Problem 35 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose that \(k\) treatments are to be applied within each of \(b\) blocks. Let \(\bar{r}_{\text {.. denote the average of the }} b k\) ranks and let \(\bar{r}_{. j}=(1 / b) r_{. j} .\) Show that the Friedman statistic given in Theorem 14.5.1 can also be written $$ g=\frac{12 b}{k(k+1)} \sum_{j=1}^{k}\left(\bar{r}_{. j}-\bar{r}_{. .}\right)^{2} $$ What analysis of variance expression does this resemble?

Step-by-Step Solution

Answer

Friedman's statistic, which is used in Friedman's test, can be rewritten in the given alternative form using algebraic manipulation and substitution. Its resemblance to an expression in ANOVA is in its similarity with the formulation of the Sum of Squares, where instead of the raw data values and their mean, ranks and the average rank are used. The sum of these squared differences has a multiplier applied to it.

Step 1: Understand the Given Information and Expressions

The given information pertains to Friedman's test. It explains that we have \(k\) treatments to be applied within each of \(b\) blocks. The formula provided, \( \bar{r}_{. j}=(1 / b) r_{. j} \), means that the mean rank of the 'jth' treatment is gotten by summing the ranks of the 'jth' treatment in each block (represented by \( r_{. j} \)) and then dividing by the number of blocks (\(b\)). The objective is to show that the Friedman's statistic can also be expressed as given in the problem statement.

Step 2: Express the Given Formula in an Alternative Form

Friedman's statistic is given by: \( F=\frac{12 b}{k(k+1)} \sum_{j=1}^{k}\left(r_{. j}- \frac{b (k+1)}{2}\right)^2 \)Replacing \( r_{. j}= b \bar{r}_{. j} \), the above becomes \( F=\frac{12 b}{k(k+1)} \sum_{j=1}^{k}\left(b \bar{r}_{. j}- \frac{b (k+1)}{2}\right)^2 \)Simplifying, we get \( F=\frac{12 b}{k(k+1)} \sum_{j=1}^{k}\left(\bar{r}_{. j}- \frac{(k+1)}{2}\right)^2 \)where \( \frac{(k+1)}{2} \) is the average of the all ranks, denoted by \( \bar{r}_{..} \). So, Friedman's statistic can be written as: \( F=\frac{12 b}{k(k+1)} \sum_{j=1}^{k}\left(\bar{r}_{. j}- \bar{r}_{..}\right)^2 \)

Step 3: Identify the ANOVA Expression it Resembles

ANOVA involves calculating the Sum of Squares (SS) which is an expression taking the form \( SS = \sum (X - \bar{X})^2 \). As you can observe, the second expression in the formula for the Friedman statistic closely resembles this Sum of Squares, where instead of having the \( X \) values we use the rank \(\bar{r}_{.j}\) and instead of \(\bar{X}\) we have the average rank \(\bar{r}_{..}\), and the sum of squares has a multiplier applied to it.

Key Concepts

Analysis of Variance (ANOVA)

The Analysis of Variance, commonly known as ANOVA, is a statistical method used to analyze differences among means of different groups. Essentially, it helps us to understand whether the means of different groups are the same or significantly different. This is crucial when comparing data across multiple groups or treatments. ANOVA works by looking at two types of variances:

• Variance within each group (which indicates how spread out the measurements are within a group).
• Variance between the groups (which reflects differences between the group means).

By comparing these variances, ANOVA provides a measure indicating whether any of these variances are larger than what might be expected due to random noise. In simple terms, ANOVA helps identify whether variations in data are due to an external factor or purely by chance.

Ranks

Ranks in statistical analysis are essentially a way of indicating the relative position of a data point within a dataset. In Friedman's test, ranks help in understanding and analyzing data in a non-parametric fashion, that is, without assuming a normal distribution. Suppose we have several groups and we assign each data point a rank based on its value. This process assigns the smallest value in a dataset a rank of one, the second smallest two, and so forth. In the context of Friedman’s test, ranked data is used to evaluate differences between groups or treatments over blocks, which allows for a comparison that accounts for variability within blocks. This method is advantageous as it deals with ordinal data seamlessly and is less affected by outliers when compared to analyzing raw score means.

Mean Rank

Mean Rank is the average of the ranks assigned to each set of observations. When performing Friedman's test, Mean Rank is crucial as it summarizes the overall ranking for a given treatment across different blocks. The formula for the mean rank of a treatment is given by \(\bar{r}_{. j} = \frac{1}{b} r_{. j}\)Here, \( r_{. j} \) signifies the total of ranks assigned to a particular treatment across different blocks, and \( b \) denotes the number of blocks. By computing the Mean Rank, we are able to evaluate if certain treatments consistently perform either better or worse than others, which is informative in applications such as clinical trials or agricultural studies. Mean Ranks provide an objective comparison across various treatments and help in identifying any significant patterns or trends.

Sum of Squares

The term 'Sum of Squares' refers to the sum of the squared deviations of data points from their mean. It is used widely in statistical analyses, particularly in ANOVA. In Friedman's test, the Sum of Squares is used to illustrate how treatment effects lead to rank differences across blocks. The formula to calculate the Sum of Squares for Friedman's test is as follows:\[SS = \sum_{j=1}^{k} \left( \bar{r}_{. j} - \bar{r}_{..} \right)^2\]In this equation, \( \bar{r}_{. j} \) is the mean rank of a specific treatment and \( \bar{r}_{..} \) is the overall mean rank. The squared terms help in emphasizing larger discrepancies from the mean, providing a comprehensive way to assess the variation or spread of ranks. This information is then scaled by certain constants (as shown in Friedman's statistic) to facilitate further analysis and determine statistical significance.