10 - One and Two Sample Non-Parametric Tests Flashcards
(16 cards)
What is a non-parametric test?
A hypothesis test that doesnβt require knowledge or assumption that the data is normally distributed.
What is the Sign Test used for?
One Sample: To test if the median of a population takes a specific value.
Two Samples: To test for preferences or differences between two populations (paired data).
What distribution is used in conjunction with the Sign Test? What is the p value?
Binomial Model with p = 0.5
What is a Paired Sign Test?
A sign test used for paired values to test whether there is a difference in the medians of two populations.
What is the Wilcoxon Signed-Rank Test?
A non-parametric test that considers the SIZE of the differences between data and the mean/median, making it more accurate than the Sign Test.
What are the assumptions for the Wilcoxon Signed-Rank Test?
Data are paired and come from the same population.
Each pair is chosen randomly and independently.
Data follows a symmetrical distribution.
How do you calculate the test statistic in a Wilcoxon Signed-Rank Test?
Find the difference between each value and the mean/median.
Rank the absolute values of the differences (ignoring zeros), using tied ranks where necessary.
Sum the ranks of the positive differences (T+).
Sum the ranks of the negative differences (T-).
Choose T+ or T- depending on the inequality, or the smallest total for a two-tailed test.
What is a Paired Wilcoxon Signed-Rank Test?
A Wilcoxon Signed-Rank Test used for paired values to test whether there is a difference in the mean/medians of two populations.
When do you use the Wilcoxon Rank-Sum Test (Mann-Whitney U Test)?
When you have unpaired data and cannot assume the data is normally distributed, and you want to test for differences between the populations from which two samples are taken.
How do you calculate the test statistic (U) for the Wilcoxon Rank-Sum Test?
Rank all the data together.
Calculate T for each sample (sum of ranks).
Calculate π = π β (π(π + 1))/2 for each sample.
The test statistic is the smaller of the two U values.
How can you check your calculated U values for the Wilcoxon Rank-Sum Test?
ππ΄ + ππ΅ = π Γ π (where m and n are the sample sizes of each group)
Hypotheses for a Wilcoxon test
Null Hypothesis (H0): Always state your null hypothesis clearly. For example, in a sign test, H0 might be: βThe median is equal to [specific value].β In a Wilcoxon test, it might be: βThere is no difference in the medians/means of the two populations.β
Alternative Hypothesis (H1): Make sure your alternative hypothesis reflects the question youβre trying to answer. Is it one-tailed (greater than or less than) or two-tailed (different from)? This will determine how you interpret your results and find the critical value.
Where to find the critical values and how to make your decision at the end of the test
Critical Values and Decision Rule:
Using Tables: The PDF mentions using tables in the formula booklet to find critical values. Make sure you know how to use these tables correctly, paying attention to the significance level, whether the test is one-tailed or two-tailed, and the sample size(s).
Decision Rule: The decision rule is based on comparing your test statistic to the critical value.
If the test statistic falls within the critical region (beyond the critical value), you reject the null hypothesis.
If the test statistic does not fall within the critical region, you fail to reject the null hypothesis. (Important: βFail to rejectβ is not the same as βaccept.β)
Types of Errors:
Type I Error (False Positive): Rejecting the null hypothesis when it is true. The probability of a Type I error is equal to the significance level (Ξ±).
Type II Error (False Negative): Failing to reject the null hypothesis when it is false.
When to use each Wilcoxon and signed test
Sign Test: Use when you have very little information about the data and need a quick and simple test. Itβs good for paired data or testing a single median.
Wilcoxon Signed-Rank Test: Use when you have paired data and want to take into account the magnitude of the differences. Itβs more powerful than the sign test when its assumptions are met (especially symmetry).
Wilcoxon Rank-Sum Test (Mann-Whitney U Test): Use when you have two independent samples (unpaired data) and cannot assume normality.
Practical Considerations:
Tied Ranks: Understand how to handle tied ranks in both the Wilcoxon Signed-Rank and Rank-Sum tests.
Zero Differences: In the Wilcoxon Signed-Rank test, you discard zero differences and adjust your sample size accordingly.
