Lecture 5 practical_guidelines_assignments
1 like819 views
The document discusses exploratory factor analysis (EFA) conducted on questionnaire data with 10 items measuring happiness from 200 respondents. EFA with principal axis factoring extracted 2 factors based on eigenvalues above 1, a scree plot analysis, and a rotated component matrix. The first factor reflected sociability and the second reflected positive feelings. Preliminary analyses like the KMO and Bartlett's test supported the factorability of the data.
Lecture 5 practical_guidelines_assignments
- 1. Introduction to Applied Statistics and Applied Statistical Methods Practical guidelines Prof. Dr. Chang Zhu page 1 Table of Contents LECTURE 5................................................................................................................................................ 2 RECAP ...................................................................................................................................................... 2 EXPLORATORY FACTOR ANALYSIS (EFA).................................................................................................. 2 OUTPUT INTEPRETATION .................................................................................................................... 4 PRELIMINARY ANALYSIS ...................................................................................................................... 4 FACTOR EXTRACTION .......................................................................................................................... 5 HOW MANY FACTORS TO EXTRACT?................................................................................................... 5 RELIABILITY ANALYSIS (Cronbach’s α) ..................................................................................................... 7 REPORTING THE RESULTS OF THE EFA .................................................................................................... 8 ASSIGNMENT 5........................................................................................................................................ 9
- 2. Introduction to Applied Statistics and Applied Statistical Methods Practical guidelines Prof. Dr. Chang Zhu page 2 LECTURE 5 EXPLORATORY FACTOR ANALYSIS (EFA) A researcher has generated a new questionnaire which is designed to measure happiness. The questionnaire that she has generated has 10 items on it and she has collected responses for this from 200 respondents. The items on the questionnaire are as follows: - I feel enthusiastic - I feel full of energy - I have lots of interesting things to do - I have lots of friends - I have a lot to look forward to - I feel excited at the start of each day - I love meeting people - I want to contact friends and family - I want to go out and party - The people at work inspire me The questionnaire is measured on a five point scale where 1 = strongly disagree and 5 = strongly agree. The data file is named Happy_measure.sav This example is taken from: http://wps.pearsoned.co.uk/ema_uk_he_dancey_statsmath_4/84/21627/5536653.cw/content/index.html RECAP Before we proceed to a worked example, it is useful to look at some criteria that we are to apply in exploratory factor analysis (EFA): To decide the sample size, the rule of thumb is that for one variable, we should have 5 to 10 participants (Field, 2009). The Kaiser-Meyer-Olkin of sampling adequacy (KMO) should be greater than .5 to be acceptable. Barlett’s test should be significant to indicate variables are relatively independent from one another. Rotation option: choose Varimax if factors are theoretically independent; Direct oblimin or Promax if factors might correlate. The community is the proportion of common variance within a variable, and should be greater than .3. Look at the table labelled Reproduced Correlations: the percentage of “nonresidual with absolute values > .05” should be less than 50%. The factor loadings should be greater than .512 with a sample size of 100, and .364 with one of 200 (Stevens, 2002). After EFA, it’s suggested that we will conduct reliability analysis for each of the factor.
- 3. Introduction to Applied Statistics and Applied Statistical Methods Practical guidelines Prof. Dr. Chang Zhu page 3 In SPSS, choose Analyse > Dimension Reduction > Factor Move all the variables to the Variables area. Click on the Descriptive button to access the dialog box. We can select all of the options to get to know all the preliminary analysis of our data. Click Continue to proceed to the next step. Access the Extraction dialog box, for Method: choose Principal components. It is advised that we choose Correlation matrix as it’s the standardized version of the Covariance matrix to overcome differences in measurement scales. Select Unrotated factor solution to see if the rotation can yield better intepretation of the factors. The Scree plot and Eigenvalues greater than 1 are the two important options we should select to come up with a decision of how many factors are retained. Click Continue to proceed to the next step. Access the Rotation option: choose Varimax (as our factors are supposed to be independent from each other). Under the Display: select Rotated solution (the result of the analysis) and Loading plot(s) (a graphical display of the cluster of variables to their corresponding factor). As default, SPSS sets the Maximum Iterations for Convergence at 25 times. However, if we have huge data set, we can reset it to 30. Click Continue to proceed to the next step. e
- 4. Introduction to Applied Statistics and Applied Statistical Methods Practical guidelines Prof. Dr. Chang Zhu page 4 Access the Scores button, we can calculate the factor scores for each participant by selecting Save as variables with the Anderson-Rubin method as our factor scores are independent. For correlated factor scores, select the Regression method. Click Continue to proceed to the next step. Click on the Options button, under the Missing Values, select Exclused cases listwise. For the Coefficient Display Format, select Sorted by size and Suppress small coefficients with Absolute value below: .4 (this value is set by us) so that we can have a clean table of results with factor loadings up to .4. Click Continue to return to the main dialog box, then select OK to run the anaysis. OUTPUT INTEPRETATION There are many output tables, but we will choose the most important tables to provide the basis for our decision making of which variables will belong to which factors. PRELIMINARY ANALYSIS The first table we should look at is labeled KMO and Barlett’s Test. The KMO value is .79 (above .05) and the Barlett’s test is significant (p < .001), which indicates that the sample is adequate for factor analysis. KMO and Bartlett’s Test Kaiser-Meyer-Olkin Measure of Sampling Adequacy. .790 Bartlett's Test of Sphericity Approx. Chi-Square 819.746 Df 45 Sig. .000 The KMO values for each variable can be found in the table labeled Anti-image Matrices, by looking at Anti-image Correlation. The values, as can be found, are well-above .5.
- 5. Introduction to Applied Statistics and Applied Statistical Methods Practical guidelines Prof. Dr. Chang Zhu page 5 FACTOR EXTRACTION If we look at the Communalities table, we can see the proportion of common variance within a variable before and after extraction. The table shows that all values are above the cut-off value of .3. Communalities Initial Extraction I feel enthusiastic 1.000 .566 I have lots of friends 1.000 .569 I love meeting people 1.000 .482 I feel full of energy 1.000 .646 I have lots of interesting things to do 1.000 .421 I have a lot to look forward to 1.000 .682 I want to contact friends & family 1.000 .701 I want to go out and party 1.000 .796 The people at work inspire me 1.000 .608 I feel excited at the start of each day 1.000 .645 Extraction Method: Principal Component Analysis. HOW MANY FACTORS TO EXTRACT? To decide how many factors to extract, we should look at 3 outputs: The Total Variance Explained table shows that there are 2 components with eigen values bigger than 1, namely 3.18 and 2.92 respectively. The first component accounts for 31.86% of variance, and the second factor 29.28%. Total Variance Explained Component Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative % 1 3.186 31.862 31.862 3.186 31.862 31.862 3.170 31.699 31.699 2 2.928 29.279 61.140 2.928 29.279 61.140 2.944 29.442 61.140 3 .757 7.569 68.710 4 .658 6.583 75.293 5 .637 6.369 81.662 6 .522 5.220 86.882 7 .429 4.290 91.171 8 .380 3.801 94.973 9 .316 3.155 98.128 10 .187 1.872 100.000 Extraction Method: Principal Component Analysis.
- 6. Introduction to Applied Statistics and Applied Statistical Methods Practical guidelines Prof. Dr. Chang Zhu page 6 The scree plot illustrates that the inflexion point where the two lines that represent the vertical and horizontal parts met is at 3, meaning that we can extract 2 factors (because we do not include the factor at the inflexion point). The information given at the end of the Reproduced Correlations table indicates that there are 20 residuals (44%) with absolute values greater than .05. Though the percentage is quite big, it does not exceeding the limit of 50%. The Component Matrix table also indicates a two-factor solution, but our final decision will be based on the table named Rotated Component Matrix. As indicated by the factor loadings, we can decide that 2 factors will be extracted. After making the decision, we will examine the underlying theme of each factor. The first factor (including items in blue) can be described as a measure of sociability (How sociable a person is) and the second factor positive feeling (or feeling good about the world). Reproduced Correlations I feel enthusiastic I have lots of friends I love meeting people I feel full of energy I have lots of interesting things to do I have a lot to look forward to I want to contact friends & family I want to go out and party The people at work inspire me I feel excited at the start of each day Reproduced Correlation I feel enthusiastic .566a .045 .041 .594 .480 .620 .080 .083 .070 .603 I have lots of friends .045 .569a .523 -.067 -.047 .018 .631 .672 .587 .019 I love meeting people .041 .523 .482a -.063 -.044 .016 .581 .618 .540 .017 I feel full of energy .594 -.067 -.063 .646a .521 .657 -.042 -.047 -.044 .639 I have lots of interesting things to do .480 -.047 -.044 .521 .421a .531 -.026 -.029 -.028 .517 I have a lot to look forward to .620 .018 .016 .657 .531 .682a .053 .054 .045 .663 I want to contact friends & family .080 .631 .581 -.042 -.026 .053 .701a .747 .653 .054 I want to go out and party .083 .672 .618 -.047 -.029 .054 .747 .796a .695 .055 The people at work inspire me .070 .587 .540 -.044 -.028 .045 .653 .695 .608a .046 I feel excited at the start of each day .603 .019 .017 .639 .517 .663 .054 .055 .046 .645a Residualb I feel enthusiastic -.023 .006 -.083 -.054 -.162 -.006 -.059 .053 -.112 I have lots of friends -.023 -.095 .013 -.005 .041 -.161 .001 -.159 -.016 I love meeting people .006 -.095 .061 -.003 .008 -.141 -.115 -.079 -.056 I feel full of energy -.083 .013 .061 -.170 -.022 -.017 .019 -.033 -.114 I have lots of interesting things to do -.054 -.005 -.003 -.170 -.131 .012 .010 .021 -.113 I have a lot to look forward to -.162 .041 .008 -.022 -.131 -.043 .040 -.056 -.047 I want to contact friends & family -.006 -.161 -.141 -.017 .012 -.043 -.006 -.033 .043 I want to go out and party -.059 .001 -.115 .019 .010 .040 -.006 -.120 -.018 The people at work inspire me .053 -.159 -.079 -.033 .021 -.056 -.033 -.120 .015 I feel excited at the start of each day -.112 -.016 -.056 -.114 -.113 -.047 .043 -.018 .015 Extraction Method: Principal Component Analysis. a. Reproduced communalities b. Residuals are computed between observed and reproduced correlations. There are 20 (44.0%) nonredundant residuals with absolute values greater than 0.05.
- 7. Introduction to Applied Statistics and Applied Statistical Methods Practical guidelines Prof. Dr. Chang Zhu page 7 Rotated Component Matrixa Component 1 sociability 2 positive feeling Q8_I want to go out and party .892 Q7_I want to contact friends & family .837 Q9_The people at work inspire me .779 Q2_I have lots of friends .754 Q3_I love meeting people .694 Q6_I have a lot to look forward to .825 Q10_I feel excited at the start of each day .802 Q4_I feel full of energy .801 Q1_I feel enthusiastic .748 Q5_I have lots of interesting things to do .647 Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization. a. Rotation converged in 3 iterations. RELIABILITY ANALYSIS (Cronbach’s α) When we develop our own questionnaire, after factor analysis, we should check the reliability of the sub-scales. Based on the factor analysis, we have 2 factors extracted or 2 sub-scales and the respective items as below: Sub-scale 1 (sociability): Q2, 3, 7, 8, and 9 Sub-scale 2 (positive feeling): Q1, 4, 5, 6, 10 We will calculate the Cronbach’s α for sub-scale 1 first. In SPSS, choose Analyse > Scale > Reliability Analysis Move the variables Q2, 3, 7, 8, and 9 to the Items area. The default model is set as Alpha, so we do not need to change it. Click on the Statistics button to access the Statistics dialog box. The important option we should select is Scale if item deleted.
- 8. Introduction to Applied Statistics and Applied Statistical Methods Practical guidelines Prof. Dr. Chang Zhu page 8 Click Continue to return to the main dialog box and OK to run the analysis. In the output, the table Reliability Statistics tells us that the internal consistency of the 5 items is measured with α = .851 (which is high). Reliability Statistics Cronbach's Alpha N of Items .851 5 If we look at the final column in the table named Item-Total Statistics, we can see that if we remove one of the items, the values of α are not any bigger than .851. Hence, the 5 items display strong internal consistency to one another. Item-Total Statistics Scale Mean if Item Deleted Scale Variance if Item Deleted Corrected Item- Total Correlation Cronbach's Alpha if Item Deleted Q2_I have lots of friends 13.3850 12.188 .612 .832 Q3_I love meeting people 13.2250 11.914 .552 .850 Q7_I want to contact friends & family 13.1800 11.173 .712 .806 Q8_I want to go out and party 13.3550 10.773 .799 .782 Q9_The people at work inspire me 13.3150 11.825 .645 .824 We can report the reliability of the first sub-scale following APA style: The sociability subscale of the Happiness questionnaire had high reliability with Cronbach’s α = .85. Or: The reliability of the sociability scale of the Happiness questionnaire was satisfactory (Cronbach’s α = .85). Repeat the same analysis for sub-scale 2 (positive feeling), we can obtain an α equal to .812. REPORTING THE RESULTS OF THE EFA Field (2009) suggests that we should report 2 things: a description of the analysis and a table of factor loadings, eigenvalues, varianced explained, and Cronbach’s α as follows. A principal component analysis (PCA) was conducted on the 10 items with orthogonal rotation (varimax). The Kaiser-Meyer-Olkin measure verified the sampling adequacy for the analysis: KMO = .79 which is good according to Field (2009). All KMO values for individual items are well above the acceptable limit of .50 (Field, 2009). Bartlett’s test of spherity χ² (45) = 819.746, p < .001, indicated that correlations between items were sufficiently large for PCA. Two components had eigenvalues over Kaiser’s criterion of 1 and in combination explained 61.14% of the variance. The scree plot also supports a two-factor structure. Table 1 shows the factor loadings after rotation. The items that cluster on the same factors suggest that factor 1 represents sociability and factor 2 positive feeling.
- 9. Introduction to Applied Statistics and Applied Statistical Methods Practical guidelines Prof. Dr. Chang Zhu page 9 Table 1 Exploratory Factor Analysis Results For The Happiness Meaure Questionnaire (N = 200) Items Rotated Factor Loadings sociability positive feeling Q8_I want to go out and party .892 Q7_I want to contact friends & family .837 Q9_The people at work inspire me .779 Q2_I have lots of friends .754 Q3_I love meeting people .694 Q6_I have a lot to look forward to .825 Q10_I feel excited at the start of each day .802 Q4_I feel full of energy .801 Q1_I feel enthusiastic .748 Q5_I have lots of interesting things to do .647 Eigenvalues 3.18 2.92 % of variance 31.86 29.28 𝜶 .85 .81 ASSIGNMENT 5 (You can work alone or in group for this assignment. If you work in group, please stay in the same group of previous assignments and indicate the group members in the submission document). The questionnaire is designed by Dr. Dian Williams to measure organizational ability. She predicted a five-factor structure measuring different dimensions of organizational ability, which are theoretically independent. The items are completed by 239 people, using a 7-point Likert scale (1 = strongly disagree, 4 = neither, 7 = strongly agree). The data file is named organization_measure.sav Conduct factor and reliability analyses, and report the result in APA style. To come up with the appropriate analysis option, and the final decision making on the number of factors that should be extracted and the report, think about these questions to help you: 1. Are the constructs independent or dependent from each other? What is the appropriate rotation method? 2. How many components/factors with eigenvalues bigger than 1? How many factors can be extracted as suggested by the scree plot and the Rotated Component Matrix? 3. What dimension (name of factor, e.g. sociability) can we attribute to each of the component? 4. Do the results from the reliability analysis suggest to remove any items that help to increase the Cronbach’s alpha?