, \( \bar{Y}_1 \) and \( \bar{Y}_2 \) represent the sample means of the first and second groups.
• \( s_1^2 \) and \( s_2^2 \) are the sample variances, and \( n_1 \) and \( n_2 \) are the sample sizes, which are equal in this case.

The key outcome is the p-value derived from the t-distribution, which tells us if the difference in means is statistically significant.

Paired t-test

The paired t-test is used when comparing two related groups. It is useful when data points are naturally paired, such as in a study measuring the same subjects at two different points in time.
In our exercise, the weights measured before and after the weight loss program for each participant form these pairs. This method utilizes the differences between paired observations to perform the test. Here's how it works:

• First, calculate the differences between paired observations: \( D_k = Y_{1k} - Y_{2k} \).
• Next, find the mean \( \bar{D} \) and standard deviation \( s_D \) of these differences.
• The test statistic is computed using: \( t = \frac{\bar{D}}{s_D/\sqrt{n}} \) where \( n \) is the number of pairs.

The paired t-test accounts for the correlation between two sets of observations, which often improves the power of the statistical test compared to using an unpaired t-test.

Statistical Assumptions

For hypothesis testing via t-tests, certain assumptions must be met to ensure valid results. Recognizing these assumptions is crucial as they guide the choice between a paired and an unpaired t-test.
In the unpaired t-test:

• Independence: The data points between groups must be independent of each other.
• Normal distribution: Each group should follow a normal distribution.
• Homogeneity of variances: The variances within the groups should be equal.

For the paired t-test:

• Normalization of difference: The distribution of the differences between pairs should be normal.
• Independence between pairs: Each pair's difference should be independent of the others.
• Presence of association: Explicit association between paired observations (e.g., repeated measures for the same individual).

Understanding and verifying these assumptions influences which test is more suitable and the reliability of the test results.

Weight Loss Study

In a weight loss study, researchers aim to evaluate the effectiveness of a program by analyzing the weight changes of participants over time. Such studies often employ t-tests to determine if observed weight loss is statistically significant.
Here's how this applies to our exercise:

• Index Time Points: Changes are measured by comparing weights before the program and twelve months after.
• Data Relationship: Since the same individuals are measured twice, a paired t-test is more appropriate because it accounts for the within-subject correlation.
• Outcome: The objective is to see if the mean weight at the baseline differs significantly from the mean weight months after the intervention, indicating true weight loss.
• Data Analysis: Both paired and unpaired t-tests offer insights, but the paired t-test effectively captures the nuances of repeated measurements on the same subjects.

This structured approach ensures that any conclusion drawn aligns with the study objectives and data characteristics.