線性結構模式下調節效果檢定之MRA修訂法

行政院國家科學委員會專題研究計畫 成果報告

線性結構模式下調節效果檢定之 MRA 修訂法

計畫類別： 個別型計畫 計畫編號： NSC91-2416-H-009-004- 執行期間： 91 年 08 月 01 日至 92 年 07 月 31 日 執行單位： 國立交通大學經營管理研究所 計畫主持人： 丁承 計畫參與人員： 陳美芳 報告類型： 精簡報告 處理方式： 本計畫可公開查詢

中 華 民 國 92 年 11 月 12 日

行政院國家科學委員會補助專題研究計畫成果報告

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※ 線性結構模式下調節效果檢定之 MRA 修訂法 ※

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計畫類別：

√

個別型計畫 整合型計畫

計畫編號：N S C 9 1 - 2 4 1 6 - H - 0 0 9 - 0 0 4

執行期間：9 1 年 8 月 1 日 至 9 2 年 7 月 3 1 日

計畫主持人：丁 承

本成果報告包括以下應繳交之附件：

□赴國外出差或研習心得報告一份

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執行單位：國立交通大學經營管理研究所

中 華 民 國 92 年 10 月 28 日

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行 政 院 國 家 科 學 委 員 會 專 題 研 究 計 畫 成 果 報 告

線性結構模式下調節效果檢定之

MRA 修訂法

Testing for Moderating Effects in SEM: A Modified MRA Approach

計畫編號：NSC 91-2416-H-009-004

執行期限：91 年 8 月 1 日至 92 年 7 月 31 日

主持人：丁 承 國立交通大學經營管理研究所

一、 中文摘要 在線性結構模式下，本計畫提出一個 檢 測 連 續 型 外 顯 調 節 變 數 調 節 效 果 的 MRA 修訂法。針對 MRA 傳統法及修訂 法，配以因素基礎分數(factor-based score) 或 基 於 驗 證 型 因 素 分 析 因 素 分 數(factor scores derived from CFA)之採用，本研究以 蒙地卡羅模擬法進行調節效果檢定力及估 計偏誤大小比較。將連續型外顯調節變數 以中位數作分類處理所導致之檢定力及估 計偏誤資訊流失亦一并檢視。研究結果顯 示，當構念指標之因素負荷量呈較大之差 異時，採用因素分數之 MRA 修訂法可獲 得最大的檢定力；然而不同方法在估計偏 誤上之差異則未見明確。此外，對連續型 調節變數作分類處理會降低調節效果檢定 力並導致估計偏誤之資訊大量流失。 關鍵詞：偏誤、驗證型因素分析、調節迴 歸分析(MRA)、調節效果、蒙地卡羅模擬、 檢定力、線性結構模式 Abstract

A modified moderated regression analysis (MRA) is proposed to detect moderating effects for continuous manifest moderators in structural equation modeling (SEM). Monte Carlo simulation is used to compare the test power and the bias associated with moderating effects by using both traditional and modified MRA with factor-based scores and factor scores derived from confirmatory factor analysis (CFA). Loss of power and bias information due to categorizing a

continuous manifest moderator by the median are also examined. The results indicated that if the factor loadings for indicators are substantially different, the modified MRA with factor scores led to the greatest power. However, how the biases resulting from different approaches differ was inconclusive. In addition, categorizing a continuous manifest moderator by the median would reduce the power and lead to much loss of the bias information.

Keywords: Bias, Confirmatory Factor

Analysis, Moderated Regression Analysis (MRA), Moderating Effect, Monte Carlo Simulation, Power, Structural Equation Modeling

二、緣由與目的

Moderators often appear in behavioral science research. It affects the form or strength of a relationship between an independent variable and a dependent variable (Baron & Kenny, 1986; Sharma, Durand, & Gur-Arie, 1981). In other words, the independent variable and the moderator interact to reflect that the effect of the independent variable on the dependent variable depends on the level of the moderator (McClelland & Judd, 1993). Moderated regression analysis (MRA), developed by Saunders (1956), has long been applied for examining moderating effects. In traditional MRA, standard multiple regression procedures are used to test for the product terms to determine whether moderating effects are statistically significant.

In social science studies involving psychological constructs, the structural equation modeling (SEM) technique has become popular. The MRA approach can be further applied to examine moderating effects on the construct relationships. However, the measurement adequacy for the constructs must be achieved before moderating effects are examined.

Confirmatory factor analysis (CFA) can be used for a priori defined constructs specified in a SEM model to assess the degree to which the measurement items are valid indicators of the constructs. CFA can be used to select the appropriate indicators of constructs (Gerbing & Anderson, 1988; Grapentine, 2000). There exists indeterminacy in obtaining estimates of latent factors since, for the same observed data, different scoring approaches would lead to different factor scores (e.g., Acito and Anderson, 1986; Lawley & Maxwell, 1971). Factor-based scores are the most commonly used scores for latent constructs. With the factor-based scoring approach, two or more indicators are summed up and divided by the number of the indicators. Alternatively, the factor scores derived from CFA (based on the regression approach) could be used, and their use can reduce the distorting effects resulting from the measurement errors on the coefficient estimates of structural models (Bollen, 1989).

The test for detecting moderating effects on the construct relationships can be assessed by power. Since how the factor scoring approaches influence the power of the test is rarely seen in the literature, the powers resulting from the two types of scores (factor-based scores and factor scores derived from CFA) will be compared in this study, for the case of continuous moderators.

Particularly, a modified MRA approach for detecting moderating effects will be proposed. The approach combines the features of the measurement model, with acceptable reliability, and traditional MRA to perform statistical inference simultaneously under the SEM frame. The main difference

between the modified MRA and the traditional MRA is that simultaneous inference is made for the former, but not for the latter. We contend that the power of the test by using the modified MRA will be relatively higher. The power comparisons between the modified and traditional MRA, together with different types of scores, will be conducted. In addition, the loss of power and bias information due to dichotomizing a continuous moderator by its median will be assessed.

三、結果與討論

3.1 Methods

In SEM, the two-step approach recommended by Anderson and Gerbing (1988) is commonly used. The first step of this approach is to use CFA to develop an acceptable measurement model. The second step deals with the structural model in the theoretical framework. When the theoretical framework includes a third manifest variable having potential moderating effect on the construct relationship, attention will be drawn to test for its significance. In contrast to the traditional MRA, a modified MRA is proposed in this article. The proposed modified MRA treats moderators and the corresponding cross-product terms with the exogenous constructs as exogenous manifest variables and keeps the multiple indicators of latent constructs. By doing so, estimates for exogenous constructs are needed for obtaining cross-product terms before simultaneous statistical inference can be conducted in SEM. Factor-based scores and the factor scores derived from CFA will be used and the resulting effects will be compared. The modified approach applies for categorical moderator through the use of dummy variables. For a dichotomous variable, introduce a dummy variable and its cross-product term with the exogenous construct into the model. To detect moderating effects, simply test if the coefficient associated with the cross-product term is significantly different from zero. If

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significant, there exist moderating effects.

3.2 Monte Carlo Simulation

Monte Carlo simulation will be used to compare the power and bias for assessing moderating effects of continuous manifest moderators between the traditional and modified MRA with factor-based scores and factor scores. Information loss due to dichotomous categorization by the moderator median will also be examined.

Without loss of generality, we proposed a simple simulation-based model, where the effect of exogenous construct FX on the

endogenous construct FY is moderated by the

continuous manifest variable M. Each construct is measured by the three indicators separately, with X1 ~ X3 measuring FX, and Y1 ~ Y3 measuring FY.

The structural model is given by FY = ß1FX

+ ß2M + ß3 FX * M + ζ, where the error term

ζ is independently normally distributed. We test H0: ß3 = 0 versus H1: ß3 ≠ 0 to detect

moderating effects. The t-test can be used by first computing the t-value (= estimate / standard error of the estimate) and then comparing |t| with zα/2, the upper 100(α/2)th

percentile of the standard normal distribution.

SAS will be used to carry out simulation.

3.2.1 Simulation Design

Jaccard and Wan (1995) gave the following power-influencing factors: (a) sample size, (b) effect size of the interaction term, (c) size of the latent variable squared multiple correlations, (d) predictor variable intercorrelation, (e) reliability of the indicators, and (f) type of estimation method. Based on Jaccard and Wan (1995), the factors considered and the factor levels are stated below:

1. Factor loadings: The factor loadings for the three indicators on their corresponding constructs were set at (0.6, 0.7, and 0.8) as well as (0.9, 0.6 and 0.5), both leading to alpha values close to 0.7.

2. Total sample size: The sample sizes of 175 and 400 were used.

3. Correlation between the exogenous construct FX and the moderator M: The

correlations of 0.2 and 0.4 were set.

4. Size of the squared multiple correlations for the structural model: The squared multiple correlations for the structural model (reflecting the explanatory power) were set as 0.3 and 0.5. Thus, the corresponding standard deviations of the error term for the structural model were 0.84 (= 1−0.3) and 0.71 (= 1−0.5).

To simulate the power functions, the moderating effect size (ES) was started from 0 (reflecting no moderating effect), where the associated power should be close to α, the significance level, through the value with power close to 1. The power functions and the bias will be compared between the traditional and the modified MRA together with factor-based scores and factor scores derived from CFA for each of the sixteen (= 2 × 2 × 2 × 2) combinations. In addition, the comparisons will be extended, for each combination, to the amount of the power and bias information loss by categorization. Since the continuous moderator Z was set to have a normal distribution, it does not make difference to categorize with the median or the mean.

3.2.2 Simulation Procedure

The simulation procedure, for each combination, is shown by the following steps (The factor loadings of 0.9, 0.6, and 0.5, the sample size of 175, the correlation of 0.2 between FX and Z, and the error variance of

0.5 are used for illustration):

Step 1. Generate data.

(1) A value of FX is randomly generated

from the standard normal distribution (denoted by N (0,1)), and then use the measurement model to obtain values of

X1 ~ X3. Under the standardized situation

specific variance, values of X1 ~ X3 can

be obtained as follows:

X1 = 0.9 *FX + 0.43589 * Z1, X2 = 0.6* FX + 0.8 * Z2,

X3 = 0.5* FX +0.866* Z3,

where Z1, Z2, Z3 are independent N (0,1)

random variables.

(2) A value of M is generated as follows (Kuan, 2000, Sec.14.3):

M= 0.2 *FX + [1 − (0.2)2]1/2 * G,

where G is an independent N (0,1) random variable.

(3) A value of FY is generated through the

structural model as follows:

FY = ES*M*FX + 0.71*H,

where ES is a specified moderating effect size and H is an independent N (0,1) random variable.

(4) Values of Y1 ~ Y3 are generated through

FY as follows:

Y1 = 0.9 *FY+ 0.43589 * Z4,

Y2 = 0.6* FY+ 0.8 * Z5,

Y3 = 0.5* FY+0.866* Z6,

where Z4, Z5, Z6 are independent N (0,1)

random variables.

Repeat Step 1(1) through Step 1(4) 175 times to obtain 175 observations for X1 ~ X3 ,

M, and Y1 ~ Y3. Then, M is categorized by

the sample median and dummy variable D is introduced to represent a dichotomous moderator.

Step 2. Estimate scores for latent constructs.

Factor-based scores for FX and FY are (X1

+ X2 + X3) / 3 and (Y1 + Y2 + Y3) / 3 ,

respectively. Factor scores derived from CFA are easy to obtain by SAS (using the PLATCOV command in PROC CALIS).

Step 3. Perform the traditional and the

modified MRA we proposed for M as well as for D.

We record the estimates of the ES and the

testing conclusions under the significance level α = 0.05 for both of the continuous and the categorized moderators.

Step 4. Repeat Step 1 through Step 3 200

repetitions to assess power and bias (The number of repetitions 200 is large enough to obtain stable results).

The power is assessed by computing r

Pˆ (reject H0) = the number of rejections /

200. rPˆ (reject H0∣H0) should be close to α

= 0.05; the higher is the rPˆ (reject H0∣H1),

the more powerful is the corresponding test.

ES were specified at 0, 0.05, 0.10, 0.15, and

0.2 to facilitate power comparison. The bias is assessed by

Estimated bias =200

1

i=

∑

( ˆESi− ES) / 200,

whereES denotes the estimate of ES for ˆi

repetition i.

3.3 Results and Discussions

To facilitate presentation, the traditional MRA with factor-based scores, the traditional MRA with factor scores, the modified MRA with factor-based scores, and the modified MRA with factor scores will be referred to as approaches (A), (B), (C), and (D), respectively.

The simulation results for the factor loadings set at 0.9, 0.6, and 0.5 for both FX

and FY indicate that the powers using the

four approaches are all close to α = 0.05 under H0, and the ranking result of the power

performance under H1 for the four

approaches is D > B > C > A.

When the moderator is categorized by its median, the power, in spite of keeping the same ranking results, will decrease by 15% ~ 30%. Therefore, categorization should not be considered when detecting moderating effects.

The simulation results for the factor loadings set at 0.6, 0.7, and 0.8 (not substantially different) for both FX and FY

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indicate that the power performance for the four approaches is about equally well (D ≈ B

≈ C ≈ A). Again, power information loss does occur if the continuous moderator is categorized by its median.

To see more about the influence of factor loading setups, we have additionally conducted the power comparison for several different sets of factor loadings such as (0.7, 0.8, 0.9) and (0.7, 0.7, 0.8) as well as their cross setups with (0.9, 0.6, 0.5) and (0.6, 0.7, 0.8) for the constructs, all leading to the Cronbach α values of at least 70%. The following results have been observed:

1. If the factor loadings are substantially different (e.g., (0.9, 0.6, 0.5)) for both exogenous and endogenous constructs, then the ranking result of the power performance is D > B > C > A.

2. If the factor loadings are substantially different for the endogenous construct only, then the ranking result of the power performance is D ≈ B ≈ C > A.

3. If the factor loadings are substantially different for the exogenous construct only, then the ranking result of the power performance is D ≈ B > C ≈ A.

4. If none of exogenous and endogenous constructs has substantially different factor loadings, then the ranking result of the power performance is D ≈ B ≈ C ≈ A.

The conclusions for the power comparisons among the four different approaches given above apply for different sample sizes (175 or 400), for different correlations (0.2 or 0.4) between the exogenous construct and the moderator, and for different sizes of the squared multiple correlations (0.3 or 0.5) for the structural model, showing the consistency of the comparative results.

When the factor loadings for indicators on their corresponding constructs show substantial difference, as can be frequently seen, the traditional MRA with factor-based scores can hardly detect trivial effect sizes. The modified MRA with factor scores

derived from CFA shows much improvement. Although the factor-based scores are meaningful when the measurement model is adequate (Gerbing & Anderson, 1988), the corresponding power is not as high as the power resulting from factor scores. In other words, factor scores could reach the same power level as factor-based scores with a smaller sample size. The influence of different scoring approaches seems stronger than that of simultaneous inference or not. The scoring coefficients for producing factor-based scores are equal while scoring coefficients for producing factor scores are differently weighted. There exists a dominant coefficient, corresponding to the one with the largest loading, for computing factor scores. Different scoring approaches will lead to different scoring results, and in turn, influence the power of the test. On the other hand, when the factor loadings are close, scoring coefficients resulting from CFA will not have dominant one. They are about equally weighted. This may explain why their corresponding simulated power functions do not show remarkable superiority.

In addition, the power loss could be clearly observed for all combinations when we categorize a continuous manifest moderator. Therefore, categorization should be avoided before testing for moderating effects. Categorization may be allowed when the moderating effects are significant and subsequent analysis is needed.

For the bias assessment, how the biases resulting from different approaches differ was inconclusive; nevertheless, categorizing a continuous manifest moderator would seriously enlarge the bias and/or change the direction of bias.

四、計畫成果自評

The findings have made contributions to empirical research by showing that, when substantial differences among factor loadings occur, the traditional MRA with factor-based score (Approach A) is the least preferable

while the proposed modified MRA with factor scores (Approach D) performs best. Moreover, it has been verified that categorizing a continuous manifest moderator by the median would reduce the power and lose much of the bias information.

Two research papers derived from this study were submitted to Asian Pacific

Management Review and British Journal of Mathematical and Statistical Psychology.

The former has been accepted for publication, and the latter is still under review.

五、參考文獻

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Anderson, J. C., & Gerbing, D. W. (1988). Structural equation modeling in practice: A review and recommended two-step approach. Psychological Bulletin, 103, 411-423.

Baron, R. M., & Kenny, D. A. (1986). The moderator-mediator variable distinction in social psychological research: conceptual, strategic, and statistical consideration.

Journal of Personality and Social Psychology, 51, 1173-1182.

Bollen, K. (1989). Structural equations with

latent variables. New York: Wiley.

Gerbing, D.W., & Anderson, J.C. (1988). An updated paradigm for scale development incorporating unidimensionality and its assessment. Journal of Marketing Research, 25, 186-192.

Grapentine, T. (2000). Path analysis vs. structural equation modeling. Marketing

Research, 12, 12-20.

Jaccard, J., & Wan, C. K. (1995). Measurement error in the analysis of interaction effects between continuous predictors using multiple regression: multiple indicator and structural equation approaches. Psychological Bulletin, 117,

348-357.

Kuan, C.-M. (2000). Statistics: concepts and

methods, Taipei, Taiwan: Hwatai.

Lawley, D. N., & Maxwell, A. E. (1971).

Factor analysis as a statistical method.

New York: American Elsevier.

McClelland, G. H., & Judd, C. M. (1993). Statistical difficulties of detecting interactions and moderator effects.

Psychological Bulletin, 114, 376-390.

Saunders, D. R. (1956). Moderator variables in prediction. Educational and

Psychological Measurement, 16, 209-222.

Sharma, S., Durand, R. D., & Gur-Arie, O. (1981). Identification and analysis of moderator variables. Journal of Marketing