Many methods for statistical inference require assumptions about the probability distribution and associated parameters of the population. Such as the normal distribution, the population mean, and standard deviation as its parameters. These methods are parametric methods, and in contrast, there's also a group of inferential methods that do not assume that the population distribution has a particular form. And these are called non-parametric methods or alternatively, distribution free methods. In this video, I'll explain when it's wise to use non-parametric methods. Let's start with an example. Imagine, tomorrow is your birthday and you have decided to give your friends and yourself a special treat on this occasion. So you go to the local confectioner to buy seven pastries. But the choice is overwhelming and everything is delicious anyway. So you asked the confectioner to select seven pieces of pastries at random. You can consider all the pastries available in the shop as the population. And the seven pieces are a sample from that population. Now each type of pastry is priced differently and it is the price that we will consider is of variable interest here. In this graph, the probability distribution for the prices of all the available pastries in the shop is given. It's the population distribution. And down there, you see the prices for the pastries the confectioner selected for you. Now you could, on the basis on your sample, estimate the mean price of the pastries in the shop, with the confidence interval. Or you could test whether the confectioner has given you pastries that are a bit on the expensive side. If you knew the amount and prices of all the pastries in the shop, and then calculate the population mean. The standard probability distribution available to answer these questions would be the T distribution. However, it makes a number of assumptions that are not exactly met. First of all, the population is not normally distributed, but it appears to be skewed towards left secondly, the price range is limited and does not exceed the value of 6 and is never lower than 1. So what happens? By still applying the t-distribution to calculate, for instance, a confidence interval. The confidence interval can still be calculated and it would suggest that in 95% of the cases, the estimated range would contain the true mean. But in reality, this confidence interval would probably be overly optimistic and would in this case, in fact, contain only the true mean in 90% of the cases. We check this by repeating the sampling of 7 pieces of pastry 100 times. It would then in only around 90 cases lead to a confidence interval which contains the true mean. So, by applying a parametric approach to calculate a confidence interval or a hypothesis test while the assumptions are violated, the numbers that you get become less accurate and may at some extreme cases, even be very wrong. Obviously you'd like to avoid making such errors and that's why you would always try to apply an analysis method for which the assumptions are met. So if you know that the assumptions of a parametric tests are violated but that there's not a parametric equivalent test available for which the assumptions are met, you would choose the latter. Another important reason to use a non parametric technique maybe a limited sample size. We have just a handful of observations, say less than 15. Violations of the assumptions in the parametric test are particularly hard to detect, yet such a violation can have large effect under these small sample conditions. In general non-parametric methods make less assumptions than parametric methods and can therefore be applied more frequently. But that's not the only reason why non-parametric methods are useful, and even more important reason maybe related to the measurement level at which the data is available. A lot of data has an ordinal measurement level, for ordinal data, with for example five incremental classes, you cannot calculate a mean and standard deviation in a meaningful way. So for that type of data, you cannot apply a T-test, an anova, or calculate the Pearson correlation coefficient. Fortunately, there are many non parametric methods that don't work with numerical data, that are specifically designed for ordinal data instead, the so-called rank based methods. With these rank based methods, you can, for example, compare the estimated median based on a sample against a theoretical median. And also calculate the correlation coefficient amongst two ordinal variables. Typically these rank based methods are insensitive to outliers in the data. And that brings us to the next property of non parametric techniques. If some assumptions for non parametric techniques are not met, they are usually fairly robust to these violations. This means that the reported confidence intervals or significance may not be very accurate, but it won't be far off the real value. For most parametric tests, an equivalent non-paramatic test. If it's not testing exactly the same population parameter then it is testing a parameter that is equivalent in practice. The two sample t-test, for example, tests for a difference among two sample means. There's a corresponding non-paramatic test. We can test for a difference in two sample medians. The underlying question usually is whether the two samples come from the same population and that question may be answered by testing for a difference in means as well as medians. The downside of non parametric method is that the power is usually lower than that of the equivalent parametric test. If the assumptions are met. This means that you wouldn't need more observations or a larger effect size to correctly reject the null hypothesis with a non-parametric test. And for that reason, parametric tests are often preferred if the assumptions are met. Let me try to summarize. Non-parametric methods are advantageous in four situations. First, the underlying probability distribution may be unknown or known to be different from what the parametric method requires. Second, the sample size may be very small so that it's impossible to test whether parametric assumptions are met. While a violation of these assumptions may have severe effects. Third, the measurement level may fall below what is required for a parametric technique and finally, there may be no parametric technique available at all for the specific question at hand. In general, a non parametric method is more robust than its parametric equivalent. Robustness means that when assumptions are violated, it doesn't heavily influence the outcome of the test. However, the power of a parametric test is always higher than an equivalent non-parametric test. And therefore, if there is a choice and assumptions are met, a parametric test is preferred.
