Friedman Test
In the landscape of Lean Six Sigma, a methodology aimed at improving business processes by reducing variability and defects, the importance of data analysis and hypothesis testing cannot be overstated. Within this domain, when dealing with normal data, there's a rich arsenal of statistical tests available. Among them, the Friedman test stands out, especially when addressing specific types of comparative analysis scenarios. This article delves into the Friedman test, particularly under the subtopic of Parametric Tests for Normal Data within the broader context of Hypothesis Testing.
Introduction to the Friedman Test
The Friedman test is a non-parametric statistical test, which might seem a bit contradictory given the mention of "Parametric Tests for Normal Data." However, it's crucial to clarify that while the Friedman test itself is non-parametric, it addresses scenarios where traditional parametric tests (like ANOVA) are not suitable due to the data not meeting all the parametric assumptions, yet the data under consideration originates from a process or distribution that is fundamentally normal. This makes the Friedman test a valuable tool in the Lean Six Sigma toolkit for analyzing ranked data derived from normal processes.
Purpose of the Friedman Test
The primary purpose of the Friedman test is to compare three or more paired groups. It is especially useful in cases where the data does not meet the assumptions necessary for parametric tests, such as the repeated measures ANOVA. The test is applied to ranked data rather than direct measurements, where the ranking occurs within each block or subject. This makes it particularly well-suited for analyzing data from repeated measures designs or matched-subjects experiments where the normality assumption may not hold.
How the Friedman Test Works
The Friedman test operates by ranking the data across each block (e.g., subjects or experiments) for all groups. Essentially, for each block, the observations across different conditions or treatments are ranked (1, 2, 3, etc.), and these ranks replace the original data values. The test then evaluates the differences in ranks across the groups to determine if there are statistically significant differences among them.
The rationale behind using ranks rather than actual data values is that it mitigates the impact of outliers and non-normal distribution shapes, focusing the analysis on the relative performance of groups across the blocks rather than their absolute measures.
Steps in Conducting the Friedman Test
Rank the Data: Within each block or subject, rank the observations from each group. If there are ties, assign the average rank for the tied values.
Calculate Test Statistics: Sum the ranks for each group across all blocks and compute the Friedman test statistic based on these rank sums. This statistic is essentially a measure of the variance in rank sums across the groups.
Determine Significance: Compare the test statistic against the critical value from the Friedman distribution or compute a p-value to assess the statistical significance. If the test statistic is beyond the critical value, or the p-value is below a predetermined significance level (commonly 0.05), the null hypothesis of no differences among the groups is rejected.
Applications in Lean Six Sigma
In Lean Six Sigma projects, the Friedman test can be particularly useful for analyzing outcomes of process improvements across multiple groups or conditions without the strict requirements for normality and homogeneity of variance. For instance, if a company wants to evaluate the effectiveness of a new training program across different departments, the Friedman test can compare the improvement scores of employees while accounting for the intrinsic differences between departments.
Conclusion
The Friedman test offers a robust and flexible approach for comparing multiple groups within Lean Six Sigma initiatives, particularly when dealing with ranked data and the assumptions for parametric tests are not met. By focusing on the ranks rather than the raw data, it provides a powerful tool for identifying significant differences across groups, thereby facilitating data-driven decisions in process improvement efforts. While it may initially seem outside the purview of "Parametric Tests for Normal Data," its application in analyzing data from fundamentally normal processes underscores its value in the comprehensive toolkit of hypothesis testing methods within Lean Six Sigma.
Friedman test Scenario
A company has launched a new training program intended to improve the productivity of its employees. To evaluate the effectiveness of this program, productivity scores (the higher, the better) are recorded for a sample of 4 employees before the training, immediately after the training, and then one month after the training. Due to various constraints, a parametric test is not suitable, and the Friedman test is chosen to analyze the data.
Data
The productivity scores for each employee (A, B, C, D) at the three different times (Before, After, One Month After) are as follows:
Rank the Data: First, rank the productivity scores within each employee. If there are ties, assign the average rank for those values. However, in our scenario, there will be no ties.
Sum of Ranks:
Before: 1+1+1+1=4
After: 2+2+2+2=8
One Month After: 3+3+3+3=12
The Friedman test statistic (Q) is calculated using the formula:
where:
N is the number of subjects (employees) = 4,
k is the number of conditions (time periods) = 3,
Rj is the sum of ranks for each condition.
Q=8
Determining Significance with the Critical Value
With k=3 conditions (time periods), N=4 subjects (employees), and using a significance level of 0.05, the critical value is 6.5.
Since our calculated Q value of 8.0 is greater than the critical value of 6.5, we can conclude that there is a statistically significant difference in the productivity scores across the three measured time periods (Before, After, One Month After). This means that we reject the null hypothesis, which posited that there are no differences among the groups.
This result suggests that the training program had a statistically significant impact on employee productivity over time, highlighting its effectiveness. With this information, the company can confidently assert that the changes observed in productivity scores are not due to random chance but are likely a result of the implemented training program.