I have some questions are needed to be answered. The questions are created based
ID: 3070775 • Letter: I
Question
I have some questions are needed to be answered. The questions are created based on Essential Statistics for public managers and policy analysts ch 8 Contingency Tables, 9 Getting Results, 10 Hypothesis Testing with Chi-Square. (Berman & Wang).
What does it mean to say that a test result is statistically significant?
When is it appropriate to use each of the following tests:
Kendall's tau-b
Kendall's tau-c
Somers' d
Give an example in your area of work or interest where each of the following tests can be used:
Kendall's tau-b
Kendall's tau-c
Somers' d
Explanation / Answer
Let me answer your question one by one
Q. What does it mean to say that a test result is statistically significant?
A. Statistical significance is the likelihood that the difference in conversion rates between a given variation and the baseline is not due to random chance.
A result of an experiment is said to have statistical significance, or be statistically significant, if it is likely not caused by chance for a given statistical significance level.
Your statistical significance level reflects your risk tolerance and confidence level. For example, if you run an A/B testing experiment with a significance level of 95%, this means that if you determine a winner, you can be 95% confident that the observed results are real and not an error caused by randomness. It also means that there is a 5% chance that you could be wrong.
It is a way of mathematically proving that a certain statistic is reliable. When you make decisions based on the results of experiments that you’re running, you will want to make sure that a relationship actually exists.
Online web owners, marketers, and advertisers have recently become interested in making sure their a/b test experiments (eg. conversion rate a/b testing, ad copy changes, email subject line tweaks) get statistical significance before jumping to conclusions.
Your statistical significance level reflects your risk tolerance and confidence level. For example, if you run an A/B testing experiment with a significance level of 95%, this means that if you determine a winner, you can be 95% confident that the observed results are real and not an error caused by randomness. It also means that there is a 5% chance that you could be wrong.
For example: Statistical significance is important because it gives you confidence that the changes you make to your website or app actually have a positive impact on your conversion rate and other metrics. Your metrics and numbers can fluctuate wildly from day to day, and statistical analysis provides a sound mathematical foundation for making business decisions and eliminating false positives.
Q. When is it appropriate to use each of the following tests:
Kendall's tau-b, Kendall's tau-c , Somers' d
A. When there are two categorical variables that are both naturally ordered, a variety of effect size measures have been proposed for such ordinal data i.e Kendall's tau-b, Kendall's tau-c, and Somers' d.
The difference among these measures lies in the power of overcoming of ties.
Kendall's tau-b ( tb ) uses a correction for ties. The rule of both variables lie on an ordinal scale for calculation Tau-b is just the same as gamma coefficient. Tau-b has also the range -1 tb 1.
The test statistics is used to test whether the degree of association of the cross tabulations when variables are measured in ordinal scale is significant. It adjusts the ties and is most appropriate for square tables what means that the number of row categories equals to the number of column categories.
Value of 1 is 100% negative association or perfect inversion whereas value of +1 is 100% positive association, or perfect agreement. A value of zero indicates no association.
If tb = ±1 then there is no ties and subjects from different cells form strict concordant and discordant pairs in these two extreme cases. When both tb = ±1 and g = ±1, it is generally concluded that tb is stronger than g . If tb = 1, then the table is diagonal and if tb = 1, the table is skewed diagonal .
Stuart's tau-c ( tc ) makes an adjustment for table size as well as a correction for ties. Tau-c is also appropriate only when both variables lie on an ordinal scale. Tau-c has the range -1 tc 1
Kendall's tau-c, also called Stuart's tau-c or Kendall-Stuart tau-c, is a special case of tau-b for larger tables. It also makes adjustments for the size of the cross table
Somers’ d(C|R) and Somers' d(R|C) are asymmetric modifications of tau-b. C|R represents that the row variable X is treated as an independent variable, whereas the column variable Y is treated as dependent. Similarly, R|C represents the reverse interpretation. Somers'd differ from tau-b in that it only makes a correction for tied pairs on the independent variable. Somers’ d can be calculated only when both variables are ordered. It varies in the range -1 d 1.
Somers’ d value of 1 is 100% negative association or perfect inversion whereas value of +1 is 100% positive association, or perfect agreement . A value of zero indicates no association. Under the null hypothesis of independence, the Somers’ d value asymptotically has standard normal distribution.
As table dimension increases, Kendall’s Tau-b, Kendall’s Tau-c and Somers’ d for such a fair and large sample present a slight different estimate of the actual degree of association. They are all underestimating the actual degree of ordinal measure of association. Meanwhile for large samples Kendall’s Tau-c is the worst no matter how large the table dimension is.
Thus, For tables in which both rows and columns contain ordered values, select Gamma (zero-order for 2-way tables and conditional for 3-way to 10-way tables), Kendall's tau-b, and Kendall's tau-c. For predicting column categories from row categories, select Somers' d.