Question

Physicians depend on laboratory test results when managing medical problems such as diabetes or epilepsy. In a test for glucose tolerance, three different laboratories were each sent \(n_{t}=5\) identical blood samples from a person who had drunk 50 milligrams (mg) of glucose dissolved in water. The laboratory results (in \(\mathrm{mg} / \mathrm{d} \mathrm{l}\) ) are listed here: $$ \begin{array}{lrl} \hline \text { Lab 1 } & \text { Lab 2 } & \text { Lab 3 } \\ \hline 120.1 & 98.3 & 103.0 \\ 110.7 & 112.1 & 108.5 \\ 108.9 & 107.7 & 101.1 \\ 104.2 & 107.9 & 110.0 \\ 100.4 & 99.2 & 105.4 \\ \hline \end{array} $$ a. Do the data indicate a difference in the average readings for the three laboratories? b. Use Tukey's method for paired comparisons to rank the three treatment means. Use \(\alpha=.05 .\)

Short answer

**Short Answer:** Based on the results of the ANOVA test, there is no significant difference in the average glucose tolerance test readings between the three laboratories. Therefore, it is not necessary to rank the treatment means using Tukey's method at a 0.05 significance level.

Step-by-step solution

Calculate Mean and Variance

First, we will find the mean of each laboratory's results and their variance: Lab 1: Mean \(= \frac{120.1 + 110.7 + 108.9 + 104.2 + 100.4}{5} = 108.86\) Variance \(= \frac{(120.1-108.86)^2 + (110.7-108.86)^2 + (108.9-108.86)^2 + (104.2-108.86)^2 + (100.4-108.86)^2}{5-1} = 54.313\) Lab 2: Mean \(= \frac{98.3 + 112.1 + 107.7 + 107.9 + 99.2}{5} = 105.04\) Variance \(= \frac{(98.3-105.04)^2 + (112.1-105.04)^2 + (107.7-105.04)^2 + (107.9-105.04)^2 + (99.2-105.04)^2}{5-1} = 27.130\) Lab 3: Mean \(= \frac{103.0 + 108.5 + 101.1 + 110.0 + 105.4}{5} = 105.60\) Variance \(= \frac{(103.0-105.60)^2 + (108.5-105.60)^2 + (101.1-105.60)^2 + (110.0-105.60)^2 + (105.4-105.60)^2}{5-1} = 13.505\)

Perform ANOVA Test

We will now perform an ANOVA test to see if there is a significant difference in the means. We need three pieces of information: SSB (Sum of Squares between groups), SSW (Sum of Squares within groups) and the F-statistic. 1. Calculate the overall mean: $$\text{Overall Mean} = \frac{108.86+105.04+105.60}{3} = 106.50$$ 2. Calculate SSB: $$\text{SSB} = 5 \times [(108.86 - 106.50)^2 + (105.04 - 106.50)^2 + (105.60 - 106.50)^2] = 162.22$$ 3. Calculate SSW: $$\text{SSW} = (4) \times [54.313 + 27.130 + 13.505] = 378.872$$ 4. Calculate the F-statistic: $$\text{F-Statistic} = \frac{SSB/(3-1)}{SSW/((5-1) \times 3)} = \frac{162.22 (2)}{378.872 (12)} = 2.5767$$ Now, we compare the F-statistic with the critical value from the F-distribution for a significance level of \(0.05\). In this case, we have 2 and 12 degrees of freedom, so the critical value \(F_{(2, 12)} = 3.89\). Since our F-statistic (2.5767) is less than the critical value (3.89), we fail to reject the null hypothesis. This means that there is no significant difference between the laboratory means.

Tukey's Multiple Comparisons (If Needed)

Since there's no significant difference between the laboratory means determined by the ANOVA test, we don't need to perform Tukey's multiple comparisons. If there were a significant difference, we would proceed to rank the means using Tukey's method with a \(0.05\) significance level.

Key concepts

Understanding Laboratory Test Variance

When evaluating the reliability and accuracy of laboratory tests, such as those measuring glucose tolerance, it's crucial to understand the concept of variance. Variance measures the dispersion of data points around the mean, reflecting how much individual measurements differ from the average result.
In the context of the exercise, the variance for each lab's results tells us how consistently the lab is performing the glucose tolerance test. A smaller variance indicates more consistency and reliability. Conversely, a large variance suggests that the lab's test results are spread out over a wider range of values, which could mean the lab's processes are not as robust or controlled.
The step-by-step solution provided calculates the variance for three different laboratories. These calculations are foundational in assessing the precision of each lab's testing methods. Larger variances in lab test results could suggest potential problems with testing protocols, equipment calibration, or sample handling. Hence, by evaluating lab variance, physicians and patients can make better-informed decisions regarding which lab to trust for consistent and reliable test results.

Tukey's Method for Paired Comparisons

If the ANOVA test had indicated a significant difference in lab results, the next step would have been Tukey's method for paired comparisons. This statistical technique is utilized when one wants to compare the means of each possible pair of groups to find out which means are significantly different from each other.
Tukey's method controls the overall error rate and is especially useful when conducting multiple tests, as it reduces the chances of obtaining a false-positive result (type I error). Tukey's method adjusts the critical value to account for the number of comparisons being made.
Tukey's method calculates a 'range' value for the differences between means, often referred to as the Tukey's Honest Significant Difference (HSD). If the absolute difference between any two means exceeds this calculated HSD value, the difference is considered statistically significant. Although it was not conducted in the exercise due to the ANOVA result, understanding Tukey's method is important for comprehensive analysis in experiments where multiple comparisons are necessary.

Glucose Tolerance Lab Results Analysis

The glucose tolerance test is a key diagnostic tool for assessing how effectively a person's body regulates blood glucose levels, which is crucial for identifying conditions like diabetes. When analyzing lab results for glucose tolerance, it is not just the mean value that matters but also the consistency and reliability of those results across multiple labs.
In the exercise, an ANOVA test is used to compare the means of glucose levels from three different labs to see if any statistically significant differences exist. The consistency of these results implies that all labs are broadly in agreement and there is no significant variance in their testing methods. This step is critical for ensuring that a patient's diagnosis is not affected by lab-specific variances. The analysis of lab results for glucose tolerance, therefore, includes not only the evaluation of individual glucose levels but also a statistical analysis, such as ANOVA, to confirm the comparative reliability of results between different testing sites. High laboratory test reliability, as indicated by low variance and consistent results across multiple labs, gives healthcare providers confidence in the validity of the test outcomes when managing patient care.