Likert Scale: Ordinal or Interval Data

For several reasons, some reputable scholars explicitly claim that Likert scales are ordinal data, not interval data. These authors are convinced that Likert scales lack equal intervals though presented in an ordered category such as Strongly Disagree, Disagree, Neutral, Agree, Strongly Agree. The authors maintained that the distance between these categories cannot be equal. The difference between “Strongly Agree” and “Agree” might not be the same as the difference between “Disagree” and “Strongly Disagree.” There is a meaningless arithmetic operation since the intervals between categories are NOT sure to be equal; performing mathematical operations such as computing for the average on the Likert value assigned in each category (e.g., 1 = Strongly Disagree, 2 = Disagree) does not hold meaning. The difference between a rating of 3 (Neutral) and 5 (Strongly Agree) does not inevitably represent twice the level of agreement. Finally, in ordinal ranking versus interval measurement, the Likert scale displays a ranking order for the response, enabling us to declare that one response conveys a more substantial level of agreement/disagreement than another. However, these authors assert that these do not provide an accurate quantitative measure of the underlying construct under measure. We cannot ultimately declare that a score of 4 is precisely twice the level of agreement to a rating of 2.

On the other side, some issues about Likert scales oppose the most common perception, being that it is ordinal data. Authors claim that Likert scales can be treated as interval data under certain conditions. For instance, when multiple Likert items were averaged to create a composite score, the central limit theorem suggests that the rating will approximate an interval scale within a sufficient number of items (20-30 sampling size). The aggregation assists in smoothing out the ordinal nature of individual items. Many psychometricians are convinced that the summed Likert scale is a reasonable approximation of an interval scale despite the ordinal nature of the Likert scale. The Likert scale is treated as interval data for convenience and to facilitate parametric statistical analyses. Increasing the number of points on a Likert scale, such as using a seven or 11-point scale, makes it more closely approximate an inter-scale by furnishing a greater number of response options and a more continuous scenario. However, the intervals still need to be guaranteed to be equal. Also, Likert scales have directionality, and the interval between response options is often considered even. However, it is not strictly an interval, which permits the response to be treated as the interval for some analytical purpose.

On the other hand, a serious consideration is needed. The Likert scale is technically ordinal, and the intervals between response options are unequal. Treating Likert scales as intervals and conventions in specific programs is convenient, but they are only sometimes accepted. Whether the data is ordinal or interval, decide the suitable statistical test. Parametric tests require interval data.

Origins of Non-Parametric statistics

In the early 18th century, John Arbuthnot’s work was the earliest basis of nonparametric statistics. In his paper “An Argument for Divine Providence, taken from the regularity observed in the birth of both sexes,” he measured birth records in London from 1629 to 1710. Mr. Arbuthnot was convinced that it was not due to chance that more males were born than females. His simple single test was considered the earliest example of a nonparametric test without relying on assumptions about the data distribution.

It was only in 1945 that the nonparametric statistics renewed a kindled interest. For instance, Frank Wilcoxon (1945) introduced the Wilcoxon signed rank test and the Wilcoxon rank-sum test. These were tested based on the ranks of observations instead of their values, allowing for distribution-free tests.

In 1947, Henry Mann and Donald Whitney independently developed the Mann-Whitney U test, similar to Wilcoxon’s rank-sum test. Earlier, Maurice Kendall, in 1938, introduced Kendall’s tau, nonparametric correlation coefficient. Following were Abraham Wald and Jacob Wolfowitz's run tests for detecting the differences between two groups in 1940 (Wald-Wolfowitz test). At the same time, the Friedman test was considered an alternative to the repeated measures of ANOVA. Similarly, in 1952, we saw the Kruskal-Wallis one-way ANOVA test. The later developments of the 1950s were the Theil-Sen Estimator, a robust nonparametric method for fitting a line to data, and the Hodges-Lehmann estimator (1963), an estimator of location. In the 1960s was the Kernel density estimation from Rosenblatt (1956) and Parzen (1962). The Bootstrap method was considered in the modern era and introduced for resampling techniques for estimating the sampling distribution of statistics.

The new computational power has led to the development and practical application of more complex nonparametric methods for resampling and permutation test techniques. The reason for embracing the development of nonparametric tests arises from the need for methods without the stringent requirement to distribute the data assumptions. The nonparametric statistics continue evolving to discover robust data analysis tools across various fields.

Several reasons to use nonparametric statistics

1. The nonparametric test does not require the assumption that data follows a normal distribution, skewed or non-normally distributed data.

2. In case sample sizes are small, nonparametric sampling is more appropriate than parametric techniques with a stricter sampling size, unlike nonparametric methods, which handle smaller sampling effectively.

3. Nonparametric test handles ordinal or nominal data that do not depend on specific population parameters.

4. Nonparametric tests are robust against outliers and heavily skewed distribution, especially when assumptions for parametric tests are violated.

Compare and Contrast Parametric and Nonparametric tests.

Aspect	Parametric test	Nonparametric test
Assumptions	Assume data follows a normal distribution.	Do not rely on assumptions about the parameters of the underlying population distribution.
Data type	Typically used for continuous data (interval, ratio)	It can be used for non-normal variables (ordinal, nominal)
Examples	1-sample t-test, 2-sample t-test, One-way ANOVA	Sign test, Mann-Whitney U test, Kruskal-Wallis test
Robustness	Less robust to outliers and non-normality	More robust against outliers and non-normality
Sample size requirements	Varies based on the specific test. For example, a 1-sample t-test requires a sample size greater than 20	Less stringent sample size requirements

Several issues arise when data is not normally distributed.

Inaccurate results for parametric tests such as t-tests and ANOVA occur. For these tests to become valid, they should come from a specific normal distribution—the risk of a Type I error (false positive) increases, leading to an incorrect conclusion. In a non-normal data parametric test, loose power cannot determine the true effect. In such a case, the nonparametric is more appropriate. An incorrect interval, p-values, and other statistical measures result from assumption violations. Hence, avoiding these issues using the Mann-Whitney U test or Kruskal –Wallis test when dealing with non-normally Distributed Data is more appropriate.

Steps to determine if a dataset follows a normal distribution

Plotting the data on a graph is to look for a symmetric bell-shaped curve centered around the mean, which converges near the center and tapers away from the central. The following is to check symmetry, which is the distribution of the mean supposed to be half spared on the left and half on the right side. Two parameters determine the normal distribution: the mean and standard deviation. The mean is the single peak of the curve. The standard deviation (SD) stretches or squeezes the curve with a small standard deviation in a narrow curve, and large are depicted in a wider curve. Finally, the empirical rule (68-95-99.7) must display that 68% of values lie within one (1) standard deviation from the mean, approximately 95% within two (2) standard deviations, and about 99.7% of values within three (3) standard deviations.

Figure 1. Normal Distribution

Figure 2. Non-normal distribution appearing data with two population

Figure 2 displays a non-normal distribution. The figure has two peaks instead of one. A distribution with two peaks conveys that different groups are mixed up in the data. For instance, the scores of students are normally distributed. However, suppose the data has a group of quiz bowl varsity students. There might be a bimodal distribution similar to the Figure 2 above.

However, the visual inspection approach to determine the distribution's normality could be more reliable and ensure that the distribution is normal. Nonetheless, the visual presentation of data may allow for the data distribution assumption. The frequency distribution (histogram), stem-and-leaf plot, boxplot, P-P plot (probability-probability plot), and Q-Q plot (quantile-quantile plot) help measure the normal distribution visually. The observed values of the frequency distribution plots furnish a visual evaluation of the bell-shaped distribution and insights into gaps in the data and outliers.