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Application of Structural Equation Model. Summarized by Ann Shih June 3, 2004 DHPR, NHRI. Sources. Structural Equation Modeling: Concepts, Issues, and Applications. Edited by Rick H. Hoyle. Chapter 8. 邱浩政 , 結構方程模式 : LISREL 的理論技術與應用 .2004

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Application of Structural Equation Model

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Application of structural equation model

Application of Structural Equation Model

Summarized by

Ann Shih

June 3, 2004

DHPR, NHRI


Sources

Sources

  • Structural Equation Modeling: Concepts, Issues, and Applications. Edited by Rick H. Hoyle. Chapter 8.

  • 邱浩政, 結構方程模式: LISREL的理論技術與應用.2004

  • Hsiu-Hung Wang, Shirley Cloutier Laffrey. A Predictive Model of Well-Being and Self-Care for Rural Elderly Women in Taiwan. Research in Nursing & Health, 2001, 24, 122-132.


Outline

Outline

  • Summarized the key points of the Chapter 8 in Book edited by Hoyle

  • Summarized the key points of applying SEM in research


Objectives of chapter 8

Objectives of Chapter 8

  • Approach to and information derived from preliminary analysis of the data

  • Treatment of data that violate the assumption of multivariate normality

  • Assessment of overall model fit

  • Identification of parameter misspecification

  • Post-hoc model fitting

  • Test for multigroup invariance


Application of structural equation model 1335360

Data

  • Beck Depression Inventory (BDI)

    • 21-item scale, self-reported

    • Cognitive, behavioral, affective, and somatic components of depression

    • Four-point Likert scale

  • 658 adolescents (grades 9-12) attending the same high school in Ottawa, Canada (male 337, female 321)


Model specification

Model Specification

1

Y1

1

Measurement model

1

X1

1

Y2

x11

11

Y3

X2

1

21

21

X3

Y4

2

Y5

Structure model

Y6

y62

2

= +  + 


Hypothetical second order model of bdi factorial structure

Hypothetical Second-order Model of BDI Factorial Structure

1

Negative

attitudes

Second-order factor loadings

11

1

2

Residual error

Depression

22

Performance

difficulties

1

2

3

33

Second-order

Somatic

elements

First-order

3


Hypothetical second order model of bdi factorial structure1

Hypothetical Second-order Model of BDI Factorial Structure

1

Negative

attitudes

1

First-order factor loadings

Measurement error associated with the item variables

11

Depression

Performance

difficulties

1

2

18

Somatic

elements

3


A prior

A Prior

  • Reponses to the BDI could be explained by 3 first-order factors and 1 second-order factor of general depression

  • Each item would have a nonzero loading on the first-order factor it was designed to measure and zero loadings on the other two first-order factors

  • Error terms associated with each item would be uncorrelated

  • Covariation among the 3 first-order factors would be explained fully by their regression onto the second-order factor


Model goodness of fit

Model Goodness-of-Fit

  • 卡方檢定

  • Index

    • GFI & AGFI

    • NFI & NNFI

    • IFI

  • Alternative Index

    • NCP

    • RMSEA

    • CFI

    • ECVI & AIC

    • CN


Application of structural equation model 1335360

卡方檢定

  • 反應SEM假設模型的導出矩陣與觀察矩陣的差異程度

  • T=(N-1)Fmin where Fmin表示以各種不同參數估計方法所的到的契合函數最小函數估計值

  • T達顯著水準表示契合度不佳 反之良好

  • C1-C4 test

  • 考慮模式複雜度後的 (= /df<2 significant)


Index for model fitting 1

Index for Model Fitting -1

  • Goodness-of-fit index (GFI) ~ R^2

    • 假設模型可以解釋觀察資料的變異數和共變數的比例

    • 愈接近1 模型契合度越高 (>0.9)

  • Adjusted GFI (AGFI) ~ adjusted R^2

    • 考慮自由度後所計算出的指數

    • 指數愈大模型愈佳(>0.9)

  • Parsimony Goodness-of-fit Index (PGFI):

    • 考慮估計參數的多寡

    • 指數愈大愈佳(>0.5)


Index for model fitting 2

Index for Model Fitting - 2

  • Normed Fit Index (NFI): 某一個假設模型比起最糟糕模型的改善情形

  • Non-normed Fit Index (NNFI): 考慮自由度但波動性大

  • Incremental Fit Index (IFI): 處理NNFI波動問題與樣本大小對NFI的影響

>0.9 is better


Alternative index

Alternative Index

  • NCP (non-centrality parameter)

    • 計算估計的卡方統計量距離理論預期的中央卡方分配的離散程度

    • 越接近0越好

  • CFI (comparative-fit index)

    • 反應假設模型與獨立模型差異度及檢驗模型與中央卡方分配的離散性

    • 越接近1越好 (>0.95)

  • RMSEA (root mean square error of approximation)

    • 不受樣本大小與模型複雜度的影響

    • 指數越小越好(<0.05)


Preliminary analysis prelis

Preliminary Analysis - PRELIS

  • Data Screening

    • Missing data: listwise deletion

    • Nonormality assumption

    • Outliers

  • Ordered, Censored data

  • Matrix for Weighted Least Square

  • Bootstrap, Monte Carlo


Sample statistics

Sample Statistics

  • Minimum and maximum frequency values

  • Single tests of zero skewness and kurtosis: z-statistics

  • Combinations of two moments:


Treatment of nonnormality

Treatment of Nonnormality

  • Asymptotic (large-sample) distribution-free methods (ADF): 2-step processes

    • PRELIS: recasts the data into asymptotic matrix form

    • WLS estimation

  • Limitation:

    • Min sample size=k(k+1)/2 where k equals the number of variables

    • This requirement offers no guarantee of good estimates of the asymptotic covariance matrix

  • SCALED


Application of structural equation model 1335360

補充觀念

  • 一個連續變項是否呈常態分配並非全有全無的概念而是程度上的問題

  • 多元常態假設沒有一個絕對公正客觀的評估方式

  • 多元常態假設僅是一種觀念性的主張或原則性的探討而不會實質加以檢測

  • 處理非常態資料的方法

    • 資料檢核與過濾: 刪除或歸為獨立樣本

    • 資料轉換: 平方根, 取對數ln, Box-Cox轉換法

    • 不同估計程序的使用:WLS, ADF, WLSM, scaled

    • CVM (<5 categorical variables) is better than ADF

    • 除非非常態問題相當嚴重(kurtosis>25)或無法經由其他方法校正時 仍使用ML


Testing the hypothesized model

Testing the Hypothesized Model

  • The appropriateness of the estimates

    • Correlations greater than 1.0

    • Standard errors that are abnormally large or small: linear dependence

    • Negative variances

  • Statistical significance of the estimates

    • Z statistic>1.96 reject the hypothesis


Post hoc model fitting

Post-hoc Model-Fitting

  • To identify misspecified parameters in the model when the fit found to be inadequate

  • Univariate and is based upon the Modification index (MI)

    • Statistical criteria only

    • Virtually any fixed parameter is eligible for testing


Application of structural equation model 1335360

補充說明 - 1

  • 模型修飾的計量策略

    • Chi-square difference test

      • 檢驗修飾後的模型卡方值是否優於未修飾前的模型卡方值

      • 兩個模型必須為nested model

    • Likelihood Ratio test

      • 模型修飾前後變動的比率是否有意義

      • LR達顯著時表示模型修飾有統計意義 反之沒有統計意義

    • LM test: so called MI (modification index)

      • 檢定個別參數逐次增加後對於模型契合度影響的檢定法

      • 僅需針對受限模型(修飾後)進行估計 且僅針對一個最顯著的參數進行估計比較模型變動前後的差異

    • Walt test

      • 檢驗當某一個參數受限後對於模型契合度降低的影響程度

      • 估計非受限模型


Application of structural equation model 1335360

補充說明 - 2

  • 注意事項

    • 指標並沒有強有力的理論基礎支持數字的意義與原則

    • 不同指標優劣比較仍具有爭議

    • SEM模型檢驗應以理論為依歸, 因此最佳指標可能只反映了技術上的最佳化而非理論上的最佳化

    • 不適當參數應事前偵測

    • 初始模型必須具有優越的理論基礎否則再多修飾也無濟於事


Testing for invariance across gender

Testing for Invariance Across Gender

  • Three restrictive hypotheses

    • Equivalency of number of underlying factors

    • Equivalency of first-order factor loadings

    • Equivalency of second-order factor loadings


Syntax for prelis 1

Syntax for PRELIS - 1

  • Main syntax:

    • DA: data specification

    • RA: raw data

    • OU: output

  • Others

    • OR: ordered variables (<16)

    • CO: continuous variables (>=16)


Syntax for prelis 2

Syntax for PRELIS - 2

  • RE variable list OLD=valuerange NEW=value list

  • NE new variable=function of old variables

  • LO variable list AL=0 BE=1

  • PO variable list AL=0 BE=1 GA=1

  • SC variable list=value (or >value or <value)

  • SC CASE=ODD (or EVEN)

  • SD variable list=value

  • SE variable list

Cut and create new variables

Select some variables or sample


Prelis example

PRELIS - Example

!Example of PRELIS command

DA NI=7 NO=300 TR=LI MI=99

LA ORDER SEX ITEM1 ITEM2 ITEM3 ITEM4 ITEM5

RA FI=PRELIS1.DAT

OR ORDER SEX

CO ITEM1 ITEM2 ITEM3 ITEM4 ITEM5

OU MA=PM SM=PRELISN.COR AC=PRELISN.COV PK ND=3


Application of structural equation model 1335360

SEM操作

  • 模型發展階段

    • 發展假設模型

    • 模型界定

  • 估計檢驗階段

    • 參數估計

    • 模型檢驗

  • 執行SEM分析

  • 報表整理與結果分析

  • 模型修飾

  • 完成SEM分析


Sem 1

SEM執行重點 - 1

  • 模型描述與設定

    • Conceptual model: path diagram

    • Statistical model: 語法指令的內容

  • 資料的準備

    • 共變數矩陣比相關係數矩陣佳

    • 輸入至小數點第三位以提高計算精確度

    • 處理類別或順序變項以產生共變矩陣

    • 變項分配特性之揭露


Sem 2

SEM執行重點 - 2

  • 報表整理與分析

    • 報告過程性的資料 & 最終解(Final solution)

    • 說明估計方策略與特殊處理

    • 報告卡方統計量及與其相關之訊息

    • 模型契合度指標

      • Absolute fit: GFI

      • Incremental fit

        • ML: NNFI for large sample, IFI for small sample

        • GLS: IFI

    • 參數報告

      • 合理性: 負值殘差變異數或超過範圍的共變數

      • 顯著性考驗:t test or p-value or standard errors

      • 標準化解: standard solution(latent) and SC(latent and observed)


Sem 3

SEM執行重點 - 3

  • 替代模型的使用

    • Deductive approach or a priori: 研究者基於理論觀點所提出的假設模型

    • Inductive approach or a post-hoc: 估計結果產生的數據所建議的修正模型

      • 樣本規模小, 變項品質差, 參數估計穩定性不佳 etc.

      • 探索性研究或應用研究

      • 測量殘差的相關應謹慎

      • 不應隨意假設其存在殘差與其他變數的相關

    • 100-400樣本數執行模型修飾有風險

    • 從absolute fit 分析等同模型(equivalent model,兩個模型具有相同的統計意義)何者比較理想


Application of structural equation model 1335360

SEM的解釋與應用

  • 因果關係論證與結果推論的限制

  • SEM的推論限制

    • SEM分析技術只是一套統計方法與分析策略並無法創造理論或知識

    • 避免過度依賴技術指標的數據與過度推論

  • SEM分析的解釋

    • 可說明某一個因果概念是可能存在的但不能據以排斥其他模型的存在除非直接檢驗其他模型

    • 分析過程透明化


An example path analysis

An Example: Path Analysis

  • 假設

    Ho:觀察數據=理論模式

    H1:觀察數據 理論模式

  • 因果關係的存在必須由研究者提出清楚合理明確的邏輯與推理程序來說明假設存在的基礎或以理論與文獻之支持來確立假設的合宜性與合理性

  • 傳統路徑分析(PA-OV) vs. 潛在變項的路徑分析(PA-LV)


Example a predictive model of well being and self care for rural elderly women in taiwan

Example: A Predictive Model of Well-being and Self-Care for Rural Elderly Women in Taiwan

  • Objective of the study is to test a theoretical model of well-being and self-care in a sample of elderly rural Taiwanese women

  • Theoretical models

    • Pender’s revised health promotion model (1996)

    • Orem’s self-care theory (1995)


Cont d

Cont’d

Perceived health

Age

Social class

Self-care agency

Self-care behavior

Social support

Marital status

Well-being


Cont d1

Cont’d

  • Descriptive statistics and model testing

    • Mean scores, SD, and ranges for variables

    • Bivariate relations (Correlation matrix)

    • Five equations

  • Model modification

    • Add two paths

    • Model trimming: GFI, AGFI, NFI, NNFI, CFI

    • Variance


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