Hypothesis Testing: When a Nonparametric Test is Appropriate

Statistical inference can be fascinating. What are the odds that tomorrow's daily return of the S&P 500 index will fall between -1% and 1%? This is a question you would try to best answer with standard parametric tests by making certain assumptions.

No, that's not a bad idea, but there actually are data series for which a parameter like the mean doesn't really make sense.

No, that's not a bad idea, but nonparametric inference isn't about circumventing parametric tests.

That's the best idea, yes!

There are a few cases in which you would want to reach into the statistics literature for a nonparametric test. The first of these is when your __data fails to meet distributional assumptions__. For example, z-scores are very useful when the underlying data distribution is normal or approximately normal. What would happen if you just used means, standard deviations, and z-scores on a distribution which was far from normally distributed?

Incorrect. You can actually put any numbers into the standard deviation calculation and come up with a number; so yes, you can calculate the standard deviation of a non-normal distribution.

That's right!

Using a calculated parameter designed for the wrong distribution will simply lead to errors in inference. This sort of error can cause dangerous miscalculations in terms of risk and returns, so a nonparametric test may be used in this instance. A second reason to consider a nonparametric test is when there are valid outliers. These will pull the mean away from the rest of the observations, so maybe the median is more appropriate; and there are nonparametric tests for the median.

A third case in which nonparametric inference is appropriate is when the __data are given in ranks__. Some data is just of an "ordinal" scale, meaning that you have ranks like first place, second place, etc. This doesn't provide you with the same information as data on measured things like currency amounts and returns. So how could you use parametric testing for data such as the rank order of performance of 20 asset managers?

No, this would lead to useless results. This is exactly the case when a nonparametric test is most appropriate.

Correct!

Which of the following seems to be the best candidate for a nonparametric test?

No, this could be describing a series of daily changes which may be distributed normally around some positive value. This is a likely candidate for a parametric test.

Yes! Matching dates with performance numbers presents an interesting challenge, and turning both of these series into an ordered ranking would be a good idea here. Then you can use a nonparametric test to look at the correlation between these rankings. There are several useful nonparametric tests available: | Tests Concerning: | Parametric | Nonparametric | |-----|-----|-----| | A single mean | t-test, z-test | Wilcoxon signed-rank test | | Differences between means | t-test | Mann-Whitney U test (Wilcoxon rank sum test) | | Mean difference (paired comparisons) | t-test | Wilcoxon signed-rank test |

No, earnings are a typical cardinal data series which often fall into a standard distributional assumption. This is a likely candidate for a parametric test.

The fourth reason you might choose to explore a nonparametric test is when your hypothesis simply doesn't concern a parameter. Perhaps you want to know whether a dataset is statistically random or not. Or maybe you would like to test the likelihood that a particular sample came from a given population. In these and many other instances, a nonparametric test may be appropriate.

To summarize this discussion: [[summary]]

But there are other situations in which something called __nonparametric inference__ is more appropriate. Based on your knowledge of parametric inference, what do you think that nonparametric inference must involve?

For a non-normal distribution, standard deviation couldn't be calculated

For a non-normal distribution, using standard deviation would lead to incorrect inference

Parametric tests wouldn't be appropriate here since the rank doesn't provide information as to the degree of performance

The ranks could be used in a standard parametric test by entering the numbers one through 20

Testing the likelihood of a negative change in an exponential growth series

Testing the connection between the performance of mutual funds and the dates in which they became operational

Testing the accuracy of earnings estimates

An inference where parameters are ignored

A superior form of inference where no parameters are needed

An inference used when parametric assumptions are invalid

Continue

Continue

Continue