Question

A randomized block design has \(k=3\) treatments, \(b=6\) blocks, with \(S S T=11.4, S S B=17.1\), and Total \(S S=42.7 . \bar{T}_{A}=21.9\) and \(\bar{T}_{B}=24.2 .\) Construct an ANOVA table showing all sums of squares, mean squares, and pertinent \(F\) -values. Then use this information to answer the questions. Find a \(95 \%\) confidence interval for the difference between \(\mu_{\mathrm{A}}\) and \(\mu_{\mathrm{B}}\).

Short answer

Answer: The 95% confidence interval for the difference between treatment A and treatment B is (-4.241, -0.359).

Step-by-step solution

Find \(SSE\)

First, find the sum of squares within the blocks (also called the error sum of squares) by subtracting \(SST\) and \(SSB\) from the total \(SS\). So, \(SSE = SS - SST - SSB = 42.7 - 11.4 - 17.1 = 14.2\).

Calculate Mean Squares

Next, calculate the mean squares for each source of variation by dividing their respective sum of squares by their degrees of freedom. The degrees of freedom for treatments (\(df_T = k-1\)), blocks (\(df_B= b-1\)), and error (\(df_E= (k-1)(b-1)\)). - \(MS_{T} = \frac{SST}{k-1} = \frac{11.4}{3-1} = 5.7\) - \(MS_{B} = \frac{SSB}{b-1} = \frac{ 17.1}{6-1} = 3.42\) - \(MS_{E} = \frac{SSE}{(k-1)(b-1)} = \frac{14.2}{(3-1)(6-1)} = 1.42\)

Calculate \(F\)-value

Calculate the \(F\)-value for treatments by dividing the Mean Square Treatment by Mean Square Error: \(F = \frac{MS_T}{MS_E} = \frac{5.7}{1.42} = 4.01\)

Construct ANOVA Table

Using the calculated values, construct the ANOVA table as follows: | Source | df | SS | MS | F | |-------------|--------------|-------|-------|------| | Treatments | (\(k-1=2\)) | 11.40 | 5.70 | 4.01 | | Blocks | (\(b-1=5\)) | 17.10 | 3.42 | - | | Error | (\((k-1)(b-1)=10\)) | 14.20 | 1.42 | - | | Total | - | 42.70 | - | - |

Calculate the confidence interval for treatment difference

Now that we have the ANOVA table, we can calculate the 95% confidence interval for the difference between \(\mu_A\) and \(\mu_B\). It is given by the formula: $$ \bar{T}_A - \bar{T}_B \pm t_{\frac{\alpha}{2}, df_E} \times \sqrt{ \frac{2(MS_E)}{b} } $$ Using the given values and the calculated \(MS_E\), we get: $$ 21.9 - 24.2 \pm t_{0.025, 10} \times \sqrt{ \frac{2(1.42)}{6} } $$ First, find \(t_{0.025, 10}\) to be \(2.228\) using a \(t\)-distribution table or statistical software. Then, calculate the confidence interval: $$ \begin{aligned} -2.3 \pm 2.228 \times \sqrt{ \frac{2(1.42)}{6} } &= -2.3 \pm 2.228 \times \sqrt{0.472} \\ &= -2.3 \pm 1.941 \\ &= (-4.241, -0.359) \end{aligned} $$ So, the 95% confidence interval for the difference between \(\mu_A\) and \(\mu_B\) is \((-4.241, \, -0.359)\).

Key concepts

Randomized Block Design

A randomized block design is a statistical approach used to reduce the potential for confounding variables—those outside influences that might affect the outcome—in an experiment. Imagine an agricultural study assessing the effect of different fertilizers on plant growth. Factors such as soil quality or sunlight could impact growth, but they are not the main focus. By grouping similar experimental units into blocks (e.g., sections of a field with comparable soil quality), we can isolate the effect of the fertilizers more effectively.

In our exercise, the blocks could represent different batches of subjects or temporal intervals, each with inherent characteristics that might influence the treatment effects. By having treatments applied randomly within each block, the design aims to balance out the potential unknown or unobserved factors across the treatments, ensuring that the differences in response are due mainly to the treatments themselves, rather than external variables.

Sum of Squares

The sum of squares (SS) in an ANOVA table represents the variance (or spread) from different sources of data. There are several types of SS: the total sum of squares (SST), which is the overall variation present in the data; the sum of squares between groups or treatments (SST), indicating how much variation exists between the different treatments; and the sum of squares within groups, also known as the error sum of squares (SSE), representing the variation within each treatment group.

To simplify, think of it as a way to dissect the total variation in the data into portions that can be attributed to the effects being studied (treatments) and the portion which is just due to random variability (error). This segregation helps us understand if our treatments really have different effects, or if what we're observing is just due to chance. In our example, we subtract the sum of squares for treatments (SST) and for blocks (SSB) from the total sum of squares (Total SS) to find the error sum of squares (SSE), which is essential in calculating other statistics like the mean squares and F-values.

Confidence Interval

The confidence interval is a range of values that is likely to contain a population parameter (such as a mean difference between groups) with a certain degree of confidence. A 95% confidence interval, which we are constructing in this exercise, implies that if we repeated our experiment many times, 95 out of 100 of those confidence intervals would contain the true mean difference. It's a way of expressing uncertainty and variability in our measurements.

How is it calculated? In the ANOVA context, we use the error term (Mean Square Error, MSE) and the sample size to estimate the standard error of the difference between means. Then, we use a t-distribution (since we're often dealing with small sample sizes) to find the critical value for our confidence level. The formula incorporates these elements to provide an interval estimate, which gives us a practical range that is likely to include the true difference between the population means. This interval not only gives us an estimate of the difference but also quantifies our uncertainty about the estimate.