Question

An article in Archaeometry involved an analysis of 26 samples of Romano- British pottery, found at four different kiln sites in the United Kingdom. \(^{8}\) Since one site only yielded two samples, consider the samples found at the other three sites. The samples were analyzed to determine their chemical composition and the percentage of iron oxide is shown next. a. What type of experimental design is this? b. Use an analysis of variance to determine if there is a difference in the average percentage of iron oxide at the three sites. Use \(\alpha=.01\). c. If you have access to a computer program, generate the diagnostic plots for this experiment. Does it appear that any of the analysis of variance assumptions have been violated? Explain.

Short answer

Answer: To determine if there is a significant difference in the average percentage of iron oxide among the three sites, we can perform a one-way ANOVA test with a significance level of 0.01. After conducting the test and checking the diagnostic plots to ensure ANOVA assumptions are met, we can either reject or accept the null hypothesis. The null hypothesis states that there is no significant difference in the average percentage of iron oxide among the three sites. If we reject the null hypothesis, there is a significant difference in the average percentage of iron oxide; otherwise, there is not enough evidence to conclude a difference.

Step-by-step solution

a. Identifying the type of experimental design

In this case, we have pottery samples taken from three different kiln sites, and we want to compare the percentage of iron oxide among these groups. This is an example of a one-way ANOVA design, where the factor of interest (kiln site) has three levels (Site A, Site B, and Site C).

b. Performing an analysis of variance (ANOVA)

To perform ANOVA, we first need to have the data for the three kiln sites. Assuming this data is provided in the format of three groups with their respective percentages of iron oxide, we can follow these steps: 1. Calculate the mean percentage of iron oxide for each site. 2. Calculate the overall mean percentage of iron oxide (total of all samples). 3. Calculate the sum of squares for between-group (SSB) variation and within-group (SSW) variation. 4. Calculate the degrees of freedom for between-group (dfB) and within-group (dfW) variations. 5. Calculate the mean square for between-group variation (MSB) and within-group variation (MSW) by dividing SSB and SSW by their respective degrees of freedom. 6. Calculate the F-statistic by dividing MSB by MSW. 7. Compare the F-statistic with the critical value at a significance level of 0.01. 8. If the F-statistic is greater than the critical value, reject the null hypothesis (no difference in average percentage of iron oxide). Otherwise, conclude that there is not enough evidence to reject the null hypothesis.

c. Checking diagnostic plots and ANOVA assumptions

Diagnostic plots are used to check for the fulfillment of ANOVA assumptions, which are: 1. Normality of the data 2. Homoscedasticity (equal variances among groups) 3. Independence of observations By using a statistical software, we can generate diagnostic plots like histogram, Q-Q plot, and residuals vs. fitted values plot. After generating these plots, we should examine them for the following: - Histogram: A symmetric bell-shaped curve indicates that the data is approximately normal. - Q-Q plot: Points follow a straight line, which indicates that the data is approximately normal. - Residuals vs. Fitted values plot: A random pattern of residuals indicates homoscedasticity. If the diagnostic plots show that the assumptions of normality, homoscedasticity, and independence are met, we can trust the results of the ANOVA test. Otherwise, the results may not be reliable.

Key concepts

Experimental Design

In the scenario provided, we are looking at an experimental design that involves samples of pottery from three different kiln sites. This is an ideal example of a **one-way ANOVA** design because it evaluates a single factor, which in this case is the kiln site. Each kiln site is a level of the factor, making it a suitable design for comparing the percentage of iron oxide across the different sites. A one-way ANOVA is used when you want to determine if there is any statistical difference between the means of three or more independent groups. It allows researchers to explore whether the factor (kiln site) has any significant effect on the outcomes (percentage of iron oxide). To set up this experimental design properly, it's crucial to ensure:

• Random sampling: Ensure each pottery sample is chosen randomly from each site to avoid bias.
• Control of variables: Keep other variables affecting the outcome constant for fair comparison.
• Replication: Have multiple samples from each site to ensure accuracy and reliability of results.

These steps help in obtaining reliable analysis and conclusions when using ANOVA.

Diagnostic Plots

Diagnostic plots are essential tools in ANOVA to verify that the data meet the necessary assumptions for valid results. These assumptions are visualized using different types of plots: - **Histogram:** Used to check the distribution of the residuals generated by the ANOVA analysis. Ideally, the histogram will display a symmetric bell-shaped curve, indicating that the data are approximately normally distributed. - **Q-Q Plot:** This quantile-quantile plot compares the quantiles of the residuals to the quantiles of a normal distribution. If the points lie along the straight line, it suggests that the residuals are approximately normal. - **Residuals vs. Fitted Values Plot:** This plot shows the residuals on the y-axis and the fitted values on the x-axis. A random scatter of points without a discernible pattern suggests homoscedasticity, or equal variance among the groups. Generating and analyzing these plots can help in identifying whether any assumptions are not met. If the assumptions are violated, it may impact the ANOVA results, necessitating data transformation or alternative statistical methods.

ANOVA Assumptions

Analyzing variance in data using ANOVA relies on three key assumptions. Understanding and checking these ensures that the results of the ANOVA are valid and meaningful. 1. **Normality:** This assumption implies that the data in each group should be normally distributed. While ANOVA is robust to deviations from normality if the sample sizes are equal or large, significant deviations can skew results. 2. **Homoscedasticity:** Also known as the assumption of equal variances, it states that the variance within each of the groups being compared should be roughly equal. This can be checked using tests like Levene's test, and visually through diagnostic plots. 3. **Independence:** Each observation should be independent of others. This primarily depends on the study design—proper randomization and careful planning in collecting data ensure independence. If these assumptions are not met, it might be necessary to transform the data or use more robust statistical methods that can handle differences in variance or normality issues. These steps are critical before making any significant conclusions from the ANOVA test.