Name three examples of parametric tests used by educational researchers

November 01, 2019


Difference Between Parametric and Nonparametric
Social researchers often construct a hypothesis, in which they assume that a certain generalized rule can be applied to a population. They test this hypothesis by using tests that can be either parametric or nonparametric. Parametric tests are usually more common and are studied much earlier as the standard tests used when performing research.

The process of performing research is relatively simple – you construct a hypothesis and assume that a certain “law” can be applied to a population. You then conduct a test and gather data that you then analyze statistically. The collected data can usually be represented as a graph and the hypothesized law as the mean value of that data. If the hypothesized law and the mean value law match, the hypothesis is confirmed.

However, in some cases, finding the mean value isn’t the most appropriate way to search for the law. A great example is the distribution of total income. If you haven’t matched the mean value, that’s probably because one or two billionaires are disturbing your mean values. However, a median will give a much more accurate result on the average income that is more likely to match your data.

In other words, a parametric test will be used when the assumptions made about the population are clear and there is a lot of available information about it. The questions will be designed to measure those specific parameters so that the data can then be analyzed as described above. A nonparametric test is used when the tested population isn’t entirely known and therefore the examined parameters are unknown as well. Additionally, while the parametric test uses mean values as its results, the nonparametric test takes the median and is therefore usually utilized when the original hypothesis doesn’t fit the data. 
What is the Parametric Test?

A parametric test is a test designed to provide the data that will then be analyzed through a branch of science called parametric statistics. Parametric statistics assume some information about the population is already known, namely the probability distribution. As an example, the distribution of body height on the entire world is described by a normal distribution model. Similar to that, any known distribution model can be applied to a set of data. However, assuming that a certain distribution model fits a dataset means that you inherently assume some additional information is known about the population, as I’ve mentioned. The probability distribution contains different parameters that describe the exact shape of the distribution. These parameters are what parametric tests provide – each question is tailored to give an exact value of a certain parameter for each interviewed individual. Combined, the mean value of that parameter is used for the probability distribution. That means that the parametric tests also assume something about the population. If the assumptions are correct, parametric statistics applied to data provided by a parametric test will give results that are much more accurate and precise than that of a nonparametric test and statistics.

What is a Nonparametric Test?
In a similar way to parametric test and statistics, a nonparametric test and statistics exist. They’re used when the obtained data is not expected to fit a normal distribution curve or ordinal data. A great example of ordinal data is the review you leave when you rate a certain product or service on a scale from 1 to 5. Ordinal data, in general, is obtained from tests that use different rankings or orders. Therefore, it doesn’t rely on numbers or exact values for the parameters that parametric tests relied on. In fact, it doesn’t utilize parameters in any way, because it doesn’t assume a certain distribution. Usually, a parametric analysis is preferred to a nonparametric one, but if the parametric test cannot be performed due to an unknown population, a resort to nonparametric tests is necessary.

Difference Between Parametric and Nonparametric Tests
1) Making assumptions
The parametric test makes assumptions about the population. It needs the parameters that are connected to the normal distribution that is used in the analysis, and the only way to know these parameters is to have some knowledge about the population. On the other hand, a nonparametric test, as the name indicates, doesn’t rely on any parameters and therefore doesn’t assume anything about the population.
2) Probability of Parametric and Nonparametric
The basis for the statistic analysis that will be performed on the data, in the case of parametric tests, is probabilistic distribution. On the other hand, the basis for nonparametric tests doesn’t exist – it’s completely arbitrary. This results in more flexibility and makes it easier to fit the hypothesis with the collected data.
3) The measure of central tendency
The measure of central tendency is a central value in a probability distribution. And although the probability distribution in the case of nonparametric statistics is arbitrary, it still exists, and therefore so does the measure of central tendency. However, those measures are different. In the case of parametric tests, it is taken to be the mean value, whereas, in the case of nonparametric tests, it is taken to be the median value.
4) Knowledge of population parameters

The first difference, information about the population varies between parametric and nonparametric tests and statistics. Namely, certain knowledge about the population is absolutely necessary for parametric analysis, because it requires population-related parameters in order to give precise results. On the other hand, a nonparametric approach can be taken without any previous knowledge of the population.
Parametric vs. Nonparametric Tests : Comparison chart

Summary of Parametric and Nonparametric
A parametric test is a test that assumes certain parameters and distributions are known about a population, contrary to the nonparametric one. The parametric test uses a mean value, while the nonparametric one uses a median value
The parametric approach requires previous knowledge about the population, contrary to the nonparametric approach. Statistics is a vital tool to provide inference in medical research. Choosing an appropriate statistical test plays a vital role in the analysis and interpretation of the research data. In the past four decades, it has been observed that use of diversified statistical methods has amplified to a greater extent in medical research publications; however, it is pertinent to mention here that the standards of reporting statistical tests and using them are very low as many shortcomings and pitfalls have been observed in the studies published in past in various biomedical journals. This is a serious problem, and it leads to misleading conclusions, wrong inferences, hazardous clinical consequences, and utter waste of resources.

With the wider availability of statistical software, performing statistical analysis has become very easy; however, the selection of an appropriate statistical test is still lacking behind which leads to wrong study findings and misleading inference. Selection of an appropriate statistical test depends on (1) nature of the data, (2) does the data follow a normal distribution or not, and (3) what is the study hypothesis. The potential source of perplexity while deciding on which statistical test to use for analyzing the data is whether the data allow for the use of parametric or nonparametric test procedures. The magnitude of this concern cannot be underrated. Before selecting the one between these two, a researcher must be aware of the underlying differences, advantages, and disadvantages of using one over the other.
Parametric Tests 
A parametric test is one which makes assumptions about the parameters of the population distribution(s) from which the sample has been drawn. In the parametric test, the assumption is made through the sample population. If the information about the population from which the sample has been drawn is completely known through its parameters than the test is called the parametric tests. The common assumptions underlying parametric tests are as follows:


The observations must be independent - independence of observation means that the data are not connected to any factor that could affect the outcome. For example, a person is hypertensive or not it does not depend on his/her choice of colour. When we talk about the independence of observations between groups, it means that the patients in both the groups under study are separate. We do not want any patient to appear in both the groups
The observation must be drawn from a normally distributed population
The data must be measured on an interval or ratio scale.
Nonparametric Tests 
Nonparametric tests are usually referred to as distribution-free tests. A nonparametric statistical test is the one that does not necessitate any conditions to be fulfilled about the parameters of the population from which the sample was drawn. Nonparametric tests can also be used when the data are nominal or ordinal. Nonparametric tests are also applied to the interval or ratio data which do not follow the normal distribution.
The dilemma of Using Parametric Versus Nonparametric Tests 
To simplify the issue, one should remember:
The scale of measurement of the data  illustrates the use of parametric or nonparametric tests according to the measurement scales
Population distribution  describes the use of parametric and nonparametric test according to the type of population distribution
Homogeneity of Variances - for applying parametric tests, it must be ensured that the variances of the population are equal. On the other hand, no such assumption is required to be fulfilled for the application of nonparametric tests
Independence of samples - for parametric tests to be used the samples drawn from the population must be independent. No such assumption is required for nonparametric tests.
Figure 1: Use of parametric or nonparametric tests according to the scale of measurement of the data.


 
Figure 2: Use of parametric versus nonparametric tests according to population distribution.
Limitations of Nonparametric Tests 
Although nonparametric tests do not require any stringent assumptions to be fulfilled for application, yet parametric tests are preferred over them due to the following limitations of nonparametric tests: Parametric tests have more statistical power than nonparametric tests; therefore, they are more likely to detect a significant difference when it really exists. 

Parametric tests can perform well with skewed and nonnormal data if the sample size is appropriate for performing the particular parametric tests. For example, for performing one-sample t-test on a nonnormal data sample size should be >20, for a two-sample t-test on nonnormal data each group should have more than 15 observations, and for performing a one-way analysis of variance on a nonnormal data having 2–9 groups, each group should have >15 observations.

Parametric tests can perform better when the (dispersion) of the groups is different. Although nonparametric tests do not follow stringent assumptions, yet one assumes that the dispersion of all the groups must be the same is difficult to be met for running nonparametric tests. If this assumption of equal dispersion is not met, nonparametric tests may result in invalid results. Parametric tests can perform better in such situations. The inference drawn from parametric tests is easy to interpret and more meaningful than that of nonparametric tests. Many nonparametric tests use rankings of the values in the data rather than using the actual data. Knowing that the difference in mean ranks between the two groups is five does not really help our intuitive understanding of the data. On the other hand, knowing that the mean systolic blood pressure of patients taking the new drug was 5 mmHg lower than the mean systolic blood pressure of patients on the standard treatment is both intuitive and useful.
Advantages of Nonparametric Tests 
The most important point while analyzing the data is to understand the fact that whether your data are better represented by mean or median. This is the key to decide whether to use a parametric or a nonparametric test. If the data are better represented by the median then use a nonparametric test. For a better understanding of the fact, let us explore an example. Suppose a researcher is interested in knowing the average income of the people in two groups and want to compare them. For this type of data, the median will be the appropriate measure of central tendency, where 50% of the people will be having income below that and 50% will be having income above that.

If we add some highly paid people in the group than those will act as outliers and mean will differ to a greater extent, however, the income of a particular person will be the same. In that case, the mean values of the two samples may differ significantly but medians will not. In such cases, using nonparametric tests is better than parametric tests.


When the sample size is small and the researcher is not sure about the normality of the data, it is better to use nonparametric tests. Because when the sample size is too small it is not possible to ascertain the normality of the data because the distribution tests will also lack sufficient power to provide meaningful results
When we have ordinal data, nominal data, or some outliers in the data that cannot be removed then nonparametric tests must be used.
Parametric Tests and Their Nonparametric Equivalents 
For all the parametric tests, there exists a parallel nonparametric equivalent. Describes, in brief, the type of situation understudy with some examples and the relevant parametric tests and their nonparametric equivalents to be used in those situations.
Table 1: Corresponding table for parametric tests and their nonparametric equivalents. How to Use the Online Calculators for Performing Mann–Whitney U-Test: A nonparametric Test.

Suppose a researcher designed a protocol to study the effectiveness of an analgesic in patients with arthritis. He/she recruits 12 participants and randomized them into two groups to receive either the new drug or a placebo. Participants are asked to record the intensity of pain on a scale of 0–10 where 0 = no pain and 10 – severe pain. The hypothetical data are shown in 



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