Question

The data were collected using a randomized block design. For each data set, use the Friedman \(F\) -test to test for differences in location among the treatment distributions using \(\alpha=.05 .\) Bound the \(p\) -value for the test using Table 5 of Appendix \(I\) and state your conclusions. $$ \begin{array}{crrrr} \hline \quad &&& {\text { Treatment }} \\ \text { Block } & 1 & 2 & 3 & 4 \\ \hline 1 & 89 & 81 & 84 & 85 \\ 2 & 93 & 86 & 86 & 88 \\ 3 & 91 & 85 & 87 & 86 \\ 4 & 85 & 79 & 80 & 82 \\ 5 & 90 & 84 & 85 & 85 \\ 6 & 86 & 78 & 83 & 84 \\ 7 & 87 & 80 & 83 & 82 \\ 8 & 93 & 86 & 88 & 90 \\ \hline \end{array} $$

Short answer

Based on the Friedman test conducted, we found a significant difference in location among the treatment distributions with a test statistic of 20.48 and a critical value of 7.81 at an alpha level of 0.05. Therefore, we reject the null hypothesis and conclude that there is a significant difference among the treatments.

Step-by-step solution

Calculate the rank for each treatment within each block

For each block row, assign a rank to each value, starting with 1 for the lowest value, 2 for the next lowest, and so on. In case of tied values, assign the average rank of the tied positions. The ranks for the given data are: $$ \begin{array}{crrrr} \hline & & & \text { Treatment } \\ \text { Block } & 1 & 2 & 3 & 4 \\ \hline 1 & 4 & 1 & 2.5 & 2.5 \\ 2 & 4 & 1 & 2.5 & 2.5 \\ 3 & 4 & 2 & 3 & 1 \\ 4 & 4 & 1 & 2.5 & 2.5 \\ 5 & 4 & 1 & 2.5 & 2.5 \\ 6 & 4 & 1 & 3 & 2 \\ 7 & 4 & 1 & 3 & 2 \\ 8 & 4 & 1 & 2 & 3 \\ \hline \end{array} $$

Calculate the sum of ranks for each treatment

Next, we sum the ranks for each treatment: $$ \begin{aligned} \text{Sum of ranks for Treatment 1: } & 4+4+4+4+4+4+4+4 = 32 \\ \text{Sum of ranks for Treatment 2: } & 1+1+2+1+1+1+1+1 = 9 \\ \text{Sum of ranks for Treatment 3: } & 2.5+2.5+3+2.5+2.5+3+3+2 = 21 \\ \text{Sum of ranks for Treatment 4: } & 2.5+2.5+1+2.5+2.5+2+2+3 = 18 \end{aligned} $$

Calculate the Friedman test statistic, \(\chi^2_r\)

We will use the following formula to compute the Friedman test statistic: $$ \chi_r^2 = \frac{12N}{(KT)(KT+1)} (\sum_{j=1}^T R_j^2 - \frac{(KT+1)^2}{4}) $$ where \(N\) is the number of blocks, \(K\) is the number of treatments, \(T\) is the number of rows, and \(R_j\) is the sum of ranks for treatment \(j\). \(\chi_r^2\) can be calculated as follows: $$ \chi_r^2 = \frac{12(8)}{(4)(4+1)}((32^2) + (9^2) + (21^2) + (18^2) - \frac{(4(4+1)+1)^2}{4}) = 20.48 $$

Determine the critical value from the table provided

The degrees of freedom for the Friedman test are \(df=T-1\). In this case, since we have 4 treatments, \(df=4-1=3\). With an alpha level of 0.05 and 3 degrees of freedom, the critical value according to Table 5 of Appendix I is 7.81.

Compare \(\chi^2_r\) with the critical value to make a conclusion

Since our calculated Friedman test statistic \(\chi^2_r=20.48\) is greater than the critical value of 7.81, we reject the null hypothesis. This means that there is a significant difference in location among the treatment distributions.

Key concepts

Randomized Block Design

Imagine a researcher who wants to avoid the variability in experiments caused by uncontrollable external factors. That’s where randomized block design comes into play. It's a statistical technique used to keep the experiment as consistent as possible, despite the presence of these external variables, commonly known as 'blocks'.

A block is a set of experimental units that are similar in some way that is expected to affect the response to the treatments. By randomizing within blocks, the experiment takes care of the variability within the blocks but not between them. For example, if we're testing the effectiveness of a new fertilizer and we know the soil quality could affect the outcome, each unique soil type would represent a different block.

In the exercise provided, the 'blocks' are the different sets of data collected under similar conditions, while the 'treatments' represent the variable being tested across those conditions. Performing tests within these blocks helps reduce the impact of variances and therefore makes the experiment more reliable.

Statistical Hypothesis Testing

How do we make decisions based on sample data? This is where statistical hypothesis testing comes into the picture. It's a framework for testing claims about a population based on the evidence in a sample. Two hypotheses are set up: the null hypothesis, which usually states there is no effect or no difference, and the alternative hypothesis, which contradicts the null.

The basic idea is to find out if the sample data is so far from what we'd expect if the null hypothesis were true, that we can declare a significant difference and thus support the alternative hypothesis. The key concept here is the p-value, the probability of observing a value as extreme as the one measured, assuming that the null hypothesis is true. If this p-value is lower than a pre-determined threshold, typically 0.05 or 5%, we reject the null hypothesis.

In the exercise, the comparison of the Friedman test statistic to the critical value is analogous to comparing the p-value to the significance level. The result suggesting a 'significant difference' is the equivalent of saying the p-value is small enough to reject the null hypothesis, indicating the treatments have different effects.

Non-parametric Statistics

When the assumptions required for traditional parametric tests are not satisfied, non-parametric statistics offer a powerful alternative. They are also known as distribution-free tests because they do not assume your data follows a specific distribution. This is particularly useful when dealing with small sample sizes or data that is not normally distributed.

The Friedman test is one such non-parametric method used for comparing three or more paired groups. It is especially beneficial when your data is ordinal or the assumptions of similar tests like repeated measures ANOVA are not met. Instead of focusing on means or variances, it works on ranks of the data.

The exercise demonstrates the step-by-step calculation for the Friedman test, showcasing the rank-based approach integral to non-parametric statistics. By using ranks, the Friedman test minimizes the impact of outliers and skewed data distributions, providing a robust assessment of the treatment effects without relying on normal distribution assumptions.