Parametric tests

Chong-ho Yu, Ph.Ds.

Restrictions of parametric tests

Conventional statistical procedures are also called parametric tests. In a parametric test a sample statistic is obtained to estimate the population parameter. Because this estimation process involves a sample, a sampling distribution, and a population, certain parametric assumptions are required to ensure all components are compatible with each other. For example, in Analysis of Variance (ANOVA) there are three assumptions:

  • Observations are independent.
  • The sample data have a normal distribution.
  • Scores in different groups have homogeneous variances.

In a repeated measure design, it is assumed that the data structure conforms to the compound symmetry. A regression model assumes the absence of collinearity, the absence of auto correlation, random residuals, linearity...etc. In structural equation modeling, the data should be multivariate normal.

Why are they important? Take ANOVA as an example. ANOVA is a procedure of comparing means in terms of variance with reference to a normal distribution. The inventor of ANOVA, Sir R. A. Fisher (1935) clearly explained the relationship among the mean, the variance, and the normal distribution: "The normal distribution has only two characteristics, its mean and its variance. The mean determines the bias of our estimate, and the variance determines its precision." (p.42) It is generally known that the estimation is more precise as the variance becomes smaller and smaller.

Put it in another way: the purpose of ANOVA is to extract precise information out of bias, or to filter signal out of noise. When the data are skewed (non-normal), the means can no longer reflect the central location and thus the signal is biased. When the variances are unequal, not every group has the same level of noise and thus the comparison is invalid. More importantly, the purpose of parametric test is to make inferences from the sample statistic to the population parameter through sampling distributions. When the assumptions are not met in the sample data, the statistic may not be a good estimation to the parameter. It is incorrect to say that the population is assumed to be normal and equal in variance, therefore the researcher demands the same properties in the sample. Actually, the population is infinite and unknown. It may or may not possess those attributes. The required assumptions are imposed on the data because those attributes are found in sampling distributions. However, very often the acquired data do not meet these assumptions. There are several alternatives to rectify this situation:

Do nothing

Ignore these restrictions and go ahead with the analysis. Hopefully your thesis advisor or the journal editor falls asleep while reading your paper. Indeed, this is a common practice. After reviewing over 400 large data sets, Micceri (1989) found that the great majority of data collected in behavioral sciences do not follow univariate normal distributions. Breckler (1990) reviewed 72 articles in personality and social psychology journals and found that only 19% acknowledges the assumption of multivariate normality, and less than 10% considered whether this assumption had been violated. Having reviewed articles in 17 journals, Keselman et al(1998) found that researchers rarely verify that validity assumptions are satisfied and they typically use analyzes that are nonrobust to assumption violations.

Monte Carlo simulations: Test of test

If you are familiar with Monte Carlo simulations (research with dummy data), you can defend your case by citing Glass et al's (1972) finding that many parametric tests are not seriously affected by violation of assumptions.

Microsoft clip art
"Let me check the weather first before we send out the USS Enterprise. If my boat has problems in sailing, then the USS Enterprise should not be deployed."

Indeed, it is generally agreed that the t-test is robust against mild violations of assumptions in many situations and ANOVA is also robust if the sample size is large. For this reason, Box (1953) mocked the idea of testing the variances prior to applying an F-test, "To make a preliminary test on variances is rather like putting to sea in a rowing boat to find out whether conditions are sufficiently calm for an ocean liner to leave port" (p.333).

In spite of these assurance, there are still some puzzling issues: "How mild a violation is acceptable? How extreme is extreme?" Unfortunately, there is no single best answer. The image on the right is a screen capture of the tests of equal variances run in JMP. JMP ran six tests simultaneously for triangulation and verification, but it also leads to confusion. Given the same data set, the Bartlett test raised a red flag by showing a p value (0.0469) below the cut-off. However, all other tests, such as the Levene test, suggested not to reject the null hypothesis that the two variances are equal.

"How large should the sample size be to make ANOVA robust?" "How much violation is acceptable?" Questions like these have been extensively studied by Monte Carlo simulations. The following table shows how a hypothetical test (Alex Yu's procedure) is tested by several combinations of factors. Because the "behaviors" of the test under different circumstances is being tested, the Monte Carlo method can be viewed as the test of test.
 

Test Conditions
Outcomes
Normality Variance Sample size Type I error Type II error Recommendation
Extremely non-normal Extremely unequal Small Acceptable Acceptable Use with caution
Extremely non-normal Slightly unequal Small Good Acceptable Use it
Extremely non-normal Extremely unequal Large Good Good Use it
Extremely non-normal Slightly unequal Large Good Good Use it
Slightly non-normal Slightly unequal Large Excellent Excellent Use it
More... More... More... More... More... Use it anyway!

Wow! Alex Yu's test appears to be a good test for all conditions. He will win the Nobel prize! Unfortunately, such a powerful test has not been invented yet. Researchers could consult Monte Carlo studies to determine whether a specific parametric test is suitable to his/her specific data structure.

Non-parametric tests

Apply non-parametric tests. As the name implies, non-parametric tests do not require parametric assumptions because interval data are converted to rank-ordered data. Examples of non-parametric tests are:
  • Wilcoxon signed rank test
  • Whitney-Mann-Wilcoxon (WMW) test
  • Kruskal-Wallis (KW) test
  • Friedman's test
Handling of rank-ordered data is considered a strength of non-parametric tests. Gibbons (1993) observed that ordinal scale data are very common in social science research and almost all attitude surveys use a 5-point or 7-point Likert scale. But this type of data are not ordinal rather than interval. In Gibbons' view, non-parametric tests are considered more appropriate than classical parametric procedures for Likert-scaled data. 1

However, non-parametric procedures are criticized for the following reasons:

  • Unable to estimate the population: Because non-parametric tests do not make strong assumptions about the population, a researcher could not make an inferene that the sample statistic is an estimate of the population parameter.

  • Losing precision: Edgington (1995) asserted that when more precise measurements are available, it is unwise to degrade the precision by transforming the measurements into ranked data. 2

  • Low power: Generally speaking, the statistical power of non-parametric tests are lower than that of their parametric counterpart except on a few occasions (Hodges & Lehmann, 1956; Tanizaki, 1997; Freidlin & Gastwirth, 2000).

  • False sense of security: It is generally believed that non-parametric tests are immune to parametric assumption violations and the presence of outliers. However, Zimmerman (2000) found that the significance levels of the WMW test and the KW test are substantially biased by unequal variances even when sample sizes in both groups are equal. In some cases the Type error rate can increase up to 40-50%, and sometime 300%. The presence of outliers is also detrimental to non-parametric tests. Zimmerman (1994) outliers modify Type II error rate and power of both parametric and non-parametric tests in a similar way. In short, non-parametric tests are not as robust as what many researchers thought.

  • Lack of software: Currently very few statistical software applications can produce confidence intervals for nonparametric tests. MINITAB and Stata are a few exceptions.

  • Testing distributions only: Further, non-parametric tests are criticized for being incapable of answering the focused question. For example, the WMW procedure tests whether the two distributions are different in some way but does not show how they differ in mean, variance, or shape. Based on this limitation, Johnson (1995) preferred robust procedures and data transformation to non-parametric tests (Robust procedures and data transformation will be introduced in the next section).
At first glance, taking all of the above shortcomings into account, non-parametric tests seem not to be advisable. However, everything that exists has a reason to exist. Despite the preceding limitations, nonparametric methods are indeed recommended in some situations. By employing simulation techniques, Skovlund and Fenstad (2001) compared the Type I error rate of the standard t-test and the WMW test, and the Welch's test (a form of robust procedure, which will be discussed later) with variations of three variables: variances (equal, unequal), distributions (normal, heavy-tailed, skewed), and sample sizes (equal, unequal). It was found that the WMW test is considered either the best or an acceptable method when the variances are equal, regardless of the distribution shape and the homogeneous of sample size. Their findings are summarized in the following table:
 

Variances

Distributions

Sample sizes

t-test

WMW test

Welch’s test

Equal

Normal

Equal

*

+

+

Unequal

*

+

+

Heavy tailed

Equal

+

*

+

Unequal

+

*

+

Skewed

Equal

_

*

_

Unequal

_

*

_

Unequal

Normal

Equal

+

_

*

Unequal

_

_

*

Heavy tailed

Equal

+

_

+

Unequal

_

_

+

Skewed

Equal

_

_

_

Unequal

_

_

_

Symbols:  * = method of choice, + = acceptable, - = not acceptable

 

Robust procedures

Employ robust procedures. The term "robustness" can be interpreted literally. If a person is robust (strong), he will be immune from hazardous conditions such as extremely cold or extremely hot weather, virus, ... etc. If a test is robust, the validity of the test result will not be affected by poorly structured data. In other words, it is resistant against violations of parametric assumptions. Robustness has a more technical definition: if the actual Type I error rate of a test is close to the proclaimed Type I error rate, say 0.05, the test is considered robust.

Several conventional tests have some degree of robustness. For example, Welch's (1938) t-test used by SPSS and Satterthwaite's (1946) t-test used by SAS could compensate unequal variances between two groups. In SAS when you run a t-test, SAS can also test the hypothesis of equal variances. When this hypothesis is rejected, you can choose the t-test adjusted for unequal variances.



Variances T DF Prob>|T|
Unequal -0.0710 14.5 0.9444
Equal -0.0750 24.0 0.9408

For H0: Variances are equal, F' = 5.32 DF = (11,13) Prob>F' = 0.0058


By the same token, for conducting analysis of variance in SAS, you can use PROC GLM (Procedure Generalized Linear Model) instead of PROC ANOVA when the data have unbalanced cells.

However, the Welch's t-test is only robust against the violation of equal variances. When multiple problems occur (welcome to the real world), such as non-normality, heterogeneous variances, and unequal sizes, the Type I error rate will inflate (Wilcox, 1998; Lix & Keselman, 1998). To deal with the problem of multiple violations, robust methods such as trimmed means and Winsorized variances are recommended. In the former, outliers in both tails are simply omitted. In the latter, outliers are "pulled" towards the center of the distribution. For example, if the data vector is [1, 4, 4, 5, 5, 5, 6, 6, 10], the values "1" and "10" will be changed to "4" and "6," respectively. This method is based upon the Winsor's principle: "All observed distributions are Gaussian in the middle." Yuen (1974) suggested that to get the best of all methods, trimmed means and Winsorized variances should be used in conjunction with Welch's t-test.

SAS/Insight can compute both Winsorized and trimmed means by pointing and clicking (under the pull down menu "Table.").

 

SAS/INSIGHT

In addition, PROC UNIVARIATE can provide the same option as well as robust measures of scale. By default, PROC UNIVARIATE does not return these statistics. "ALL" must be specified in the PROC statement to request the following results.

 

PROC UNIVARIATE ALL

Mallows and Tukey (1982) argued against the Winsor's principle. In their view, since this approach pays too much attention to the very center of the distribution, it is highly misleading. Instead, he recommended to develop a way to describe the umbrae and penumbrae around the data. In addition, Keselman and Zumo (1997) found that the nonparametric approach has more power than the trimmed-mean approach does. Nevertheless, Wilcox (2001) asserted that the trimmed-mean approach is still desirable if 20 percent of the data are trimmed under non-normal distributions.

Regression analysis also requires several assumptions such as normally distributed residuals. When outliers are present, this assumption is violated. To rectify this situation, join a weight-loss program! Robust regression (Lawrence & Arthur, 1990) can be used to down-weight the influence of outliers. The following figure shows a portion of robust regression output in NCSS (NCSS Statistical Software, 2010). The weight range is from 0 to 1. Observations that are not extreme have the weight as "1" and thus are fully counted into the model. When the observations are outliers and produce large residuals, they are either totally ignored ("0" weight) or partially considered (low weight). The down-weighted observations are marked with an asterisk (*) in the following figure.

 

Downweight in robust regression

Besides NCSS, Splus and SAS can also perform robust regression analysis (e.g. PROC ROBUSTREG) (Schumacker, Monahan, & Mount, 2002). The following figure is an output from Splus (TIBCO, 2010). Notice that the outlier is not weighted and thus the regression line is unaffected by the outlier.

 

Robust regression in Splus Robust regression in Splus

In addition to robust regression, SAS provides the users with several other regression modeling techniques to deal with poorly structured data. The nice thing is that you don't need to master SAS to use those procedures. SAS Institute (2012) produces a very user-friendly package called JMP. Users can access some of the SAS procedures without knowing anything about SAS.

 

When data for ANOVA cannot meet the parametric assumptions, one can convert the grouping variables to dummy variables (1, 0) and run a robust regression procedure (When a researcher tells you that he runs a dummy regression, don't think that he is a dummy researcher). As mentioned before, robust regression down-weights extreme scores. When assumption violations occur due to extreme scores in one tail (skew distribution) or in two tails (wide dispersion, unequal variances), robust regression is able to compensate for the violations (Huynh & Finch, 2000).

Cliff (1996) was skeptical to the differential data-weighting of robust procedures. Instead he argued that data analysis should follow the principle of "one observation, one vote." Nevertheless, robust methods and conventional procedures should be used together when outliers are present. Two sets of results could be compared side by side in order to obtain a thorough picture of the data.

Data transformation

Employ data transformation methods suggested by exploratory data analysis (EDA) (Behrens, 1997; Ferketich & Verran, 1994). Data transformation is also named data re-expression. Through this procedure, you may normalize the distribution, stabilize the variances or/and linearize a trend. The transformed data can be used in different ways. Because data transformation is tied to EDA, the data can be directly interpreted by EDA methods. Unlike classical procedures, the goal of EDA is to unveil the data pattern and thus it is not necessary to make a probabilistic inference. Alternatively, the data can be further examined by classical methods if they meet parametric assumptions after the re-expression. Parametric analysis of transformed data is considered a better strategy than non-parametric analysis because the former appears to be more powerful than the latter (Rasmussen & Dunlap, 1991). Vickers (2005) found that ANCOVA was generally superior to the Mann-Whitney test in most situations, especially where log-transformed data were entered into the model.
 

Isaiah said, "Every valley shall be exalted, and every mountain and hill shall be made low, and the crooked shall be made straight, and the rough places plain."

Today Isaiah could have said: "Every datum will be normalized, every variance will be made low. The rough data will be smoothed, the crooked curve will be made straight. And the pattern of the data will be revealed. We will all see it together."

Isaiah's Lips Anointed with Fire
Source: BJU Museum and Gallery


However, it is important to note that log transformation is not the silver bullet. If the data set has zeros and negative values, log transformation doesn't work at all.

Resampling

Use resampling techniques such as randomization exact test, jackknife, and bootstrap. Robust procedures recognize the threat of parametric assumption violations and make adjustments to work around the problem. Data re-expression converts data to ensure the validity of using of parametric tests. Resampling is very different from the above remedies for it is not under the framework of theoretical distributions imposed by classical parametric procedures. Robust procedures and data transformation are like automobiles with more efficient internal combustion engines but resampling is like an electrical car. The detail of resampling will be discussed in the next chapter.

Data mining

Fisherian parametric tests are classified by data miners Nisbet, Elder, and Miner (2009) as first generation statistical methods. While parametric tests are efficient to handle relatively small experimental data sets in academic settings, business and industry, which use huge data sets, admitted that "analysts could bring computers to their 'knees' with the processing of classical statistical analyses" (Nisbet, Elder, & Miner, 2009, p.30). As a remedy, a new approach to decision making was created based on artificial intelligence (AI), which modeled on the human brain rather than on Fisher’s parametric approach. As a result, a new set of non-parametric tools, including neural nets, classification trees, and multiple auto-regressive spine (MARS), was developed for analyzing huge data sets. This cluster of tools is called data mining. Unlike conventional parametric tests that emphasize theoretical explanation, data mining is primarily used by business for prediction. This paradigm shift is reflected by the renaming of SPSS in 2009. After IBM acquires SPSS, SPSS became Predictive Analytical Software (PASW) because data mining and text mining tools had been tightly integrated into formerly SPSS's parametric procedures.  But later IBM reverted the name to SPSS. For more information about data mining, please read this write-up.

Multilevel modeling

In social sciences, the assumption of independence, which is required by ANOVA and many other parametric procedures, is always violated to some degree. Take Trends for International Mathematics and Science Study (TIMSS) as an example. The TIMSS sample design is a two-stage stratified cluster sampling scheme. In the first stage, schools are sampled with probability proportional to size. Next, one or more intact classes of students from the target grades are drawn at the second stage (Joncas, 2008). Parametric-based ordinary Least Squares (OLS) regression models are valid if and only if the residuals are normally distributed, independent, with a mean of zero and a constant variance. However, TMISS data are collected using a complex sampling method, in which data of one level are nested with another level (i.e. students are nested with classes, classes are nested with schools, schools are nested with nations), and thus it is unlikely that the residuals are independent of each other. If OLS regression is employed to estimate relationships on nested data, the estimated standard errors will be negatively biased, resulting in an overestimation of the statistical significance of regression coefficients. In this case, hierarchical linear modeling (HLM) (Raudenbush & Bryk, 2002) should be employed to specifically tackle the nested data structure. To be more specific, instead of fitting one overall model, HLM takes this nested data structure into account by constructing models at different levels, and thus HLM is also called multilevel modeling.

The merit of HLM does not end here. For analyzing longitudinal data, HLM is considered superior to repeated measures ANOVA because the latter must assume compound symmetry whereas HLM allows the analyst specify many different forms of covariance structure (Littell & Milliken, 2006). Readers are encouraged to read Shin's (2009) concise comparison of repeated measures ANOVA and HLM.


What should we do?

No doubt parametric tests have limitations. Unfortunately, many people select the first solution--do nothing. They always assume that all tests are "ocean liners." In my experience, many researchers do not even know what a "parametric test" is and what specific assumptions are attached to different tests. To conduct responsible research, one should contemplate the philosophical paradigms of different schools of thought, the pros and cons of different techniques, the research question, as well as the data structure. The preceding options are not mutually exclusive. Rather than they can be used together to compliment each other and to verify the results. For example, Wilcox (1998, 2001) suggested that the control of Type I error can be improved by resampling trimmed means.


Notes

  1. Today very seldom researchers use a single Likert scale as a variable. Instead, many items are combined as a composite score if Cronbach Alpha verifies that the items are internally consistent and factor analysis confirms that all items could be loaded into one single dimension. By using a composite score, some social scientists believe that the ordinal-scaled data based upon a Likert-scale could be converted into a form of pseudo-interval-scaled data. To be specific, when 50 five-point Likert-scaled items are totaled as a composite score, the possible range of data value would be from 1 to 250. In this case, a more extensive scale could form a wider distribution. Nonetheless, this argument is not universally accepted.

    The issue regarding the appropriateness of ordinal-scaled data in parametric tests was unsettled even in the eyes of Stevens (1951), the inventor of the four levels of measurement: "As a matter of fact, most of the scales used widely and effectively by psychologists are ordinal scales ... there can be involved a kind of pragmatic sanction: in numerous instances it leads to fruitful results." (p.26) Based on the central limit theorem and Monte Carlo simulations, Baker, Hardyck, and Petrinovich (1966) and Borgatta and Bohrnstedt (1980) argued that for typical data, worrying about whether scales are ordinal or interval doesn't matter.

    Another argument against not using interval-based statistical techniques for ordinal data was suggested by Tukey (1986). In Tukey's view, this was a historically unfounded overreaction. In physics before precise measurements were introduced, many physical measurements were only approximately interval scales. For example, temperature measurement was based on liquid-in-glass thermometers. But it is unreasonable not to use a t-test to compare two groups of such temperatures. Tukey argued that researchers painted themselves into a corner on such matters because we were too obsessed with "sanctification" by precision and certainty. If our p-values or confidence intervals are to be sacred, they must be exact. In the practical world, when data values are transformed (e.g. transforming y to sqrt(y), or logy), the p values resulted from different expressions of data would change. Thus, ordinal-scaled data should not be banned from entering the realm of parametric tests. For a review of the debate concerning ordinal- and interval- scaled data, please consult Velleman and Wilkinson (1993).
     

  2. Harrell (1999) disagreed with Edgington, "Edgington's comment is off the mark in most cases. The efficiency of the Wilcoxon-Mann-Whitney test is 3/pi (0.96) with respect to the t-test IF THE DATA ARE NORMAL. If they are non-normal, the relative efficiency of the Wilcoxon test can be arbitrarily better than the t-test. Likewise, Spearman's correlation test is quite efficient (I think the efficiency is 9/pi3) relative to the Pearson r test if the data are bivariate normal. Where you lose efficiency with nonparametric methods is with estimation of absolute quantities, not with comparing groups or testing correlations. The sample median has efficiency of only 2/pi against the sample mean if the data are from a normal distribution."


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Last updated: 2013


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