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asympTest: A Simple R Package for
Classical Parametric Statistical Tests and
Confidence Intervals in Large Samples
by J.-F. Coeurjolly, R. Drouilhet, P. Lafaye de Micheaux
An important point to be noticed is that stu- dents are usually told that mean tests are robust tonon-normality for large samples as indicated by the Abstract:
asymptotic N (0, 1) distribution in the last two cells menting large sample tests and confidence in- tervals. One and two sample mean and vari- could think that this also occurs for variance tests. In- ance tests (differences and ratios) are considered.
deed, many practitioners use the classical chi-square The test statistics are all expressed in the same single variance test or Fisher’s two variances test, form as the Student t-test, which facilitates their even if the Gaussian assumption fails. This could presentation in the classroom. This contribution lead to heavy errors, even for large samples, as
also fills the gap of a robust (to non-normality) alternative to the chi-square single variance test this situation as "catastrophic".
for large samples, since no such procedure is im- To have a better idea of the type I error in the plemented in standard statistical software.
classical single variance test, let us test for example 0 : σ = 1 versus H1 : σ < 1, by simulating 10000 Introduction
samples of size 1000 from an E (1) distribution (i.e.
under H0) and using α = 5%. We obtained a percent- It is sometimes desirable to compare two variances age of rejection of the null of 21.53%, thus showing rather than two averages. To cite a few examples a type I error far greater than α. The percentage for the asymptotic test (described later) is 9.05% which is would like two college professors grading exams to not too far from α. For a U ([0, 5]), the classical single have the same variation in their grading; in order for variance test leads to a type I error far lesser than α a lid to fit a container, the variation in the lid and the (0.44%). Our test still behaves correctly with a type container should be the same; a supermarket might I error near α (5.39%). This is mainly due to the de- be interested in the variability of check-out times for parture of the kurtosis of the distribution from 3 (for more theoretical details see e.g. Section 2.2 of Now usually, a first course on statistical inference presents mean tests in both Gaussian and asymptoti- Note that the problem of the robustness (to de- cal frameworks (Table 1), but variance tests are often partures from normality) of tests for comparing two presented only in the Gaussian case (Table 2).
(or more) variances has been widely treated in theliterature, see e.g. ences therein. These authors built specific test statis- tics. Note also that in the one sample (non Gaussian) case, to the best of our knowledge, no statistical tool is available to compare a population variance to a ref- p. 492), that a common method for construct- ing a large sample test statistic may be based on an estimator that has an asymptotic normal distribu- tion. Suppose we wish to test a hypothesis about a parameter θ, and ˆθn is some estimator of θ based on 0 : µ = µre f for both the Gaussian a sample of size n. If we can prove some form of thecentral limit theorem to show that, as n → +∞, where ˆσˆ is the usual standard error, which is a con- vergent (in probability) estimate of σˆ = then one has the basis for an approximate test.
Chi−square variance test
Fisher's ratio of variances test
Figure 1: P-value Plots (see under H0 of m = 10000 replications of teststatistics of the chi-square variance test (top) and Fisher’s ratio of variances test (bottom) in the large sample Gaussian context. The parameters of the simulation are: n = n 1 = n2 = 500, Y = Y1 = Y2 ∼ χ (5) (resp. E (1), resp. U [0, 5]). The dotted lines are 45◦ lines.
This approach can be used to complete Table 2 for framework, with no additional difficulty, to test var- the large sample case, shown in Table 3 for the single ious parameters such as the mean, the variance, and the difference or ratio of means or variances (forlarge samples). This approach also allows the direct derivation of asymptotic confidence intervals. Note ilar asymptotic approach, with a refinement based on a variance stabilizing transformation, to obtain asymptotic confidence intervals, solely for the single variance and ratio of variances cases. Table 4 gives a summary of the various parameters we can test and the R functions we have implemented to computethe standard error ˆσˆ of ˆθ: The case of a (large sample) test for a difference in scale parameters (possibly weighted by a factor also of interest as suggested by the availability of re- lated procedures in R (to compute Ansari-Bradley’s and Mood’s tests for example). The standard error The point to be noted here is that this general ap- Table 4: Various parameters we can test and available very similar to the classical t-test from a mathemat- R functions to compute standard error ˆσˆ.
ical point of view. Proofs, which are not very com-plicated, are provided in the report just cited. The These functions can be used in conjunction with details are not fully expounded here but lead us to to obtain p-values for various tests. For a simple propose a more complete, homogeneous teaching example, if you want to use a sample contained in Table 5: Type I error in terms of n for the test H This contribution also solves the problem of pro- viding an implemented “robust” (to departure of the i.i.d. large sample distribution from normality) al-ternative to the chi-square single variance test for large samples. Indeed, we did not find any such of n for m = 10000 replications of the distribution procedure in standard statistical software and so it is highly likely that practitioners would incorrectly usea chi-square test on a single variance. It also provides a very simple alternative to the (ratio of variances) Fisher test in large samples. Some other “robust” alternative procedures to the Fisher test in the case of non Gaussian (not necessary large) samples are implemented in R: the Bartlett test (bartlett.test), the Fligner test (fligner.test) and the Levene test levene.test available in the lawstat package). R of n for m = 10000 replications of the distribution functions, Ansari-Bradley’s and Mood’s two-sample rank-based tests for a difference in scale parameters.
The purpose of this paper is not to compare our teststo their competitors in terms of power. We neverthe- Using asympTest
less conduct two short simulation studies (limited tothe probability of Type I error): first for the problem of testing a variance (Table 5), comparing the clas- tion asymp.test and six auxiliary ones designed to single variance test to our procedure, and compute standard errors of estimates of different pa- second for the problem of comparing (the differences rameters, see Table 4. The auxiliary functions will not be the most useful ones for the user, except if σ2 of) two variances (Tables 6, 7 and 8), comparing the classical Fisher test to our procedure, as well as he/she wants to compute the confidence interval Ansari-Bradley’s test and Mood’s test. These sim- himself/herself. The function asymp.test has been ulations were based on the three distributions used written in the same spirit as the standard R functions earlier in Figure 1. The simulations show that the t.test or var.test. The arguments of asymp.test level α is quite correct (when n increases) for our and the resulting outputs are also inspired from these procedure in the case of testing a single variance and functions. In particular, the function asympt.test re- for all three alternative tests (ours, Ansari-Bradley’s turns an object of class "htest" (which is the general and Mood’s tests) for testing two variances.
This asymp.test function has several arguments, similar to those of the t.test function, whose description can be obtained using the command In order to illustrate this function, let us con- sider the Digitalis Investigation Group NHLBI of n for m = 10000 replications of the distribution NHLBI. Note that statistical processes such as per- mutations within treatment groups were used to completely anonymize the data; therefore, inferences derived from the teaching dataset may not be valid.
The DIG Trial was a randomized, double-blind, multicenter trial with more than 300 centers in the United States and Canada participating. The pur- pose of the trial was to examine the safety and effi-cacy of Digoxin in treating patients with congestive We can see that var.test, not to be used due to the unlikely normality of the data, significantly shows a Diastolic BP (DIABP, mmHg) is a known risk fac- difference in variances (at a 5% level). We don’t ob- tor of cardiovascular diseases. In this case, it is de- tain the same conclusion with our test.
sirable to compare the variability of this quantity for We can also place ourselves in a fictitious case by placebo (TRTMT=0) and treatment (TRTMT=1) groups, both our test and the classical chi-square test to show Reading of the data
> asymp.test(x, par = "var", alt = "gr", Comparing the two variances
Shapiro-Wilk normality test performed by the func- tion shapiro.test() indicates that the two samples seem to be far from the Gaussian distribution. Thus, this should prevent us from using the following > var.test(DIABP ~ TRTMT, data = DIGdata, > pchisq(chisq.stat, n-1, lower.tail = F) For the above generated sample x, we respectivelyfound the following p-values: 0.0398 and 0.120. In this case, we can thus see that our proposition cor- F = 0.9295, num df = 3399, denom df = 3394 rectly accepts H1 (at the 5% level) but not the chi- true ratio of variances is not equal to 1 Conclusion
This paper has introduced a new package called procedures available. It is interesting firstly in the fact that it provides a unified teaching frameworkto present classical parametric tests (based on the > asymp.test(DIABP ~ TRTMT, data = DIGdata, Central Limit Theorem). These tests are made read- na.action = na.omit, parameter = "dVar") ily available in R through an easy to use functioncalled asymp.test. This function resembles t.test Two-sample asymptotic diff. of variances test or var.test, so students will not be confused. Sec-ondly, it also makes available in R a robust (to non- normality) alternative to the classical chi-square sin- gle variance test. In the future, we also plan to pro- vide tools similar to the power.t.test function in the true diff. of variances is not equal to 0 Bibliography
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