Question

More on Exercise and Stress Exercise 6.219 on page 465 introduces a study showing that exercise appears to offer some resiliency against stress, and Exercise 8.19 on page 556 follows up on this introduction. In the study, mice were randomly assigned to live in an enriched environment (EE), a standard environment (SE), or an impoverished environment (IE) for several weeks. Only the enriched environment provided opportunities for exercise. Half the mice then remained in their home cage \((\mathrm{HC})\) as control groups while half were subjected to stress (SD). The researchers were interested in how resilient the mice were in recovering from the stress. One measure of mouse anxiety is amount of time hiding in a dark compartment, with mice who are more anxious spending more time in darkness. The amount of time (in seconds) spent in darkness during one trial is recorded for all the mice and the means and the results of the ANOVA analysis are shown. There are eight mice in each of the six groups. \(\begin{array}{l}\text { Group: } & \text { IE:HC } \\ \text { Mean: } & 192 & 196 & 205 & 392 & 438 & \text { SE:HC } & \text { EE:HC } & \text { IE:SD } & \text { SE:SD EE:SD } \\ \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } & \\ \text { Light } & 5 & 481776 & 96355.2 & 39.0 & 0.000 & \\ \text { Error } & 42 & 177835 & 2469.9 & & \\\ \text { Total } & 47 & 659611 & & & \end{array}\) (a) Is there a difference between the groups in the amount of time spent in darkness? Between which two groups are we most likely to find a difference in mean time spent in darkness? Between which two groups are we least likely to find a difference? (b) By looking at the six means, where do you think the differences are likely to lie? (c) Test to see if there is a difference in mean time spent in darkness between the IE:HC group and the EE:SD group (that is, impoverished but not stressed vs enriched but stressed).

Short answer

Yes, there is a statistically significant difference in the amount of time spent in darkness among the groups. The most significant difference is likely between the IE:HC and EE:SD groups while the least significant difference is probably between IE:HC and SE:HC groups. However, an exact t-test can't be performed for IE:HC and EE:SD groups without necessary information.

Step-by-step solution

Interpreting the ANOVA Table

The analysis of variance (ANOVA) table gives us details about the variability within our data. The 'source' row tells us where the variability comes from: either differences within each group (error) or between groups (light). The p-value is associated with an F-test which compares the variability between and within groups. If the p-value is less than 0.05, we can reject the null hypothesis that all group means are equal.

Identifying Differences Between Groups

The P value given in the ANOVA table is 0.000, which is less than 0.05. This suggests that there is a statistically significant difference between the group means. To determine which specific groups are likely to be different, we need to look at the means. The groups with the minimum mean (IE:HC, 192 seconds) and the maximum mean (EE:SD, 438 seconds) are likely to be most different. Similarly, the groups with the most similar means (IE:HC, 192 seconds and SE:HC, 196 seconds) are least likely to be different.

Conducting a t-test for IE:HC and EE:SD

We already know from ANOVA that there is a significant difference among means. However, to compare mean time in darkness between the IE:HC and EE:SD groups, we can perform a t-test. This test compares the means of two groups and tells us if they are different from each another. To perform this test, we need the mean, the standard error, and the number of observations from each group, which are not provided. So, the actual test cannot be performed in this case, but this would be the necessary step if provided.

Key concepts

Exercise and Stress Study

Understanding how exercise influences stress resilience can provide valuable insights into behavioral health. A unique study was designed to examine this relationship using mice as model organisms. The mice were placed in different housing conditions—enriched (EE), standard (SE), or impoverished (IE)—with variations in exercise opportunities.

The study proceeded to introduce stress (SD) to half of the mice, aiming to assess their recovery based on their environment and exercise levels. This study is crucial as it mirrors human lifestyle conditions, where some individuals have access to rich environments with plentiful resources and exercise opportunities, while others do not.

By monitoring the time these mice spent hiding in a dark compartment—a behavior suggestive of anxiety—the researchers could infer levels of stress and anxiety. In this context, mice who spent more time hidden were inferred to be more anxious than their peers.

Measures of Mouse Anxiety

Behavioral Assessment

In the study's methodology, mouse anxiety was quantified by the duration of time each mouse spent hiding in a dark compartment. This measure is based on the behavior that anxious mice exhibit a tendency to avoid open and brightly illuminated areas.

Interpreting the Measures

To ensure accurate interpretation, researchers must control external variables such as noise levels, light intensity, and human interaction, as these can affect the mice's behavior and, consequently, the results.

It's also crucial to consider the variability within these anxiety measures. For example, time spent in darkness is one aspect, but one could also consider heart rate, cortisol levels, and other physiological parameters for a more comprehensive understanding of the stress response.

Statistical Significance in ANOVA

ANOVA, short for Analysis of Variance, is a statistical test used to analyze the differences among group means in a sample. In the context of the mice study, ANOVA helped determine if the time spent in darkness by mice varied significantly across different housing and stress conditions.

Deciphering the P-Value

The statistical significance of an ANOVA is often reported as a p-value. This study yielded a p-value of 0.000, indicating that at least one group mean is significantly different from the others when considering the level of anxiety as measured by time spent hiding. When the p-value is below a predefined threshold, typically 0.05, it suggests that the observed differences in means are unlikely to have occurred by random chance.

Importance of Multiple Comparisons

After establishing significance with ANOVA, further post hoc tests are necessary to pinpoint where these differences lie between specific groups. This is because ANOVA informs us about the variance across all groups collectively but does not identify which exact group pairs are significantly different. Correcting for multiple comparisons is essential to avoid type I errors, which is falsely declaring a difference when there is none.