Question

In a study of the dietary treatment of anemia in cattle, researchers randomly divided 144 cows into four treatment groups. Group A was a control group, and groups \(\mathrm{B}, \mathrm{C},\) and \(\mathrm{D}\) received different regimens of dietary supplementation with selenium. After a year of treatment, blood samples were drawn and assayed for selenium. The accompanying table shows the mean selenium concentrations \((\mu g / \mathrm{d} l) .^{36}\) The MS(within) from the ANOVA was 2.071 . $$ \begin{array}{|ccc|} \hline \text { Group } & \text { Mean } & n \\ \hline \text { A } & 0.8 & 36 \\ \text { B } & 5.4 & 36 \\ \text { C } & 6.2 & 36 \\ \text { D } & 5.0 & 36 \\ \hline \end{array} $$ (a) Compute three Bonferroni-adjusted intervals comparing diets \(\mathrm{B}, \mathrm{C},\) and \(\mathrm{D}\) to the control (diet \(\mathrm{A}\) ) using \(\alpha_{e w}=0.05 .\) (Note: This is an example of a situation for which the Bonferroni comparisons may be preferred over the Tukey HSD comparisons since not all comparisons are considered-we are only interested in comparing the control to each of the other three treatments. (b) In the context of the problem, interpret the Bonferroni interval computed in part (a) that compares the control (group A) to the group that is most different from it.

Short answer

Compute each Bonferroni-adjusted interval using the standard error, critical t-value, and mean differences. Interpret the largest interval that does not contain zero as the group most different from the control.

Step-by-step solution

Calculate the Bonferroni Correction

Given that the overall significance level is \(\alpha = 0.05\) for the comparisons, and there are three comparisons to be made, the Bonferroni correction will adjust the significance level by dividing \(\alpha\) by the number of comparisons. In this case, it is three comparisons. So the experiment-wise error rate for each individual comparison is \(\alpha_{ew} = \frac{\alpha}{3} = \frac{0.05}{3} = 0.01667\).

Determine the Critical Value for the t-Distribution

With \(\alpha_{ew} = 0.01667\) and \(df = n_{total} - n_{groups} = 144 - 4 = 140\), we look up the critical value for the t-distribution, which will approximately be based on the t-table for a one-tailed test at the 0.01667 significance level.

Calculate the Standard Error

The standard error (SE) for the difference in means is calculated using the MS(within) from ANOVA and the sample size of each group. \[SE = \sqrt{\frac{MS(within)}{n}} = \sqrt{\frac{2.071}{36}}\].

Compute the Confidence Intervals

Use the standard error and the critical t-value to create the Bonferroni-adjusted confidence intervals for the difference in means between the control group and each of the other groups. \[CI = \text{Mean difference} \pm (t_{critical} \times SE)\]. The mean differences are Group B - Group A, Group C - Group A, and Group D - Group A.

Interpret the Interval for the Group Most Different from Control Group

Compare and interpret the interval for the group (between B, C, and D) that is most different from the control group, based specifically on the range of differences and if zero is included in that range.

Key concepts

ANOVA

Analysis of Variance (ANOVA) is a statistical method used to compare the means of three or more samples to determine if at least one of the sample means is significantly different from the others. It does so by analyzing the variations within each group and between the groups. In the context of the dietary treatment of anemia in cattle, ANOVA has been utilized to see if different selenium supplements have varying effects on the selenium concentrations in the blood of the cows.

The ANOVA results in two types of mean squares: MS(between), which reflects the variation among the different treatment groups, and MS(within), which reflects the variation within the treatment groups. In this study, the MS(within) value of 2.071 is used as an essential component in calculating the standard error for subsequent post-hoc comparisons.

t-Distribution

The t-distribution is a type of probability distribution that is symmetrical and bell-shaped, similar to the normal distribution, but has fatter tails. It's used commonly in statistics when the sample size is small and the population standard deviation is unknown. The t-distribution allows for finding the critical value that corresponds with a particular confidence level for the estimation or hypothesis testing..

In the cattle study, the critical value from the t-distribution is needed for constructing Bonferroni-adjusted intervals. This critical value will adjust for multiple comparisons and control the Type I error rate when comparing the selenium levels between the control group and the treated groups.

Standard Error

Standard Error (SE) measures the accuracy with which a sample represents a population. In other words, it tells us the precision of our sample mean estimate. A small standard error indicates that the sample mean is likely to be a good estimate of the population mean.

In the cattle study, SE is calculated from the MS(within) from ANOVA and the number of cows in each group to understand the variability within our mean differences. SE is crucial for constructing confidence intervals and for any inferential statistics that allow us to draw conclusions about the population from our sample.

Dietary Treatment of Anemia in Cattle

Dietary treatment of anemia in cattle involves addressing nutritional deficiencies that may lead to reduced red blood cell counts. One such nutrient is selenium, which is important for multiple bodily functions including the immune system and thyroid function. Anemia can be detrimental to the health and productivity of cattle, so understanding the optimal level of nutrients like selenium is critical.

In the given study, researchers are evaluating the effectiveness of different regimens of dietary selenium supplementation. By measuring blood selenium concentrations, they can assess which treatment, if any, is most effective in addressing selenium deficiencies that may be related to anemia.

Selenium Supplementation

Selenium supplementation is a key focus in the dietary treatment of cattle for preventing and addressing anemia. Selenium is an essential trace mineral that plays a critical role in metabolism and thyroid function, and it can impact red blood cell production in cattle.

The study examines different levels of selenium supplementation and their effects. Appropriate selenium levels are necessary; both deficiency and excess can lead to health issues. Therefore, carefully designed supplement regimens are tested to identify whether selenium can correct anemia-related issues and to understand the appropriate dosage for maximum benefit without toxicity.