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Chapter 11: Problem 2

It is thought that stress may increase susceptibility to illness through suppression of the immune system. In an experiment to investigate this theory, 48 rats were randomly allocated to four treatment groups: no stress, mild stress, moderate stress, and high stress. The stress conditions involved various amounts of restraint and electric shock.The concentration of lymphocytes (cells/ml \(\times 10^{-6}\) ) in the peripheral blood was measured for each rat, with the results given in the accompanying table. \(^{4}\) Calculations based on the raw data yielded SS(between) \(=89.036\) and \(\mathrm{SS}(\) within \()=340.24\) $$ \begin{array}{|lrrrr|} \hline & \begin{array}{c} \text { No } \\ \text { stress } \end{array} & \begin{array}{c} \text { Mild } \\ \text { stress } \end{array} & \begin{array}{c} \text { Moderate } \\ \text { stress } \end{array} & \begin{array}{c} \text { High } \\ \text { stress } \end{array} \\ \hline \bar{y} & 6.64 & 4.84 & 3.98 & 2.92 \\ s & 2.77 & 2.42 & 3.91 & 1.45 \\ n & 12 & 12 & 12 & 12 \\ \hline \end{array} $$ (a) Construct the ANOVA table and test the global null hypothesis at \(\alpha=0.05 .\) (b) Calculate the pooled standard deviation, \(s_{\text {pooled }}\)

Short Answer

Expert verified
The ANOVA table must be constructed by calculating the total sum of squares, degrees of freedom, mean squares, and F-statistic, then comparing the F-statistic with the critical F-value to decide on the null hypothesis. The pooled standard deviation is the square root of MS(within).

Step by step solution

01

Calculate the Total Sum of Squares (SST)

First, calculate the total sum of squares (SST), which is the sum of SS(between) and SS(within). Use the formula: SST = SS(between) + SS(within).
02

Calculate the Degrees of Freedom (df)

Calculate df(between) by subtracting 1 from the number of treatment groups, and calculate df(within) by subtracting the number of treatment groups from the total number of subjects. The df(total) is the sum of df(between) and df(within).
03

Calculate Mean Squares (MS)

Compute the Mean Square Between (MS(between)) by dividing SS(between) by df(between), and the Mean Square Within (MS(within)) by dividing SS(within) by df(within).
04

Compute the F-Statistic

Divide MS(between) by MS(within) to get the F-statistic.
05

Determine the Critical F-Value

Using an F-distribution table and the degrees of freedom, determine the critical F-value for \(\alpha=0.05\).
06

Compare the F-Statistic with the Critical F-Value and Make a Decision

If the F-statistic is greater than the critical F-value, reject the null hypothesis. Otherwise, do not reject the null hypothesis.
07

Calculate the Pooled Standard Deviation

Use the formula \(s_{\text{pooled}} = \sqrt{MS(within)}\) to calculate the pooled standard deviation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Stress and the Immune System
The study of how stress affects the immune system is a vital research area within life sciences. Chronic stress is believed to suppress immune function, which can increase susceptibility to illnesses. The theories behind this relationship are grounded in the field of psychoneuroimmunology, which examines how the nervous system and immune system interact.

When an individual experiences stress, their body responds by releasing various hormones, such as cortisol and adrenaline. These hormones can lead to a decrease in the production and efficacy of lymphocytes, which are white blood cells that play a crucial role in defending the body against infections. Reduced lymphocyte concentration may signify a weakened immune response, providing a possible explanation for increased illness during periods of high stress.

Empirical research, using controlled experiments, often measures changes in lymphocyte concentrations under different stress levels to understand these complex dynamics. By examining such relationships, it becomes possible to establish links between stress and immune function and develop potential interventions to mitigate adverse health outcomes associated with stress.
Lymphocyte Concentration Measurement
In life science research, quantifying the concentration of lymphocytes in peripheral blood is a common method to assess immune function. Lymphocytes, including T cells, B cells, and natural killer cells, play a significant role in the immune response. Measuring their concentration highlights changes in immune activity that could be related to various conditions, including stress.

The process of measuring lymphocyte concentration typically involves collecting blood samples and using a technique known as flow cytometry. This technology allows scientists to identify different types of cells by their physical and chemical properties. However, for practical reasons such as simplicity and cost, other methods like hemocytometers or automated cell counters may also be used.

The collected data on lymphocyte concentrations is then used in statistical analyses to infer the status of the immune system. As seen in experimental designs involving stress, these concentrations become critical endpoints or dependent variables that researchers focus on to determine the physiological impact of different stress levels.
F-Statistic Calculation in ANOVA
Analysis of variance (ANOVA) is a powerful statistical method used to compare the means of three or more groups to determine if at least one group mean is statistically different from the others. Within the context of life sciences, it can be applied to evaluate changes in lymphocyte concentration across varying stress levels.

The F-statistic is the test statistic in ANOVA which measures how much the group means differ (variability between groups) compared to the variability within groups. It is calculated by dividing the mean square between (MS(between)) by the mean square within (MS(within)).

MS(between) and MS(within)

The MS(between) represents variance between the group means, reflecting the effect of the independent variable (in our case, stress levels). Whereas MS(within) represents the variance within each group, attributed to random variation or error.

The computed F-statistic is then compared to a critical value from the F-distribution table. A significant F-statistic indicates that the experiment's treatment has had a statistically significant effect on the dependent variable, justifying further post hoc analyses to identify specific group differences.
Pooled Standard Deviation
Understanding the pooled standard deviation is essential when analyzing experiments in the life sciences. When dealing with ANOVA, the pooled standard deviation offers insight into the variability within groups across different treatments or conditions.

In the context of our stress experiment, the pooled standard deviation, denoted as \(s_{\text{pooled}}\), is used to summarize the within-group variability of lymphocyte concentration across all stress levels. It is calculated as the square root of the mean square within (MS(within)), which in turn is derived from the sum of squares within (SS(within)) divided by its respective degrees of freedom (df(within)).

This statistic serves as an estimate of the common within-group standard deviation, assuming that the variances in each group are equal (homogeneity of variance). As a measure of dispersion, \(s_{\text{pooled}}\) aids in interpreting the results of the experiment and is a critical component in the calculation of effect sizes and post hoc tests, which could further delineate how different levels of stress may affect lymphocyte concentrations.

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Most popular questions from this chapter

(Continuation of Exercise 11.8.4) Consider the null hypothesis that flooding has no effect on ATP level in river birch. This hypothesis could be tested in two ways: as a contrast (using the method of Section 11.8), or with a two-sample \(t\) test (as in Exercise 7.2.11). Answer the following questions; do not actually carry out the tests. (a) In what way or ways do the two test procedures differ? (b) In what way or ways do the conditions for validity of the two procedures differ? (c) One of the two procedures requires more conditions for its validity, but if the conditions are met, then this procedure has certain advantages over the other one. What are these advantages?

A botanist used a completely randomized design to allocate 45 individually potted eggplant plants to five different soil treatments. The observed variable was the total plant dry weight without roots \((\mathrm{gm})\) after 31 days of growth. The treatment means were as shown in the following table. \({ }^{35}\) The MS(within) was \(0.2246 .\) Use Fisher's LSD intervals to compare all pairs of means at \(\alpha_{c_{\mathbb{Y}}}=0.05 .\) Present your results in a summary table similar to Table 11.9.4. (Hint: Take note that all sample sizes are equal; thus the calculated margin of error need only be calculated once for all comparisons. There are a total of 10 comparisons possible.) $$ \begin{array}{|llllll|} \hline \text { Treatment } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \text { Mean } & 4.37 & 4.76 & 3.70 & 5.41 & 5.38 \\ n & 9 & 9 & 9 & 9 & 9 \\ \hline \end{array} $$

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.

In an experiment to understand the impact of fish grazing on invertebrate populations in streams, researchers established nine observation channels in three streams (three channels per stream). Each of the three channels within a stream received one of three treatments: No fish were added, Galaxias fish were added, or Trout fish were added. (The channels were constructed with mesh to prevent fish from entering or leaving.) Twelve days after establishing the channels, the number of Deleatidium mayfly nymphs present in a specified region in the center of the channel were counted. The number of nymphs for each treatment in each creek follows. (a) Identify the blocking, treatment (i.e., the explanatory variable of interest), and response variables in this study. (b) In the context of this problem, explain to someone who has never taken a statistics course how blocking may help better identify treatment differences should they exist.

In an experiment to investigate the effects of different soil nutrient amendments on pomegranate fruit yield (number of fruits per tree) researchers first selected 24 crop rows within a pomegranate farm to follow the irrigation lines. Nine trees in each row were selected and were randomly assigned one of three soil nutrients (three trees per nutrient): potassium, \(\mathrm{KNO}_{3}\); magnesium, \(\mathrm{MgSO}_{4}\); or zinc, \(\mathrm{ZnSO}_{4}\). All of the fruit from each the treated trees were counted. In all 216 trees \((24 \times 9)\) were observed. \(^{17}\) (a) Identify the blocking, treatment, and response variables in this study. (b) Why do you think the blocks were chosen in the way they were in this study? (c) What were the researchers hoping to accomplish by using blocks? (d) If it is found that there are statistically significant differences between the mean number of fruits produced across the treatments, will the design of this study allow the researchers to conclude that the nutrients contribute to this difference?
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