1 / 42

570 likes | 1.28k Vues

Non-Parametric Tests. Non Parametric Tests. Do not make as many assumptions about the distribution of the data as the t test. Do not require data to be Normal Good for data with outliers Non-parametric tests based on ranks of the data

Télécharger la présentation
## Non-Parametric Tests

**An Image/Link below is provided (as is) to download presentation**
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.
Content is provided to you AS IS for your information and personal use only.
Download presentation by click this link.
While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

**Non Parametric Tests**• Do not make as many assumptions about the distribution of the data as the t test. • Do not require data to be Normal • Good for data with outliers • Non-parametric tests based on ranks of the data • Work well for ordinal data (data that have a defined order, but for which averages may not make sense).**We’ll cover three non-parametric tests:***The Kruskal-Wallis Test won’t be discussed further, but explanation can be found in Rosner §12.7**Paired data example: body image**Children in an orthodontia study were asked to rate how they felt about their teeth on a 5 point scale. Survey administered before and after treatment. How do you feel about your teeth? • Wish I could change them • Don’t like, but can put up with them • No particular feelings one way or the other • I am satisfied with them • Consider myself fortunate in this area**Paired data example: body image**These data are • Ordinal They have a definite order, but averages may not have a clear interpretation. • Paired Two observations (before and after treatment) are made on each child. How do you feel about your teeth? • Wish I could change them • Don’t like, but can put up with them • No particular feelings one way or the other • I am satisfied with them • Consider myself fortunate in this area**Sign Test**• Used for paired data • Can be ordinal or continuous • Very simple and easy to interpret • Makes no assumptions about distribution of the data • Not very powerful**Sign Test: null hypothesis**• The null hypothesis for the sign test is • To evaluate H0 we only need to know the signs of the differences • If half the differences are positive and half are negative, then the median = 0 (H0 is true). • If the signs are more unbalanced, then that is evidence against H0. H0: the median difference is zero**Example: Body image data**• Use the sign test to evaluate whether these data provide evidence that ortho tx improves children’s image of their teeth.**Example: Body image data**• Use the sign test to evaluate whether these data provide evidence that ortho tx improves children’s image of their teeth. • First, for each child, compute the diffference between the two ratings**Example: Body image data**• The sign test looks at the signs of the differences**Example: Body image data**• The sign test looks at the signs of the differences • 15 children felt better about their teeth (+ difference in ratings)**Example: Body image data**• The sign test looks at the signs of the differences • 15 children felt better about their teeth (+ difference in ratings) • 1 child felt worse (- diff.)**Example: Body image data**• The sign test looks at the signs of the differences • 15 children felt better about their teeth (+ difference in ratings) • 1 child felt worse (- diff.) • 4 children felt the same (difference = 0)**Example: Body image data**• The sign test looks at the signs of the differences • 15 children felt better about their teeth (+ difference in ratings) • 1 child felt worse (- diff.) • 4 children felt the same (difference = 0) • Looks like good evidence • Need a p-value**P-value for sign test**• The p-value is the probability of an outcome as or more extreme (under H0 ) than that observed. • We observed 15 positives and 1 negative. • If H0 were true we’d expect an equal number of positive and negative differences. • More extreme outcomes would be • more than 15 positives • or less than 1 positives**P-value for sign test**• P-value = P(X> 15) + P(X < 1) • X is the number of positive differences • Under H0, X is Binomial(n = 16, p = 0.5) • n = 16 because the sign test disregards the zero differences • Compute P-value using Binomial tables**Wilcoxon Signed-rank test**• Wilcoxon Signed-rank test is another non-parametric test used for paired data. • It uses the magnitudes of the differences • the sign test does not • More powerful than the sign test • More difficult to interpret than the sign test**Example: Body image data**• Use the Wilcoxonsigned-rank test to evaluate whether these data provide evidence that orthodontic treatment improves children’s image of their teeth.**Example: Body image data**• Use the Wilcoxonsigned-rank test to evaluate whether these data provide evidence that orthodontic treatment improves children’s image of their teeth. • Work with the differences • Remove those with zero difference**Example: Body image data**• To compute the test we need to**Example: Body image data**• To compute the test we need to • note the signs of the differences**Example: Body image data**• To compute the test we need to • note the signs of the differences • get magnitudes of the differences**Example: Body image data**• To compute the test we need to • note the signs of the differences • get magnitudes of the differences • reorder the data by magnitude**Example: Body image data**• To compute the test we need to • note the signs of the differences • get magnitudes of the differences • reorder the data by magnitude • assign ranks to the observations**Example: Body image data**• Note that since there are many ties in the magnitudes we had to assign average ranks.**Example: Body image data**For example, the 2nd through 5th differences all have the same magnitude, so we give them all the average of the 2nd through 5th rank (2+3+4+5)/4 = 3.5**Example: Body image data**The statistic for the signed-rank test is the sum of the ranks of the positive differences**Example: Body image data**The statistic for the signed-rank test is the sum of the ranks of the positive differences**R1: What does it mean?**• With 16 observations R1 could range from 0 (all differences are negative) to 136 (all differences are positive). • If H0 were true we’d expect R1 to be near the middle of the range, in this case, 68. • R1= 132.5 appears to be evidence against H0 • Need a p-value**Signed-rank test p-value**For n > 15, can use a normal approximation where ti are the numbers of ties in each group of ties (note that if ti = 1 then the term is 0), and n is the number of non-zero differences The two-sided p-value is given by**p-value for body image example**There are 4 people tied with difference 2, 8 with difference 3, and 3 tied with difference 4. So And so,**p-value for signed-rank test**• If n< 15 then should not use Normal approximation, but instead use an “exact” p-value. • See §13.2 in text for example of calculating an exact p-value. • In body image example, exact p-value is 0.00015.**Wilcoxon Rank Sum Test**• Used to compare two independent samples • Equivalent to Mann-Whitney U test. • Like the Signed-rank test, the Rank-Sum test is based on the ranks of the data.**Example: Shear Strengths**• Wish to compare two methods of preparing ceramics in terms of product strength. • Two methods of preparation • Press (n=10) • Layer (n=10) • Note outliers in “press” group • T test not appropriate • Large outliers in press group • Data likely not Normal**Computing the Wilcoxon Rank-Sum Test Statistic**• Assign ranks to the combined sample • Sum the ranks in one of the groups • Which group does not matter**Computing the Wilcoxon Rank-Sum Test Statistic**• Assign ranks to the combined sample • Sum the ranks in one of the groups • Which group does not matter • We’ll choose the “press” group**Computing the Wilcoxon Rank-Sum Test Statistic**• Assign ranks to the combined sample • Sum the ranks in one of the groups • We’ll choose the “press” group • The “rank sum” is**Interpretation of rank sum**• Like the signed-rank statistic, the rank sum does not have an obvious interpretation. • It will depend on the numbers of observations in the entire sample and in the chosen group. • In this case (total = 20, number in group = 10), R1 could range from 55 to 155. • If the groups are equal we’d expect R1to be in the middle, around 105. • R1 = 77 seems rather on the low end**Rank-sum p-value**• The null hypothesis is that the two distributions are equivalent • The distribution of R1 under H0 is all possible rank sums that could occur when 10 ranks are randomly chosen from 20.**Rank-sum p-value**• The p-value is the percentage of possible combinations that result in a result as extreme as R1 = 77 • 2-sided (exact) p-value is p = .0178 + .0178 = .0356 p = .0178 p = .0178**Normal approximation p-value**• Exact p-value is difficult to compute. • Can use Normal approximation when both groups have at least 10 observations • See text §13.3 for computation details • In this example, the Normal approximation p-value is p=0.0376.

More Related