Question

For the two-way interaction model, \((9.5 .15)\), show that the following decomposition of sums of squares is true: $$ \begin{aligned} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{\ldots}\right)^{2}=& b c \sum_{i=1}^{a}\left(\bar{X}_{i . .}-\bar{X}_{\ldots .}\right)^{2}+a c \sum_{j=1}^{b}\left(\bar{X}_{. j .}-\bar{X}_{\ldots}\right)^{2} \\ &+c \sum_{i=1}^{a} \sum_{j=1}^{b}\left(\bar{X}_{i j .}-\bar{X}_{i . .}-\bar{X}_{. j .}+\bar{X}_{\ldots}\right)^{2} \\ &+\sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{i j .}\right)^{2} \end{aligned} $$ that is, the total sum of squares is decomposed into that due to row differences, that due to column differences, that due to interaction, and that within cells.

Short answer

The total variance (the sum of squares) can be decomposed into the sum of the variance due to row differences, variance due to column differences, variance due to the interaction of rows and columns, and the variance within cells. This is proofed by substituting sums with definitions (mean across rows, columns, individual observations) and simplifying using the principles of algebra.

Step-by-step solution

Express each term

First, let's translate each of the terms into English to better understand what they represent. The term \( \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{\ldots}\right)^{2} \) is the total variance in the data. \( bc \sum_{i=1}^{a}\left(\bar{X}_{i . .}-\bar{X}_{\ldots .}\right)^{2} \) represents the variance attributed to differences in the rows. \( ac \sum_{j=1}^{b}\left(\bar{X}_{. j .}-\bar{X}_{\ldots}\right)^{2} \) represents the variance attributed to differences in columns. \( c \sum_{i=1}^{a} \sum_{j=1}^{b}\left(\bar{X}_{i j .}-\bar{X}_{i . .}-\bar{X}_{. j .}+\bar{X}_{\ldots}\right)^{2} \) is the interaction term. This measures the difference when changing both row and column at the same time. \( \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{i j .}\right)^{2} \) represents the variance that can't be explained by any of the above, what is often referred to as 'error' or 'residual' variance.

Understand the components

Following the translation, the goal is to understand that total variance can be thought of as variance caused by cells, rows, columns and the interaction effect of rows and columns together. This is a fundamental concept of statistics, particularly in analysis of variance (ANOVA).

Prove through algebra

The actual proof of this formula involves a lot of algebra and unpacking the notation. Keep in mind that the bar notation \(\bar{X}\) represents the mean, and that each of the subscript indices i, j, k represent the rows, columns, and individual observations in those cells, respectively. Therefore, to tackle this we would need to go through each term one at a time and explain where they originate from through the principles of algebra.

Applying these principles for each term

To break it down, start by expanding the formula, replacing each sum with the definition of what it represents (mean across rows, columns, individual observations), and simplifying. Continue until all the terms have been simplified and the sum of squares decomposition has been proven. Remember that the i, j, and k in the sums and the X values are indexes for the rows, columns and individual observations in those cells, respectively. This will help you understand the algebra.

Key concepts

ANOVA

Analysis of Variance (ANOVA) is a statistical method used to compare means across several groups to determine if at least one of the means is statistically different from the others.

Imagine you are an agronomist researching three different types of fertilizer and their effect on plant growth. You've applied each fertilizer to a distinct set of plants and observed their growth over time. To ascertain whether the growth variations are due to the fertilizers or just random chance, you use ANOVA.

In the context of our exercise, a two-way ANOVA is used, which evaluates the effects of two independent categorical variables (factors) and the interaction between them on a continuous outcome.

• If our fertilizers are applied under different temperatures, ANOVA can tell us not only if different fertilizers have different impacts, but also if temperature matters, and crucially, if the combination of a specific fertilizer and temperature is significant as well.

Sum of Squares

Sum of Squares is a key concept used in statistical analysis to measure the variation within a set of observations.

Think of it as a way to quantify the total variation, or 'spread' of data points around their mean. In our fertilization example, some plants may grow more than others, but by calculating the sum of squares, we can determine just how much the growth differs from the average growth across all plants.

The equation we're examining decomposes the total sum of squares into several components that each represent a certain source of variance (rows, columns, interaction, and error).

• By breaking down the total variance in this way, we learn not just the total variability but also how much each factor contributes to the overall differences in plant growth.

Statistical Interaction

Statistical interaction occurs when the effect of one variable on the outcome depends on the level of another variable.

In simpler terms, if our two variables are 'type of fertilizer' and 'temperature', an interaction would suggest that the effect of the fertilizer varies depending on what the temperature is — the two factors don't work independently when impacting plant growth.

This component of variance is crucial to two-way ANOVA and adds complexity to our understanding, as real-world phenomena often involve such interactions that must be considered for accurate analyses.

• Identifying an interaction in the two-way ANOVA leads to more nuanced conclusions about our data. It can also inform decisions, like identifying the optimal combination of fertilizer and temperature for plant growth.

Variance Decomposition

Variance decomposition is the process of breaking down the total variance observed in the data into its constituent parts.

Returning to our fertilizer scenario, this means that instead of just knowing that there's variation in plant growth, we can determine how much of that variation is due to different fertilizers, different temperatures, their interaction, or simply unexplained randomness (error).

The beauty of this process is that it helps us understand the underlying structure of the data and the relative importance of factors and their interaction.

• This insights gained are critical for improving future experiments — perhaps by controlling for certain factors — and for making informed decisions based on the data.