Question

Do lizards play a role in spreading plant seeds? Some research carried out in South Africa would suggest so ("Dispersal of Namaqua Fig (Ficus cordata cordata) Seeds by the Augrabies Flat Lizard (Platysaurus broadleyi)," Journal of Herpetology [1999]: \(328-330\) ). The researchers collected 400 seeds of this particular type of fig, 100 of which were from each treatment: lizard dung, bird dung, rock hyrax dung, and uneaten figs. They planted these seeds in batches of 5 , and for each group of 5 they recorded how many of the seeds germinated. This resulted in 20 observations for each treatment. The treatment means and standard deviations are given in the accompanying table. $$\begin{array}{lccc} \text { Treatment } & \boldsymbol{n} & \overline{\boldsymbol{x}} & \boldsymbol{s} \\ \hline \text { Uneaten figs } & 20 & 2.40 & .30 \\ \text { Lizard dung } & 20 & 2.35 & .33 \\ \text { Bird dung } & 20 & 1.70 & .34 \\ \text { Hyrax dung } & 20 & 1.45 & .28 \end{array}$$ a. Construct the appropriate ANOVA table, and test the hypothesis that there is no difference between mean number of seeds germinating for the four treatments. b. Is there evidence that seeds eaten and then excreted by lizards germinate at a higher rate than those eaten and then excreted by birds? Give statistical evidence to sup- port your answer.

Short answer

In an analysis of variance (ANOVA) test, we calculate an F-statistic and find its P-value to decide whether or not the null hypothesis should be rejected. If rejected, it suggests a significant difference exists among the treatments. Then, pairwise comparisons between treatments where we're interested in are conducted to find out where the differences occur.

Step-by-step solution

Tabulate the Data

Organize the data in a suitably accessible manner. This means creating a table displaying the treatment means and standard deviations, along with their counts. The table will be as follows: \[\begin{array}{lccc} \text {Treatment} & n & \overline{x} & s \ \hline \text{Uneaten figs} & 20 & 2.40 & .30 \ \text{Lizard dung} & 20 & 2.35 & .33 \ \text{Bird dung} & 20 & 1.70 & .34 \ \text {Hyrax dung} & 20 & 1.45 & .28 \end{array}\]

Analyze the Variance

Compute the total, between-treatments, and within-treatments sum of squares (SS). Then calculate the degree of freedom (df), the mean square (MS), F-statistic and P-value to test the null hypothesis (H0, there is no difference in the mean germination rate among the four treatments). Use statistical software or calculator for the calculations.

Pairwise Comparisons

Based on the outcome of the F-test in step 2, if there is a significant difference between the treatment means, perform pairwise comparisons to determine where the differences lie. In this case, the interest lies in comparing the germination rates of seeds passed by lizards and those passed by birds. Calculate the difference of the means and using the standard deviations, compute a t-statistic and find the corresponding P-value to test the null hypothesis (H0, there is no difference in the mean germination rates between seeds passed by lizards and those passed by birds).

Key concepts

Statistical Hypothesis Testing

Understanding statistical hypothesis testing is crucial for analyzing experimental data. It is a method of making decisions using data, whether from a controlled experiment or an observational study. The process begins with formulating two opposing hypotheses: the null hypothesis \( H_0 \) and the alternative hypothesis \( H_a \). The null hypothesis often represents the idea of no effect or no difference, while the alternative hypothesis represents the idea that there is an effect or a difference.

In the context of an ANOVA, the null hypothesis typically states that all groups are drawn from populations with the same mean, and we use variance to assess this. ANOVA compares the ratio of variability between groups to the variability within groups. To determine if there is a statistically significant difference between the groups, we calculate the F-statistic and compare it to a critical value from the F-distribution. If the calculated F is greater than the critical value, we have reason to reject the null hypothesis, suggesting that at least one group is different.

To perform hypothesis testing properly, it is essential to understand concepts such as significance level \( \alpha \), p-values, and the type of errors (Type I and Type II) that can occur during testing. This framework allows researchers to make informed decisions about their hypotheses based on the data collected and the statistical evidence.

Germination Rate

The germination rate of seeds is a key metric in botanical studies and agriculture, indicating the percentage of seeds that begin to grow over a certain period. High germination rates are usually desirable, indicating that the seeds are healthy and have a high potential for developing into mature plants. In the exercise provided, the germination rates are of particular interest as they might be affected by various treatments, such as passage through the digestive system of different animals.

Evaluating the germination rate is vital when examining the roles animals play in seed dispersal and propagation. By comparing germination rates from various treatments—such as lizard dung, bird dung, or uneaten figs—researchers can draw conclusions about which methods of seed dispersal are most effective. In this scenario, statistical analysis helps to isolate the impact of each treatment on the germination rate, providing insights into the interactions between wildlife and plant life.

Sum of Squares

The sum of squares (SS) is a statistical tool that quantifies the variation within a dataset. The concept divides into two main types: the between-group sum of squares and the within-group sum of squares.

The between-group sum of squares measures the variance of the group means from the grand mean (the mean of all data points regardless of group) and is associated with the effect that the treatments may have on the data. The within-group sum of squares, on the other hand, measures the variation of individual data points around their respective group means, capturing random error or noise in the data.

In an ANOVA table, these components are crucial for calculation of mean squares (MS), which are the sums of squares divided by their corresponding degrees of freedom (df). Mean square for treatments (MS treatment) and mean square for error (MS error), are then used in the F-ratio, which is the primary statistic of ANOVA. A higher F-ratio indicates a larger proportion of variance accounted for by the treatment effects, suggesting that the treatments indeed have a significant impact on the outcome.