5. Factor Analysis

The Factor Analysis model aims to describe the variability of observed correlated variables in a smaller number of unobserved variables (linearly related to the original variables). variables (linearly related to the original variables), called common factors or latent variables.

Example:

A study was conducted to assess life satisfaction. A questionnaire was administered to a random sample of 100 adults. The questionnaire contains 10 questions to measure job satisfaction (WORK1, WORK2, WORK3), leisure satisfaction (LEISURE1, LEISURE2), home satisfaction (HOME1, HOME2, HOME3) and general satisfaction in other areas of life (GENERAL1, GENERAL2). The aim of the study is to understand the relationship between the different spheres of satisfaction. We will use Factor Analysis by the Factor Score Method of Regression and the Factor Method of Solutions by Principal Factors. In addition, we will use Orthogonal Rotation by Max. Variance Factors (Varimax).

We will upload the data to the system.

WORK1 WORK2 WORK3 LEISURE1 LEISURE2 HOME1 HOME2 HOME1 GENERAL1 GENERAL2
105.13 101.66 115.06 101 95.18 100.28 101.67 85.55 104.03 110.28
77.05 72.93 77.49 72.74 61.56 93.85 95.39 88.61 70.12 72
86.02 82.21 78.89 77.95 91.71 86.77 108.07 93.35 86.02 70.69
91.42 106.11 95.64 90.9 111.47 100.25 86.08 93.82 101.22 82.67
113.71 92.03 99.08 79.28 98.42 104.01 83.27 69.62 82.82 70.02
86.61 87.82 67.66 93.66 78 99.82 97.28 108.62 91.4 79.78
95.07 94.5 98.08 94.51 97.42 93.69 99.18 96.4 90.73 86.71
113.5 104.61 105.57 101.01 102.27 87.43 96.66 86.58 93.06 112.7
104.55 97.3 94.07 88.54 98.11 97.79 99.59 99.76 99.4 105.91
104.64 97.91 85.82 82.49 90.45 104.69 95.08 99.69 77.63 62.03
102.06 87.01 94.69 79.2 68.48 79 86.43 78.82 89.73 87.55
109.43 94.94 104.4 119.29 112.99 122.93 114.82 102.11 116.34 101.36
89.99 77.39 100.77 97.03 111.11 107.66 116.86 114.73 88.71 76.5
89.98 91.26 89.22 81.25 100.52 94.9 96.72 94.41 88.42 89.67
96.95 103.04 97.49 98.52 96.83 92.78 110.07 95.16 102.37 105.52
93.21 93.84 87.35 104.26 115.32 114.29 111.62 111.45 95.7 105.72
70.91 65.9 74.23 60.11 47.09 79.81 94.8 88.35 54.66 58.24
114.85 98.08 105.63 90.24 101.01 98.95 87.92 88.91 91.3 92.22
100.31 92.52 87.61 103.27 88.65 108.4 103.56 82.49 102.33 97.57
111.36 104.37 101.85 108.08 121.88 99.95 96.69 89.37 102.15 114.45
108.8 97.88 91.46 99.55 110.49 104.38 95.03 85.51 94.53 91.8
74.72 105.62 102.25 90.17 93.52 86 94.31 106.23 85.31 87.67
93.88 90.58 95.49 86.06 92.44 107.83 97.74 105.22 87.35 89.17
77.94 94.99 98.59 100.32 100.91 96.46 95.41 95.11 104.18 115.85
77.51 100.67 102.3 99.83 106.57 109.1 116.68 105.65 106.5 107.62
99.12 110.27 120.53 115.78 118.91 109.75 108.69 119.02 112.25 120
92.47 94.97 94.38 101.19 95.08 105.82 102.77 116.44 98.45 91.18
92.77 107.08 94.02 111.29 146.45 111.86 112.55 119.95 121.33 113.35
76.86 83.32 67.79 54.6 66.49 80.83 76.36 81.02 48.98 50.06
87.31 85.48 82.82 90.37 89.68 98.09 108.59 123.27 88.39 92.86
90.11 98.8 93.92 102.66 117.8 113.12 100.33 108.16 100.14 99.2
107.98 105.77 114.25 132.67 152.62 121.61 132.88 128.48 145.24 147.19
118.09 96.85 93.45 96.84 74.38 95.04 104.45 106.56 105.7 94.97
79.57 91.44 92.24 100.24 91.13 117.61 112.11 108.55 102.02 107.75
94.41 97.26 95.57 91.77 98.6 85.13 101.34 93.76 93.62 85.33
121.24 114.29 113.35 118.7 122.61 112.35 110.94 123.05 117.64 123.77
102.72 111.1 121.56 83.28 107.95 84.16 76.39 80.05 93.21 87.84
91.07 105.77 101.44 103.06 123.07 116.93 112.74 106.48 113.19 105.96
94.63 81.73 96.75 102.31 95.43 99.81 111.43 110.64 108.01 93.04
114.47 115.63 115.66 121.38 127.68 106.55 113.19 111.58 127.93 128.9
76.79 83.17 95.77 69.32 76.99 85.36 101.62 100.1 80.36 80.74
69.57 99.52 107.38 100.47 109.37 109.88 105.85 105.62 100.43 99.73
70.79 84.38 88.21 97.1 89.06 114.37 119.38 105.06 99.28 100.67
109.55 94.93 96.31 84.74 89.19 95.44 96.34 95.85 92.39 80.25
86.77 87.33 93.17 98.38 90.79 108.82 102.66 95.36 108.16 78.94
104.99 103.9 101.99 102.75 90.12 106.99 108.6 90.36 96.69 89.32
77.76 87.28 97.76 72.13 82.1 98.93 87.41 94.1 83.24 73.81
92.25 96.78 95.17 91.03 84.05 93.97 103.56 104.66 94.38 71.17
81.8 85.98 88.06 75.11 87.42 97.98 107.11 112.57 85.93 93.27
114.7 108.36 111.09 101.6 92.98 90.26 97.36 94.04 93.87 106.84
102.79 102.02 109.05 111.17 123.89 98.76 99.79 100.41 110.04 99.85
89.38 102.8 96.61 109.89 123.46 97.02 112.45 93.96 111.32 108.93
125.77 109.85 106.55 124.27 122.3 133.98 115.39 137.48 129.44 118.87
76.15 105.53 80.2 96.99 98.89 83.3 82.74 86.43 83.94 89.58
134.92 117.95 119.93 107.14 117.86 94.83 97.93 98.66 106.67 126.6
135.75 131.61 129.22 155.27 175.97 116.73 127.48 122.77 150.76 154.44
103.72 106.39 107.8 95.97 107.75 100.65 97.3 108.4 107.53 97.14
109.18 100.09 108.38 108.25 96.75 92.8 91.32 84 90.2 100.33
121.69 118.61 122.43 115.22 117.93 91.83 95.28 104.82 122.89 132.75
102.94 110.61 113.92 93.78 116.07 84.12 86.14 97.92 108.81 110.08
86.12 91.01 92.1 94.26 90.08 102.55 92.36 95.08 85.29 95.51
75.04 82.26 84.46 79.01 70.26 83.45 96.98 101.71 80.22 79.61
97.52 84.22 86.32 79.58 78.94 92.44 100.88 99.69 80.68 78.19
100.24 93.87 103.68 81.76 74.41 98.87 101.64 100.05 94.31 96.23
106.61 100.96 89.52 104.83 94.86 84.6 92.44 99.35 104.33 89.8
78.04 88.49 81.61 78.79 89.1 101.23 100.77 96.65 79.79 84.73
121.88 113.57 119.69 116.08 116.78 84.78 99.98 105.45 113.73 102.11
105.97 99.71 118.53 120.74 108.25 99.07 108.06 111.61 119.71 119.25
84.32 95.96 89.3 111.16 92.27 121.79 113.1 122.86 103.59 89.87
94.73 85.15 104.49 110.74 109.11 117.41 110.7 135.66 105.44 105.77
116.48 109.43 98.44 107.3 111.4 98.52 97.82 91.98 109.64 105.2
89.04 95.97 100.01 96.32 96.34 102.88 96.21 109.84 91.25 97.44
114.54 107.96 115.35 123.42 127.91 119.34 114.2 103.73 122.52 140.4
80.54 88.53 84.75 83.96 81.53 97.08 83.44 97.83 70.69 68.01
95.2 93.74 103.6 102.3 85.74 102.66 115.06 105.07 102.91 110.89
100.77 99.3 109.25 105.06 105.28 115.16 107.8 107.26 110.86 90.84
97.77 95.37 93.22 111.59 118.08 102.03 106.76 111.07 100.05 108.51
100.58 93.54 103.02 93.34 106.17 97.02 91.82 92.7 113.16 90.67
94.75 93.46 105.87 86.91 81.34 99.8 94.37 94.86 85.79 93.36
60.88 77.23 85.27 79.93 78.1 96.81 97.32 90.01 64.52 94.27
94.04 98.5 96.18 100.33 99.11 82.09 91.75 94.19 91.96 105.17
92.62 105.25 108.01 92.75 111.62 83.55 101.46 105.24 104.07 111.37
104.01 99.03 106.89 102.44 108.11 110.37 96.97 95.51 98.34 99.09
81.97 98.56 90.1 84.23 60.21 73.76 85.85 96.65 84.12 81.74
83.15 89.82 89.16 77.56 81.56 79.53 96.67 83.72 75.5 73.73
105.21 105.87 107.16 100.95 113.39 94.39 94.26 100.85 100.74 95.5
95.85 94.18 98.73 113.27 104.91 110.97 115.82 120.51 116.96 109.18
83.57 120.81 107.75 109.01 126.91 108.98 116.46 119.52 118.12 124.6
90.6 103.57 93.01 106.71 70.59 90.87 97.01 96.41 94.81 94.22
115.87 102.89 113.81 98.88 94.38 93.39 111.29 99.67 109.88 105.77
129.71 113.59 118.34 116.97 117.67 96.54 102.37 114.41 121.06 129.09
80.34 84.65 77.01 78.94 86.52 86.97 81.36 77.07 80.32 69.89
92.68 99.48 83.76 104.18 89.88 98.66 119.79 107.8 105.07 111.8
103.19 96.17 83.33 86.12 89.27 108.6 94.22 95.07 81.03 95.01
89.76 99.26 101.21 74.02 72.02 86.05 79.98 79.89 79.87 83.7
94.62 117.87 103.16 113.34 122.24 118.59 117.45 109.06 115.11 115.6
84.76 97.65 91.38 99.43 103.38 116.61 116.93 115.45 106.27 109.38
138.88 117.43 112.19 132.83 128.1 99.54 114.69 114.63 138.35 149.03
95.37 89.84 107.43 105.64 106.76 92.32 109.71 110.57 108.45 105.66
106.05 120.71 119.81 101.85 94.96 75.86 93.17 93.38 109.36 83.79

We will upload the data to the system.

Configuring as shown in the figure below to perform the Factorial analysis.

Then click Calculate to get the results. You can also generate the analyses and download them in Word format.

The tables of results are extensive, so if you want to see them, download the Word document from the system. The graphs obtained are:

Descriptive Analysis

Variable Sample zise Mean Standard deviation Median Trimean Median Absolute deviation (MAD)
WORK1 1 100 97.0295 15.521 94.91 96.3722 15.1003
WORK2 2 100 98.1688 11.2675 97.895 97.9752 10.8749
WORK3 3 100 98.9432 12.4851 98.26 98.9699 13.0765
LEISURE1 4 100 98.0251 15.9367 99.69 97.808 13.9142
LEISURE2 5 100 100.1076 19.9407 98.265 99.8651 17.769
HOME1 6 100 99.5083 11.9781 98.9 99.2841 12.4019
HOME2 7 100 101.6032 11.0615 100.55 101.7372 11.1417
HOME3 8 100 101.3716 12.7349 99.905 101.0177 11.3048
GENERAL1 9 100 99.1631 16.9673 100.095 98.9388 14.7074
GENERAL2 10 100 98.2239 19.1113 96.685 97.4972 17.1833

Continuation of descriptive analysis

Minimum Maximum Range Skew Kurtosis Standard error
WORK1 60.88 138.88 78 0.3583 -0.0755 1.5521
WORK2 65.9 131.61 65.71 0.0932 0.2766 1.1268
WORK3 67.66 129.22 61.56 -0.0447 -0.2851 1.2485
LEISURE1 54.6 155.27 100.67 0.2119 0.9815 1.5937
LEISURE2 47.09 175.97 128.88 0.4597 1.4411 1.9941
HOME1 73.76 133.98 60.22 0.1875 -0.3636 1.1978
HOME2 76.36 132.88 56.52 0.0616 -0.2154 1.1061
HOME3 69.62 137.48 67.86 0.2922 0.1084 1.2735
GENERAL1 48.98 150.76 101.78 0.0903 0.9245 1.6967
GENERAL2 50.06 154.44 104.38 0.3816 0.5178 1.9111

Analysis Result

Factorial Analysis data Simulated Factorial Analysis data Principal componente data simulated principal Components data Uniqueness Communality Loads
WORK1 5.7841 0.8642 6.1183 0.8642 0.4322 0.5678 0.7522
WORK2 1.1885 0.4112 1.8007 0.4112 0.2140 0.7860 0.8827
WORK3 0.0500 0.2932 0.4729 0.2932 0.3055 0.6945 0.8247
LEISURE1 0.0306 0.1763 0.4080 0.1763 0.1113 0.8887 0.7252
LEISURE2 0.0202 0.0649 0.3171 0.0649 0.2694 0.7306 0.7024
HOME1 -0.1208 -0.0251 0.2933 -0.0251 0.4522 0.5478 0.1165
HOME2 -0.1973 -0.1067 0.1958 -0.1067 0.1793 0.8207 0.1406
HOME3 -0.2468 -0.1876 0.1705 -0.1876 0.3661 0.6339 0.1703
GENERAL1 -0.3294 -0.2643 0.1380 -0.2643 0.0849 0.9151 0.7582
GENERAL2 -0.3948 -0.3634 0.0853 -0.3634 0.2129 0.7871 0.7230

Analysis Result

Multiple R² Mean square of residual Mean square of adjusted residual Mean square root of adjusted residual ® Lower Limit ® Upper Limit ®
PA1 0.9245 0.0207 0.0272 0.0000 0.0000 0.0881
PA2 0.8942 0.0207 0.0272 0.0000 0.0000 0.0881

Model General analysis

Value
Objective 0.2785
Tucker Lewes Index 0.9999
Statistics 26.0436
P-value 0.4607
Degrees of freedom 26
BIC -93.6909
Fitted Factorial model 0.9789
Fitted factorial Model without diagonal 0.9988
Nº de Observations 100

Load Factors

PA1 PA2
WORK1 0.7522 0.0443
WORK2 0.8827 0.0827
WORK3 0.8247 0.1199
LEISURE1 0.7252 0.6023
LEISURE2 0.7024 0.4871
HOME1 0.1165 0.7309
HOME2 0.1406 0.8950
HOME3 0.1703 0.7778
GENERAL1 0.7582 0.5833
GENERAL2 0.7230 0.5141

Ressiduals matrix

WORK1 WORK2 WORK3 LEISURE1 LEISURE2 HOME1 HOME2 HOME3 GENERAL1 GENERAL2
WORK1 0.4322 -0.0203 0.0269 0.0259 -0.0289 0.0228 -0.0003 -0.0247 0.0151 -0.0177
WORK2 -0.0203 0.2140 -0.0060 -0.0014 0.0375 -0.0200 -0.0163 0.0213 -0.0089 0.0040
WORK3 0.0269 -0.0060 0.3055 -0.0334 -0.0077 -0.0201 0.0150 0.0209 0.0027 0.0127
LEISURE1 0.0259 -0.0014 -0.0334 0.1113 0.0019 0.0117 -0.0067 -0.0091 0.0032 0.0092
LEISURE2 -0.0289 0.0375 -0.0077 0.0019 0.2694 0.0681 -0.0388 -0.0161 -0.0057 -0.0025
HOME1 0.0228 -0.0200 -0.0201 0.0117 0.0681 0.4522 -0.0127 0.0017 -0.0162 -0.0352
HOME2 -0.0003 -0.0163 0.0150 -0.0067 -0.0388 -0.0127 0.1793 0.0107 0.0150 0.0316
HOME3 -0.0247 0.0213 0.0209 -0.0091 -0.0161 0.0017 0.0107 0.3661 0.0031 -0.0052
GENERAL1 0.0151 -0.0089 0.0027 0.0032 -0.0057 -0.0162 0.0150 0.0031 0.0849 -0.0067
GENERAL2. -0.0177 0.0040 0.0127 0.0092 -0.0025 -0.0352 0.0316 -0.0052 -0.0067 0.2129

Analysis Result**

Estatístics P-value Degree of Freedom
897.3544757562457 6.349220050666458e-159 45