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結構方程模式(SEM)
Structure Equation Modeling

方法簡介
範例F-5(2)
影音教學
維基百科(英文)

此方法最早由基因學家Sewall Wright(1921)提出，經濟學家Trygve Haavelmo(1943)與認知科學家Herbert A. Smith(1953)亦有相關之研究，最後由美國電腦暨統計學家Judea Pearl(2000)定名為SEM。SEM主要由兩部分組成，測量模式(measurement model)與結構模式(structure model)。在測量模式中使用驗證性因素分析(Confirmatory Factor Analysis, CFA)建構觀察變數與”潛在變數(latent variable)”間的關係；在結構模式中則利用徑向分析(Path Analysis)來探討變數間的因果關係，以建立模式架構。SEM已大量應用於社會科學、行為科學等領域，及市場調查研究領域上等，成為主流的研究方法之一。

本方法使用之R相關套件與參考文獻：
相關套件：stats、base、lavaan、stringr、semPlot、semTools、mvnormtest
參考文獻：(依套件名稱排序)

R Core Team (2013). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL：http://www.R-project.org/.
Yves Rosseel (2012). lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 48(2), 1-36. URL：http://www.jstatsoft.org/v48/i02/.
Hadley Wickham (2012). stringr: Make it easier to work with strings.. R package version 0.6.2. URL：http://CRAN.R-project.org/package=stringr
Sacha Epskamp (2013). semPlot: Path diagrams and visual analysis of various SEM packages' output. R package version 0.3.3. URL：http://CRAN.R-project.org/package=semPlot
Sunthud Pornprasertmanit, Patrick Miller, Alexander Schoemann and Yves Rosseel (2013). semTools: Useful tools for structural equation modeling.. R package version 0.4-0. URL：http://CRAN.R-project.org/package=semTools
Slawomir Jarek (2012). mvnormtest: Normality test for multivariate variables. R package version 0.1-9. URL：http://CRAN.R-project.org/package=mvnormtest

範例F-5：

數學家高斯曾說過”數學為科學之母”，數學的希臘語辭源為”學問的基礎”(取自維基百科)，由此可見數學的重要性。某中學老師欲了解學生的數學能力，以Holzinger & Swineford (1939)之問卷對該校七、八年級學生進行智力測驗分析。問卷含五種構念I (空間、語言、速度、記憶及數學能力)並以24道試題呈現。隨機抽取145名七、八年級學生為施測對象協助完成測驗。資料包含性別、年齡(年)、年齡(月)、24道試題成績與年級，記錄如下表，資料說明於表後。

表：智力測驗分析表

編號	性別	年齡(年)	年齡(月)	t1	t2	t3	t4	t5	...	t23	t24	年級
1	1	13	0	23	19	13	4	46	...	11	17	7
2	2	11	10	33	22	12	17	43	...	31	32	7
3	1	12	6	34	24	14	22	36	...	21	18	7
：	：	：	：	：	：	：	：	：	：	：	：	：
145	1	13	5	26	24	16	27	51	...	31	29	8

性別：男性(1)、女性(2)
年級：七年級(7)、八年級(8)
註I：題1-4衡量空間概念、題5-9衡量語言、題10-13衡量速度、題14-19衡量記憶、題20-24衡量數學能力

Q2：在以往文獻中的研究認為影響數學能力的因素包含空間概念、語言、速度及記憶等四種能力，他們的關係為何，該如何建構分析模式?
問題解析資料中空間概念、語言、速度及記憶等四項因素並無法直接觀測，為潛在(latent)變數，須利用問卷中的試題予以量測，並使用因素分析(Factor Analysis) 類型的統計方法。欲探討四種能力對於數學能力的影響，適合迴歸分析類型的統計方法。
統計方法綜合問題解析中的說明，所使用的統計方法須包含具有量測能力的因素分析及能討論變數間因果關係的迴歸分析等方法。可採用分析方法：結構方程模式(SEM)。

結構方程模式 - 分析結果

分析方法：結構方程模式
資料名稱：範例F-5
觀察變數名稱：ID, gender, ageyr, agemo, t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11, t12, t13, t14, t15, t16, t17, t18, t19, t20, t21, t22, t23, t24, grade
計算時間：5.568秒

樣本敘述統計量^I：

觀察變數訊息(連續變數)

變數名稱
Variable 樣本數
Count 平均數
Mean 中位數
Median 最小值
Minimum 最大值
Maximum 標準差
Std. dev.
ID 145 73 73 1 145 42.002
gender 145 1.5034 2 1 2 0.5017
ageyr 145 12.7241 13 11 16 0.9681
agemo 145 5.3448 5 0 11 3.4809
t1 145 29.5793 30 11 51 6.9138
t2 145 24.8 25 9 37 4.445
t3 145 14.3034 14 6 23 2.8218
t4 145 15.9655 15 3 36 8.317
t5 145 44.8483 44 16 84 11.6579
t6 145 9.9517 9 1 19 3.3754
t7 145 18.8483 19 4 28 4.6493
t8 145 28.2 28 10 43 5.3575
t9 145 17.2828 16 2 41 7.9475
t10 145 90.1793 89 30 149 23.7822
t11 145 68.5931 67 19 118 16.6635
t12 145 109.7655 110 61 200 20.995
t13 145 191.7793 191 112 333 37.0354
t14 145 176.2759 177 130 198 10.8285
t15 145 89.3724 89 73 112 7.5424
t16 145 103.4 103 87 119 6.8011
t17 145 7.2069 7 0 26 4.5688
t18 145 9.4759 9 1 20 4.4738
t19 145 15.2414 16 4 20 3.5964
t20 145 30.3448 30 -18 87 19.7802
t21 145 14.4552 15 2 24 4.8319
t22 145 27.731 27 7 50 9.8243
t23 145 18.7517 19 1 40 9.3605
t24 145 25.8276 26 11 35 4.7131
grade 145 7.4552 7 7 8 0.4997

觀察變數共變異數矩陣：

ID gender ageyr agemo t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15 t16 t17 t18 t19 t20 t21 t22 t23 t24 grade
ID 1764.1667 -1.8611 24.9653 -7.4653 14.6042 6.6111 17.9444 30.9167 112.0694 11.9792 19.0417 37.7292 16.6458 273.6458 95.4375 165.8611 350.6181 43.0486 24.9861 26.625 21.7986 8.8889 9.8125 42.1597 24.3958 9.0625 -18.5694 14.9931 18.1042
gender -1.8611 0.2517 -0.0893 0.0752 -0.134 -0.0514 -0.3483 -0.5728 -0.2286 0.3231 0.2297 -0.0042 0.3705 0.8119 1.4285 -1.1242 0.4244 0.256 0.1237 0.1375 0.4229 0.1338 0.3012 -1.022 -0.2585 -0.197 -0.0616 0.2818 -0.0224
ageyr 24.9653 -0.0893 0.9373 -0.8834 -0.228 -0.0833 0.3273 -0.0165 1.2148 -0.1592 -0.438 -0.3889 -0.5812 4.1123 2.3523 5.664 4.8693 1.1808 1.4437 0.1042 0.0366 0.3058 -0.2802 -1.4598 0.0431 -0.665 -1.5065 -0.3534 0.3
agemo -7.4653 0.0752 -0.8834 12.1164 2.4725 0.4306 -1.2929 -0.6477 -1.7459 -1.136 -0.2459 -0.2153 -0.7926 3.0975 -6.5254 0.2134 0.1252 -4.2 -0.421 -1.8611 -1.3566 0.8903 0.4579 3.9567 -0.1928 -0.5524 2.6695 2.1432 -0.033
t1 14.6042 -0.134 -0.228 2.4725 47.801 10.0125 7.2605 25.7979 26.408 7.9726 9.9357 12.0778 17.4253 17.1315 35.2512 44.6507 124.6565 9.7071 11.6161 19.7181 5.3376 11.4932 6.6439 49.2433 12.3803 28.1013 30.3254 8.9061 0.5192
t2 6.6111 -0.0514 -0.0833 0.4306 10.0125 19.7583 2.3875 15.4167 14.2542 3.4208 3.2958 3.7139 6.8764 7.0153 11.175 15.675 40.8028 3.9375 4.5194 8.7403 0.2222 5.1514 1.7639 25.8819 6.3069 10.2375 14.1931 4.3125 0.3208
t3 17.9444 -0.3483 0.3273 -1.2929 7.2605 2.3875 7.9628 8.5869 10.1575 2.48 3.4839 5.0431 5.8372 8.6188 11.6452 11.7244 40.6508 7.6379 3.9557 5.8917 1.6451 3.3199 1.5721 11.7627 4.5276 4.4433 7.6523 1.5874 0.2706
t4 30.9167 -0.5728 -0.0165 -0.6477 25.7979 15.4167 8.5869 69.1724 36.8975 9.2067 11.092 16.9167 22.9543 14.7632 25.1248 41.6585 114.7632 14.5235 12.874 16.3333 5.278 12.9887 5.3903 54.8036 16.5575 31.4212 33.3247 9.0982 0.5852
t5 112.0694 -0.2286 1.2148 -1.7459 26.408 14.2542 10.1575 36.8975 135.9074 24.4648 35.4421 35.8431 66.7376 87.1038 66.5281 51.4503 148.2649 32.9935 19.2652 14.0125 11.3302 13.6908 8.2035 92.2332 17.9793 50.9242 47.3579 22.8348 2.1251
t6 11.9792 0.3231 -0.1592 -1.136 7.9726 3.4208 2.48 9.2067 24.4648 11.3935 11.2773 9.4125 19.1665 16.7657 20.2233 7.3567 39.3087 10.4648 6.3375 6.6167 4.2531 2.5092 3.0465 28.9195 4.293 12.9105 13.6615 6.8875 0.2791
t7 19.0417 0.2297 -0.438 -0.2459 9.9357 3.2958 3.4839 11.092 35.4421 11.2773 21.6157 15.7597 25.321 28.0691 19.2087 19.3114 61.2302 11.7366 5.5152 6.3458 5.3233 3.8227 4.0369 42.1638 7.0626 18.2922 17.7537 9.6403 0.5834
t8 37.7292 -0.0042 -0.3889 -0.2153 12.0778 3.7139 5.0431 16.9167 35.8431 9.4125 15.7597 28.7028 22.8528 37.7833 26.2139 32.6583 80.3708 14.0764 6.8625 10.8847 6.625 6.0708 5.0347 45.5972 9.3389 19.1167 24.7792 9.8125 0.6375
t9 16.6458 0.3705 -0.5812 -0.7926 17.4253 6.8764 5.8372 22.9543 66.7376 19.1665 25.321 22.8528 63.1625 33.7684 38.0256 20.2126 79.9934 21.5048 12.7829 12.7542 10.3508 7.8437 7.9035 70.9782 10.4468 38.3891 36.8832 15.6463 0.5371
t10 273.6458 0.8119 4.1123 3.0975 17.1315 7.0153 8.6188 14.7632 87.1038 16.7657 28.0691 37.7833 33.7684 565.5926 185.2957 293.1257 368.4357 40.4224 26.905 22.1569 32.7404 33.1363 17.012 76.153 46.1748 36.1875 56.8295 59.3367 4.7859
t11 95.4375 1.4285 2.3523 -6.5254 35.2512 11.175 11.6452 25.1248 66.5281 20.2233 19.2087 26.2139 38.0256 185.2957 277.6736 147.64 325.3124 58.4464 29.9651 35.6222 27.1889 24.688 17.3628 67.0649 31.9157 51.8967 39.1413 32.2905 1.6101
t12 165.8611 -1.1242 5.664 0.2134 44.6507 15.675 11.7244 41.6585 51.4503 7.3567 19.3114 32.6583 20.2126 293.1257 147.64 440.7919 410.8229 29.6415 25.8865 18.3097 26.6739 32.161 8.1195 104.5953 36.1977 41.8532 69.4344 40.9385 2.9061
t13 350.6181 0.4244 4.8693 0.1252 124.6565 40.8028 40.6508 114.7632 148.2649 39.3087 61.2302 80.3708 79.9934 368.4357 325.3124 410.8229 1371.6176 77.5821 38.5689 69.7833 32.2543 53.6335 33.6023 181.3752 77.1081 105.5652 132.9657 62.3089 5.3303
t14 43.0486 0.256 1.1808 -4.2 9.7071 3.9375 7.6379 14.5235 32.9935 10.4648 11.7366 14.0764 21.5048 40.4224 58.4464 29.6415 77.5821 117.2567 31.5979 28.1042 18.4078 9.09 8.5163 66.1889 9.5402 26.533 25.4231 15.0271 0.9777
t15 24.9861 0.1237 1.4437 -0.421 11.6161 4.5194 3.9557 12.874 19.2652 6.3375 5.5152 6.8625 12.7829 26.905 29.9651 25.8865 38.5689 31.5979 56.8881 16.0653 11.9363 10.3424 4.972 39.2526 8.4613 18.1911 17.8153 5.7174 0.5446
t16 26.625 0.1375 0.1042 -1.8611 19.7181 8.7403 5.8917 16.3333 14.0125 6.6167 6.3458 10.8847 12.7542 22.1569 35.6222 18.3097 69.7833 28.1042 16.0653 46.2556 10.5486 10.7875 6.2083 52.9931 11.6361 19.4556 22.5861 8.6597 0.8028
t17 21.7986 0.4229 0.0366 -1.3566 5.3376 0.2222 1.6451 5.278 11.3302 4.2531 5.3233 6.625 10.3508 32.7404 27.1889 26.6739 32.2543 18.4078 11.9363 10.5486 20.8736 9.0953 5.3803 25.1712 4.1691 13.2783 12.8295 7.3831 0.4468
t18 8.8889 0.1338 0.3058 0.8903 11.4932 5.1514 3.3199 12.9887 13.6908 2.5092 3.8227 6.0708 7.8437 33.1363 24.688 32.161 53.6335 9.09 10.3424 10.7875 9.0953 20.015 5.7871 26.8278 7.65 14.5942 11.0287 8.5687 0.518
t19 9.8125 0.3012 -0.2802 0.4579 6.6439 1.7639 1.5721 5.3903 8.2035 3.0465 4.0369 5.0347 7.9035 17.012 17.3628 8.1195 33.6023 8.5163 4.972 6.2083 5.3803 5.7871 12.9344 12.1037 5.7574 12.2529 10.0395 6.3336 0.2713
t20 42.1597 -1.022 -1.4598 3.9567 49.2433 25.8819 11.7627 54.8036 92.2332 28.9195 42.1638 45.5972 70.9782 76.153 67.0649 104.5953 181.3752 66.1889 39.2526 52.9931 25.1712 26.8278 12.1037 391.2553 38.8558 92.6837 95.3848 34.4071 1.3072
t21 24.3958 -0.2585 0.0431 -0.1928 12.3803 6.3069 4.5276 16.5575 17.9793 4.293 7.0626 9.3389 10.4468 46.1748 31.9157 36.1977 77.1081 9.5402 8.4613 11.6361 4.1691 7.65 5.7574 38.8558 23.3469 17.7552 20.2318 10.204 0.6247
t22 9.0625 -0.197 -0.665 -0.5524 28.1013 10.2375 4.4433 31.4212 50.9242 12.9105 18.2922 19.1167 38.3891 36.1875 51.8967 41.8532 105.5652 26.533 18.1911 19.4556 13.2783 14.5942 12.2529 92.6837 17.7552 96.5174 46.7036 17.495 0.6094
t23 -18.5694 -0.0616 -1.5065 2.6695 30.3254 14.1931 7.6523 33.3247 47.3579 13.6615 17.7537 24.7792 36.8832 56.8295 39.1413 69.4344 132.9657 25.4231 17.8153 22.5861 12.8295 11.0287 10.0395 95.3848 20.2318 46.7036 87.6185 19.0194 0.2943
t24 14.9931 0.2818 -0.3534 2.1432 8.9061 4.3125 1.5874 9.0982 22.8348 6.8875 9.6403 9.8125 15.6463 59.3367 32.2905 40.9385 62.3089 15.0271 5.7174 8.6597 7.3831 8.5687 6.3336 34.4071 10.204 17.495 19.0194 22.2131 0.5929
grade 18.1042 -0.0224 0.3 -0.033 0.5192 0.3208 0.2706 0.5852 2.1251 0.2791 0.5834 0.6375 0.5371 4.7859 1.6101 2.9061 5.3303 0.9777 0.5446 0.8028 0.4468 0.518 0.2713 1.3072 0.6247 0.6094 0.2943 0.5929 0.2497

觀察變數相關係數矩陣：

ID gender ageyr agemo t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15 t16 t17 t18 t19 t20 t21 t22 t23 t24 grade
ID 1 -0.0883 0.614 -0.0511 0.0503 0.0354 0.1514 0.0885 0.2289 0.0845 0.0975 0.1677 0.0499 0.2739 0.1364 0.1881 0.2254 0.0946 0.0789 0.0932 0.1136 0.0473 0.065 0.0507 0.1202 0.022 -0.0472 0.0757 0.8626
gender -0.0883 1 -0.1839 0.0431 -0.0386 -0.023 -0.246 -0.1373 -0.0391 0.1908 0.0985 -0.0016 0.0929 0.068 0.1709 -0.1067 0.0228 0.0471 0.0327 0.0403 0.1845 0.0596 0.1669 -0.103 -0.1066 -0.04 -0.0131 0.1192 -0.0894
ageyr 0.614 -0.1839 1 -0.2621 -0.0341 -0.0194 0.1198 -0.0021 0.1076 -0.0487 -0.0973 -0.075 -0.0755 0.1786 0.1458 0.2787 0.1358 0.1126 0.1977 0.0158 0.0083 0.0706 -0.0805 -0.0762 0.0092 -0.0699 -0.1662 -0.0775 0.6202
agemo -0.0511 0.0431 -0.2621 1 0.1027 0.0278 -0.1316 -0.0224 -0.043 -0.0967 -0.0152 -0.0115 -0.0287 0.0374 -0.1125 0.0029 0.001 -0.1114 -0.016 -0.0786 -0.0853 0.0572 0.0366 0.0575 -0.0115 -0.0162 0.0819 0.1306 -0.019
t1 0.0503 -0.0386 -0.0341 0.1027 1 0.3258 0.3721 0.4486 0.3276 0.3416 0.3091 0.3261 0.3171 0.1042 0.306 0.3076 0.4868 0.1297 0.2228 0.4193 0.169 0.3716 0.2672 0.3601 0.3706 0.4137 0.4686 0.2733 0.1503
t2 0.0354 -0.023 -0.0194 0.0278 0.3258 1 0.1903 0.417 0.2751 0.228 0.1595 0.156 0.1947 0.0664 0.1509 0.168 0.2479 0.0818 0.1348 0.2891 0.0109 0.259 0.1103 0.2944 0.2936 0.2344 0.3411 0.2058 0.1444
t3 0.1514 -0.246 0.1198 -0.1316 0.3721 0.1903 1 0.3659 0.3088 0.2604 0.2655 0.3336 0.2603 0.1284 0.2477 0.1979 0.389 0.25 0.1859 0.307 0.1276 0.263 0.1549 0.2107 0.3321 0.1603 0.2897 0.1194 0.1919
t4 0.0885 -0.1373 -0.0021 -0.0224 0.4486 0.417 0.3659 1 0.3805 0.3279 0.2869 0.3797 0.3473 0.0746 0.1813 0.2386 0.3726 0.1613 0.2052 0.2888 0.1389 0.3491 0.1802 0.3331 0.412 0.3846 0.4281 0.2321 0.1408
t5 0.2289 -0.0391 0.1076 -0.043 0.3276 0.2751 0.3088 0.3805 1 0.6217 0.6539 0.5739 0.7203 0.3142 0.3425 0.2102 0.3434 0.2614 0.2191 0.1767 0.2127 0.2625 0.1957 0.4 0.3192 0.4446 0.434 0.4156 0.3648
t6 0.0845 0.1908 -0.0487 -0.0967 0.3416 0.228 0.2604 0.3279 0.6217 1 0.7186 0.5205 0.7145 0.2089 0.3595 0.1038 0.3144 0.2863 0.2489 0.2882 0.2758 0.1662 0.251 0.4331 0.2632 0.3893 0.4324 0.4329 0.1654
t7 0.0975 0.0985 -0.0973 -0.0152 0.3091 0.1595 0.2655 0.2869 0.6539 0.7186 1 0.6327 0.6853 0.2539 0.2479 0.1978 0.3556 0.2331 0.1573 0.2007 0.2506 0.1838 0.2414 0.4585 0.3144 0.4005 0.408 0.4399 0.2511
t8 0.1677 -0.0016 -0.075 -0.0115 0.3261 0.156 0.3336 0.3797 0.5739 0.5205 0.6327 1 0.5367 0.2965 0.2936 0.2903 0.4051 0.2426 0.1698 0.2987 0.2707 0.2533 0.2613 0.4303 0.3608 0.3632 0.4941 0.3886 0.2381
t9 0.0499 0.0929 -0.0755 -0.0287 0.3171 0.1947 0.2603 0.3473 0.7203 0.7145 0.6853 0.5367 1 0.1787 0.2871 0.1211 0.2718 0.2499 0.2132 0.236 0.2851 0.2206 0.2765 0.4515 0.272 0.4917 0.4958 0.4177 0.1352
t10 0.2739 0.068 0.1786 0.0374 0.1042 0.0664 0.1284 0.0746 0.3142 0.2089 0.2539 0.2965 0.1787 1 0.4676 0.5871 0.4183 0.157 0.15 0.137 0.3013 0.3114 0.1989 0.1619 0.4018 0.1549 0.2553 0.5294 0.4027
t11 0.1364 0.1709 0.1458 -0.1125 0.306 0.1509 0.2477 0.1813 0.3425 0.3595 0.2479 0.2936 0.2871 0.4676 1 0.422 0.5271 0.3239 0.2384 0.3143 0.3571 0.3312 0.2897 0.2035 0.3964 0.317 0.2509 0.4112 0.1934
t12 0.1881 -0.1067 0.2787 0.0029 0.3076 0.168 0.1979 0.2386 0.2102 0.1038 0.1978 0.2903 0.1211 0.5871 0.422 1 0.5283 0.1304 0.1635 0.1282 0.2781 0.3424 0.1075 0.2519 0.3568 0.2029 0.3533 0.4137 0.277
t13 0.2254 0.0228 0.1358 0.001 0.4868 0.2479 0.389 0.3726 0.3434 0.3144 0.3556 0.4051 0.2718 0.4183 0.5271 0.5283 1 0.1935 0.1381 0.277 0.1906 0.3237 0.2523 0.2476 0.4309 0.2901 0.3836 0.357 0.288
t14 0.0946 0.0471 0.1126 -0.1114 0.1297 0.0818 0.25 0.1613 0.2614 0.2863 0.2331 0.2426 0.2499 0.157 0.3239 0.1304 0.1935 1 0.3869 0.3816 0.3721 0.1876 0.2187 0.309 0.1823 0.2494 0.2508 0.2944 0.1807
t15 0.0789 0.0327 0.1977 -0.016 0.2228 0.1348 0.1859 0.2052 0.2191 0.2489 0.1573 0.1698 0.2132 0.15 0.2384 0.1635 0.1381 0.3869 1 0.3132 0.3464 0.3065 0.1833 0.2631 0.2322 0.2455 0.2523 0.1608 0.1445
t16 0.0932 0.0403 0.0158 -0.0786 0.4193 0.2891 0.307 0.2888 0.1767 0.2882 0.2007 0.2987 0.236 0.137 0.3143 0.1282 0.277 0.3816 0.3132 1 0.3395 0.3545 0.2538 0.3939 0.3541 0.2912 0.3548 0.2702 0.2362
t17 0.1136 0.1845 0.0083 -0.0853 0.169 0.0109 0.1276 0.1389 0.2127 0.2758 0.2506 0.2707 0.2851 0.3013 0.3571 0.2781 0.1906 0.3721 0.3464 0.3395 1 0.445 0.3274 0.2785 0.1889 0.2958 0.3 0.3429 0.1957
t18 0.0473 0.0596 0.0706 0.0572 0.3716 0.259 0.263 0.3491 0.2625 0.1662 0.1838 0.2533 0.2206 0.3114 0.3312 0.3424 0.3237 0.1876 0.3065 0.3545 0.445 1 0.3597 0.3032 0.3539 0.332 0.2634 0.4064 0.2317
t19 0.065 0.1669 -0.0805 0.0366 0.2672 0.1103 0.1549 0.1802 0.1957 0.251 0.2414 0.2613 0.2765 0.1989 0.2897 0.1075 0.2523 0.2187 0.1833 0.2538 0.3274 0.3597 1 0.1701 0.3313 0.3468 0.2982 0.3737 0.151
t20 0.0507 -0.103 -0.0762 0.0575 0.3601 0.2944 0.2107 0.3331 0.4 0.4331 0.4585 0.4303 0.4515 0.1619 0.2035 0.2519 0.2476 0.309 0.2631 0.3939 0.2785 0.3032 0.1701 1 0.4065 0.4769 0.5152 0.3691 0.1323
t21 0.1202 -0.1066 0.0092 -0.0115 0.3706 0.2936 0.3321 0.412 0.3192 0.2632 0.3144 0.3608 0.272 0.4018 0.3964 0.3568 0.4309 0.1823 0.2322 0.3541 0.1889 0.3539 0.3313 0.4065 1 0.374 0.4473 0.4481 0.2587
t22 0.022 -0.04 -0.0699 -0.0162 0.4137 0.2344 0.1603 0.3846 0.4446 0.3893 0.4005 0.3632 0.4917 0.1549 0.317 0.2029 0.2901 0.2494 0.2455 0.2912 0.2958 0.332 0.3468 0.4769 0.374 1 0.5079 0.3778 0.1241
t23 -0.0472 -0.0131 -0.1662 0.0819 0.4686 0.3411 0.2897 0.4281 0.434 0.4324 0.408 0.4941 0.4958 0.2553 0.2509 0.3533 0.3836 0.2508 0.2523 0.3548 0.3 0.2634 0.2982 0.5152 0.4473 0.5079 1 0.4311 0.0629
t24 0.0757 0.1192 -0.0775 0.1306 0.2733 0.2058 0.1194 0.2321 0.4156 0.4329 0.4399 0.3886 0.4177 0.5294 0.4112 0.4137 0.357 0.2944 0.1608 0.2702 0.3429 0.4064 0.3737 0.3691 0.4481 0.3778 0.4311 1 0.2517
grade 0.8626 -0.0894 0.6202 -0.019 0.1503 0.1444 0.1919 0.1408 0.3648 0.1654 0.2511 0.2381 0.1352 0.4027 0.1934 0.277 0.288 0.1807 0.1445 0.2362 0.1957 0.2317 0.151 0.1323 0.2587 0.1241 0.0629 0.2517 1

模式訊息：

模式方程式
spatial =~ t1 + t2 + t3 + t4
verbal =~ t5 + t6 + t7 + t8 + t9
speed =~ t10 + t11 + t12 + t13
memory =~ t14 + t15 + t16 + t17 + t18 + t19
math_ability =~ t20 + t21 + t22 + t23 + t24
math_ability ~ spatial + verbal + speed + memory
spatial ~~ verbal
spatial ~~ speed
spatial ~~ memory
verbal ~~ speed
verbal ~~ memory
speed ~~ memory

模式適合度分析
觀察值個數：145
估計方法：ML
疊代次數：273

最小化函數檢定統計：
模式
model 卡方檢定統計量
Chi-square statistic 自由度
d.f. p-值
p-value
基準模式^I 1638.466 276 0.000
使用模式 358.789 242 0.000
I：基準模式又稱獨立模式

概似訊息準則：
模式
model 概似值
likelihood
虛無模式(H0) -11453.195
未限制模式(H1) -11273.801

模式方程式參數估計(未標準化)
潛在變數項：
潛在變數
latent 顯性變數
endogenous 估計值
estimate 標準差
std. err. Z-值
z-value p-值
p-value
spatial t1 1.000
spatial t2 0.460 0.090 5.094 0.000
spatial t3 0.297 0.057 5.167 0.000
spatial t4 1.181 0.179 6.611 0.000
verbal t5 1.000
verbal t6 0.293 0.027 10.994 0.000
verbal t7 0.414 0.036 11.368 0.000
verbal t8 0.398 0.044 8.993 0.000
verbal t9 0.717 0.062 11.570 0.000
speed t10 1.000
speed t11 0.716 0.106 6.740 0.000
speed t12 0.950 0.136 7.000 0.000
speed t13 1.707 0.241 7.092 0.000
memory t14 1.000
memory t15 0.690 0.156 4.420 0.000
memory t16 0.745 0.151 4.925 0.000
memory t17 0.511 0.103 4.980 0.000
memory t18 0.515 0.102 5.059 0.000
memory t19 0.325 0.074 4.381 0.000
math_ability t20 1.000
math_ability t21 0.234 0.035 6.594 0.000
math_ability t22 0.497 0.073 6.847 0.000
math_ability t23 0.526 0.070 7.469 0.000
math_ability t24 0.232 0.035 6.689 0.000
迴歸項：
依變數
dependent 解釋變數
explanatory 估計值
estimate 標準差
std. err. Z-值
z-value p-值
p-value
math_ability spatial 0.821 0.345 2.382 0.017
math_ability verbal 0.492 0.129 3.812 0.000
math_ability speed 0.132 0.081 1.636 0.102
math_ability memory 0.696 0.290 2.395 0.017
共變異數項：
變數1
variable 1 變數2
variable 2 估計值
estimate 標準差
std. err. Z-值
z-value p-值
p-value
spatial verbal 26.544 5.795 4.580 0.000
spatial speed 43.125 10.607 4.066 0.000
spatial memory 16.907 4.403 3.840 0.000
verbal speed 68.057 17.330 3.927 0.000
verbal memory 25.944 7.020 3.696 0.000
speed memory 52.350 13.926 3.759 0.000
變異數項：
變數
variable 估計值
estimate 標準差
std. err.
t1 24.391 3.893
t2 14.734 1.900
t3 5.874 0.761
t4 36.528 5.683
t5 47.344 6.775
t6 3.793 0.552
t7 6.458 0.980
t8 14.617 1.889
t9 17.692 2.762
t10 310.774 44.454
t11 147.193 21.447
t12 211.533 32.622
t13 631.077 99.951
t14 85.945 11.079
t15 41.977 5.398
t16 29.020 3.996
t17 12.773 1.780
t18 11.782 1.675
t19 9.632 1.234
t20 221.324 29.064
t21 14.056 1.815
t22 54.621 7.172
t23 40.824 5.722
t24 13.076 1.698
spatial 23.080 5.461
verbal 87.626 15.341
speed 250.919 60.528
memory 30.503 10.297
math_ability 25.028 10.965

模式方程式參數估計(標準化)
潛在變數項：
潛在變數
latent 顯性變數
endogenous 估計值
estimate 標準差
std. err. Z-值
z-value p-值
p-value
spatial t1 0.6972726 NA NA NA
spatial t2 0.4991314 0.09798015 5.094209 3.502003e-07
spatial t3 0.5071163 0.09814220 5.167158 2.376803e-07
spatial t4 0.6842930 0.10350973 6.610905 3.819767e-11
verbal t5 0.8057477 NA NA NA
verbal t6 0.8153595 0.07416608 10.993698 0.000000e+00
verbal t7 0.8361608 0.07355106 11.368440 0.000000e+00
verbal t8 0.6980080 0.07761329 8.993408 0.000000e+00
verbal t9 0.8473226 0.07323623 11.569719 0.000000e+00
speed t10 0.6683704 NA NA NA
speed t11 0.6828068 0.10131005 6.739774 1.586331e-11
speed t12 0.7188709 0.10269074 7.000348 2.553291e-12
speed t13 0.7326042 0.10330071 7.091957 1.322276e-12
memory t14 0.5118084 NA NA NA
memory t15 0.5069482 0.11469188 4.420088 9.866065e-06
memory t16 0.6068362 0.12321932 4.924846 8.442668e-07
memory t17 0.6195218 0.12439004 4.980478 6.342744e-07
memory t18 0.6381723 0.12615138 5.058782 4.219436e-07
memory t19 0.5001172 0.11415170 4.381163 1.180473e-05
math_ability t20 0.6560467 NA NA NA
math_ability t21 0.6274959 0.09516851 6.593524 4.295075e-11
math_ability t22 0.6558617 0.09579106 6.846794 7.552403e-12
math_ability t23 0.7285836 0.09755383 7.468528 8.104628e-14
math_ability t24 0.6381529 0.09539869 6.689326 2.242007e-11
迴歸項：
依變數
dependent 解釋變數
explanatory 估計值
estimate 標準差
std. err. Z-值
z-value p-值
p-value
math_ability spatial 0.3048281 0.12798384 2.381770 0.0172296544
math_ability verbal 0.3562889 0.09345437 3.812437 0.0001376032
math_ability speed 0.1616176 0.09880978 1.635644 0.1019141595
math_ability memory 0.2971493 0.12404893 2.395420 0.0166013377
共變異數項：
變數1
variable 1 變數2
variable 2 估計值
estimate 標準差
std. err. Z-值
z-value p-值
p-value
spatial verbal 0.5902316 0.07597511 7.768750 7.993606e-15
spatial speed 0.5666926 0.08548525 6.629127 3.376788e-11
spatial memory 0.6371861 0.08329891 7.649394 2.020606e-14
verbal speed 0.4589757 0.08068770 5.688299 1.283112e-08
verbal memory 0.5018150 0.08085958 6.206005 5.434833e-10
speed memory 0.5983741 0.07984999 7.493729 6.683543e-14
變異數項：
變數
variable 估計值
estimate 標準差
std. err.
t1 0.5138110 0.08201070
t2 0.7508679 0.09685058
t3 0.7428331 0.09623285
t4 0.5317430 0.08272742
t5 0.3507707 0.05019575
t6 0.3351889 0.04874370
t7 0.3008351 0.04564884
t8 0.5127848 0.06625510
t9 0.2820444 0.04403177
t10 0.5532810 0.07914299
t11 0.5337749 0.07777308
t12 0.4832246 0.07452091
t13 0.4632911 0.07337704
t14 0.7380522 0.09514385
t15 0.7430035 0.09554220
t16 0.6317498 0.08699598
t17 0.6161927 0.08588791
t18 0.5927361 0.08427212
t19 0.7498828 0.09609783
t20 0.5696028 0.07479967
t21 0.6062489 0.07826235
t22 0.5698454 0.07482238
t23 0.4691660 0.06576084
t24 0.5927608 0.07698067
spatial 1.0000000 0.23663233
verbal 1.0000000 0.17506852
speed 1.0000000 0.24122352
memory 1.0000000 0.33756051
math_ability 0.1496583 0.06556625

方程式路徑圖(彩色圖示暨標準化係數)

方程式路徑圖(黑白圖示暨標準化係數)

配適指標：

絕對配適指標
指標
index 值
value 判斷準則
criterion 說明
interpretation
F_min 1.2372
NCP 116.7894 越小越好 Non-Central Parameter
GFI 0.8288 > 0.95 Goodness of Fit Index
AGFI 0.7878 > 0.95 調整GFI
RMR 10.9195 < 0.05 Root Mean Squared Residual
SRMR 0.0674 < 0.08 標準化RMR
RMSEA 0.0577 < 0.06 Root Mean Squared Error Approximation
RMSEA P-值 0.1571 > 0.05
ECVI 3.2744 越小越好 Expected Cross-Validation Index

增量配適指標
指標
index 值
value 判斷準則
criterion 說明
interpretation
CFI 0.9143 > 0.95 Comparative Fit Index
TLI 0.9022 > 0.95 Tucker-Lewis Index，同Non-Normed Fit Index(NNFI)
NFI 0.781 > 0.95 Normed Fit Index

精簡配適指標
指標
index 值
value 判斷準則
criterion 說明
interpretation
PNFI 0.6848 > 0.5 Parsimonious Normed Fit Index
PGFI 0.6686 > 0.5 Parsimonious Goodness of Fit Index
AIC 23022.3909 越小越好 Akaike Information Criterion
BIC 23195.0414 越小越好 Bayesian Information Criterion
SABIC 23011.5088 越小越好 Sample-Size Adjustment BIC

結構信度與平均萃取變異
潛在變數
Latent variable 結構信度^I
CR 平均萃取變異^II
AVE
spatial 0.6919 0.3652
verbal 0.8999 0.6437
speed 0.7943 0.4916
memory 0.7376 0.3214
math_ability 0.7956 0.4385
I：一般建議CR > 0.7較合適
II：一般建議AVE > 0.5較合適，AVE必須小於CR

修正指標(> 3.84)
左側
LHS 關係式
operator 右側
RHS 修正指標
MI EPC^I
t10 ~~ t24 20.220979 26.98860595
spatial =~ t13 16.345395 3.42705132
spatial =~ t10 13.237412 -1.99630711
t10 ~~ t12 12.551946 107.66530632
spatial =~ t17 11.857517 -0.44472100
speed =~ t24 11.298942 0.12109709
spatial =~ t24 9.634760 -0.55104735
t14 ~~ t18 9.111084 -9.51533876
speed =~ t21 8.273187 0.10676768
verbal =~ t21 7.842220 -0.19080144
t1 ~~ t13 7.498814 35.87362772
speed =~ t20 7.492335 -0.41040398
t1 ~~ t10 7.392686 -23.94582071
speed =~ t8 7.146754 0.07357608
verbal =~ t12 6.653365 -0.49492344
math_ability =~ t8 6.611190 0.12603871
t6 ~~ t12 6.598589 -7.39229577
t6 ~~ t11 6.587695 6.03018866
spatial =~ t16 6.220882 0.47979484
t10 ~~ t13 6.172634 -133.78417060
memory =~ t11 5.988082 0.84351103
t11 ~~ t23 5.834516 -18.36798049
t11 ~~ t12 5.671219 -50.91466405
t14 ~~ t15 5.321954 12.81179067
t13 ~~ t17 5.286894 -20.71235036
t2 ~~ t17 5.141536 -2.87319942
t8 ~~ t9 4.890968 -3.76989395
spatial =~ t8 4.686180 0.23464545
t12 ~~ t16 4.523558 -16.46094197
t19 ~~ t20 4.448097 -8.82173044
t18 ~~ t23 4.432901 -4.51324725
t3 ~~ t13 4.390706 12.40382886
t17 ~~ t21 4.379930 -2.61319950
memory =~ t8 4.309314 0.17419047
t5 ~~ t9 4.286154 7.15993631
t12 ~~ t19 4.228339 -8.89685462
t9 ~~ t22 4.154071 6.29035204
t3 ~~ t22 4.125851 -3.31384264
speed =~ t9 4.098412 -0.06844909
speed =~ t22 4.055922 -0.14998835
t1 ~~ t16 4.022474 5.32962474
t6 ~~ t7 3.970328 1.16808139
t6 ~~ t18 3.939074 -1.30572483
spatial =~ t18 3.860518 0.24845804
I：expected parameter change

多變量常態分配檢定
虛無假設：資料服從多變量常態分配
檢定方法
method w統計量
w-statistic p-值
p-value
Shapiro-Wilk常態性檢定 0.7703 8.5226e-14

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變數名稱 Variable	樣本數 Count	平均數 Mean	中位數 Median	最小值 Minimum	最大值 Maximum	標準差 Std. dev.
ID	145	73	73	1	145	42.002
gender	145	1.5034	2	1	2	0.5017
ageyr	145	12.7241	13	11	16	0.9681
agemo	145	5.3448	5	0	11	3.4809
t1	145	29.5793	30	11	51	6.9138
t2	145	24.8	25	9	37	4.445
t3	145	14.3034	14	6	23	2.8218
t4	145	15.9655	15	3	36	8.317
t5	145	44.8483	44	16	84	11.6579
t6	145	9.9517	9	1	19	3.3754
t7	145	18.8483	19	4	28	4.6493
t8	145	28.2	28	10	43	5.3575
t9	145	17.2828	16	2	41	7.9475
t10	145	90.1793	89	30	149	23.7822
t11	145	68.5931	67	19	118	16.6635
t12	145	109.7655	110	61	200	20.995
t13	145	191.7793	191	112	333	37.0354
t14	145	176.2759	177	130	198	10.8285
t15	145	89.3724	89	73	112	7.5424
t16	145	103.4	103	87	119	6.8011
t17	145	7.2069	7	0	26	4.5688
t18	145	9.4759	9	1	20	4.4738
t19	145	15.2414	16	4	20	3.5964
t20	145	30.3448	30	-18	87	19.7802
t21	145	14.4552	15	2	24	4.8319
t22	145	27.731	27	7	50	9.8243
t23	145	18.7517	19	1	40	9.3605
t24	145	25.8276	26	11	35	4.7131
grade	145	7.4552	7	7	8	0.4997

	ID	gender	ageyr	agemo	t1	t2	t3	t4	t5	t6	t7	t8	t9	t10	t11	t12	t13	t14	t15	t16	t17	t18	t19	t20	t21	t22	t23	t24	grade
ID	1764.1667	-1.8611	24.9653	-7.4653	14.6042	6.6111	17.9444	30.9167	112.0694	11.9792	19.0417	37.7292	16.6458	273.6458	95.4375	165.8611	350.6181	43.0486	24.9861	26.625	21.7986	8.8889	9.8125	42.1597	24.3958	9.0625	-18.5694	14.9931	18.1042
gender	-1.8611	0.2517	-0.0893	0.0752	-0.134	-0.0514	-0.3483	-0.5728	-0.2286	0.3231	0.2297	-0.0042	0.3705	0.8119	1.4285	-1.1242	0.4244	0.256	0.1237	0.1375	0.4229	0.1338	0.3012	-1.022	-0.2585	-0.197	-0.0616	0.2818	-0.0224
ageyr	24.9653	-0.0893	0.9373	-0.8834	-0.228	-0.0833	0.3273	-0.0165	1.2148	-0.1592	-0.438	-0.3889	-0.5812	4.1123	2.3523	5.664	4.8693	1.1808	1.4437	0.1042	0.0366	0.3058	-0.2802	-1.4598	0.0431	-0.665	-1.5065	-0.3534	0.3
agemo	-7.4653	0.0752	-0.8834	12.1164	2.4725	0.4306	-1.2929	-0.6477	-1.7459	-1.136	-0.2459	-0.2153	-0.7926	3.0975	-6.5254	0.2134	0.1252	-4.2	-0.421	-1.8611	-1.3566	0.8903	0.4579	3.9567	-0.1928	-0.5524	2.6695	2.1432	-0.033
t1	14.6042	-0.134	-0.228	2.4725	47.801	10.0125	7.2605	25.7979	26.408	7.9726	9.9357	12.0778	17.4253	17.1315	35.2512	44.6507	124.6565	9.7071	11.6161	19.7181	5.3376	11.4932	6.6439	49.2433	12.3803	28.1013	30.3254	8.9061	0.5192
t2	6.6111	-0.0514	-0.0833	0.4306	10.0125	19.7583	2.3875	15.4167	14.2542	3.4208	3.2958	3.7139	6.8764	7.0153	11.175	15.675	40.8028	3.9375	4.5194	8.7403	0.2222	5.1514	1.7639	25.8819	6.3069	10.2375	14.1931	4.3125	0.3208
t3	17.9444	-0.3483	0.3273	-1.2929	7.2605	2.3875	7.9628	8.5869	10.1575	2.48	3.4839	5.0431	5.8372	8.6188	11.6452	11.7244	40.6508	7.6379	3.9557	5.8917	1.6451	3.3199	1.5721	11.7627	4.5276	4.4433	7.6523	1.5874	0.2706
t4	30.9167	-0.5728	-0.0165	-0.6477	25.7979	15.4167	8.5869	69.1724	36.8975	9.2067	11.092	16.9167	22.9543	14.7632	25.1248	41.6585	114.7632	14.5235	12.874	16.3333	5.278	12.9887	5.3903	54.8036	16.5575	31.4212	33.3247	9.0982	0.5852
t5	112.0694	-0.2286	1.2148	-1.7459	26.408	14.2542	10.1575	36.8975	135.9074	24.4648	35.4421	35.8431	66.7376	87.1038	66.5281	51.4503	148.2649	32.9935	19.2652	14.0125	11.3302	13.6908	8.2035	92.2332	17.9793	50.9242	47.3579	22.8348	2.1251
t6	11.9792	0.3231	-0.1592	-1.136	7.9726	3.4208	2.48	9.2067	24.4648	11.3935	11.2773	9.4125	19.1665	16.7657	20.2233	7.3567	39.3087	10.4648	6.3375	6.6167	4.2531	2.5092	3.0465	28.9195	4.293	12.9105	13.6615	6.8875	0.2791
t7	19.0417	0.2297	-0.438	-0.2459	9.9357	3.2958	3.4839	11.092	35.4421	11.2773	21.6157	15.7597	25.321	28.0691	19.2087	19.3114	61.2302	11.7366	5.5152	6.3458	5.3233	3.8227	4.0369	42.1638	7.0626	18.2922	17.7537	9.6403	0.5834
t8	37.7292	-0.0042	-0.3889	-0.2153	12.0778	3.7139	5.0431	16.9167	35.8431	9.4125	15.7597	28.7028	22.8528	37.7833	26.2139	32.6583	80.3708	14.0764	6.8625	10.8847	6.625	6.0708	5.0347	45.5972	9.3389	19.1167	24.7792	9.8125	0.6375
t9	16.6458	0.3705	-0.5812	-0.7926	17.4253	6.8764	5.8372	22.9543	66.7376	19.1665	25.321	22.8528	63.1625	33.7684	38.0256	20.2126	79.9934	21.5048	12.7829	12.7542	10.3508	7.8437	7.9035	70.9782	10.4468	38.3891	36.8832	15.6463	0.5371
t10	273.6458	0.8119	4.1123	3.0975	17.1315	7.0153	8.6188	14.7632	87.1038	16.7657	28.0691	37.7833	33.7684	565.5926	185.2957	293.1257	368.4357	40.4224	26.905	22.1569	32.7404	33.1363	17.012	76.153	46.1748	36.1875	56.8295	59.3367	4.7859
t11	95.4375	1.4285	2.3523	-6.5254	35.2512	11.175	11.6452	25.1248	66.5281	20.2233	19.2087	26.2139	38.0256	185.2957	277.6736	147.64	325.3124	58.4464	29.9651	35.6222	27.1889	24.688	17.3628	67.0649	31.9157	51.8967	39.1413	32.2905	1.6101
t12	165.8611	-1.1242	5.664	0.2134	44.6507	15.675	11.7244	41.6585	51.4503	7.3567	19.3114	32.6583	20.2126	293.1257	147.64	440.7919	410.8229	29.6415	25.8865	18.3097	26.6739	32.161	8.1195	104.5953	36.1977	41.8532	69.4344	40.9385	2.9061
t13	350.6181	0.4244	4.8693	0.1252	124.6565	40.8028	40.6508	114.7632	148.2649	39.3087	61.2302	80.3708	79.9934	368.4357	325.3124	410.8229	1371.6176	77.5821	38.5689	69.7833	32.2543	53.6335	33.6023	181.3752	77.1081	105.5652	132.9657	62.3089	5.3303
t14	43.0486	0.256	1.1808	-4.2	9.7071	3.9375	7.6379	14.5235	32.9935	10.4648	11.7366	14.0764	21.5048	40.4224	58.4464	29.6415	77.5821	117.2567	31.5979	28.1042	18.4078	9.09	8.5163	66.1889	9.5402	26.533	25.4231	15.0271	0.9777
t15	24.9861	0.1237	1.4437	-0.421	11.6161	4.5194	3.9557	12.874	19.2652	6.3375	5.5152	6.8625	12.7829	26.905	29.9651	25.8865	38.5689	31.5979	56.8881	16.0653	11.9363	10.3424	4.972	39.2526	8.4613	18.1911	17.8153	5.7174	0.5446
t16	26.625	0.1375	0.1042	-1.8611	19.7181	8.7403	5.8917	16.3333	14.0125	6.6167	6.3458	10.8847	12.7542	22.1569	35.6222	18.3097	69.7833	28.1042	16.0653	46.2556	10.5486	10.7875	6.2083	52.9931	11.6361	19.4556	22.5861	8.6597	0.8028
t17	21.7986	0.4229	0.0366	-1.3566	5.3376	0.2222	1.6451	5.278	11.3302	4.2531	5.3233	6.625	10.3508	32.7404	27.1889	26.6739	32.2543	18.4078	11.9363	10.5486	20.8736	9.0953	5.3803	25.1712	4.1691	13.2783	12.8295	7.3831	0.4468
t18	8.8889	0.1338	0.3058	0.8903	11.4932	5.1514	3.3199	12.9887	13.6908	2.5092	3.8227	6.0708	7.8437	33.1363	24.688	32.161	53.6335	9.09	10.3424	10.7875	9.0953	20.015	5.7871	26.8278	7.65	14.5942	11.0287	8.5687	0.518
t19	9.8125	0.3012	-0.2802	0.4579	6.6439	1.7639	1.5721	5.3903	8.2035	3.0465	4.0369	5.0347	7.9035	17.012	17.3628	8.1195	33.6023	8.5163	4.972	6.2083	5.3803	5.7871	12.9344	12.1037	5.7574	12.2529	10.0395	6.3336	0.2713
t20	42.1597	-1.022	-1.4598	3.9567	49.2433	25.8819	11.7627	54.8036	92.2332	28.9195	42.1638	45.5972	70.9782	76.153	67.0649	104.5953	181.3752	66.1889	39.2526	52.9931	25.1712	26.8278	12.1037	391.2553	38.8558	92.6837	95.3848	34.4071	1.3072
t21	24.3958	-0.2585	0.0431	-0.1928	12.3803	6.3069	4.5276	16.5575	17.9793	4.293	7.0626	9.3389	10.4468	46.1748	31.9157	36.1977	77.1081	9.5402	8.4613	11.6361	4.1691	7.65	5.7574	38.8558	23.3469	17.7552	20.2318	10.204	0.6247
t22	9.0625	-0.197	-0.665	-0.5524	28.1013	10.2375	4.4433	31.4212	50.9242	12.9105	18.2922	19.1167	38.3891	36.1875	51.8967	41.8532	105.5652	26.533	18.1911	19.4556	13.2783	14.5942	12.2529	92.6837	17.7552	96.5174	46.7036	17.495	0.6094
t23	-18.5694	-0.0616	-1.5065	2.6695	30.3254	14.1931	7.6523	33.3247	47.3579	13.6615	17.7537	24.7792	36.8832	56.8295	39.1413	69.4344	132.9657	25.4231	17.8153	22.5861	12.8295	11.0287	10.0395	95.3848	20.2318	46.7036	87.6185	19.0194	0.2943
t24	14.9931	0.2818	-0.3534	2.1432	8.9061	4.3125	1.5874	9.0982	22.8348	6.8875	9.6403	9.8125	15.6463	59.3367	32.2905	40.9385	62.3089	15.0271	5.7174	8.6597	7.3831	8.5687	6.3336	34.4071	10.204	17.495	19.0194	22.2131	0.5929
grade	18.1042	-0.0224	0.3	-0.033	0.5192	0.3208	0.2706	0.5852	2.1251	0.2791	0.5834	0.6375	0.5371	4.7859	1.6101	2.9061	5.3303	0.9777	0.5446	0.8028	0.4468	0.518	0.2713	1.3072	0.6247	0.6094	0.2943	0.5929	0.2497

模式 model	卡方檢定統計量 Chi-square statistic	自由度 d.f.	p-值 p-value
基準模式^I	1638.466	276	0.000
使用模式	358.789	242	0.000

模式 model	概似值 likelihood
虛無模式(H0)	-11453.195
未限制模式(H1)	-11273.801

潛在變數 latent	顯性變數 endogenous	估計值 estimate	標準差 std. err.	Z-值 z-value	p-值 p-value
spatial	t1	1.000
spatial	t2	0.460	0.090	5.094	0.000
spatial	t3	0.297	0.057	5.167	0.000
spatial	t4	1.181	0.179	6.611	0.000
verbal	t5	1.000
verbal	t6	0.293	0.027	10.994	0.000
verbal	t7	0.414	0.036	11.368	0.000
verbal	t8	0.398	0.044	8.993	0.000
verbal	t9	0.717	0.062	11.570	0.000
speed	t10	1.000
speed	t11	0.716	0.106	6.740	0.000
speed	t12	0.950	0.136	7.000	0.000
speed	t13	1.707	0.241	7.092	0.000
memory	t14	1.000
memory	t15	0.690	0.156	4.420	0.000
memory	t16	0.745	0.151	4.925	0.000
memory	t17	0.511	0.103	4.980	0.000
memory	t18	0.515	0.102	5.059	0.000
memory	t19	0.325	0.074	4.381	0.000
math_ability	t20	1.000
math_ability	t21	0.234	0.035	6.594	0.000
math_ability	t22	0.497	0.073	6.847	0.000
math_ability	t23	0.526	0.070	7.469	0.000
math_ability	t24	0.232	0.035	6.689	0.000

依變數 dependent	解釋變數 explanatory	估計值 estimate	標準差 std. err.	Z-值 z-value	p-值 p-value
math_ability	spatial	0.821	0.345	2.382	0.017
math_ability	verbal	0.492	0.129	3.812	0.000
math_ability	speed	0.132	0.081	1.636	0.102
math_ability	memory	0.696	0.290	2.395	0.017

變數1 variable 1	變數2 variable 2	估計值 estimate	標準差 std. err.	Z-值 z-value	p-值 p-value
spatial	verbal	26.544	5.795	4.580	0.000
spatial	speed	43.125	10.607	4.066	0.000
spatial	memory	16.907	4.403	3.840	0.000
verbal	speed	68.057	17.330	3.927	0.000
verbal	memory	25.944	7.020	3.696	0.000
speed	memory	52.350	13.926	3.759	0.000

變數 variable	估計值 estimate	標準差 std. err.
t1	24.391	3.893
t2	14.734	1.900
t3	5.874	0.761
t4	36.528	5.683
t5	47.344	6.775
t6	3.793	0.552
t7	6.458	0.980
t8	14.617	1.889
t9	17.692	2.762
t10	310.774	44.454
t11	147.193	21.447
t12	211.533	32.622
t13	631.077	99.951
t14	85.945	11.079
t15	41.977	5.398
t16	29.020	3.996
t17	12.773	1.780
t18	11.782	1.675
t19	9.632	1.234
t20	221.324	29.064
t21	14.056	1.815
t22	54.621	7.172
t23	40.824	5.722
t24	13.076	1.698
spatial	23.080	5.461
verbal	87.626	15.341
speed	250.919	60.528
memory	30.503	10.297
math_ability	25.028	10.965

指標 index	值 value	判斷準則 criterion	說明 interpretation
F_min	1.2372
NCP	116.7894	越小越好	Non-Central Parameter
GFI	0.8288	> 0.95	Goodness of Fit Index
AGFI	0.7878	> 0.95	調整GFI
RMR	10.9195	< 0.05	Root Mean Squared Residual
SRMR	0.0674	< 0.08	標準化RMR
RMSEA	0.0577	< 0.06	Root Mean Squared Error Approximation
RMSEA P-值	0.1571	> 0.05
ECVI	3.2744	越小越好	Expected Cross-Validation Index

潛在變數 Latent variable	結構信度^I CR	平均萃取變異^II AVE
spatial	0.6919	0.3652
verbal	0.8999	0.6437
speed	0.7943	0.4916
memory	0.7376	0.3214
math_ability	0.7956	0.4385

左側 LHS	關係式 operator	右側 RHS	修正指標 MI	EPC^I
t10	~~	t24	20.220979	26.98860595
spatial	=~	t13	16.345395	3.42705132
spatial	=~	t10	13.237412	-1.99630711
t10	~~	t12	12.551946	107.66530632
spatial	=~	t17	11.857517	-0.44472100
speed	=~	t24	11.298942	0.12109709
spatial	=~	t24	9.634760	-0.55104735
t14	~~	t18	9.111084	-9.51533876
speed	=~	t21	8.273187	0.10676768
verbal	=~	t21	7.842220	-0.19080144
t1	~~	t13	7.498814	35.87362772
speed	=~	t20	7.492335	-0.41040398
t1	~~	t10	7.392686	-23.94582071
speed	=~	t8	7.146754	0.07357608
verbal	=~	t12	6.653365	-0.49492344
math_ability	=~	t8	6.611190	0.12603871
t6	~~	t12	6.598589	-7.39229577
t6	~~	t11	6.587695	6.03018866
spatial	=~	t16	6.220882	0.47979484
t10	~~	t13	6.172634	-133.78417060
memory	=~	t11	5.988082	0.84351103
t11	~~	t23	5.834516	-18.36798049
t11	~~	t12	5.671219	-50.91466405
t14	~~	t15	5.321954	12.81179067
t13	~~	t17	5.286894	-20.71235036
t2	~~	t17	5.141536	-2.87319942
t8	~~	t9	4.890968	-3.76989395
spatial	=~	t8	4.686180	0.23464545
t12	~~	t16	4.523558	-16.46094197
t19	~~	t20	4.448097	-8.82173044
t18	~~	t23	4.432901	-4.51324725
t3	~~	t13	4.390706	12.40382886
t17	~~	t21	4.379930	-2.61319950
memory	=~	t8	4.309314	0.17419047
t5	~~	t9	4.286154	7.15993631
t12	~~	t19	4.228339	-8.89685462
t9	~~	t22	4.154071	6.29035204
t3	~~	t22	4.125851	-3.31384264
speed	=~	t9	4.098412	-0.06844909
speed	=~	t22	4.055922	-0.14998835
t1	~~	t16	4.022474	5.32962474
t6	~~	t7	3.970328	1.16808139
t6	~~	t18	3.939074	-1.30572483
spatial	=~	t18	3.860518	0.24845804

虛無假設：資料服從多變量常態分配
檢定方法 method	w統計量 w-statistic	p-值 p-value
Shapiro-Wilk常態性檢定	0.7703	8.5226e-14