The paper "Trends in Blood Lead Levels and Blood Lead...

The paper "Trends in Blood Lead Levels and Blood Lead Testing among U.S. Children Aged 1 to 5 Years" (Pediatrics [2009]: e376-e385) gave data on blood lead levels (in $\mathrm{mg} / \mathrm{dL}$ ) for samples of children living in homes that had been classified either at low, medium, or high risk of lead exposure, based on when the home was constructed. After using a multiple comparison procedure, the authors reported the following: 1. The difference in mean blood lead level between low-risk housing and medium-risk housing was significant. 2. The difference in mean blood lead level between low-risk housing and high-risk housing was significant. 3. The difference in mean blood lead level between mediumrisk housing and high-risk housing was significant. Which of the following sets of T-K intervals (Set $1,2,$ or 3$)$ is consistent with the authors' conclusions? Explain your choice. $\mu_{L}=$ mean blood lead level for children living in low-risk housing $\mu_{M}=$ mean blood lead level for children living in mediumrisk housing $\mu_{H}=$ mean blood lead level for children living in high-risk housing

The paper "Trends in Blood Lead Levels and Blood Lead Testing among U.S. Children Aged 1 to 5 Years" (Pediatrics [2009]: e376-e385) gave data on blood lead levels (in $\mathrm{mg} / \mathrm{dL}$ ) for samples of children living in homes that had been classified either at low, medium, or high risk of lead exposure, based on when the home was constructed. After using a multiple comparison procedure, the authors reported the following:
1. The difference in mean blood lead level between low-risk housing and medium-risk housing was significant.
2. The difference in mean blood lead level between low-risk
housing and high-risk housing was significant.
3. The difference in mean blood lead level between mediumrisk housing and high-risk housing was significant.
Which of the following sets of T-K intervals (Set $1,2,$ or 3$)$ is consistent with the authors' conclusions? Explain your choice.
$\mu_{L}=$ mean blood lead level for children living in low-risk housing $\mu_{M}=$ mean blood lead level for children living in mediumrisk housing $\mu_{H}=$ mean blood lead level for children living in high-risk housing

Key Concepts

Multiple Comparison Procedures

This concept involves methods used to compare several groups simultaneously while controlling the overall Type I error rate. In statistics, when multiple pairwise comparisons are performed, there is a risk that the probability of incorrectly rejecting at least one true null hypothesis (false positive) increases; hence techniques such as Tukey's method are used to adjust the confidence levels appropriately.

Tukey-Kramer Intervals (and Confidence Intervals in Multiple Comparisons)

Tukey-Kramer intervals are a form of confidence interval specifically designed for pairwise comparisons among group means. The intervals are adjusted to account for the fact that multiple comparisons are being made simultaneously. In this context, if a confidence interval for a difference between two group means does not contain zero, it indicates that the difference is statistically significant.

Significance of Pairwise Differences

In the context of ANOVA and multiple group comparisons, significance is determined by whether the confidence intervals for differences in means overlap zero. When using a procedure like the Tukey-Kramer method, if all pairwise differences yield intervals that do not include zero, it implies that each pair of groups has a statistically significant difference in their means. This is pivotal for concluding distinct differences in any group comparisons.

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