Testing equality of two mean vectors with unequal sample sizes for populations with correlation

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Testing equality of two mean vectors with unequal sample sizes for populations with correlation Aya Shinozaki Naoya Okamoto 2 and Takashi Seo Department of Mathematical Information Science Tokyo University of Science Japan 2 Department of Food Sciences Tokyo Seiei College Japan In order to test the equality of mean vectors with unequal sample sizes when populations have correlation we propose a new T 2 type statistic. In this paper we consider observed data as missing data as described by Seko Yamazaki and Seo (202) and derive the approximate upper percentile for the distribution of the proposed statistic. Furthermore we evaluate the accuracy of the approximate upper percentile using Monte Carlo simulation for selected values of parameters namely sample sizes dimensions and covariance matrices. Keywords: Hotelling T 2 type statistic; Paired T 2 statistic; Upper percentile Introduction Testing equality of mean vectors has been considered by many authors. Hotelling s T 2 type statistic is a generalization of Student s t statistic. This statistic is important and is discussed for normal and non-normal populations. Asymptotic distribution of Hotelling s T 2 statistic in elliptical population has been discussed by Iwashita (997) and Iwashita and Seo (2002). The non-normal case has been discussed by Kano (995) Fuikoshi (997) Kakizawa and Iwashita (2008) and Kakizawa (2008). Test for equality of mean vectors in multivariate normal populations that have correlation was introduced by Morrison (2005) and discussed for equal sample sizes. Morrison (2005) introduced a test statistic usually called paired T 2 statistic which is Hotelling s T 2 type statistic. Shinozaki Okamoto and Seo (204) discussed testing equality of mean vectors for the case of k-sample problem and derived approximate upper percentiles for the distribution of paired T 2 statistic under elliptical populations with correlation. In this paper we expand this discussion for the case of unequal sample sizes. Even though we consider observed data as two-step monotone missing data practically this data is not missing data. In the case of two-step monotone missing data Anderson and Olkin (985) derived maximum-likelihood estimators (MLEs) for mean vectors µ and covariance matrix Σ. Kanda and Fuikoshi (998) discussed the distributions for the MLEs of ˆµ and ˆΣ based on monotone missing data. We propose a new T 2 type statistic using sample mean vectors and MLE of covariance matrix and derive the approximate upper percentile for the distribution of T 2 type statistic using the method described by Seko Yamazaki and Seo (202). Finally the accuracy of approximate E-mail : 43704@ed.tus.ac.p

upper percentile and type I error for the new test statistic is evaluated using Monte Carlo simulation. 2 Paired T 2 statistic for equal sample sizes Let x (i) x(i) 2... x(i) N (i = 2) represent N independent random sample vectors distributed as the i-th p-dimensional normal population with mean vector µ (i) and covariance matrix Σ ii where the first and second populations have correlation. Let x = (x () x (2) ) ( = 2... N) represent N independent random sample vectors distributed as a 2p-dimensional normal population with mean vector µ and covariance matrix Σ given by ( µ (l) µ = µ (m) ) ( Σll Σ Σ = lm Σ lm Σ mm respectively. The sample mean vector and sample covariance matrix are defined as N x = N = ( x () = S = = x (2) x ) N (x x)(x x) N = ( ) S S 2 S 2 S 22 respectively. In order to test the equality of mean vectors we set a hypothesis as ) H 0 : µ () = µ (2) vs. H : µ () µ (2). The test statistic for testing the hypothesis H 0 is given by Under H 0 T 2 pd = N(x() x (2) ) (S + S 22 S 2 S 2) (x () x (2) ). F pn p = N p (N )p T 2 pd where F pn p is an F distribution with p and N p degrees of freedom. 2

3 Paired T 2 statistic for unequal sample sizes In this section we expand the discussion for the case of unequal sample sizes. Here the first and second populations have correlation. Subect Condition Condition 2 x () x (2)... N 2 x () N 2 x (2) N 2.. N x () N Let x (i) (i = 2 = 2... N i ) represent N i independent random sample vectors distributed as a p-dimensional multivariate normal population with mean vector µ (i) and covariance matrix Σ ii. Then x = (x () x (2) ) ( = 2... N 2 ) represent N 2 independent random sample vectors distributed as a 2p-dimensional multivariate normal population with mean vector µ and covariance matrix Σ where ( µ () µ = µ (2) ) ( Σ Σ Σ = 2 Σ 2 Σ 22 respectively. The sample mean vector and sample covariance matrix are defined as respectively. x () = N 2 x () 2 = N 2 = N N 2 = x () x (2) = N 2 x (2) N 2 x = N 2 x N 2 S () = S (2) = = N 2 N 2 = N =N 2 + N N 2 x () (x x)(x x) N =N 2 + (x () x () ) 2 )(x() 3. T 2 type test statistic and its upper percentile 2 ) x () In order to test the equality of mean vectors we set a hypothesis as follows: H 20 : µ () = µ (2) vs. H 2 : µ () µ (2). 3

The test statistic for testing the hypothesis H 20 is given by T 2 pd = y {Ĉov(y)} y () where y = { } N 2 x () + (N N 2 )x () 2 x (2) N E (y) = N {N 2 µ () + (N N 2 )µ ()} µ (2) = µ () µ (2) Cov (y) = N Σ N Σ 2 N Σ 2 + N 2 Σ 22. Hence the MLE for covariance matrix of y is obtained using the method proposed by Kanda and Fuikoshi (998) as where Ĉov(y) = N ˆΣ N ˆΣ2 N ˆΣ2 + N 2 ˆΣ22 ˆΣ = N (W () + W (2) ) ˆΣ 2 = ˆΣ (W () ) W () 2 ˆΣ 22 = W () 22 N + ˆΣ ˆΣ ˆΣ 2 2 2 ( W () = (N 2 )S () = W () W () 2 W () 2 W () 22 W (2) = (N N 2 )S (2) + N 2(N N 2 ) N (x () x (2) )(x () x (2) ) W () 22 = W () 22 W () 2 (W () ) W () 2. We consider that the upper percentile of Tpd 2 lies between the upper percentiles of 2p-dimensional data of N and N 2. Therefore we propose a formula to calculate the approximate upper percentile of Tpd 2 given by t 2 (α) = 2 ) ( t 2 pdn2 (α) + t 2 pdn (α) ) (2) where t 2 pdn 2 (α) is the upper percentile of T 2 pd for N = N 2 and t 2 pdn (α) is the upper percentile of T 2 pd for N = N. 4 Simulation study and conclusion In order to evaluate the accuracy of the obtained approximation Monte Carlo simulation for the upper percentile t 2 (α) as shown in (2) was implemented for varied parameters. Some of the results are listed in this paper. As a numerical experiment we carried out 000000 replications. The asymptotic distribution of Tpd 2 is χ2 distribution with p degrees of freedom for N and a fixed N 2. These values are listed in the Tables. 4

For considering the correlation between the two populations we put the covariance matrix in two patterns i. e. pattern : ( ) Σ Σ Σ = 2 Σ 2 Σ 22 where and pattern 2: ρ ρ ρ ρ ρ ρ ρ Σ =..... Σ ρ ρ ρ 2 =..... Σ 22 = βσ ρ ρ ρ ρ ρ Σ = ρ ρ 2 ρ 2p ρ ρ ρ 2p 2 ρ 2 ρ ρ 2p 3......... ρ 2p ρ 2p 2 ρ Tables -4 give the simulated and approximated values for the upper percentile of Tpd 2 as shown in () for the following parameters: p = 2 5 0; α = 0.05; N : N 2 = 5 : N : N 2 = 2 : N : N 2 = 5 : 4. Tables 2 3 and 4 illustrate the cases Σ: pattern ρ = 0 and β = ; Σ: pattern ρ = 0.5 and β = ; Σ: pattern ρ = 0.5 and β = 2; and Σ: pattern 2 ρ = 0.5 respectively. t and χ 2 represent upper percentiles of the distributions for Tpd 2 and χ 2 p respectively. t 2 represents the approximate upper percentile t 2 pd (α). type I error (χ2 ) and type I error (t 2 ) represent the actual type I error for χ 2 and t 2 respectively. Tables -4 show that the simulated value t 2 is closer to the upper percentile of the χ 2 distribution with p degrees of freedom and the approximate accuracy increases as N and N 2 increase. Further the upper percentile t 2 is better than the upper percentile of χ 2 distribution with p degrees of freedom when N and N 2 are small and the value of t 2 rapidly converges with the simulated value t. When p is small the approximate accuracy is better than when p is large. When p = 2 there is not much difference in the unbalance of the sample size. However approximate accuracy is affected by the unbalance of the sample size for a large p. We also simulated a case where ρ = 0.3 and ρ = 0.8; however the approximate accuracy is not affected to a great extent by the difference in correlation ρ. In conclusion the upper percentile t 2 (α) for the test statistic Tpd 2 proposed in this paper is a good approximation even for a small sample size and it is useful for the test for equality of mean vectors with unequal sample sizes for populations with correlation.. 5

References [] Anderson T. W. and Olkin I. (985). Maximum-likelihood estimation of the parameters of a multivariate normal distribution. Linear Algebra and Its Applications 70 47-7. [2] Fuikoshi Y. (997). An asymptotic expansion for the distribution of Hotelling s T 2 - statistic under nonnormality. Journal of Multivariate Analysis 6(2) 87-93. [3] Iwashita T. (997). Asymptotic null and nonnull distribution of Hotelling s T 2 -statistic under the elliptical distribution. Journal of Statistical Planning and Inference 6() 85-04. [4] Iwashita T. and Seo T. (2002). An application of Lévy s inversion formula for higher order asymptotic expansion. Journal of Statistical Planning and Inference 04(2) 403-46. [5] Kakizawa Y. (2008). Asymptotic expansions for the distributions of maximum and sum of quasiindependent Hotelling s T 2 statistics under nonnormality. Communications in Statistics - Theory and Methods 37() 97-20. [6] Kakizawa Y. and Iwashita T. (2008). Hotelling s one-sample and two-sample T 2 tests and the multivariate Behrens-Fisher problem under nonnormality. Journal of Statistical Planning and Inference 38() 3379-3404. [7] Kanda T. and Fuikoshi Y. (998). Some basic properties of the MLE s for a multivariate normal distribution with monotone missing data. American Journal of Mathematical and Management Sciences 8(-2) 6-90. [8] Kano Y. (995). An asymptotic expansion of the distribution of Hotelling s T 2 -statistic under general distributions. American Journal of Mathematical and Management Sciences 5(3-4) 37-34. [9] Morrison D. F. (2005). Multivariate Statistical Methods 4th edition. Duxbury. [0] Seko N. Yamazaki A. and Seo T. (202). Tests for mean vector with two-step monotone missing data. SUT Journal of Mathematics 48() 3-36. [] Shinozaki A. Okamoto N. and Seo T. (204). Test for equality of mean vectors and profile analysis. (Japanese) Bulletin of the Computational Statistics of Japan 27() 29-47. 6

Table : Σ: pattern ρ = 0 and β = p = 2 N N 2 t 2 χ 2 t 2 type I error (χ 2 ) type I error (t 2 ) 40 8 9.2 5.99 9.33 0.7 0.047 80 6 7.45 5.99 7.6 0.085 0.056 60 32 6.70 5.99 6.50 0.068 0.054 320 64 6.32 5.99 6.23 0.058 0.052 20 0 8.9 5.99 8.77 0.0 0.042 40 20 6.99 5.99 7.08 0.074 0.048 80 40 6.46 5.99 6.48 0.062 0.050 60 80 6.22 5.99 6.23 0.056 0.050 320 60 6.0 5.99 6. 0.053 0.050 20 6 7.86 5.99 7.76 0.095 0.052 40 32 6.8 5.99 6.76 0.070 0.05 80 64 6.38 5.99 6.35 0.060 0.05 60 28 6.8 5.99 6.7 0.055 0.050 320 256 6.08 5.99 6.08 0.052 0.050 p = 5 N N 2 t 2 χ 2 t 2 type I error (χ 2 ) type I error (t 2 ) 40 8 23.43.07 59.50 0.247 0.00 80 6 5.88.07 7.08 0.42 0.039 60 32 3.36.07 3.2 0.095 0.052 320 64 2.7.07 2.0 0.072 0.053 20 0 9.59.07 3.9 0.20 0.009 40 20 4.57.07 6. 0.8 0.034 80 40 2.66.07 3.08 0.082 0.044 60 80.8.07.98 0.065 0.047 320 60.42.07.5 0.057 0.048 20 6 8.9.07 20. 0.98 0.04 40 32 4.08.07 4.3 0. 0.047 80 64 2.37.07 2.48 0.076 0.048 60 28.68.07.73 0.062 0.049 320 256.34.07.39 0.055 0.049 p = 0 N N 2 t 2 χ 2 t 2 type I error (χ 2 ) type I error (t 2 ) 80 6 34.59 8.3 6.86 0.30 0.003 60 32 24.66 8.3 26.22 0.59 0.037 320 64 2.35 8.3 2.3 0.02 0.05 40 20 29.87 8.3 42.36 0.243 0.009 80 40 23.04 8.3 25.8 0.32 0.032 60 80 20.45 8.3 2.5 0.086 0.042 320 60 9.33 8.3 9.62 0.067 0.046 20 6 52.30 8.3 79.04 0.5 0.00 40 32 28.46 8.3 30.25 0.23 0.038 80 64 22.30 8.3 22.84 0.20 0.044 60 28 20.05 8.3 20.33 0.079 0.046 320 256 9.4 8.3 9.27 0.064 0.048 7

Table 2: Σ: pattern ρ = 0.5 and β = p = 2 N N 2 t 2 χ 2 t 2 type I error (χ 2 ) type I error (t 2 ) 40 8 9.3 5.99 9.33 0.22 0.050 80 6 7.50 5.99 7.6 0.086 0.056 60 32 6.70 5.99 6.50 0.067 0.054 320 64 6.32 5.99 6.23 0.058 0.052 20 0 8.3 5.99 8.77 0.04 0.044 40 20 7.05 5.99 7.08 0.076 0.049 80 40 6.5 5.99 6.48 0.063 0.05 60 80 6.24 5.99 6.23 0.056 0.050 320 60 6.0 5.99 6. 0.053 0.050 20 6 7.88 5.99 7.76 0.096 0.052 40 32 6.85 5.99 6.76 0.07 0.052 80 64 6.39 5.99 6.35 0.060 0.05 60 28 6.9 5.99 6.7 0.055 0.050 320 256 6.08 5.99 6.08 0.052 0.050 p = 5 N N 2 t 2 χ 2 t 2 type I error (χ 2 ) type I error (t 2 ) 40 8 25.35.07 59.50 0.274 0.002 80 6 6.55.07 7.08 0.55 0.045 60 32 3.53.07 3.2 0.099 0.055 320 64 2.22.07 2.0 0.073 0.054 20 0 20.75.07 3.9 0.222 0.0 40 20 4.99.07 6. 0.29 0.038 80 40 2.86.07 3.08 0.086 0.047 60 80.93.07.98 0.067 0.049 320 60.49.07.5 0.058 0.050 20 6 8.87.07 20. 0.20 0.04 40 32 4.20.07 4.3 0.4 0.049 80 64 2.46.07 2.48 0.078 0.050 60 28.72.07.73 0.063 0.050 320 256.37.07.39 0.056 0.050 p = 0 N N 2 t 2 χ 2 t 2 type I error (χ 2 ) type I error (t 2 ) 80 6 37.00 8.3 6.86 0.340 0.004 60 32 25.58 8.3 26.22 0.77 0.045 320 64 2.65 8.3 2.3 0.07 0.054 40 20 3.37 8.3 42.36 0.274 0.02 80 40 23.77 8.3 25.8 0.47 0.037 60 80 20.78 8.3 2.5 0.092 0.046 320 60 9.5 8.3 9.62 0.070 0.049 20 6 52.42 8.3 79.04 0.57 0.0 40 32 28.59 8.3 30.25 0.237 0.039 80 64 22.50 8.3 22.84 0.24 0.046 60 28 20.9 8.3 20.33 0.082 0.048 320 256 9.20 8.3 9.27 0.065 0.049 8

Table 3: Σ: pattern ρ = 0.5 and β = 2 p = 2 N N 2 t 2 χ 2 t 2 type I error (χ 2 ) type I error (t 2 ) 40 8 0.4 5.99 9.33 0.34 0.060 80 6 7.78 5.99 7.6 0.092 0.06 60 32 6.82 5.99 6.50 0.070 0.057 320 64 6.38 5.99 6.23 0.060 0.053 20 0 8.77 5.99 8.77 0.3 0.050 40 20 7.23 5.99 7.08 0.080 0.053 80 40 6.6 5.99 6.48 0.065 0.053 60 80 6.28 5.99 6.23 0.057 0.05 320 60 6.2 5.99 6. 0.053 0.050 20 6 8.06 5.99 7.76 0.099 0.055 40 32 6.9 5.99 6.76 0.073 0.053 80 64 6.43 5.99 6.35 0.06 0.052 60 28 6.9 5.99 6.7 0.055 0.05 320 256 6.09 5.99 6.08 0.053 0.050 p = 5 N N 2 t 2 χ 2 t 2 type I error (χ 2 ) type I error (t 2 ) 40 8 33.27.07 59.50 0.346 0.009 80 6 8.03.07 7.08 0.77 0.059 60 32 4.00.07 3.2 0.08 0.06 320 64 2.4.07 2.0 0.077 0.057 20 0 23.99.07 3.9 0.265 0.02 40 20 5.85.07 6. 0.45 0.047 80 40 3.2.07 3.08 0.093 0.052 60 80 2.08.07.98 0.070 0.052 320 60.55.07.5 0.060 0.05 20 6 9.94.07 20. 0.27 0.049 40 32 4.54.07 4.3 0.20 0.053 80 64 2.60.07 2.48 0.08 0.052 60 28.79.07.73 0.064 0.05 320 256.4.07.39 0.057 0.050 p = 0 N N 2 t 2 χ 2 t 2 type I error (χ 2 ) type I error (t 2 ) 80 6 46.55 8.3 6.86 0.435 0.07 60 32 27.44 8.3 26.22 0.208 0.06 320 64 22.27 8.3 2.3 0.9 0.062 40 20 35.52 8.3 42.36 0.332 0.024 80 40 24.92 8.3 25.8 0.68 0.048 60 80 2.25 8.3 2.5 0.0 0.05 320 60 9.73 8.3 9.62 0.074 0.052 20 6 59.46 8.3 79.04 0.569 0.09 40 32 29.87 8.3 30.25 0.258 0.047 80 64 22.96 8.3 22.84 0.33 0.05 60 28 20.4 8.3 20.33 0.085 0.05 320 256 9.32 8.3 9.27 0.066 0.05 9

Table 4: Σ: pattern 2 ρ = 0.5 p = 2 N N 2 t 2 χ 2 t 2 type I error (χ 2 ) type I error (t 2 ) 40 8 9.3 5.99 9.33 0.22 0.050 80 6 7.5 5.99 7.6 0.086 0.057 60 32 6.7 5.99 6.50 0.068 0.055 320 64 6.32 5.99 6.23 0.058 0.052 20 0 8.43 5.99 8.77 0.07 0.045 40 20 7.09 5.99 7.08 0.077 0.050 80 40 6.5 5.99 6.48 0.063 0.05 60 80 6.24 5.99 6.23 0.056 0.050 320 60 6.2 5.99 6. 0.053 0.050 20 6 8.00 5.99 7.76 0.098 0.054 40 32 6.88 5.99 6.76 0.072 0.053 80 64 6.42 5.99 6.35 0.06 0.052 60 28 6.9 5.99 6.7 0.055 0.05 320 256 6.08 5.99 6.08 0.052 0.050 p = 5 N N 2 t 2 χ 2 t 2 type I error (χ 2 ) type I error (t 2 ) 40 8 24.84.07 59.50 0.268 0.002 80 6 6.38.07 7.08 0.5 0.043 60 32 3.5.07 3.2 0.099 0.054 320 64 2.23.07 2.0 0.073 0.054 20 0 20.89.07 3.9 0.225 0.02 40 20 5.07.07 6. 0.30 0.039 80 40 2.87.07 3.08 0.086 0.047 60 80.94.07.98 0.067 0.049 320 60.47.07.5 0.058 0.049 20 6 9.5.07 20. 0.2 0.045 40 32 4.35.07 4.3 0.7 0.05 80 64 2.50.07 2.48 0.079 0.050 60 28.75.07.73 0.063 0.050 320 256.38.07.39 0.056 0.050 p = 0 N N 2 t 2 χ 2 t 2 type I error (χ 2 ) type I error (t 2 ) 80 6 36. 8.3 6.86 0.327 0.004 60 32 25.30 8.3 26.22 0.7 0.042 320 64 2.58 8.3 2.3 0.06 0.053 40 20 3.2 8.3 42.36 0.268 0.02 80 40 23.64 8.3 25.8 0.44 0.036 60 80 20.75 8.3 2.5 0.092 0.045 320 60 9.50 8.3 9.62 0.070 0.048 20 6 54.97 8.3 79.04 0.537 0.03 40 32 29.09 8.3 30.25 0.244 0.042 80 64 22.6 8.3 22.84 0.27 0.048 60 28 20.24 8.3 20.33 0.082 0.049 320 256 9.24 8.3 9.27 0.065 0.050 0

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