Jth Group Research Articles

A Heteroscedastic Method for Comparing Regression Lines at Specified Design Points When Using a Robust Regression Estimator

It is well known that the ordinary least squares (OLS) regression estimator is not robust. Many robust regression estimators have been proposed and inferential methods based on these estimators have been derived. However, for two independent groups, let θj (X) be some conditional measure of location for the jth group, given X, based on some robust regression estimator. An issue that has not been addressed is computing a 1 − confidence interval for θ1(X) − θ2(X) in a manner that allows both within group and between group hetereoscedasticity. The paper reports the finite sample properties of a simple method for accomplishing this goal. Simulations indicate that, in terms of controlling the probability of a Type I error, the method performs very well for a wide range of situations, even with a relatively small sample size. In principle, any robust regression estimator can be used. The simulations are focused primarily on the Theil-Sen estimator, but some results using Yohai’s MM-estimator, as well as the Koenker and Bas sett quantile regression estimator, are noted. Data from the Well Elderly II study, dealing with measures of meaningful activity using the cortisol awakening response as a covariate, are used to illustrate that the choice between an extant method based on a nonparametric regression estimator, and the method suggested here, can make a practical difference.

ANCOVA: A Global Test Based on a Robust Measure of Location or Quantiles When There Is Curvature

For two independent groups, let Mj(x) be some conditional measure of location for the jth group associated with some variable Y, given that some covariate X = x. The goal is to test H0: M1(x) = M2(x) for ∀x ∈ {x1,...,xp} when using a robust or quantile location estimator

ANCOVA: a heteroscedastic global test when there is curvature and two covariates

For two independent groups, let $$M_j(\mathbf {X})$$Mj(X) be some conditional measure of location for the jth group associated with some random variable Y given $$\mathbf {X}=(X_1, X_2)$$X=(X1,X2). Let $$\Omega =\{\mathbf {X}_1, \ldots , \mathbf {X}_K\}$$Ω={X1,ź,XK} be a set of K points to be determined. An extant technique can be used to test $$H_0$$H0: $$M_1(\mathbf {X})=M_2(\mathbf {X})$$M1(X)=M2(X) for each $$\mathbf {X} \in \Omega $$XźΩ without making any parametric assumption about $$M_j(\mathbf {X})$$Mj(X). But there are two general reasons to suspect that the method can have relatively low power. The paper reports simulation results on an alternative approach that is designed to test the global hypothesis $$H_0$$H0: $$M_1(\mathbf {X})=M_2(\mathbf {X})$$M1(X)=M2(X) for all $$\mathbf {X} \in \Omega $$XźΩ. The main result is that the new method offers a distinct power advantage. Using data from the Well Elderly 2 study, it is illustrated that the alternative method can make a practical difference in terms of detecting a difference between two groups.

Decomposition of Prediction Error in Multilevel Models

ABSTRACT We present a decomposition of prediction error for the multilevel model in the context of predicting a future observable y *j in the jth group of a hierarchical dataset. The multilevel prediction rule is used for prediction and the components of prediction error are estimated via a simulation study that spans the various combinations of level-1 (individual) and level-2 (group) sample sizes and different intraclass correlation values. Additionally, analytical results present the increase in predicted mean square error (PMSE) with respect to prediction error bias. The components of prediction error provide information with respect to the cost of parameter estimation versus data imputation for predicting future values in a hierarchical data set. Specifically, the cost of parameter estimation is very small compared to data imputation.

Pairwise comparisons of dependent groups based on medians

For J ⩾ 2 dependent groups, let θ j be the 0.5 quantile associated with the marginal distribution of the jth group. The paper focuses on two methods for testing H 0 : θ j = θ k for all j < k . The first uses an estimate of the standard error of the estimated difference, with the estimates of the 0.5 quantiles based on a single order statistic. In simulations it is found to perform well with a sample size greater than or equal to 30. The second method uses the usual sample median in conjunction with a so-called bias adjusted percentile bootstrap method and performs well in simulations with sample sizes as small as 11.

Using Trimmed Means to Compare K Measures Corresponding to Two Independent Groups

Consider two independent groups with K measures for each subject. For the jth group and kth measure, let μtjk be the population trimmed mean, j = 1, 2; k = 1, ..., K. This article compares several methods for testing H 0 : u1k = t2k such that the probability of at least one Type I error is, and simultaneous probability coverage is 1 - α when computing confidence intervals for μt1k - μt2k . The emphasis is on K = 4 and α = .05. For zero trimming the problem reduces to comparing means, but it is well known that when comparing means, arbitrarily small departures from normality can result in extremely low power relative to using say 20% trimming. Moreover, when skewed distributions are being compared, conventional methods for comparing means can be biased for reasons reviewed in the article. A consequence is that in some realistic situations, the probability of rejecting can be higher when the null hypothesis is true versus a situation where the means differ by a half standard deviation. Switching to robust measures of location is known to reduce this problem, and combining robust measures of location with some type of bootstrap method reduces the problem even more. Published articles suggest that for the problem at hand, the percentile t bootstrap, combined with a 20% trimmed mean, will perform relatively well, but there are known situations where it does not eliminate all problems. In this article we consider an extension of the percentile bootstrap approach that is found to give better results.

Comparing the slopes of two independent regression lines when there is complete heteroscedasticity

For two independent groups, let (X1j, Y1j), …, (Xnjj, Ynjj) be a random sample from some bivariate distribution corresponding to the jth group, and assume that Yij = β1jXij + β0j + εij. Type I heteroscedasticity is defined to be a situation where for fixed j, ɛij does not have a common variance. That is, the variance of εij depends on the value of X. Type II heteroscedasticity is taken to mean that the variance differs between groups. That is, the variance of εi1 is not equal to the variance of εi2 even when Xi1 = Xi2. Complete heteroscedasticity refers to a situation where there is both type I and II heteroscedasticity. This paper considers the problem of computing a confidence interval for β11 — β12, the difference between the slopes, when there is complete heteroscedasticity. Some results on computing a confidence interval for the difference between the intercepts are reported as well. The one method found to give good results is used to reanalyse some data from a study dealing with pygmalion in the classroom.

ANCOVA based on comparing a robust measure of location at empirically determined design points

For two independent groups, let (Yij), Xij), i = 1, …, nj; J = 1, 2 be a random sample of nj observations from the jth group. Suppose that for the jth group, the population trimmed mean of Y, given that X = x, is given by some unknown function mj(x). The paper describes a method of comparing the two regression lines at q design points determined empirically. The proposed procedure is based in part on a simple application of the running interval smoother which allows mj(x) to be modelled non‐parametrically. In contrast to extant methods, it is not assumed that the error term associated with the jth group has constant variance, nor is it assumed that the variance of Y, given X, is the same for both groups. The power of the new method compares well to the conventional method when the standard assumptions of normality, parallel regression lines, and homogeneity of variance are true. The new method can have substantially higher power when standard assumptions are violated, it provides good control over the probability of a Type I error for a much wider range of situations, and it provides a more detailed indication of where regression lines differ and by how much.

Pairwise Comparisons Using Trimmed Means or M‐Estimators when Working with Dependent Groups

AbstractFor J dependent groups, let θj, j = 1, …, J, be some measure of location associated with the jth group. A common goal is computing confidence intervals for the pairwise differences, θj — θk, j &lt; k, such that the simultaneous probability coverage is 1 — α. If means are used, it is well known that slight departures from normality (as measured by the Kolmogorov distance) toward a heavy‐tailed distribution can substantially inflate the standard error of the sample mean, which in turn can result in relatively low power. Also, when distributions differ in shape, or when sampling from skewed distributions with relatively light tails, practical problems arise when the goal is to obtain confidence intervals with simultaneous probability coverage reasonably close to the nominal level. Extant theoretical and simulation results suggest replacing means with trimmed means. The Tukey‐McLaughlin method is easily adapted to the problem at hand via the Bonferroni inequality, but this paper illustrates that practical concerns remain. Here, the main result is that the percentile t bootstrap method, used in conjunction with trimmed means, gives improved probability coverage and substantially better power. A method based on a one‐step M‐estimator is also considered but found to be less satisfactory.

グループ分けの新しい基準

The grouping of n objects into g groups is equivalent to the assignment of the every membership α_<ij> of ith object to jth group. When these α_<ij>'s are un-known, a criterion to get optimal α_<ij>'s from obtained date is proposed. Making use of the hypothesis decomposition for the contingency table, the change in the amount of information by grouping I_g is defined as I_g=n log n+Σ__iΣ__j α_<ij>logα_<ij>-Σ__j n_jlogn_j, 0≦α_<ij>≦1,Σ__j α__ij=1,where n_j is the number of members in jth group. On the other hand, we also have the parallel hypothesis decomposition with that for the contingency table, if the object data is assumed to be drawn from the normal population. Then from a geometrical viewpoint a new criterion FGC (Fuzzy Group Criterion) is proposed in order to search for an optimal grouping such as FGC=S_W/S_T+I_g/I_n, where S_T and S_W denote the total S.S. and the within S.S., respectively, and I_n=nlog n which denotes the change in the amount of information by grouping of n objects into n non-fuzzy groups. An optimal grouping is then selected to have the minimum FGC. This grouping strategy is intended to have the small within S.S. without a clear-cut grouping, that is, with small I_g. It may lead to a fuzzy grouping optimal. The application of this strategy to the fuzzy k-means method is described, where the optimal number of clusters and the parameter value which controls the degree of separation between clusters are searched for. An example to two artificial two-dimensional data sets is also given in which we get a result that the fuzzy clusters are optimal if the boundary between clusters seems obscure and that the non-fuzzy ones are optimal if it seems clear.

Comparing robust regression lines corresponding to two independent groups

Let X and Y be random variables, and for two independent groups, suppose that for the jth group where E(∈j) =0, j=1,2. Suppose that the β's are estimated with Welsch's robust procedure, and that the goal is to test Hoβ01 =β02 and Hoβ11 =β12. A solution to this problem is to use the influence function estimate of the standard errors of the estimates of the β's, and attempt to control the probability of a Type I error via the percentile - t bootstrap procedure. This paper uses simulations to assess the small - sample performance of this procedure. A few results are also reported on a procedure for testing Ho: E(Y1|X) =E(Y2|X).

Non‐parametric analysis of covariance based on predicted medians

A common problem is comparing two independent treatment groups in terms of some random variable Y when there is some covariate X. Typically the comparison is made in terms of E(Y|X). However, highly skewed distributions occur in psychology (Micceri, 1989), and so a better method might be to compare the two groups in terms of the median of Y given X, say M(Y|X). For the jth group, assume that M(Y|X) = βjX + αj. The βs and αs can be estimated with the Brown‐Mood procedure, but there are no results on how one might test H0:β1 = β2 or H0:α1 = α2. Let and be the estimates of α and β, respectively. One of the more obvious approaches is to use a jackknife estimate of the variance of and then assume that the resulting test statistics have a standard normal distribution. This approach was found to be unsatisfactory. Some alternative procedures were considered, some of which gave good results when testing H0:α1 = α2, but they were too conservative, in terms of Type I errors, when testing H0:β1 = β2. Still another procedure was considered and found to be substantially better than all others. The new procedure is based on a modification of the Brown‐Mood procedure and a bootstrap estimate of the standard errors. Some limitations of the new method are noted.

One-node closed multichannel service system: several types of customers and service rates, and random pick-up from the waiting line

A closed queueing system has Q service channels and a waiting line. There are Ni customers of type i in the system, i = 1, ···, m, ∑ m i=1 Ni = N &amp;gt; Q. Q customers are served and Q 0 = N – Q stay in the waiting line. Q channels are partitioned into n groups with Qj channels in the jth group, j = 1, ···, n. The service time of the ith type customer by a channel of the jth group is τ ij ~ Exp (λ ij ). When a customer leaves the channel, it is immediately replaced by another one picked up randomly from the waiting time. The customer which has cleared service joins the waiting line without delay. Let Xij be the number of ith type customers served by jth group channels in equilibrium. An explicit formula for P(Xij = kij , i = 1, ···, m; j = 1, ···, n) is found. It is shown in a form of a local limit theorem that the asymptotic distribution of {Xij } is a multidimensional normal, if Qj/N and Qj/N have positive limits as N → ∞. Formulas for mean values and covariances are given. It turns out that the means of Xij and covariances between Xij and Xrk can be found, using an efficient iterative algorithm, from the deterministic version of the system. A numerical example demonstrates that the normal approximation is rather accurate.

One-node closed multichannel service system: several types of customers and service rates, and random pick-up from the waiting line

A closed queueing system has Q service channels and a waiting line. There are Ni customers of type i in the system, i = 1, ···, m, ∑mi=1Ni = N &gt; Q. Q customers are served and Q0 = N – Q stay in the waiting line. Q channels are partitioned into n groups with Qj channels in the jth group, j = 1, ···, n. The service time of the ith type customer by a channel of the jth group is τij ~ Exp (λij). When a customer leaves the channel, it is immediately replaced by another one picked up randomly from the waiting time. The customer which has cleared service joins the waiting line without delay. Let Xij be the number of ith type customers served by jth group channels in equilibrium. An explicit formula for P(Xij = kij, i = 1, ···, m; j = 1, ···, n) is found. It is shown in a form of a local limit theorem that the asymptotic distribution of {Xij} is a multidimensional normal, if Qj/N and Qj/N have positive limits as N → ∞. Formulas for mean values and covariances are given. It turns out that the means of Xij and covariances between Xij and Xrk can be found, using an efficient iterative algorithm, from the deterministic version of the system. A numerical example demonstrates that the normal approximation is rather accurate.

Estimation of acidic and physical properties of solid acids based on adsorption of basic molecules in non-polar solvents

The adsorption equilibrium among a solid acid having both acidic and physically adsorptive sites, a base and a Hammett indicator have been investigated theoretically to derive a new surface acidity function SA and a physical adsorptivity function SP. A system composed of silica–alumina as a solid acid, pyridine as a base and each Hammett indicator have been experimentally treated for application of SA and SP. SA and SP are defined by SA=–log (Σmi= 1αi[Ai]) where αi and [Ai] are the strength and amount of the ith group of vacant acid sites, respectively, and SP=–log (Σnj= 1µj and [Pj]), where µj and [Pj] are the strength and amount of the jth group of vacant physical adsorptive sites, respectively. Silica–alumina dispersed in benzene or benzene–cyclohexane was neutralized to various degrees with pyridine in the presence of one of five Hammett indicators and the solid and liquid phases were subjected to u.v.–visible spectrophotometric measurements. A data analysis based on SA has revealed that the silica–alumina contains 0.11 m mol g–1 acid sites of –log α=–3.9 ± 0.6 as referenced to –logβ= 1.5, assigned as a standard for 4-(phenylazo)diphenylamine, a Hammett indicator, where β is the strength of a base in a similar sense to α. Relative strengths have been evaluated for the five Hammett indicators as basic and physical adsorbates to the silica-alumina. An experimental criterion has been set up for the application of SA to the determination of the strength–amount distribution for a solid acid.

Combining Intra and Inter Block Analysis of Group Divisible Designs

Introduction: Kapadia and Weeks [2] suggested an alternative method of computing the adjusted treatment sum of squares for Group-Divisible Partially Balanced Incomplete Block Designs. The purpose of this note is to present a similar method for combining intra and inter block analysis for such designs. Method: It is assumed that readers are familiar with the results given by Bose, Clatworthy, Shrikhande [1 (hereafter referred to as BCS)]. The notations used here are exactly the same as those given there except that Q, , , P, G' , t are replaced by Qii , Qi , Pii , GI and ti' (i = 1 *... , j = 1 .. m) respectively, where each one of these is defined if the ith treatment belongs to the jth group. Further, let G6, = G;' if the ith treatment belongs to the jth group. For the combined intra and inter block analysis there is no exact test of hypothesis of the equality of treatment means, but when this hypothesis is true, the quantity ,. 3. t^'iPi is approximately distributed as x2 with v - 1 degrees of freedom. It will be shown that the Table 2.7 given by BCS (p. 25) is enough to compute , , m tiP,i which will be shown to be equal to the proposed formula:

On two methods of bias reduction in the estimation of ratios

SUMMARY Quenouille's method of bias reduction, based on splitting the sample at random into groups, is applied to estimation of ratios and the optimum choice of the number of groups, g, is investigated assuming a model. It is shown that both the bias and variance of the estimator are decreasing functions of g. The estimator is compared with a modified ratio estimator assuming the same model. Durbin (1959) has applied Quenouille's (1956) method of bias reduction to the estimation of ratios. Suppose r denotes the customary ratio estimator of the form r = y/x based on n observations whose bias is en-' + O(n-2) where c is a constant and x and x are unbiased estimators of the population mean of the characters 'y' and 'x' respectively. Let the sample be divided at random into g groups, each of size p, where n = pg. Let rj denote the ratio estimator calculated from the sample after omitting the jth group (i.e. based on p(g -1) observations). Then the estimator (g-1) a 1 g rQ=ygr- 2 ~ rJ=- rQj, 1 Y j=i Y j=i

A numerical method for the solution of the kinetic equation of a reactor

The phase density N( r, v, t) of neutrons in a reactor obeys Boltzmann's linearized equation. The solution of the equation is complicated and in many applied problems we can be satisfied by knowing how the total number n( t) of neutrons in the reactor depends on time. Under certain conditions, the phase density of the neutrons can be found in the form N = N 0( r, v) n( t). We then obtain the following integro-differential equation for n( t) (see [1], [2]) from the Boltzmann equation: dn dt =(K (t)− β l) n(t)+ ∑ j β jλ j l ∫ −∞ t n(t′)e −λ j(t−t′) dt′ . where K(t) = (k eff − 1) lk eff , l is the average life of a neutron in the reactor, λ j is the decay constant of predecessors of the jth group of delayed neutrons, β j is the effective fraction of delayed neutrons in the jth group relative to the total number of fission neutrons, and β = ∑ j β j . The quantities k eff , β j and l depend, in general, on time. However, in many cases the fact that β j and l are not constant can be ignored [1]. The work which follows is concerned with a numerical method for the solution of equation (1) where β j = const, l = const and K( t) is an arbitrary function from a sufficiently wide class. Since (1) is equivalent to a system of ordinary linear differential equations no difficulties will arise as far as the existence and uniqueness of its solution are concerned. In finding a numerical method to solve equation (1) we have been principally interested in the possibilities of controlling the accuracy of the computation internally, and of choosing the optimal size of the time step. We have chosen the latter in the following way. Suppose that A = A( h) is some parameter to be worked out in the approximate solution of the problem on a step of length h. We stipulate that h shall satisfy the conditions ε − ⩽ A ( h) ⩽ ε +, where ge − and ge su+ are pre-set constants. Since A( h) depends on the behaviour of the reactivity K( t) and the solution n( t) within the step, h is, generally speaking, a variable function of the step number. If A( h) is related in some way to the accuracy of the computation in the given step, then any change in the step length will automatically keep the error in the approximate method introduced during each step within constant limits, depending on the values of the constants ge + and ge −. It might happen for some values of ge + and ge − that the step h is near its optimal value h 0, for which the calculation error of the unknown function at the end of the step will be least (or will vanish altogether). In the latter case a small change in the step length can lead to a change in the sign of the error in this step, thus preventing the build-up of errors from step to step. This procedure for choosing the step length is easily programmed. Actual calculations have confirmed the considerations discussed above to be valid, and have shown that a very high degree of accuracy in the solution of equation (1) with variable reactivity can be obtained with a moderate increase in machine time.