The main purpose of the factor analysis, if possible, is to explain the correlation relations between a number of variables in terms of several unobservable random quantities that are called factors. Suppose variables can be categorized by their correlations. That is, all the variables of a particular category have a high correlation among themselves, while having relatively low correlation with the variables of other categories. In this case, it can be assumed that each group of variables represents an examined compound or agent, which explains the observed correlations .
To perform factor analysis, there should be a reasonable correlation between variables (questionnaire questions). Not many correlations between variables should be less than 0.3, because then observations would not be suitable for performing factor analysis due to low correlation. Likewise, not many correlations should exceed 0.8 as this would create a multiple linear relationship between the variables . By examining the correlation matrix between the variables, correlation between the variables was appropriate; therefore, observations were made to perform a relevant factor analysis (Table 1).
Table 1:
Correlation coefficient of the questions
Question
|
Q1
|
Q2
|
Q3
|
Q4
|
Q5
|
Q6
|
Q7
|
Q8
|
Q9
|
Q10
|
Q1
|
1.000
|
|
|
|
|
|
|
|
|
|
Q2
|
.262
|
1.000
|
|
|
|
|
|
|
|
|
Q3
|
-.019
|
.209
|
1.000
|
|
|
|
|
|
|
|
Q4
|
.163
|
.361
|
.530
|
1.000
|
|
|
|
|
|
|
Q5
|
.108
|
.282
|
.171
|
.088
|
1.000
|
|
|
|
|
|
Q6
|
-.169
|
.084
|
.127
|
.032
|
.268
|
1.000
|
|
|
|
|
Q7
|
.952
|
.195
|
-.013
|
.139
|
.108
|
-.157
|
1.000
|
|
|
|
Q8
|
.184
|
.857
|
.277
|
.379
|
.350
|
.100
|
.225
|
1.000
|
|
|
Q9
|
.040
|
.301
|
.810
|
.683
|
.228
|
.069
|
.023
|
.324
|
1.000
|
|
Q10
|
.140
|
.334
|
.508
|
.844
|
.161
|
.043
|
.137
|
.373
|
.548
|
1.000
|
To determine if the data are suitable for Factor Analysis, the KMO Index is used to examine the Bartlett Spread Sampling and Testing Capability. The minimum acceptable value for the KMO index is 0.5 and the closer it is to one, the better. Bartlett's test is also used to determine whether the matrix of correlation between the opposite variables of the matrix is the same or not (Hamani is a matrix whose members are equal to 1 for the main diameter and zero to the other members). The zero assumption of this test shows the equality of the matrix of correlation between variables with the Hamani matrix. If the zero assumption is rejected, it is concluded that there is a reasonable correlation between the variables; therefore, it is permissible to perform factor analysis on the data in question. According to the results of these tests, as presented in Table 2, it is seen that according to the above indicators, the available observations for performing factor analysis are sufficient and the factor analysis is justifiable.
Table 2:
Bartlett Test and KMO Indicator
KMO Indicator
|
.516
|
Bartlett
Test
|
χ 2
|
1182.802
|
Df
|
45
|
P
|
<0.001
|
Selecting the analysis method and the number of factors
In this research, factor analysis by main component method and by Varimax rotation were used to evaluate the governing structure of 10 questions of the research questionnaire. In addition to choosing the method of factor analysis and type of rotation, it is also important to decide on the number of factors.
One way to determine the number of factors is to plot the special values chart against the number of factors that is called the pebble or rocks graph (Fig. 1). The manner of specifying the number of factors with this graph is that the point at which the chart begins to flatten is considered as the number of factors. The disadvantage of this diagram is that in this method, t selection of the number of factors is performed intuitively, depending on the individual judgment of the analyst.
A more precise method to determine the number of factors is to check the special value of the extracted factors. Each agent has a special value that measures the variance explained by that agent. Therefore, higher special values indicate higher importance of this factor. Using the Kaiser criterion, if the special value of an agent is greater than 1, then the agent is extracted; otherwise, it is eliminated. In this study, it was found that 3 factors had a specific value greater than 1. Therefore, taking this criterion into account, there are up to 3 factors which can be extracted potentially (Figure 1).
Another important factor to be considered in a factor analysis is to examine the percentage of the variance explained by each factor and the cumulative percentage of variance explained by the extracted factors. Table 3 illustrates this issue. According to this Table, the three extracted factors together account for approximately 71.84% of the total variance, which is a perfectly good value.
Table 3:
Specific values and variance explained by factors (after rotation)
Factors
|
Eigenvalue
|
% of variances
|
Cumulative of variances
|
1
|
3.029
|
30.289
|
30.289
|
2
|
2.113
|
21.126
|
51.415
|
3
|
2.042
|
20.422
|
71.836
|
One of the criteria of examining the questions in the questionnaire in exploration factor analysis is the amount of their common extraction. This value represents the percentage of the variance of each question that is justified by the extracted factors. Table 4 shows the amount of the common extraction of the questions in the questionnaire. As you can see, the questions are in a good position and there is no need to remove one.
Table 4:
Extracted Commonalities of the Questions in the Questionnaire
Questions
|
Communalities Extraction
|
Q1
|
.916
|
Q2
|
.732
|
Q3
|
.696
|
Q4
|
.810
|
Q5
|
.460
|
Q6
|
.493
|
Q7
|
.903
|
Q8
|
.774
|
Q9
|
.784
|
Q10
|
.716
|
The matrix of the rotated factor loads is shown for each of the questions on the extraction agents. Factor loads are, in fact, correlation coefficients between the questions and factors. In addition, the power of the relationship between the factors and questions is shown by factor load.
The result of factor analysis by principal component method and Varimax rotation with 3 factors is presented in Table 5. To demonstrate which factor each question of the questionnaire belongs to, we highlighted the most significant factor load of that question in another color in the Table. It should be noted that the cutting threshold of 0.4 is considered to be the minimum acceptable factor load. It can be seen that the factor loads of each question on its own factor, in addition to being greater than the cutting threshold of 0.4, is due to other factor loads of that question on other factors.
Table 5:
Matrix of the rotated factor loads with main component method and Varimax rotation with three factors
Questions
|
Factor 1
|
Factor 2
|
Factor 3
|
Q4
|
.880
|
.148
|
.120
|
Q9
|
.871
|
-.049
|
.155
|
Q10
|
.821
|
.136
|
.152
|
Q3
|
.817
|
-.123
|
.113
|
Q1
|
.030
|
.949
|
.115
|
Q7
|
.019
|
.943
|
.112
|
Q8
|
.293
|
.200
|
.805
|
Q2
|
.254
|
.238
|
.782
|
Q5
|
.052
|
-.025
|
.676
|
Q6
|
-.022
|
-.407
|
.475
|
Eigenvalue
|
3.029
|
2.113
|
2.042
|
% of variances
|
30.289
|
21.126
|
20.422
|
Cumulative of variances
|
30.289
|
51.415
|
71.836
|
Table 5 can also be presented in a different way, so that we do not report values lower than 0.4 to better understand the structure of the obtained factors (Table 6).
Table 6:
Matrix of factor loads greater than 0.4 by the principal component and Varimax rotation
Questions
|
Factor 1
|
Factor 2
|
Factor 3
|
Q4
|
.880
|
|
|
Q9
|
.871
|
|
|
Q10
|
.821
|
|
|
Q3
|
.817
|
|
|
Q1
|
|
.949
|
|
Q7
|
|
.943
|
|
Q8
|
|
|
.805
|
Q2
|
|
|
.782
|
Q5
|
|
|
.676
|
Q6
|
|
|
.475
|
Eigenvalue
|
3.029
|
2.113
|
2.042
|
% of variances
|
30.289
|
21.126
|
20.422
|
Cumulative of variances
|
30.289
|
51.415
|
71.836
|
According to the results, three categories of questions can be considered as the explanations for 3 factors, as presented in Table 7. It is noticeable that the three extracted factors together account for approximately 71.84% of the total variance. (Table 7)
Table 7:
Extracted factors and related questions with principal component method and Varimax rotation
Factors
|
Questions
|
1
|
10-9-4-3
|
2
|
7-1
|
3
|
8-6-5-2
|
Reliability by using Cronbach's alpha (for internal consistency assessment):
Given the extraction factors and related questions, reliability is obtained using the Cronbach's alpha method, as follows. It is observed that the alpha value for the extracted factors is well suited. (Table 8)
Table 8.
Internal consistency with Cronbach's alpha
Factors
|
Questions
|
Cronbach's alpha
|
1
|
10-9-4-3
|
0.882
|
2
|
7-1
|
0.975
|
3
|
8-6-5-2
|
0.637
|
Investigating the relationship between each item and the entire questionnaire:
The correlation coefficient of each question with the total score of the questionnaire is presented in Table 9.
Table 9:
Correlation coefficient of each question with the total score of the questionnaire
Questions
|
Correlation coefficient
|
P.value
|
q1
|
.501**
|
<0.001
|
q2
|
.649**
|
<0.001
|
q3
|
.603**
|
<0.001
|
q4
|
.726**
|
<0.001
|
q5
|
.504**
|
<0.001
|
q6
|
.529**
|
<0.001
|
q7
|
.492**
|
<0.001
|
q8
|
.680**
|
<0.001
|
q9
|
.680**
|
<0.001
|
q10
|
.707**
|
<0.001
|
Investigating the relationship between the score of each factor and that of the total questionnaire:
The correlation coefficient of each extracted factor with the total score of the questionnaire is presented in Table 10.
Table 10:
Correlation coefficient of each extracted factor with the total score of the questionnaire
factor
|
Correlation coefficient
|
P.value
|
f1
|
.791**
|
<0.001
|
f2
|
.503**
|
<0.001
|
f3
|
.723**
|
<0.001
|
Reliability by test-retest method:
The reliability of the test was obtained 0.940 through test-retest method, which is a good value.