Abstract
Parametric hypothesis testing associated with two independent samples arises frequently in several applications in biology, medical sciences, epidemiology, reliability and many more. In this paper, we propose robust Wald-type tests for testing such two sample problems using the minimum density power divergence estimators of the underlying parameters. In particular, we consider the simple two-sample hypothesis concerning the full parametric homogeneity as well as the general two-sample (composite) hypotheses involving some nuisance parameters. The asymptotic and theoretical robustness properties of the proposed Wald-type tests have been developed for both the simple and general composite hypotheses. Some particular cases of testing against one-sided alternatives are discussed with specific attention to testing the effectiveness of a treatment in clinical trials. Performances of the proposed tests have also been illustrated numerically through appropriate real data examples.
1 Introduction
Testing of parametric hypothesis is an important paradigm of statistical inference. In many real life applications like medical sciences, biology, epidemiology, sociology, reliability etc., we need to compare data from two independent samples through appropriate two-sample tests of hypotheses. Examples include, but are not limited to, comparing the means of any biomarker or success of any treatment between control and treatment groups, comparing lifetime of two populations in reliability, etc.
Mathematically, let
On the basis of two independent random samples
The classical test statistics for testing eq. (1) are the likelihood ratio test, Wald test and Rao test, where the unknown parameters are estimated by the maximum likelihood estimators (MLEs). Some alternative test statistics have also been presented in the literature based on divergence measures; see, for instance, Basu et al. [1] and Pardo [2]. It is well-known that the MLE is a BAN estimator, i.e., asymptotically efficient, but at the same time it has serious lack of robustness against data contamination and model misspecification. In order to avoid the robustness problem, appropriate testing procedures have been developed in the statistical literature based on suitable robust estimators. For example, Basu et al. [3] have introduced a family of test statistics for testing eq. (1) based on the density power divergence (DPD) measure between
Note that, if the problem considered in eq. (1) has been solved, we will be able to apply it for the particular (and most common) normal populations with
But there are other interesting problems of testing for partial homogeneity, for instance, to test
when the variances are the same but unknown; this is a particular case of the general composite hypotheses involving two samples. This particular problem with normal population has been considered in [5] on the basis of a family of test statistics based on the DPD measure and by estimating the unknown parameters using the MDPDE. The results presented in their paper have been excellent in relation to the robustness and efficiency trade-off; for some suggested members of their proposed test family the loss in efficiency based on the size and the power under pure data was not really significant but the improvement in terms of robustness under contaminated data was highly significant. Although their approach can theoretically be extended beyond the simple case of normal populations, from a practical point of view, it is often not very easy to compute the density power divergence measure between
The rest of the paper is organized as follows: In Section 1.1 we present some important background results and definitions in relation to the MDPDE that will be necessary for the rest of the paper. Section 2 is devoted to developing the family of Wald-type tests for solving the problem of complete homogeneity given by eq. (1). We study its asymptotic distribution as well as the theoretical robustness properties with examples in the same section. In Section 3, we present a family of Wald-type tests for the more general composite hypotheses in the two sample context. We again derive their asymptotic distributions and robustness properties. Illustrations are provided for the special case of testing partial homogeneity in presence of nuisance parameters like, for example, testing equality of two normal means with unknown (nuisance) variances. In Section 4, we briefly describe the extensions for testing the two-sample hypotheses against one-sided alternatives. Section 5 presents several real life applications of our proposal with interesting data from applied sciences like medical science, biology, reliability etc. Appropriate simulation studies with some comments on the choice of the robustness tuning parameters are presented in Section 6. The paper ends with a short concluding remark in Section 7. For brevity in presentation, the proofs of all the results have been moved to Appendix 8.
1.1 The minimum density power divergence estimator: Asymptotic properties and robustness
Given any two densities
The divergence corresponding to
Let G represent the distribution function corresponding to the underlying true density g that generates the data and we want to model it by the parametric model density
where
Notice that
the expression in eq. (3) can be written as
The functional
where
where
Next, the influence function (IF) can be used to study the robustness of the MDPDE. Note that, if the influence function is bounded, the corresponding estimator or test statistic is said to have local robustness against infinitesimal contamination. More simply, the influence function
where
2 A simple two-sample problem
Let
In general notation, let us assume that
with
with
Based on eq. (8), the classical Wald test statistic for testing
is given by
We can observe that, when r = p we have
where
As an example, in the case of two normal populations, with known variances
Although it has several nice optimum properties, it is highly non-robust in presence of outliers even in any one sample. Here, we will generalize this classical Wald test to make it robust through replacing the non-robust MLEs by the corresponding robust MDPDEs. In the following we will present the results for r = p, i.e., to test for the hypothesis in eq. (1). The case r = p can be studied in a similar way.
Let us assume
Note that, at
2.1 Asymptotic properties
In order to perform any statistical test, we first need to derive the asymptotic distribution of the test statistics under
Assumption (A):
The asymptotic variance-covariance matrix
Theorem 2.1
Suppose the model density satisfies the Lehmann and Basu et al. conditions, and Assumption (A) holds. Then the asymptotic distribution of
The asymptotic null distribution of the test in [3] is a linear combination of chi-square distribution and hence it is somewhat difficult to obtain the critical values of their test in practice. On the contrary, our proposed tests have a simple chi-square limit under the null hypothesis and hence are much easier to perform. Our proposal provides, in this sense, an advantageous procedure for testing.
However, when the null hypothesis is not correct, i.e.,
Theorem 2.2
Suppose the model density satisfies the Lehmann and Basu et al. conditions, and Assumption (A) holds. Then, as
where
This theorem leads to an approximation to the power function
Corollary 2.3
Under the assumption of Theorem 2.2, we have
for a sequence of distributions
The corollary also helps us to determine the sample size requirement for our proposed test to achieve any pre-specified power level. Further, we have
Corollary 2.4
Under the assumption of Theorem Theorem 2.2, the proposed Wald-type test is consistent in the Fraser’s sense.
Next, we look at the performance of the proposed test under contiguous alternatives. Now, in case of two sample problem, we can have different types of contiguous alternatives. For example, we can assume
for some fixed
Theorem 2.5
Suppose the model density satisfies the Lehmann and Basu et al. conditions and the assumption (A) holds. Then the asymptotic distribution of
We can easily obtain the asymptotic power
Example 2.1 (Testing equality of two Normal means with known equal variances)
We first present the simplest possible case of testing two normal means with known equal variance
and it has
Clearly, these tests are consistent for any
with
Asymptotic contiguous power of the proposed Wald-type test at 95% level for testing equality of two normal means as in Example 2.1 with known common
| β | |||||||
|---|---|---|---|---|---|---|---|
| WΔ1,Δ2 | 0 | 0.1 | 0.3 | 0.5 | 0.7 | 0.9 | 1 |
| 0 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 |
| 1 | 0.170 | 0.169 | 0.160 | 0.150 | 0.140 | 0.131 | 0.127 |
| 2 | 0.516 | 0.511 | 0.484 | 0.449 | 0.413 | 0.380 | 0.364 |
| 3 | 0.851 | 0.847 | 0.821 | 0.784 | 0.742 | 0.698 | 0.677 |
| 5 | 0.999 | 0.999 | 0.998 | 0.996 | 0.992 | 0.985 | 0.981 |
2.2 Influence function of the wald-type test statistics
The robustness of any two sample test is relatively complicated compared to the one sample case because, here, one may have contamination in either of the two sample or even in both the samples. Let us first derive the Hampel’s influence function (IF) of the two sample Wald-type test statistics to study the robustness of the proposed test. Consider the set-up of previous subsection and denote
where
Now consider the contaminated distributions
Similarly, if there is contamination only in the second sample, then the corresponding IF is given by
Finally, if we assume that the contamination is in both the samples, Hampel’s IF turns out to be
where
Therefore, we need to consider the second order influence function for this case of two sample problem. When there is contamination only in the first sample, the corresponding second order IF is given by
For the particular case of null distribution
Similarly, if the contamination is in the second sample only, then the second order IF simplifies to
Note that these two IFs are bounded with respect to the contamination points x or y if and only if the IF of the corresponding MDPDE used is bounded; but it is the case for all
Finally for the case of contamination in both samples, the corresponding second order IF is given by
In particular, at the null hypothesis
Note that if x = y then
Second order influence function of the proposed Wald-type test statistics and corresponding gross error sensitivity
Example 2.2 (Continuation of Example 2.1)
Let us again consider the previous example on testing two normal means as in Example 2.1. We have seen that the proposed Wald-type tests are consistent for all
Clearly, the first order IFs of the test statistics will always be zero. For contamination only in the first sample, the second order IF of the test statistic
It clearly shows that the robustness of our proposed test statistics increases as
Similar inferences can also be drawn for contamination only in the second sample.
Next consider the case when there is contamination in both the samples. In this case, the second order IF is given by
The plot of
Second order influence function of the proposed Wald-type test statistics under contamination in both the samples for testing equality of two normal means as in Example 2.1 with known common
2.3 Power and level influence functions
The robustness of a test statistic, although necessary, may not be sufficient in all the cases since the performance of any test is finally measured through its level and power. In this section, we consider the effect of contamination on the asymptotic power and level of the proposed Wald-type tests. Due to consistency, the asymptotic power against any fixed alternative will be one. So, we again consider the contiguous alternatives
for the level and power calculations respectively along with the usual uncontaminated distributions for the second population. Then the corresponding level influence function (LIF) and the power influence function (PIF) at the null
Similarly, when contamination is assumed to be only in the second sample, then we take the uncontaminated distributions for the first population and the contaminated distribution for the second population as
for the level and power calculations respectively. Corresponding LIF and PIF at the null
Finally, while considering contamination in both the samples with above contaminated distributions, we define the corresponding LIF and PIF as
First let us derive the asymptotic distribution of the proposed Wald-type test statistics
Theorem 2.6
Suppose the model density satisfies the Lehmann and Basu et al. conditions and Assumption (A) holds. Then the asymptotic distribution of
From the above theorem, we get the asymptotic power of the proposed Wald-type tests under the contaminated contiguous alternatives as
Using infinite series expansion of a non-central chi-square distribution function [14], we get
In particular, substituting
Further, substituting
Now we can define the power influence functions of our proposed tests which is nothing but
Theorem 2.7
Suppose the model density satisfies the Lehmann and Basu et al. conditions, and Assumption (A) holds. Then the power influence functions of our proposed Wald-type tests are given by
where
Note that the PIFs are also a function of the influence function of the MDPDE used and hence they are bounded whenever
Note that, although there is no direct relationship between the IF of test statistics with the corresponding PIF in general, in this present case they are seen to be related indirectly via the IF of the MDPDE. So, using a robust MDPDE with
Finally, we can find the level influence function of the proposed Wald-type tests either starting from
provided the corresponding IF of
Example 2.3 (Continuation of Examples 2.1 and 2.1)
Let us again consider the problem of testing for normal means as in Examples 2.1 and 2.6. As seen above, the level influence function is always zero implying the level robustness of our proposed Wald-type test for all
Power influence functions of the proposed Wald-type test statistics at 95% level for testing equality of two normal means as in Example Example 2.2 (Continuation of Example 2.1) with known common
3 General composite hypotheses with two samples
In the previous section, we have considered the simplest two sample problem which tests for equality of all the model parameters. However, in practice, we need to test many different complicated hypotheses which cannot be solved just by considering the Wald-type test statistic
In this section, we will consider a general class of hypotheses involving two independent samples, which would include most of the above real life testing problems. Suppose
In particular, the problem of testing normal mean with unknown variance can be seen as a particular case of the above general set-up with
Now, to define a robust Wald-type test statistics for this general set-up, we again consider the MDPDEs of
where
Note that, at
Interestingly, although the general hypothesis contains the hypothesis eq. (9) as its special case, the Wald-type test statistics
Assumption (B):
3.1 Asymptotic properties
We again start with the asymptotic null distribution of the proposed Wald-type test statistics
Theorem 3.1
Suppose the model density satisfies the Lehmann and Basu et al. conditions and Assumptions (A) and (B) hold. Then, under the null hypothesis in eq. (15),
Therefore, the level-
Next, in order to consider an approximation to the asymptotic power for this general test based on
Theorem 3.2
Suppose the model density satisfies the Lehmann and Basu et al. conditions and Assumptions (A)-(B) hold. Then, whenever
Note that, from the above theorem, we can easily obtain an approximation to the power function of the proposed level-
for a sequence of distributions
In such cases, it can be easily checked that
Corollary 3.3
Under the assumptions of Theorem Theorem 3.2, the proposed Wald-type tests based on
Now, let us study the performance of the proposed general two-sample Wald-type tests under the contiguous alternative hypotheses. As discussed in the previous section, there could be different choices for the contiguous alternative hypotheses for any general null hypothesis. Here, following the similar idea as in the alternatives in eq. (12), we consider the general form of the contiguous alternatives given by
for some fixed
Theorem 3.4
Suppose the model density satisfies the Lehmann and Basu et al. conditions and Assumptions (A)-(B) hold. Then the asymptotic distribution of
with
The above theorem directly helps us to obtain the asymptotic power
3.2 Robustness properties
Let us now study the robustness properties of the proposed general two-sample Wald-type tests based on
where
Theorem 3.5
Consider the notations of Section 2.2. Under the null hypothesis in eq. (15) with
For contamination only in the i-th sample (i = 1,2) at the point
For contamination in both the samples
with
Clearly, as in the previous case of simple two sample problem in Section 2.2, here also the first order IF of the test statistics are always zero and hence non-informative about their robustness. However, their second order IFs are clearly bounded whenever the IF of the corresponding MDPDE is bounded which holds for all
Next, we consider the effect of contamination on the asymptotic power and level of the proposed general Wald-type tests based on
Theorem 3.6
Suppose the model density satisfies the Lehmann and Basu et al. conditions and Assumptions (A)-(B) hold. Then, the asymptotic distribution of the general Wald-type test statistics
The above theorem can be used to get the asymptotic power of the proposed general two-sample Wald-type tests under the contiguous contaminated alternatives in terms of an infinite series following Section 2.3 (arguments after 2.6). This can be also simplified by substituting
Theorem 3.7
Suppose the model density satisfies the Lehmann and Basu et al. conditions, and Assumptions (A)–(B) hold. Then we have the following results for the proposed Wald-type test functional
The power influence functions are given by
where
Provided the IF of the MDPDE
Note that for the general two-sample hypothesis eq. (15) also, the LIFs and the PIFs of our proposed test are bounded whenever the influence function of the MDPDE used is bounded which holds for all
3.3 Special case: Testing partial homogeneity with nuisance parameters
Let us consider a simplified and possibly the most common special case of the general hypothesis in eq. (15), where we test for partial homogeneity of the two samples assuming some parameters to be nuisance. Mathematically, let us consider the partition of the parameters
with
Note that the hypothesis eq. (18) is indeed a special case of the general hypothesis in eq. (15) with
where
for some fixed
Next, for examining robustness properties, we define the corresponding test functional following Section 3.2 as given by
where
and the same for contamination in both samples is given by
with
where
Example 3.1 (Testing equality of two Normal means with unknown and unequal variances)
We again consider the example of comparing two normal means (say
with
with
whose null asymptotic distribution is
where
We can now study the asymptotic and robustness properties of these proposed Wald-type tests following the theoretical results derived in this section. However, due to the asymptotic independence of the MDPDEs of
4 The cases of one-sided alternatives
As we have mentioned in the introduction (Section 1), majority of common practical applications of the two-sample problems are in comparing the treatment and control groups in any experimental or clinical trials or any observational studies among two such groups of population. However, in most of such cases, researchers want to test weather there is any improvement in the treatment group over the control groups due to the treatment effects. For example, one might be interested to test if the success rate of cure (modeled by binomial probability model) is reduced, or if the number of attacks of a disease (modeled by Poisson model) decreases in the treat group, or some continuous biomarkers like blood pressure etc. (modeled by normal model) changes in the targeted direction from control to treatment group. All of them lead to the one-sided alternatives in contrast to the omnibus two-sided alternatives considered so far in this paper. Although the case of general one-sided alternatives with vector parameters are much difficult to define and dealt with and hence need more targeted future research, our proposal of robust Wald-type tests in this paper can be easily extended for comparing any scalar parameters with one-sided alternatives. Noting that all the above motivating practical scenarios indeed deal with scalar parameter comparison, in this section we extend our proposal to these particular one sample problems.
In general, we consider the class of one-sided version of eq. (15) with r = 1. So,
Note that the one sided version of the simple two-sided hypothesis in eq. (9) with scalar parameters (p = 1), that contains the motivating examples for Poisson and binomial models and normal model with known variances, belong to this general class eq. (23). Also, this general class of hypotheses contains many more useful cases like testing for increase (or decrease) in normal means with unknown variances.
For testing the one sided hypothesis eq. (23), we define the corresponding robust Wald-type test statistics by taking a signed square-root of our two-sided Wald-type test statistics
where sgn(
Theorem 4.1
Under the assumptions of Theorem Theorem 3.1, the asymptotic null distribution of the one-sided test statistics
Following the above theorem, the level-
Further, as in the case of two-side alternatives, we can also derive an power approximation of these proposed Wald-type tests at any fixed alternative
for a sequence of distributions
Now, since
Next to study the contiguous power of the proposed Wald-type tests, we can consider the class of contiguous alternatives in eq. (17) but now with
with
Theorem 4.2
Under the assumptions of Theorem 3.4, the asymptotic distribution of
Now we can also derive the robustness properties of the proposed Wald-type tests against one-sided alternatives by defining the corresponding statistical function as
Then, under the assumptions of Theorem Theorem 3.5 with contamination in only i-th sample at the point
and the same for contamination in both the samples is given by
with
In order to derive the corresponding level and power influence functions, we consider the same set of hypothesis as in Section 3.2 but now with the restriction
Theorem 4.3
Under the assumptions of Theorem 3.2, the asymptotic distribution of
Using above theorem and following the arguments similar to those for the two-sided alternatives in Section 3.2, we can get the power influence functions for this case of one-sided alternatives also, which is presented in the next theorem.
Theorem 4.4
Under the assumptions of Theorem 3.7, the power influence functions of our proposed Wald-type test functional
Note that, the nature of these PIFs with respect to the contamination points x and y are exactly same as those in the case of two-sided alternatives except for a multiplicative constant. In particular, they are bounded whenever the influence function of the MDPDE used is bounded, i.e., at
Finally, we can get the level influence functions from the above theorem by substituting
For illustration, we will again present the case of normal model with one-sided alternatives in the following example. Other motivating models with relevant data examples will be provided in the next section.
Example 4.1 (Comparing two Normal means against one-sided alternatives)
Let us again consider the two-sample problem under normal model with unknown and unequal variances as in Example 3.1, but now with the one-sided alternatives so that our target hypothesis is
with the variance parameters
which has standard normal asymptotic distribution under the null. Clearly this statistic also coincides with the corresponding classical Wald test statistic at
Asymptotic contiguous power of the proposed Wald-type tests at 95% level for testing equality of two normal means against one-sided alternatives as in Example 4.1.
| β | |||||||
|---|---|---|---|---|---|---|---|
| d | 0 | 0.1 | 0.3 | 0.5 | 0.7 | 0.9 | 1 |
| 0 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 |
| 1 | 0.260 | 0.258 | 0.247 | 0.233 | 0.219 | 0.207 | 0.201 |
| 2 | 0.639 | 0.634 | 0.608 | 0.574 | 0.538 | 0.503 | 0.487 |
| 3 | 0.912 | 0.909 | 0.891 | 0.865 | 0.833 | 0.798 | 0.780 |
| 5 | 1.000 | 1.000 | 0.999 | 0.998 | 0.997 | 0.994 | 0.991 |
Further, the influence function of the proposed Wald-type test statistics in this case of one-sided alternatives simplifies to
and
Note that these influence functions are square roots of the corresponding influence functions under two-sided alternatives in Example Example 2.2 (Continuation of Example 2.1) except for a multiplicative constant. Further, by the general theory developed above, the corresponding PIFs and LIFs in this case can be shown to be also a constant multiplication of the corresponding PIFs in the two-sided case presented in Example Example 2.2 (Continuation of Example 2.1). Therefore, the boundedness nature of all these influence functions for the one-sided alternative will be similar to those presented in Figure 1a and Figure 3, i.e., bounded at
5 Real life applications
5.1 Poisson model for clinical trial: Adverse events data
In our first example we will consider the application of the proposed Wald-type tests with Poisson model to the adverse event data in an Asthma clinical trial conducted by Kerstjens et al. [15][Table 3]. In this two phase randomized controlled trials, 912 patients having asthma and receiving inhaled glucocorticoids and LABAs had been divided into treatment and control groups of the two trials and were randomly assigned a total dose of 5 g tiotropium (treatment group) or suitable placebo (control group) once daily for 48 weeks. Then, Kerstjens et al. [15] investigated the effect of this combined treatment on patient’s lung function and exacerbations.
No of Different adverse events reported in Trial 2 of the Kerstjens et al. [15] clinical trail study.
| Treatment | 91 | 49 | 19 | 12 | 12 | 3 | 13 | 10 | 6 | 3 | 3 | 7 | 6 | 5 | 4 | 4 | 3 | 2 | 0 |
| Control | 109 | 58 | 20 | 13 | 10 | 10 | 6 | 4 | 5 | 7 | 5 | 1 | 2 | 4 | 4 | 5 | 2 | 2 | 1 |
Here we will consider the data on 19 reported adverse effect on the patients in trail 2 of this study, presented in Table 3, that can be modeled by a Poisson distribution with mean
We have applied our proposed Wald-type tests for this problem, as developed in Section 4, to both the full dataset and after deleting the first two outliers from both the groups; the resulting p-values are presented in Figure 4a. Clearly, the classical Wald test results in completely different inference due to the inclusion of these outlying observations – it’s p-value becomes significant from non-significant inference without them (at 95% level). On the other hand, proposed MDPDE based robust Wald-type tests with
MDPDEs of Poisson parameter
| (β) | ||||||||
|---|---|---|---|---|---|---|---|---|
| Group | 0 | 0.1 | 0.3 | 0.5 | 0.7 | 0.9 | 1 | |
| With | Treatment | 15.00 | 7.25 | 6.94 | 6.35 | 5.86 | 6.05 | 5.70 |
| Outlier | Control | 18.47 | 8.25 | 7.75 | 7.56 | 7.53 | 7.41 | 7.81 |
| Without | Treatment | 8.82 | 7.47 | 6.44 | 6.20 | 6.14 | 5.58 | 6.58 |
| Outlier | Control | 9.65 | 7.97 | 7.63 | 7.61 | 7.56 | 7.68 | 7.75 |
5.2 Poisson model for experimental trial: Drosophila data
We next consider another application to the Poisson model with data from an controlled experimental trial with Drosophila flies producing occasional spurious counts. The dataset contains two independent samples on the numbers of recessive lethal mutations observed among the daughters of male flies who are exposed either to a certain degree of chemical to be screened (treatment group) or to control conditions. This dataset has been previously analyzed by many statisticians including Woodru et al. [17], Simpson [16], Basu et al. [3] who have shown that the response data can be modeled by Poisson distribution, but there are two outlying observations in one sample that affects the likelihood based inference and so the classical Wald test. See Basu et al. [3][Table 7] for the dataset and the MDPDEs of the Poisson parameters.
Here, we will apply the proposed Wald-type tests for comparing the Poisson parameters for the two samples, say
P-values of the proposed Wald-type tests under the real data examples with outliers (solid line) and without outliers (doted line).
5.3 Normal model for clinical trial: Infant platelet count data
We will now present another clinical trial example from Karpatkin et al. [18] to illustrate the applications under the normal model. This clinical trial was conducted to study if the infant platelet count can be increased by giving steroids to the mothers with autoimmune thrombocytopenia during pregnancy. The study consists of 19 mothers with 12 being given steroid (treatment group) and 7 not given steroid (control group) and the corresponding infant platelet counts (in thousands, per mm
Infant Platelet count after delivery (in thousands, per mm
| Treatment | 120 | 124 | 215 | 90 | 67 | 126 | 95 | 190 | 180 | 135 | 399 | 65 |
| Control | 12 | 20 | 112 | 32 | 60 | 40 | 18 |
The p-values for this testing problem obtained by applying the proposed Wald-type tests, as described in Example 4.1, are presented in Figure 4c for different
5.4 Normal model for health study: Hair Zn content data
Two-sample test under the normal model has many possible applications from which we now present a health study to examine the impact of polluted urban environment over individual health in Sri Lanka. The dataset consist of the zinc (Zn) content of the hair of two independent samples taken from urban (polluted) and rural (unpolluted) Sri Lanka and our target is to check if the Zn content is more for polluted urban residents impacting their health conditions. The dataset was presented in Basu et al. [5][Table 6] and it has been shown their that each sample can be modeled by normal distributions with means
We have applied the proposed MDPDE based Wald-type test for this problem following Example 4.1 and the resulting p-values are presented in Figure 4d. Clearly, the significance increase of the zinc contents in urban residents cannot be identified by the classical Wald-test in presence of outliers, but our proposal with
5.5 Normal model for quality control: Cloth manufacturing data
Our third and final example with normal model will be in the context of quality control based on the data from the Levi-Strauss clothing manufacturing plant. The dataset consists of 22 measurements on run-up (a percentage measure of wastage in cloth) for each of two particular mills supplying cloths to the plant [5][Table 1]. To control the quality of the cloths, the plant want to test for the consistency of the run-up measures from the two mills. Since the sample from each mill can be modeled by normal distribution with mean
with
5.6 Exponential model for reliability testing: Components life-time data
We will end this section with an example of exponential model used in reliability testing between two sets of products’ lifetimes. We will use the (simulated) data from Perng [19] which consist of the lifetimes (in thousand of hours) of a particular electronic components produced by two different processes (see Table 6). Each sample can be then modeled by exponential distributions with mean
Lifetimes (in thousand of hours) of a particular electronic components produced by two different processes [19].
| Process 1 | 0.044 | 0.134 | 0.142 | 0.158 | 0.216 | 0.625 | 0.649 | 0.658 | 1.062 | 1.140 | 1.159 | 1.238 |
| Process 2 | 0.060 | 0.174 | 0.237 | 0.272 | 0.335 | 0.391 | 0.670 | 0.902 | 1.543 | 1.615 | 2.013 | 2.309 |
Since there is no outliers in this dataset, in order to study the robustness aspect of our proposal we add one outlying value of 20 (assuming a decimal is misplaced by one digit from 2.0) in the second sample. The resulting p-values obtained by the proposed Wald-type tests for both the pure data and with this artificial outlier are presented in Figure 4f for different
6 Simulation study and the choice of tuning parameter β
Finally, to examine the finite sample performances of our Wald-type tests, we have performed several simulation studies with all the models considered in the previous section for real datasets. However, noting the similarity of the results for different models, for brevity, here we will report the results from only one simulation study under normal model with two-sided alternatives.
We simulate 1000 pairs of samples, each of size n = 50, independently drawn from
Empirical sizes and powers of the proposed Wald-type tests for testing equality of two normal means with both the known and unknown variance case at sample size n = 50 under pure data (solid line) and with contamination of 10% (dash-doted line), 15% (doted line) and 20% (dashed line).
It can be easily observed from Figure 5 that the size and power of the proposed Wald-type tests under pure data change (increase and decrease, respectively) only very slightly with increasing
To further illustrate the advantages of our proposed tests compared to the non-parametric Wilcoxon rank-sum test, we have repeated the above simulation exercise to derive the corresponding empirical sizes and powers of the Wilcoxon test. This Wilcoxon test is equivalent to the two-sample Mann-Whitney test and is the most commonly used default method for robust two-sample tests of hypotheses. The resulting values of its empirical sizes and powers are reported in Table 6 along with the same for the classical Wald test and the proposed MDPDE based Wald-type tests at some particular
Empirical sizes and powers for the classical Wald test, the non-parametric Wilcoxon rank-sum test and the proposed MDPDE based Wald-type tests at different
| Cont. | Wald | Wilcoxon | MDPDE based Wald-Type tests with β | |||||
|---|---|---|---|---|---|---|---|---|
| Prop. | Test | Test | 0.1 | 0.3 | 0.5 | 0.7 | 1 | |
| Size | 0% | 0.049 | 0.047 | 0.049 | 0.053 | 0.052 | 0.052 | 0.058 |
| 10% | 0.209 | 0.116 | 0.163 | 0.104 | 0.079 | 0.069 | 0.064 | |
| 15% | 0.466 | 0.248 | 0.374 | 0.228 | 0.155 | 0.106 | 0.075 | |
| 20% | 0.652 | 0.395 | 0.556 | 0.408 | 0.289 | 0.187 | 0.119 | |
| Power | 0% | 1.000 | 1.000 | 1.000 | 1.000 | 0.999 | 0.993 | 0.979 |
| 10% | 0.628 | 0.908 | 0.747 | 0.904 | 0.961 | 0.970 | 0.959 | |
| 15% | 0.292 | 0.728 | 0.416 | 0.681 | 0.859 | 0.926 | 0.937 | |
| 20% | 0.138 | 0.492 | 0.209 | 0.403 | 0.641 | 0.793 | 0.874 | |
Throughout all our example and simulations above, we have notices that the tuning parameter
Although this ad hoc empirical choice of
over
Although there is no direct choice for
Histograms for optimally chosen tuning parameter
We have implemented this proposal for the above simulation study with normal model to check its effectiveness. Figure 6 presents the histograms of the 1000 selected optimum
7 Concluding remarks
In this paper, we have considered the problem of testing with two independent samples of i.i.d. observations and proposed a class of robust Wald-type tests for both simple and composite hypothesis testing. These Wald-type tests are constructed using the robust minimum density power divergence estimators of the underlying parameters in each sample. The asymptotic and robustness properties of the proposed Wald-type tests have been discussed along with their applications to several important real-life problems like clinical trial, medical experiment, reliability testing and many more.
Our focus in this paper has been on robust two sample tests. Nonparametric methods and robust methods share some common goals, yet robust methods are inherently different from nonparametric methods as they are essentially parametric, although they allow the parametric model to be only approximately true. It is well known that when a parametric model does hold, the parametric procedures are much more efficient compared to the nonparametric methods. However, when the parametric model holds only approximately, the robust methods are still often substantially more efficient in doing inference about the major component of the data generating distribution compared to nonparametric methods. This has been amply demonstrated by the simulations reported in Table 7. And while parametric models may never “exactly” fit the data, they often provide reasonable “approximate” fits to many practical data sets. So we expect that our method will have a better scope of application in real problems compared to classical parametric methods, and will have greater efficiency in many cases compared to nonparametric methods; in either case, our method will have better robustness properties.
Although we have discussed all possible types of general two-sample hypotheses, in this paper, we have restricted our attention to the cases where each of the two independent samples is identically distributed. The natural extension of this work will be to develop robust tests for hypotheses involving two independent samples from non-homogeneous populations; this also has many practical applications including comparing the regression lines between two groups of patients in a fixed design clinical trial. Also, one could further explore the possibility of robust hypothesis testing using the minimum density power divergence estimators for two paired samples or for more than two sample cases. we hope to pursue some of this possible extensions in our future research.
8 Proof of Results
8.1 Proof of Theorem 2.1
Using the asymptotic distribution of
and
Hence under
Further, under
8.2 Proof of Theorem 2.2
Note that,
Then, the theorem follows from the above expression by noting that
as
8.3 Proof of Theorem 2.5
Using the asymptotic distribution of
and
Hence, under
from which the theorem follows immediately. □
8.4 Proof of 2.6
We will only prove the case
Let us denote
Similarly, we have
Combining all these, we get
and
Hence, under
and hence the theorem follows immediately. □
8.5 Proof of Theorem 3.1
Using suitable Taylor series expansion, we get
Now, from the asymptotic distribution of
and
Hence under
Finally, by the consistency of the MDPDEs and the continuity of the matrices
8.6 Proof of Theorem 3.2
Using an appropriate Taylor series expansion, we get
Then, the theorem follows from the asymptotic distributions of the MDPDEs
8.7 Proof of 3.4
Using the asymptotic distribution of
and
Hence, following the proof of Theorem 2.5, we get under
from which the theorem follows immediately. □
8.8 Proof of Theorems 3.2 and 3.7
These proofs are similar to that of Theorems 2.6 and 2.7 and hence omitted. □
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