Multiple Comparisons: Tukey-Kramer Test: Videos & Practice Problems

After conducting an ANOVA test and rejecting the null hypothesis, the Tukey Kramer test is employed to identify which means differ among groups. This post hoc test compares each pair of means, requiring the calculation of a q statistic for each pair against a critical value derived from the studentized range distribution table. The critical value is determined using degrees of freedom and an alpha level of 0.05. The process involves calculating the mean squares due to error and applying the formula for the q statistic, allowing for conclusions about the differences in means.

Tukey-Kramer Test

When conducting an ANOVA test, if the null hypothesis is rejected, it indicates that at least one group mean is different from the others. However, it does not specify which means differ, leading to the necessity of post hoc tests. One such test is the Tukey Kramer test, which allows for pairwise comparisons between group means to identify specific differences.

The Tukey Kramer test operates by comparing each possible pair of means. For instance, with three groups, there are three pairs to evaluate. Although this may seem overwhelming, the process involves a few initial steps that are consistent across all comparisons, making it similar to t-tests previously learned.

To begin, ensure that the null hypothesis from the ANOVA test has been rejected, confirming that there is a difference among the means. Set the significance level (alpha) for the Tukey Kramer test, typically at 0.05. The critical value needed for comparisons is obtained from the studentized range distribution table, also known as the q table. This table is similar to the f table used in ANOVA, but it requires the degrees of freedom, calculated as the total number of observations minus the number of groups. For example, with 30 observations across three groups, the degrees of freedom would be 27. Using these values, the critical value can be determined, which in this case is 3.05.

In the Tukey Kramer test, each pair's test statistic, known as the q statistic, is compared against the critical value. If the q statistic exceeds the critical value, the null hypothesis for that pair is rejected, indicating a significant difference between the means. Conversely, if the q statistic is less than the critical value, the null hypothesis is not rejected, suggesting no significant difference.

The q statistic is calculated using the formula:

\[q = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{MSE}{n_1} + \frac{MSE}{n_2}}}\]

where \(\bar{X}_1\) and \(\bar{X}_2\) are the means of the two groups being compared, \(MSE\) is the mean squares due to error from the ANOVA output, and \(n_1\) and \(n_2\) are the sample sizes of the respective groups.

For example, when comparing the average study times of grades 10 and 11, if the calculated q statistic is 1.949, which is less than the critical value of 3.05, we fail to reject the null hypothesis, concluding that the average study times are the same. In contrast, when comparing grades 10 and 12, if the q statistic is 4.498, which exceeds the critical value, we reject the null hypothesis, indicating a significant difference in average study times.

By systematically applying the Tukey Kramer test to each pair of means, one can effectively determine which specific means differ, providing clarity following the initial ANOVA analysis.

Tukey-Kramer Test Example 1

In this analysis, we explore the effectiveness of three diet plans (A, B, and C) by employing the Tukey Kramer test following an ANOVA test. The initial step in this process is to confirm that the ANOVA test has been conducted and the null hypothesis has been rejected, indicating that at least one diet plan leads to a different average weight loss. With this established, we proceed to the Tukey Kramer test to identify which specific pairs of means differ.

The first task is to determine the critical value, denoted as q, using the studentized range distribution table. For an alpha level of 0.05, we identify the number of groups (k = 3) and calculate the degrees of freedom as the total number of observations (n = 15) minus the number of groups (3), resulting in 12 degrees of freedom. This yields a critical value of q = 3.773.

Next, we calculate the test statistic for each pair of diet plans. The formula for the test statistic is given by:

q = \frac{|\bar{x}_1 - \bar{x}_2|}{\sqrt{\frac{s^2}{n_1} + \frac{s^2}{n_2}}}

where s² is the mean square error from the ANOVA output, and n represents the sample sizes for each group. In this case, the mean square error is 1.9, and the sample sizes for each diet plan are all 5.

We begin by comparing diet plans A and B. The null hypothesis states that the average weight loss for plan A is equal to that for plan B (μ_A = μ_B), while the alternative hypothesis posits that they are different (μ_A ≠ μ_B). Plugging the means (8 for A and 4.8 for B) into the formula, we calculate:

q = \frac{|8 - 4.8|}{\sqrt{\frac{1.9}{5} + \frac{1.9}{5}}} = 5.19

Since 5.19 exceeds the critical value of 3.773, we reject the null hypothesis, concluding that the average weight loss for plans A and B is significantly different.

Next, we compare plans B and C. The null hypothesis here is μ_B = μ_C, and the alternative is μ_B ≠ μ_C. Using the means (4.8 for B and 11 for C), we find:

q = \frac{|4.8 - 11|}{\sqrt{\frac{1.9}{5} + \frac{1.9}{5}}} = 10.06

Again, since 10.06 is greater than 3.773, we reject the null hypothesis, indicating a significant difference in average weight loss between plans B and C.

Finally, we compare plans A and C. The null hypothesis is μ_A = μ_C, and the alternative is μ_A ≠ μ_C. Using the means (8 for A and 11 for C), we calculate:

q = \frac{|8 - 11|}{\sqrt{\frac{1.9}{5} + \frac{1.9}{5}}} = 4.87

Since 4.87 also exceeds the critical value of 3.773, we reject the null hypothesis, concluding that the average weight loss for plans A and C is significantly different.

In summary, the Tukey Kramer test reveals that all pairs of diet plans (A vs. B, B vs. C, and A vs. C) exhibit significant differences in average weight loss, providing valuable insights into the effectiveness of these diet plans.

Here’s what students ask on this topic:

What is the Tukey-Kramer test and when is it used?

The Tukey-Kramer test is a post hoc statistical test used after conducting an ANOVA test and rejecting the null hypothesis. It helps identify which specific group means differ from each other. This test compares all possible pairs of means and determines if the differences are statistically significant. It is particularly useful when analyzing data with multiple groups, such as comparing average study times across different grades. The test uses a q statistic derived from the mean squares due to error and compares it to a critical value from the studentized range distribution table. If the q statistic exceeds the critical value, the null hypothesis for that pair is rejected, indicating a significant difference between the means.

How do you calculate the q statistic in the Tukey-Kramer test?

The q statistic in the Tukey-Kramer test is calculated using the formula:

$q=\frac{|M_1-M_2|}{\sqrt{\frac{s^2}{2\times \frac{1}{n_1}+\frac{1}{n_2}}}}$

Here, |M₁ - M₂| is the absolute difference between the two group means, s² is the mean squares due to error from the ANOVA readout, and n₁ and n₂ are the sample sizes of the two groups being compared. The denominator accounts for variability within the groups. Once calculated, the q statistic is compared to the critical value from the studentized range distribution table to determine significance.

What is the role of the critical value in the Tukey-Kramer test?

The critical value in the Tukey-Kramer test serves as the threshold for determining whether the q statistic indicates a significant difference between group means. It is obtained from the studentized range distribution table, which depends on the alpha level (commonly 0.05), the number of groups, and the degrees of freedom. If the calculated q statistic for a pair of means exceeds the critical value, the null hypothesis for that pair is rejected, indicating a significant difference. Conversely, if the q statistic is less than the critical value, the null hypothesis is not rejected, suggesting no significant difference between the means. This comparison ensures the test maintains the desired level of statistical rigor while accounting for multiple comparisons.

How does the Tukey-Kramer test handle multiple comparisons?

The Tukey-Kramer test addresses multiple comparisons by evaluating all possible pairs of group means individually while controlling the overall Type I error rate. It uses the studentized range distribution to adjust for the increased likelihood of false positives that arise when conducting multiple tests. For each pair, the q statistic is calculated and compared to the critical value. This systematic approach ensures that the conclusions drawn about differences between means are statistically valid, even when analyzing numerous groups. By doing so, the Tukey-Kramer test provides a reliable method for identifying specific differences without inflating the risk of incorrect rejections of the null hypothesis.

What are the assumptions of the Tukey-Kramer test?

The Tukey-Kramer test relies on several assumptions to ensure valid results: (1) The data in each group should be normally distributed. (2) The variances across groups should be approximately equal (homogeneity of variance). (3) The observations within each group should be independent of each other. These assumptions are similar to those of ANOVA, as the Tukey-Kramer test is a post hoc analysis following ANOVA. Violations of these assumptions can lead to inaccurate conclusions, so it is important to check them before applying the test. If assumptions are not met, alternative methods or adjustments may be necessary.