交互作用的統計研究

國 立 交 通 大 學

統計學研究所

碩士論文

交互作用的統計研究

A Statistical Study of Interactions

研 究 生 : 張家榕 指導教授 : 陳鄰安 博士

交互作用的統計研究

A Statistical Study of Interactions

研 究 生：張家榕 Student：Jia-Rong Chang

指導教授：陳鄰安 Advisor：Lin-An Chen

國 立 交 通 大 學

統計學研究所

碩 士 論 文

A Thesis

Submitted to Institute of Statistics College of Science

National Chiao Tung University in Partial Fulfillment of the Requirements

for the Degree of Master

in Statistics June 2012

Hsinchu, Taiwan, Republic of China

i

交互作用的統計研究

研究生：張家榕 指導教授：陳鄰安 博士 國立交通大學統計學研究所 摘要 我們引入統計觀點來制訂 isobole，並用它來制訂正(synergistic)效果和負(antagonistic) 效果。isobole 的點估計也被引入，且通過點估計鑑定交互作用檢定力的模擬研究已經 完成。模擬結果顯示該方法是統計上的滿意。 關鍵字：協同作用; 交互作用; isobole

ii

A Statistical Study of Interactions

Student: Jia-Rong Chang Adviser: Lin-An Chen

Institute of Statistics National Chiao Tung University

SUMMARY

We introduce the statistical formulation of isobole and use it to formulate

synergistic interaction effect and antagonistic interaction effect. Point estimation of isobole is also introduced and simulation study for power of identification of interaction through this point estimator has been performed.

The simulation results show that this approach is statistically satisfactory.

iii

誌謝

本論文的完成承蒙指導教授陳鄰安老師的諄諄教誨與指導，在此致上最誠摯的謝 意。從一開始老師就很有耐心的指導，一步一步帶領著我們去完成這未知的論文， 遇到問題時老師總是不厭其煩的為我解惑。不但如此，老師更是常告誡我們一些人 生道理，教導我們人生要有目標和態度，非常榮幸自己能成為陳鄰安教授的學生。 也感謝許文郁老師、彭南夫老師、蕭金福老師，在我的口試時，提供了許多寶 貴的見解，你們的意見使本論文得以表現的更加完善。 還要感謝我的研究所同學們，不論是課業的討論、心情上的紓解、抑或是未來 的發展，你們都是我前進的動力，讓我在學習中得到更多靈感。 最後，感謝家人在碩班兩年來給予我最大的支持，你們總是關心我的生活，希 望我不要給自己太大的壓力。在未來我會繼續認真朝著自己的目標邁進，希望能成 為你們的驕傲。 張家榕 謹誌于 國立交通大學統計研究所 中華民國一百零一年六月

Contents

摘要………. i

Summary………. ii

誌謝………. iii

1. Introduction………. 1

2. Statistical Model for Isobole and Interaction Analysis… 2

3. Isobole based Interaction Identification……….. 6

4. Data Analysis……….. 10

References……… 13

Figures……….. 16

List of Tables

1. Correctness and Errors in interaction detection….……….. 8

2. Power performance for interaction detection through

Estimation………..8

3. Power performance for interaction detection through

Estimation………..9

4. Interaction study for arterial blood pressure data …………11

1. Introduction

The toxicological research has long been devoted to assess the risk with exposure to single chemicals in the environment. However, organisms are rarely environmentally exposed to single chemicals in isolation. More typ-ically, exposures occur to multiple chemicals simultaneously. It has long understood that the behavior of one chemical in the body is aected by other chemicals. Recently most researches in the literature have been in-vestigated on the important area of toxicology of mixed chemicals. One very important study in chemical mixtures is the detection for existence of interactions and characterization of an interaction being synergistic or antagonistic eect.

There is widespread confusion about the concept and methods for eval-uating a possible interaction in biological or environmental system. Among the popular approaches, analysis of variance (ANOVA) is designed with re-striction of zero sum interactions between levels. This technique can detect the existence of interactions, however, there are no descriptions of signs and magnitudes of the interaction to be given. The linear regression approach considers the presence of interactions when product terms xc1

1 x

c2

2 existed

in the statistical model that is criticized for several grounds (see Rothman (1974), Rothman, Greenland and Walker (1980) and Geenland (1993)). For one concern, the presence or absence of interaction with the usage of linear regression practically depends on the model one chooses. Hence, it often happen in analyzing one real data that interaction exists when one model is applied but not exist when the others are used. For another concern, Maud-erly and Samet (2009) pointed out that statistical tests for the presence of interaction have low statistical power.

The isobologram, popularized by Loewe (1928, 1953), is presently the most widely used method as an alternative method for the study of chemi-cal or biologichemi-cal interactions. An isobole forms a dose-response relation for chemical mixtures to obtain the same eect. Results of chemical mixtures are considered to be performed through deterministic experiment (Beren-baum (1981), Rider and LeBlane (2005), Ei-Masri, Reardon and Yang (1997)).

TypesetbyA M S-T E X 1

Unfortunately this very convincing technique require experimental iterations which is not only labor extensive and require a large number of animal ex-periments. Eorts of mathematical formulation for isobole has been done by many authors (see, for examples, Ei-Masri, Reardon and Yang (1997), Lam (1993) and Suhnel (1992)). Incorporating the experimental variation, response surface model has widely been applied for isobole study (Greco, Bravo and Parsons (1995), Sorensen et al. (2007)). This requires models for various chemical mixtures again sharing the disadvantage of experimental labor cost. To get rid of these disadvantages, we consider an isobole of in-terest as an unknown (unique) curve in terms of parameters of distribution of response and independent variables.

The aim in this paper is modeling the isobole from a probability distri-bution involving the chemical variables and response variable. This draws us a mechanism to dene the so-called zero interaction through concept of statistical independence and use it to present interactions. Modeling the interaction under the normal distribution is introduced in Section 2 and methods of detection of interaction and power simulation of these methods are provided in Section 3. Analyses of two real data sets are given in Section 4.

2. Statistical Model for Isobole and Interaction Analysis

We consider two chemicals A and B for study of their combined eect. In experiment, each combination (x1x2) of dosages of A and B generates

a combined eect yx1x2. The plot of magnitude of combined eect as a

function of dosages of two chemicals is a three dimensional surface. The plot of dosages of two chemicals that produces a xed single point of eect magnitude (eect level) is a two dimensional curve, called an isobole. Isobole may forms a straight line or other curves. Given an eect level`and choosing the two end points (x1`0) and (0x2`) such that yx

1`

0 = y0x

2` = `, the

following straight line IB0(`) = f ax1` (1;a)x 2`  : 0a1g (2.1)

is called the \line of additivity" or the no-interaction isobole. Now, an isobole is the curve of combination (x1x2)'s of equl eect ` as

IB(`) = f x1 x2  :yx1x2 =`x 1 >0x2 >0 g (2.2)

the sizes (x1x2) of chemicals A and B that produce equal eect `. Roughly

it is conjectured that there are three types of isoboles showing in Figure 1. Figure 1 is here

We say that there is synergistic eect if the isobole lies below the line of no-interaction and there is antagonistic eect if the isobole lies above the line of no-interaction.

Systematic eect investigation of chemical mixtures is usually done through laboratory study to control the combined eect without uncertainty. This suers the construction of isobole with requiring a very large number of combinations in experiment and is not practical to investigate interaction eect when uncertainty through other uncontrolled factors exists in the en-vironment or workplace (Ei-masri, Reardon and Yang (1997)).

We consider statistical approach of isobole that allows the combined eect with uncertainty from uncontrolled factors. LetX1 andX2 be independent

variables representing magnitudes of chemicals and Y be the response vari-able with normal distribution as

0 @ Y X1 X2 1 A N( 0 @ y 1 2 1 A 0 @ 2 y y1 y2 1y  2 1  12 2y 21  2 2 1 A):

With statistical model, the mean combined eect is represented as the con-ditional mean given X1 =x1 and X2 =x2 as

(x1x2) =y + (y1y2) 2 1  12 21  2 2  ;1 x1 ; 1 x2 ; 2  : (2.3)

Given value `, the eect level `isobole is dened as IB(`) =f x1 x2  :(x1x2) =` g:

In the classical isobole study, eect level`may be determined by the analyst when the experiment of sampling is done in laboratory. But in environment study, it is determined by distributional parameters, generally unknown. Extended from the classical isobole of (2.1), we solve statistical isobole's endpoints by solving x1` =x1 andx2` =x2 from(x10) =(0x2) =`for

x1 and x2 given that

x1` = y2 2 1 ;y 112 y1 2 2 ;y 212 2+ 2 1 2 2 ; 2 12 y1 2 2 ;y 212 (`;y) + 1 and (2.4) x2` = y1 2 2 ;y 212 y2 2 1 ;y 112 1+ 2 1 2 2 ; 2 12 y2 2 1 ;y 112 (`;y) + 2:

We then have derived a statistical isobole.

Theorem 2.1.

With x1` and x2` in (2.4), an isobole is a straight line that

may be represented as IB(`) = f ax1` (1;a)x 2`  : 0a1g: (2.5)

We say that this isobole contains combinations x1

x2  = ax1` (1;a)x 2`  0 a1.

The denition of synergism and antagonism depends on how the concept of no-interaction is dened. Among the three basic criteria (summation, independence and isobole) of interaction evaluation (Suhnel (1992)), we are allowed to joining with this isobole approach with the independence ap-proach that, under the normal assumption, the no-interaction (zero interac-tion) isobole is the one with 12 = 0. By letting x1`0 =x1` and x2`0 =x2`

in (2.4) subjected to 12 = 0, then we have the following theorem.

Theorem 2.2.

. The no-interaction line may be formulated as IB0(`) = f ax1`0 (1;a)x 2`0  : 0a1g =f 0 @ ay2 2 1 y1 2 2 2+ 2 1 y1(` ;y) + 1] (1;a) y1 2 2 y2 2 1 1+ 2 2 y2(` ;y) + 2] 1 A: 0 a1g (2.6)

We follow the denition of interaction eect for classical isobole to this statistical model.

Denition 2.3.

We say that there is synergistic eect if IB(`) lies below IB0(`) and there is antagonistic eect ifIB(`) lies above IB0(`).

However, Both isoboles IB(`) and IB0(`) are generally unknown since

they involve unknown distributional parameters. Hence statistical inferences should be done for detection of interaction that require observation from the underlying distribution. For observations 0 @ yi x1i x2i 1

Ai= 1:::n, we denote mean estimate

0 @ ^ y ^ 1 ^ 2 1 A= 0 @  y  x1  x2 1 A where yx

1 and x2 are, respectively, the sample means of

vari-ables yx1 and x2 and covariance matrix estimate 0 @ ^ 2 y ^y1 ^y2 ^ 1y ^ 2 1 ^ 12 ^ 2y ^21 ^ 2 2 1 A = 0 @ s2 y sy1 sy2 s1y s 2 1 s 12 s2y s21 s 2 2 1 A = 1 n;1 Pn i=1 0 @ yi;y x1i ;x 1 x2i ;x 2 1 A 0 @ yi;y x1i ;x 1 x2i ;x 2 1 A 0 . We then esti-mate the conditional mean by

^ (x1x2) = y+ (sy1sy2) s2 1 s 12 s21 s 2 2  ;1 x1 ;x 1 x2 ;x 2  : We dene estimates of x1` and x2`, respectively, as

^ x1` = sy2s 2 1 ;sy 1s12 sy1s 2 2 ;sy 2s12  x2 + s2 1s 2 2 ;s 2 12 sy1s 2 2 ;sy 2s12 (`;y) + x 1 and (2.7) ^ x2` = sy1s 2 2 ;sy 2s12 sy2s 2 1 ;sy 1s12  x1 + s2 1s 2 2 ;s 2 12 sy2s 2 1 ;sy 1s12 (`;y) + x 2

Then the estimator of population isobole IB is ^ IB(`) =f ax^1` (1;a)^x 2`  : 0a1g: (2.8)

and the no-interaction line may be formulated as ^ IB0(`) = f ax^1`0 (1;a)^x 2`0  : 0a1g =f 0 @ asy2s 2 1 sy1s 2 2  x2+ s2 1 sy1(` ;y) + x 1] (1;a) sy1s 2 2 sy2s 2 1  x1+ s2 2 sy2(` ;y) + x 2] 1 A: 0 a1g (2.9) where ^x1`0 = ^x1` and ^x2`0 = ^x2` by setting s12 = 0.

3. Isobole based Interaction Identication

With statistical formulation of isobole, three interaction detection prob-lems may be considered. First, given an eect level`, is the unknown isobole IB(`) synergistic or antagonistic? Unlike the laboratory test, the eect level `is determined in practical environment that is altered not only in location but also in time. This leads to the second problem of prediction at mean (12) and the third problem of prediction at sample point (x1x2).

We start from the rst problem. Since isoboles IB(`) and IB0(`) are

straight lines, dierences of two end points as c1` = x1` ;x

1`0 and c2` =

x2` ;x

2`0 may be used to detect interaction. With careful re-arrangements,

we have c1` = 12( 2 y2 2 1 ; 2 y1 2 2) (y1 2 2 ;y 212)y1 2 2 2+ ( 2 1 2 2 ; 2 12 y1 2 2 ;y 212 ; 2 1 y1 )(`;y) and (3.1) c2` = 12( 2 y1 2 2 ; 2 y2 2 1) (y2 2 1 ;y 112)y2 2 1 1+ ( 2 1 2 2 ; 2 12 y2 2 1 ;y 112 ; 2 2 y2 )(`;y):

We call c1` and c2` the interaction indices. In case that c1` = c2`, two

isoboles IB(`) andIB0(`) are parallel.

Based on Denition 2.3 and (3.1), interaction detection for combinations of x1 and x2 in isobole IB(`) can be described below:

(a) Isobole IB(`) contains combinations with synergistic eects if c1` <

0 and c2` <0.

(b) Isobole IB(`) contains combinations with antagonistic eects if c1` >

When it ocuurs thatc1` >0 and c2` <0 orc1` <0 and c2`>0, interactions

exist for all combinations of x1 and x2 on the isobole besides the single

intersection point. However, each combination (x1x2) to be synergistic or

antagonistic requires to be further veried.

Pictures of isoboles are displayed in the following gure. Figure 2 of synergistic eect and antagonistic eect

The above procedure can detect the case that IB(`) is above or below IB0(`). When the isoboleIB(`) andIB0(`) intersect somewhere, the

detec-tion is slightly complicated and we suggest to conduct the detecdetec-tion through the next approach.

With modeling the interactions through probability distribution, it is the interesting to propose statistical methods for interaction of identication. Among the statistical inferences, the point estimation is the basic technique and can be derived straight forward through the formulas in (3.1).

With ecient estimator of conditional mean (x1x2), we expect that

corresponding estimates of interaction indices c1` and c2` dened as

^ c1` = s12(s 2 y2s 2 1 ;s 2 y1s 2 2) (sy1s 2 2 ;sy 2s12)sy1s 2 2  x2+ ( s2 1s 2 2 ;s 2 12 sy1s 2 2 ;sy 2s12 ; s2 1 sy1 )(`;y) (3.2) ^ c2` = s12(s 2 y1s 2 2 ;s 2 y2s 2 1) (sy2s 2 1 ;sy 1s12)sy2s 2 1  x1+ ( s2 1s 2 2 ;s 2 12 sy2s 2 1 ;sy 1s12 ; s2 2 sy2 )(`;y):

may provide ecient tool in interaction detection. Induced rule for identi-cation of interactions based on interaction index estimates is as follows: (a) There is synergistic eect if ^c1` <0 and ^c2` <0.

(b) There is antagonistic eect if ^c1` >0 and ^c2` >0.

The advantage of interpreting the isobole's concept of interaction through statistical model is that it requires only a data for estimation of distribu-tional parameters.

With the above rule of interaction identication, there are four categories for the result computed from one data set that are displayed in Table 1.

Table 1.

Correctness and Errors in interaction detection sign(^c1`) = sign(c1`) sign(^c1`)

6 = sign(c1`) ;; ;; sign(^c2`) = sign(c2`) j Correct Error I sign(^c2`) 6 = sign(c2`) j Error I Error II

Error II is the most serious one since two signs of interactions are predicted with error.

To evaluate the power of correctness in detection, we consider the follow-ing settfollow-ing of joint distribution:

0 @ Y X1 X2 1 A  N 3( 0 @ 1 2 3 1 A 0 @ 2 0:7 0:7 0:7 2 12 0:7 12 2 1 A)

and set c1` = c2` = c for some values c, positive and negative for being,

respectively, antagonistic and synergistic. With replication number m, the power of identication of interaction is dened as

= 1_mXm

j=1

I(sign(^c1`) = sign(c1`)sign(^c2`) = sign(c2`)):

With m = 1000 and some sample sizes, the simulated results of power are displayed in Tables 2 and 3.

Table 2.

Power performance for interaction detection through estimation

c=;1:029 c=;1:286 c=;1:929 12 = ;0:9 n= 30 0:517 0:601 0:736 n= 50 0:635 0:748 0:860 n= 100 0:816 0:875 0:959 12 = 0:9 n= 30 0:641 0:695 0:774 n= 50 0:679 0:775 0:899 n= 100 0:829 0:918 0:977

c= 0:286 c= 0:571 c= 0:857 c= 1:143 12 = ;0:4 n= 30 0:398 0:690 0:791 0:805 n= 50 0:506 0:806 0:892 0:904 n= 100 0:651 0:945 0:971 0:974 12 = 0:4 c= ;0:286 c=;0:571 c=;0:857 c=;1:143 n= 30 0:350 0:630 0:747 0:790 n= 50 0:428 0:771 0:866 0:902 n= 100 0:594 0:907 0:975 0:970

We conclude the following from Tables 2 and 3:

(a) The results show that the detection power is increasing when sample size increases.

(b) It also shows that the power increases when value c lies away of zero. (c) The power performance showing in these two tables is satisfactory.

Let (t1t2) be xed values for (X1X2). We consider a process below for

detection of interaction at this obaservation:

(a) The predicted interaction level of the isobole that this sample point lies is `= ^(t1t2).

(b) Given this predicted eect level `, a combination (X1X2) = (t1t20)

lying on no-interaction isobole of level `may be solved as t20 = ; sy1s 2 2 sy2s 2 1 (t1 ;x 1) + x2+ s2 2 sy2 (`;y): (3.3)

(c) Rule for predicting the interaction eect on this isobole: There is synergistic eect at (t1t2) if t2 < t20

There is no-interaction eect at (t1t2) if t2= t20

There is antagonistic eect at (t1t2) if t2 > t20

For mean interaction detection, let (t1t20) = (x1x2). The rule in (c)

above predicts the interaction eect when (X1X2) = (12). For

inter-action detection at an observation, let (t1t20) = (x1x2). The rule in (c)

4. Data Analyses

A data of sizen= 20 for studying blood pressure with some explanatory variables has been considered in Daniel (1999, p484-485) where variables in this data are listed below:

Y = mean arterial blood pressure (mm Hg) X1 = age (years)

X2 = weight (kg)

X3 = body surface area (sq m)

X4 = duration of hypertension (years)

X5 = basal pulse (beats/ min)

X6 = measure of stress.

The sample mean and covariance matrix are displayed below:

0 @ ^ y ^ 3 ^ 5 1 A= 0 @ 114:0 1:998 69:6 1 A^ = 0 @ 31:02 0:675 15:67 0:675 0:019 0:253 15:67 0:253 15:22 1 A

Next, we display the estimated interaction indices ^c1` and ^c2` as functions

of ` in Figure 3.

Figure 3 is here

For ` > 185:4352, there are antagonistic eect since ^c1` >0 and ^c2` > 0.

However, we can not conclude the eect for that ` <185:4352 since signs of ^

c1` altered. We then draw a picture of ^IB and ^IB0 for `= 160 and 200 in

Figure 4.

Figure 4 is here

Both level `= 160 and 200 correspond to isobole of antagonistic eect that may not be seen in Figure 3 based on ^c1` and ^c2`.

We consider the problem of interaction prediction(detection) when an observation (x3x5) of independent variables (X3X5) is given. For this

example, we evaluate all observations ofn= 20 samples. So, for each obser-vation point (x3x5), we conduct the process of interaction prediction and

we display in Table 4 the predicted interaction level `and the quantityx50

such that (x3x50) lies on the level `no-interaction isobole. The results (S

for synergistic eect and A for antagonistic eect) of interaction detections for all sample points (X3X5) are also displayed.

Table 4.

Interaction study for arterial blood pressure data

prediction ` x3(x5) x5(x3) Eect (x3x5) 1:7563 103:49 1:890 67:68 S 2:170 116:9 2:072 69:08 A 1:9872 114:9 1:952 71:08 A 2:0173 116:2 1:963 71:42 A 1:8972 112:4 1:882 71:74 A 2:2571 121:5 2:176 68:54 A 2:2569 120:4 2:202 67:42 A 1:966 109:2 1:968 68:28 S 1:8369 109:1 1:874 70:48 S 2:0764 112:6 2:127 65:91 S 2:0774 118:4 1:996 71:55 A 1:9871 114:3 1:965 70:52 A 2:0568 114:4 2:059 68:31 S 1:9267 110:3 1:970 68:70 S 2:1976 122:8 2:064 71:80 A 1:9869 113:1 1:991 69:39 S 1:8762 106:1 1:997 66:24 S 1:970 111:5 1:916 70:54 S 1:8871 111:6 1:887 71:25 S 2:0975 119:6 1:999 71:97 A (x3x5) x3 x50 1:99869:6 114 1:998 69:6 N

We have several comments for the displayed results:

(a) The predicted interaction levels`(column 2) for all observations (x3x5)

are larger than the sample mean, 114:0, of response variable y indicates that independent variables X3 and X5 make signicant contribution on the

mean conditional eect of y given (x3x5). Their dierences between `'s

and 4978:4 give the sizes of contributions.

(b) Some observations show synergistic eects and some show antagonistic eects. However, the sample mean (x1x2) gives no-interaction eect.

In concerning the need for hospital labor, Bowerman and O'Connell (1990) conduct this analysis on a data set of size n = 17 through linear regression model. The response variable Y represents the monthly labor hours. Among the explanatory variables, we rst choose X1, the average

daily patient load, and X5, the average length of patients' stay in days for

analysis. We rst consider explanatory variables X1 and X5. The mean

estimate and covariance estimate are, respectively,

0 @ ^ y ^ 1 ^ 5 1 A= 0 @ 4978:4 148:27 5:89 1 A^ = 0 @ 32852004 937770 5414 937770 27554 181:92 5414 181:92 2:666 1 A

For this set of observations, we also compute their corresponding pre-dicted interaction eects that are displayed in Table 5.

Table 5.

Interaction study for monthly labor hours data

prediction ` x1(x5) x5(x1) Eect (x1x5) 15:574:45 762:7 110:5 6:041 S 44:026:92 521:2 ;43:94 5:445 A 20:424:28 1034 128:6 6:094 S 18:743:9 1173 155:4 6:190 S 49:25:5 1468 68:62 5:825 S 44:924:6 1784 131:6 6:053 S 55:485:62 1640 66:52 5:085 S 59:285:15 2032 106:0 5:934 S 94:396:18 2804 67:29 5:725 A 128:06:15 4082 106:6 5:791 A 965:88 3023 91:64 5:806 A 131:44:88 4883 205:9 6:129 S 127:25:5 4396 154:6 5:959 S 252:97:0 8318 180:3 5:785 A 409:210:78 12181 68:31 5:067 A 463:77:05 16204 409:1 6:135 A 510:26:35 18321 513:0 6:398 S (x1x5) x1 x50 148:275:89 4978 148:2 5:893 N

This analysis shows that most observations give synergistic eects and only a few give antagonistic eects.

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Figure 1. Classical interaction by isobole

17

Figure 3. Lines ˆc₁ and ˆc₂

Figure 4. Isobole IB  and IB0  (a) =160 (b) =200 2 ˆc 1 ˆc IB  IB  0 IB  0 IB  1 1 ˆ 0 ,if <185.4352 ˆ 0, ,if >185.4352 c c    _ 