Question

The partially completed ANOVA table for a randomized block design is presented here: $$ \begin{array}{lcl} \text { Source } & d f & \text { SS } & \text { MS } \quad F \\ \hline \text { Treatments } & 4 & 14.2 & \\ \text { Blocks } & & 18.9 & \\ \text { Error } & 24 & & \\ \hline \text { Total } & 34 & 41.9 & \end{array} $$ a. How many blocks are involved in the design? b. How many observations are in each treatment total? c. How many observations are in each block total? d. Fill in the blanks in the ANOVA table. e. Do the data present sufficient evidence to indicate differences among the treatment means? Test using \(\alpha=.05\) f. Do the data present sufficient evidence to indicate differences among the block means? Test using \(\alpha=.05\)

Short answer

a. There are 7 blocks involved in the design. b. There are 7 observations in each treatment total. c. There are 5 observations in each block total. d. The completed ANOVA table is: $$ \begin{array}{lcl} \text { Source } & d f & \text { SS } & \text { MS } & F \\\ \hline \text { Treatments } & 4 & 14.2 & 3.55 & 9.68\\\ \text { Blocks } & 6 & 18.9 & 3.15 & 8.59\\\ \text { Error } & 24 & 8.8 & 0.367 & \\\ \hline \text { Total } & 34 & 41.9 & & \end{array} $$ e. The data present sufficient evidence to indicate differences among the treatment means. f. The data present sufficient evidence to indicate differences among the block means.

Step-by-step solution

a. How many blocks are involved in the design?

The df (degrees of freedom) for the blocks is missing in the table. You can find the total df (34) by adding the dfs for treatments, blocks, and error. As the treatments have a df of 4, then it must be that the df for the error is 24, so the df for the blocks must be 6: \(34 = 4 + 6 + 24\) So, there are 7 blocks involved in the design (since df for blocks = number of blocks -1).

b. How many observations are in each treatment total?

To find the number of total observations in each treatment, first, note that there are 35 total observations (since df_total = 34, so total observations = df_total + 1). The design has 7 blocks (as determined in step a) and 5 treatments (since df_treatments = 4, so total treatments = df_treatments + 1). Therefore, the number of total observations in each treatment is: \(\frac{35}{5}=7\) There are 7 observations in each treatment total.

c. How many observations are in each block total?

To find the number of total observations in each block, divide the total number of observations by the total number of blocks: \(\frac{35}{7}=5\) There are 5 observations in each block total.

d. Fill in the blanks in the ANOVA table.

To fill in the blanks in the ANOVA table, first note that we know the df for treatments, blocks, and the total. Using the sum of squares (SS), we can calculate the mean squares (MS) for treatments and blocks by dividing the SS by their respective dfs: MS_Treatments = \(\frac{14.2}{4} = 3.55\) MS_Blocks = \(\frac{18.9}{6} = 3.15\) To find the SS for errors, we can subtract the SS_Treatments and SS_Blocks from SS_Total: SS_Error = \(41.9 - 14.2 - 18.9 = 8.8\) Now, we can calculate the MS for errors: MS_Error = \(\frac{8.8}{24} = 0.367\) Now, we can calculate the F values for treatments and blocks: F_Treatments = \(\frac{3.55}{0.367} = 9.68\) F_Blocks = \(\frac{3.15}{0.367} = 8.59\) The completed ANOVA table is: $$ \begin{array}{lcl} \text { Source } & d f & \text { SS } & \text { MS } & F \\\ \hline \text { Treatments } & 4 & 14.2 & 3.55 & 9.68\\\ \text { Blocks } & 6 & 18.9 & 3.15 & 8.59\\\ \text { Error } & 24 & 8.8 & 0.367 & \\\ \hline \text { Total } & 34 & 41.9 & & \end{array} $$

e. Test for differences among treatment means.

Using the F_Treatments value (9.68), we perform an F-test to determine if there are differences among the treatment means at the given significance level (\(\alpha=.05\)). We compare the calculated F-value with the critical F-value, F(4, 24). Using a F-distribution table, the critical F-value is found to be 2.76. Since 9.68 > 2.76, we reject the null hypothesis that there are no differences among the treatment means. The data present sufficient evidence to indicate differences among the treatment means.

f. Test for differences among block means.

Using the F_Blocks value (8.59), we perform an F-test to determine if there are differences among the block means at the given significance level (\(\alpha=.05\)). We compare the calculated F-value with the critical F-value, F(6, 24). Using a F-distribution table, the critical F-value is found to be 2.51. Since 8.59 > 2.51, we reject the null hypothesis that there are no differences among the block means. The data present sufficient evidence to indicate differences among the block means.

Key concepts

Degrees of Freedom

In a statistical context, degrees of freedom (df) represent the number of independent values or quantities which can be assigned to a statistical distribution. In simpler terms, degrees of freedom in an analysis refer to the number of values within a calculation that are free to vary.

For example, when considering an ANOVA (Analysis of Variance) randomized block design, degrees of freedom are crucial for calculating various components of the ANOVA table, such as the error term. They also affect the shape of various statistical distributions which are used to estimate probability and critical values in hypothesis testing. In the provided exercise, the degrees of freedom for treatments, blocks, and error were calculated to fill in the missing pieces of the ANOVA table. It's important to remember that for treatments and blocks, the degrees of freedom are one less than the number of treatments and blocks since one value is constrained by the need for the total to remain constant.

Sum of Squares

The sum of squares (SS) is a measure of the total variability within a dataset. It's the sum of the squared differences from the mean, and in the context of ANOVA, it helps us understand how much variation there is within treatment groups and within blocks.

The total SS is partitioned into the treatment SS, block SS, and error SS. Understanding how this sum of squares is calculated and how it is partitioned is critical when interpreting ANOVA results. For instance, the treatment sum of squares measures variability due to differences between treatment levels, whereas the block sum of squares captures variability caused by differences in blocks. The error sum of squares represents the variation within treatment groups that cannot be explained by treatment effects or block effects.

F-test

The F-test in ANOVA is used to calculate statistical significance in experiments with two or more group means. It compares the model with no predictors to the model with predictors, essentially testing whether the group means are drawn from populations with the same mean.

The test is based on an F-distribution and requires two sets of degrees of freedom (df1 for the between-group variability, and df2 for the within-group variability). An F-value is calculated by dividing the mean square due to treatments (or blocks) by the mean square due to error. If the F-value is higher than a critical value from the F-distribution, we conclude there is a statistically significant difference between means. The exercise provided demonstrates how to calculate F-values and compare them against critical values to make a statistical decision.

Null Hypothesis

The null hypothesis is a fundamental concept in hypothesis testing. It generally represents the statement of no effect or no difference and serves as the baseline assumption that the observed effects are due to random variation.

In ANOVA, the null hypothesis typically states that all group means are equal, implying that any observed differences are simply due to random chance. Rejecting the null hypothesis, which occurs when the calculated F-value is greater than the critical value from the F-distribution, suggests there is evidence of significant effects or differences. Conversely, failing to reject the null hypothesis suggests that any differences in group means are not statistically significant. The exercise example required the rejection of the null hypothesis based on the F-test results for both treatments and blocks.

Mean Squares

The mean squares (MS) in an ANOVA table represent the average of sums of squares divided by their corresponding degrees of freedom.

It's effectively the variance (SS divided by df) associated with each source of variation in the model. There are usually two mean squares calculated in ANOVA: the mean square due to treatments (MS treatments) and the mean square due to error (MS error). The MS for treatments quantifies the average variability between the groups, while the MS for error represents the average variability within the groups. To find the F-statistic used for the F-test, the MS for treatments is divided by the MS for error. The exercise demonstrated the calculations required to determine the mean squares and emphasized their role in evaluating the differences by treatments and blocks.