Question

In addition to monitoring weight gain, food consumed, and activity level, the study measured stress levels in the mice by measuring corticosterone levels in the blood (higher levels indicate more stress). Conditions for ANOVA are met and computer output for corticosterone levels for each of the three light conditions is shown. \(\begin{array}{lrrr}\text { Level } & \mathrm{N} & \text { Mean } & \text { StDev } \\ \text { DM } & 10 & 73.40 & 67.49 \\ \text { LD } & 8 & 70.02 & 54.15 \\ \text { LL } & 9 & 50.83 & 42.22\end{array}\) (a) What is the conclusion of the analysis of variance test? (b) One group of mice in the sample appears to have very different corticosterone levels than the other two. Which group is different? What aspect of the data explains why the ANOVA test does not find this difference significant? How is this aspect reflected in both the summary statistics and the ANOVA table?

Short answer

Without the ANOVA data, a general conclusion can't be made but differences in group means suggest the 'LL' group as being significantly different. However, large within-group variances, reflected by high standard deviations, might cause ANOVA not to detect this significance.

Step-by-step solution

Understanding the ANOVA test

Analyze the result of the ANOVA test. The Analysis of Variance (ANOVA) is a statistical method used to test if there are significant differences between the means of multiple groups. The ANOVA test first calculates the sum of squares (variance) between each group's mean and the total mean of all measurements (SSB), and the sum of squares within each group (SSW). The F-statistic is then calculated as the ratio of mean square between to mean square within. The p-value is the probability of observing the F-statistic given the null hypothesis.

Interpreting the conclusion of ANOVA test

As no numerical data is given for ANOVA, it is not possible to draw a concrete conclusion. However, generally, if the p-value from the ANOVA test is less than the significance level (conventionally 0.05), the null hypothesis is rejected, indicating that there is a statistical difference in corticosterone levels across the three different light conditions.

Identifying the different group

Analyzing the provided sample means and standard deviations for each group suggests that the 'LL' group might be the significantly different group. They have a noticeably lower mean corticosterone level than the 'DM' and 'LD' groups.

Reasoning the insignificance in ANOVA

The reason that ANOVA test might not find the difference in corticosterone levels of the 'LL' group significant could be due to large within-group variance (high standard deviation values), which increases the F value and hence the p-value. This could be reflected in the ANOVA table as high variability within each group, leading to inability to significantly differentiate groups.

Key concepts

Analysis of Variance

The Analysis of Variance (ANOVA) test is a cornerstone of statistical analysis when comparing the means of multiple groups. It helps determine if there is any statistically significant difference between the means. It does this by separating the total variation observed in the data into two parts: variation between the groups and variation within the groups. By comparing these variances, ANOVA can tell us if the differences among group means are more than what could be expected by chance alone. For example, in studying the corticosterone levels in mice under various light conditions, ANOVA can establish if the variation in stress levels is associated with the light exposure or if it's likely due to random variability.

Statistical Significance

Statistical significance is a term used to indicate that the results of an analysis are unlikely to have occurred by chance. In the context of ANOVA, when we say that the differences between group means are statistically significant, we mean there's a high confidence that these differences are real and not just random noise. This concept hinges on a predefined threshold called the alpha level (often 0.05), which is the probability limit we set to reject the null hypothesis. If the calculated p-value from ANOVA is below this level, we conclude that there's sufficient evidence to say that not all group means are equal, suggesting a significant effect of the independent variables (like light conditions) on the dependent variable (corticosterone levels).

F-statistic

The F-statistic is the test statistic calculated in an ANOVA test, which is used to determine whether the group means are significantly different. It's a ratio of the variance between the groups to the variance within the groups. A higher F-statistic indicates a greater degree of variance between group means than within groups, suggesting that the variable being tested for (like the intensity of the light conditions) has a significant impact on the outcome variable (stress levels in the mice). Nevertheless, its interpretation is not in isolation and should always be considered in conjunction with the p-value to make informed conclusions about the statistical significance of the results.

p-value

The p-value is a crucial concept in statistical hypothesis testing, and it represents the probability of observing the results, or more extreme results, assuming that the null hypothesis is true. In the context of ANOVA, the null hypothesis typically states that all group means are equal, and the alternative is that at least one group mean is different. A p-value lower than the chosen significance level (usually 0.05) empowers us to reject the null hypothesis with confidence, endorsing the presence of significant differences between the groups. In the mice study, if the p-value is below 0.05 for the light conditions, we would assert that the type of light environment influences stress levels in mice as measured by corticosterone.

Within-group variance

Within-group variance is a measure of dispersion that reflects how spread out the individual observations are around their group mean. High within-group variance can obscure differences between group means because it indicates that the data points within each group are widely scattered. This could be the case in our mice study, where the standard deviation, a measure of within-group variance, may be large. If the within-group variances are substantially higher, it can increase the p-value, making it more difficult to find a statistically significant difference between the group means. Hence, the ANOVA's ability to detect a true difference is hampered, illustrating the importance of not only looking at the means but also considering the variability within each group.