# Adjusting Structural Equation Modelling Of Spiritual Coping Scale: Use Of The Sattora-Bentler Method As An Alternative To Maximum Likelihood Estimation

M Khodayari, A Abadi, H Majd, M Rassouli, H Sadeghi-Bazargani

###### Keywords

likert scale, maximum likelihood estimator, measurement model, psychometrics, robust sattora-bentler estimator., spiritual coping scale, structural equation modelling, structural model

###### Citation

M Khodayari, A Abadi, H Majd, M Rassouli, H Sadeghi-Bazargani. *Adjusting Structural Equation Modelling Of Spiritual Coping Scale: Use Of The Sattora-Bentler Method As An Alternative To Maximum Likelihood Estimation*. The Internet Journal of Epidemiology. 2012 Volume 10 Number 2.

###### Abstract

^{2} method showed the model to have a better fit to the data (^{χ2
})/ df)

### Introduction

Nearly everyone working in research fields of psychology and neuropsychiatric diseases is acquainted with the Likert-type scale. The Likert scale was invented by a psychologist called Likert Rensis ^{1}. Although general applicability of Likert scales has been often questioned, this type of measurement remains an extremely popular methodology in the fields of psychology, public health and nursing research^{2-6}. In fact, it is so widely used in scaling responses in surveys that it is sometimes used interchangeably with rating scales, even though the two are not synonymous. Analysis of data measured on Likert scales is another issue requiring special notation. A variable measured using Likert type questions exists on an ordinal scale- generally one limited to a few levels. This poses the question of whether we can safely apply statistical methods that rely on assumptions of normality. The issue gets even more problematic when the sample size is small and a complex model needs to be developed to assess a latent variable representative of a health-related phenomenon.

Structural equation modelling (SEM) is a statistical methodology that takes a confirmatory (i.e. hypothesis-testing) approach to the analysis of a structural theory bearing on some phenomenon. Typically, this theory represents “causal” processes that generate observations on multiple variables^{7}.

Review of SEM applications during the past 15 years (in psychological research, at least) reveals most measurements to be based on Likert scaled data with estimation of parameters done using maximum likelihood (ML) procedures^{8}.

When the number of categories is large and the data approximate a normal distribution, failure to address the ordinal form of the data is likely to have negligible consequences; however, this may not be the case in many studies^{9}. In psychology and other social sciences, data are often collected through questionnaireswhich use a Likert scale. Multivariate normality is an essential assumption that may not hold for this kind of data.

In an experiment on assessing spiritual coping, the aim of our study was to illustrate the problems of using the common maximum likelihood (ML) method, and to assess a way for dealing with structural equation models when variables are in categorical form and don’t follow a normal and continuous distribution.

### Material and methods

The study population where the data come from contained 120 adolescents in State Welfare Organizations of Tehran database. Data was derived from institutionalized orphan adolescents between 14-20 years of age enrolled in nineteen protector centres of Tehran.

A structural model was developed and specified for spiritual coping. The developed scale was called “Institutionalized adolescents spiritual coping scale”. The structural statistical model was fitted and investigated using common fit indices of ML, and Sattora-bentler estimators for small samples and non-normal Likert scale data.

### Sattora-bentler method ()

ML methods produce parameter estimators to ensure that observed sample probability is maximized. This method assumes that observed variables have multi-normal distribution. The likelihood function will be:

In which:

Quantity of A cause estimator function minimized.

Assuming that π shows parameters, we have:

In which:

When mean structure was unknown, assume

An alternative hypothesis under this offered model is

Regarding that:

Assuming that π̑ showed the ML estimator value given the null hypothesis, the chi square likelihood proportion for fitting model _{0}
_{1}
_{ML}(π̑)

Bentler (2005) noted that adjusting test statistics for small samples may help to achieve better estimators when analysing Likert data for the purpose of hypothesis testing.

When observed variables that do not have multivariate normality, ^{2}

In which

In which

In which W is a weighted matrix and its optimum value is

This study was approved as a thesis research project for a degree of master in biostatistics registered in Tarbiat Modarres University in Tehran, Iran.

### Results

The model for spiritual coping, offered by researchers, contained seven variables. They were spiritual coping, spiritual attitude and subjective norm, which were taken as latent variables; and hopefulness, self-respect, self-efficacy and age, which were taken as observational variables. Observational variable scores were computed from the mean of their items. Because of being three latent variables in the structural model, there were three observational models. After reducing nonrelated items, spiritual attitude, subjective norms and spiritual coping were computed by 11, 10 and 16 items by respectively. Exploratory factor analyses had given the best SEM model. Figure 1 shows the structural model, which describes spiritual coping. As shown in Fig. 1, the measurement model related to subjective norms used second order factor analyses.

Table 1 shows the results of confirmatory factor analyses for measurement models. Results show that fit indices and parameters encountered the under-estimation problem by using the common ML estimator method. Indices show that application of a robust SB-estimator produced a better fit.

The Mardia index shows that none of the variables had normal distribution. Results show that when variables aren’t normal, it is better to use Robust SB-xi square method. Figure 2 shows the measurement model of subjective norms drawn by EQS software package.

Measurement models evaluation show the efficiency of Robust Sattora-Bentler correction. Fit indices for total SEM model confirm the same results. Table 2 compares SEM model fit indices in both continuous and categorical status. Study shows that fit indices and parameters encounter the under-estimation problem by using the common ML estimator. Using a Robust SB-chi square method produces a model which better fits data of this type.

##### Figure 16

### Discussion

An important disadvantage of all normal theory methods, including ML, is that they can yield distorted conclusions about model adequacy when there are violations of the requisite distributional assumption- here, multivariate normality. That is, when data are not normal, the test statistic

This study was an example of SEM with small sample size and non-normal distribution. Many studies use Likert scale data with a small sample size in psychology, psychiatry, and social sciences. As found in our experience, in case of non- normal distribution and small sample size, fit indices get problematic and under-estimation problem is encountered. This study used the robust Sattora-bentler method for adjusting under-estimated fit indices. Results show that using this method could be beneficial for researches who need to handle categorical Likert scale data similar to that used here in spiritual coping assessment.

Crawford and Henry evaluated the Positive and Negative Affect Schedule(PANAS) in a large sample of 1003 cases. Data had high positive skew. PANAS is measured with 20 Positive and Negative items, offered by Watson and Clark. Results produced a ^{2}
^{2}
^{10} using the Sattora-Bentler method.

Consistent with our results, indices garnered from another study by Crawford and Henry (2003) on the Depression Anxiety Stress Scales (DASS) demonstrated that the ^{2}
^{2}
^{11}.

The Personal Disturbance Scale was evaluated by Henry (2002) with regards to anxiety and depression scores being computed from 14 Delusions-Symptoms-States items. Data was derived from 758 adolescents between the ages of 16 and 19. Based on the Kolmogorov-Smirnoff test, the normality hypothesis was rejected, and the ^{2}

Beck Depression Inventory (BDI) has also been evaluated by fitting second order factor analysis model. Results of the study showed very different values between the ML method and SB estimators. CFI index for the unadjusted ML was equal to 0.76, while it was equal to 0.93 for SB robust method^{12}.

In the present study, the CFA model fit indices for spiritual coping with 11 items showed evidence of moderate fitting of the model to the data. The reason was that all the variables used for measuring spiritual coping were categorical variables, yet were applied as continues variables ( ^{2}

Consistent with previous research on other types of psychological scales, we found in our study on spiritual coping that, when variables are categorical, using the ^{2}