Wednesday, July 2, 2025

Structural Equation Modeling (SEM) using R-Project-based Lavaan, and Online calculator (R-Project-Shiny-based)

This online calculator has been developed using R-Project (4.4.1), RStudio 2025.05.0 Build 496. It is made using packages, including lavaan 0.6-19; Shiny; semPlot; shinethemes, and DT Page-1 Usman Zafar Paracha 2.1643 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1642 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1641 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1640 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1639 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1638 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1637 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1635 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1632 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1628 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1626 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1624 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1623 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1622 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1621 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1620 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1619 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1617 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1616 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1615 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1613 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 1 Usman Zafar Paracha Usman Zafar Paracha Structural Equation Modeling (SEM) Structural Equation Modeling (SEM) Structural Equation Modeling (SEM) The SEM is a statistical technique that can help in looking at different variables and their relationship at the same time, rather than spending time on these variables and their relationship one by one The SEM is a statistical technique that can help in looking a... The SEM is a statistical technique that can help in looking at different variables and their relationship at the same time, rather than spending time on these variables and their relationship one by one Patreon and LinkedIn links LinkedIn Profile /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile /uzparacha /uzparacha /uzparacha Patreon Usman Zafar Paracha 2 Usman Zafar Paracha Usman Zafar Paracha Page Break 1 Moreover, SEM can help in working on different types of equations, which can be considered important in assessing the relationship of variables, at the same time. Moreover, SEM can help in working on different types of equat... Moreover, SEM can help in working on different types of equations, which can be considered important in assessing the relationship of variables, at the same time. This technique can also help in assessing different types of relationships, such as the effect of a single variable on another variable or the combined effect of more than one variable/s on some other variable/s This technique can also help in assessing different types of ... This technique can also help in assessing different types of relationships, such as the effect of a single variable on another variable or the combined effect of more than one variable/s on some other variable/s Structural Equation Modeling (SEM).1047 Structural Equation Modeling (SEM) Structural Equation Modeling (SEM) ML – Maximum Likelihood ML – Maximum Likelihood GLS - Generalized Least Squares WLS -... · ML – Maximum Likelihood· GLS - Generalized Least Squares· WLS - Weighted Least Squares· ULS - Unweighted Least Squares· DWLS - Diagonally Weighted Least Squares Different estimation methods to find the best-fitting parameters for a model Different estimation methods to find the best-fitting paramet... Different estimation methods to find the best-fitting parameters for a model Estimators include Estimators include Estimators include Patreon and LinkedIn links.1051 LinkedIn Profile.1052 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1054 /uzparacha /uzparacha /uzparacha Patreon Page Break 12 ML - Maximum Likelihood ML - Maximum Likelihood ML - Maximum Likelihood Multivariate normality of observed variables Multivariate normality of observed variables Multivariate normality of observed variables Each option is suited for different types of data and assumptions about the distribution. Each option is suited for different types of data and assumpt... Each option is suited for different types of data and assumptions about the distribution. Continuous variables with approximately normal distribution Continuous variables with approximately normal distribution Continuous variables with approximately normal distribution Most common estimator Most common estimator Most common estimator Assumes 1 Assumes Assumes Best for 1 Best for Best for Properties 1 Properties Properties Arrow 4 Arrow 5 Arrow 6 Patreon and LinkedIn links.1070 LinkedIn Profile.1071 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1073 /uzparacha /uzparacha /uzparacha Patreon Page Break 13 Cohen's d.1077 GLS - Generalized Least Squares GLS - Generalized Least Squares Multivariate normality Multivariate normality Multivariate normality When error structure is known or can be modeled more accurately than ML When error structure is known or can be modeled more accurate... When error structure is known or can be modeled more accurately than ML Similar to ML but uses a different weight matrix. Similar to ML but uses a different weight matrix. Less robust... Similar to ML but uses a different weight matrix.Less robust to violations of assumptions than ML. Assumes 2 Assumes Assumes Best for 2 Best for Best for Properties 2 Properties Properties Arrow 7 Arrow 8 Arrow 9 Patreon and LinkedIn links.1088 LinkedIn Profile.1089 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1091 /uzparacha /uzparacha /uzparacha Patreon Page Break 14 WLS - Weighted Least Squares WLS - Weighted Least Squares WLS - Weighted Least Squares Large sample size Large sample size Large sample size Ordinal/categorical data (though rarely used now due to DWLS) Ordinal/categorical data (though rarely used now due to DWLS) Ordinal/categorical data (though rarely used now due to DWLS) Uses full weight matrix (inverse of asymptotic covariance matrix of sample statistics). Uses full weight matrix (inverse of asymptotic covariance mat... Uses full weight matrix (inverse of asymptotic covariance matrix of sample statistics).Requires large samples; may fail or be unstable in small samples. Assumes 3 Assumes Assumes Best for 3 Best for Best for Properties 3 Properties Properties Arrow 10 Arrow 11 Arrow 12 Patreon and LinkedIn links.1106 LinkedIn Profile.1107 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1109 /uzparacha /uzparacha /uzparacha Patreon Page Break 15 ULS - Unweighted Least Squares ULS - Unweighted Least Squares ULS - Unweighted Least Squares No distributional assumptions (distribution-free) No distributional assumptions (distribution-free) No distributional assumptions (distribution-free) Exploratory purposes or when robustness is desired Exploratory purposes or when robustness is desired Exploratory purposes or when robustness is desired Doesn't use a weight matrix. Doesn't use a weight matrix. Often leads to underestimation o... Doesn't use a weight matrix.Often leads to underestimation of standard errors. Assumes 4 Assumes Assumes Best for 4 Best for Best for Properties 4 Properties Properties Arrow 13 Arrow 14 Arrow 15 Patreon and LinkedIn links.1124 LinkedIn Profile.1125 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1127 /uzparacha /uzparacha /uzparacha Patreon NORM.DIST Function.1129 DWLS - Diagonally Weighted Least Squares DWLS - Diagonally Weighted Least Squares DWLS - Diagonally Weighted Least Squares Suitable for ordinal/categorical data Suitable for ordinal/categorical data Suitable for ordinal/categorical data Ordinal variables or non-normal continuous data Ordinal variables or non-normal continuous data Ordinal variables or non-normal continuous data Uses only diagonal elements of the weight matrix (simpler, more stable than WLS). Uses only diagonal elements of the weight matrix (simpler, mo... Uses only diagonal elements of the weight matrix (simpler, more stable than WLS).Recommended in lavaan for ordinal data (e.g., Likert scales).Also returns robust standard errors and a mean- and variance-adjusted test statistic. Assumes Assumes Assumes Best for Best for Best for Properties Properties Properties Arrow 16 Arrow 17 Arrow 18 Patreon and LinkedIn links.1142 LinkedIn Profile.1143 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1145 /uzparacha /uzparacha /uzparacha Patreon Page Break 17 The model syntax The model syntax The model syntax It is the main area of how you define measurement models (like CFA) and structural models (like SEM). It is the main area of how you define measurement models (lik... It is the main area of how you define measurement models (like CFA) and structural models (like SEM). Latent variables (also called factors) Latent variables (also called factors) Regression paths Covar... · Latent variables (also called factors)· Regression paths· Covariances· Intercepts or means· (Optional) Constraints Helps in specifying Helps in specifying Helps in specifying Arrow 2 =~ =~ =~ Latent variable definition (factor loading) Latent variable definition (factor loading) Latent variable definition (factor loading) F1 =~ x1 + x2 + x3 F1 =~ x1 + x2 + x3 F1 =~ x1 + x2 + x3 Operator Operator Operator Meaning Meaning Meaning Example Example Example ~ ~ ~ Regression (dependent ~ independent) Regression (dependent ~ independent) Regression (dependent ~ independent) y ~ x1 + x2 y ~ x1 + x2 y ~ x1 + x2 ~~ ~~ ~~ Covariance or variance Covariance or variance Covariance or variance x1 ~~ x2 or x1 ~~ x2 or x1 ~~ x1 x1 ~~ x2 or x1 ~~ x1 ~1 ~1 ~1 Intercept or mean Intercept or mean Intercept or mean x1 ~ 1 x1 ~ 1 x1 ~ 1 == == == Constraint Constraint Constraint x1 ~~ x2 == 0 x1 ~~ x2 == 0 x1 ~~ x2 == 0 * * * Fix or label a parameter Fix or label a parameter Fix or label a parameter F1 =~ 1*x1 or F1 =~ 1*x1 or F1 =~ a*x2 F1 =~ 1*x1 or F1 =~ a*x2 Patreon and LinkedIn links.1180 LinkedIn Profile.1181 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1183 /uzparacha /uzparacha /uzparacha Patreon Page Break 2 ability =~ item1 + item2 + item3 + item4 ability =~ item1 + item2 + item3 + item4 ability =~ item1 + item2 + item3 + item4 Confirmatory Factor Analysis (CFA) Confirmatory Factor Analysis (CFA) Confirmatory Factor Analysis (CFA) syntax example syntax example syntax example Patreon and LinkedIn links.1198 LinkedIn Profile.1199 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1201 /uzparacha /uzparacha /uzparacha Patreon Page Break 3 One-factor model One-factor model One-factor model Multiple Factors.1212 verbal =~ v1 + v2 + v3 math =~ m1 + m2 + m3 verbal =~ v1 + v2 + v3math =~ m1 + m2 + m3 Multiple Latent Variables Multiple Latent Variables Multiple Latent Variables syntax example 2 syntax example syntax example Patreon and LinkedIn links.1215 LinkedIn Profile.1216 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1218 /uzparacha /uzparacha /uzparacha Patreon Page Break 4 Multiple Factors Multiple Factors Multiple Factors stress =~ s1 + s2 + s3 stress =~ s1 + s2 + s3 anxiety =~ a1 + a2 + a3 stress =~ s1 + s2 + s3anxiety =~ a1 + a2 + a3 Structural Equation Model (SEM) Structural Equation Model (SEM) Structural Equation Model (SEM) syntax example 4 Patreon and LinkedIn links.1236 LinkedIn Profile.1237 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1239 /uzparacha /uzparacha /uzparacha Patreon Page Break 5 Measurement model Measurement model Measurement model anxiety ~ stress anxiety ~ stress anxiety ~ stress syntax example 3 syntax example syntax example Structural model Structural model Structural model y ~ x1 + x2.1247 y ~ x1 + x2 y ~ x1 + x2 Regression between Observed Variables Regression between Observed Variables Regression between Observed Variables Patreon and LinkedIn links.1250 LinkedIn Profile.1251 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1253 /uzparacha /uzparacha /uzparacha Patreon Page Break 6 syntax example 5 syntax example syntax example x1 ~~ x2 x1 ~~ x2 x1 ~~ x2 Covariances between Variables Covariances between Variables Covariances between Variables syntax example 6 syntax example syntax example Patreon and LinkedIn links.1263 LinkedIn Profile.1264 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1266 /uzparacha /uzparacha /uzparacha Patreon Page Break 7 Covariance between x1 and x2 Covariance between x1 and x2 Covariance between x1 and x2 x1 ~~ x1 x1 ~~ x1 x1 ~~ x1 syntax example 7 syntax example syntax example Variance of x1 Variance of x1 Variance of x1 factor1 ~~ factor2 factor1 ~~ factor2 factor1 ~~ factor2 Covariance between latent variables Covariance between latent variables Covariance between latent variables syntax example 8 syntax example syntax example F1 =~ 1*x1 + x2 + x3 F1 =~ 1*x1 + x2 + x3 F1 =~ 1*x1 + x2 + x3 Fix a Loading or Set Equality Constraints Fix a Loading or Set Equality Constraints Fix a Loading or Set Equality Constraints syntax example 9 Patreon and LinkedIn links.1280 LinkedIn Profile.1281 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1283 /uzparacha /uzparacha /uzparacha Patreon Page Break 8 Fix loading of x1 to 1 (often done for scale) Fix loading of x1 to 1 (often done for scale) Fix loading of x1 to 1 (often done for scale) F1 =~ a*x2 + a*x3 F1 =~ a*x2 + a*x3 F1 =~ a*x2 + a*x3 syntax example 10 syntax example syntax example Equal loading constraint (labeled 'a') Equal loading constraint (labeled 'a') Equal loading constraint (labeled 'a') x1 ~ 1.1290 x1 ~ 1 x1 ~ 1 Means/Intercepts of Observed Variables Means/Intercepts of Observed Variables Means/Intercepts of Observed Variables syntax example 11 syntax example syntax example Patreon and LinkedIn links.1293 LinkedIn Profile.1294 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1296 /uzparacha /uzparacha /uzparacha Patreon Page Break 9 Estimate intercept/mean of x1 Estimate intercept/mean of x1 Estimate intercept/mean of x1 y ~ x1 + x2.1300 y ~ x1 + x2 x1 ~~ x2 y ~ x1 + x2x1 ~~ x2 Path Model with Observed Variables Only Path Model with Observed Variables Only Path Model with Observed Variables Only syntax example 12 syntax example syntax example Patreon and LinkedIn links.1303 LinkedIn Profile.1304 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1306 /uzparacha /uzparacha /uzparacha Patreon Page Break 10 F1 =~ x1 + x2 + x3.1310 F1 =~ x1 + x2 + x3 F1 =~ x1 + x2 + x3 Constraints on Parameters Constraints on Parameters Constraints on Parameters syntax example 13 Patreon and LinkedIn links.1313 LinkedIn Profile.1314 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1316 /uzparacha /uzparacha /uzparacha Patreon Page Break 11 x1 ~~ x2 == 0.1320 x1 ~~ x2 == 0 x1 ~~ x2 == 0 syntax example 14 syntax example syntax example Force zero covariance Force zero covariance Force zero covariance Test statistic: 85.306 | Degrees of freedom = 24 | P-value (Chi-square) = 0.000 Test statistic: 85.306 | Degrees of freedom = 24 | P-value (C... Test statistic: 85.306 | Degrees of freedom = 24 | P-value (Chi-square) = 0.000 Chi-square Test of Model Fit – User Model Chi-square Test of Model Fit – User Model Chi-square Test of Model Fit – User Model Arrow 19 Patreon and LinkedIn links.1326 LinkedIn Profile.1327 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1329 /uzparacha /uzparacha /uzparacha Patreon Page Break 19 This test checks if the model-implied covariance = sample covariance. This test checks if the model-implied covariance = sample cov... This test checks if the model-implied covariance = sample covariance. Significant p-value (p < 0.001) means poor fit if taken strictly, but Chi-square is sensitive to large sample sizes. Significant p-value (p < 0.001) means poor fit if taken stric... Significant p-value (p < 0.001) means poor fit if taken strictly, but Chi-square is sensitive to large sample sizes. Don't rely on this alone — look at other fit indices. Don't rely on this alone — look at other fit indices. Don't rely on this alone — look at other fit indices. Test statistic: 918.852 | Degrees of freedom = 36 | P-value (Chi-square) = 0.000 Test statistic: 918.852 | Degrees of freedom = 36 | P-value (... Test statistic: 918.852 | Degrees of freedom = 36 | P-value (Chi-square) = 0.000 Model Test - Baseline Model Model Test - Baseline Model Model Test - Baseline Model Arrow 20 Patreon and LinkedIn links.1338 LinkedIn Profile.1339 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1341 /uzparacha /uzparacha /uzparacha Patreon Page Break 20 The baseline model assumes no correlations between variables. The baseline model assumes no correlations between variables. The baseline model assumes no correlations between variables. This model is much better than the baseline (918 vs. 85). This model is much better than the baseline (918 vs. 85). This model is much better than the baseline (918 vs. 85). Comparative Fit Index (CFI)= 0.931 | Tucker-Lewis Index (TLI) = 0.896 Comparative Fit Index (CFI)= 0.931 | Tucker-Lewis Index (TLI)... Comparative Fit Index (CFI)= 0.931 | Tucker-Lewis Index (TLI) = 0.896 Incremental Fit Indices Incremental Fit Indices Incremental Fit Indices Arrow 21 Patreon and LinkedIn links.1351 LinkedIn Profile.1352 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1354 /uzparacha /uzparacha /uzparacha Patreon Page Break 21 CFI and TLI > 0.90 is good fit, > 0.95 is excellent fit. CFI and TLI > 0.90 is good fit, > 0.95 is excellent fit. CFI and TLI > 0.90 is good fit, > 0.95 is excellent fit. CFI is good to excellent, TLI slightly below good. CFI is good to excellent, TLI slightly below good. CFI is good to excellent, TLI slightly below good. It is a method to measure how well a statistical model fits the data. It is a method to measure how well a statistical model fits t... It is a method to measure how well a statistical model fits the data. RMSEA = 0.092 | 90% CI: [0.071, 0.114] RMSEA = 0.092 | 90% CI: [0.071, 0.114] P(RMSEA ≤ 0.05) = 0.00... RMSEA = 0.092 | 90% CI: [0.071, 0.114]P(RMSEA ≤ 0.05) = 0.001 | P(RMSEA ≥ 0.08) = 0.840 Root Mean Square Error of Approximation (RMSEA) Root Mean Square Error of Approximation (RMSEA) Root Mean Square Error of Approximation (RMSEA) Arrow 22 Patreon and LinkedIn links.1365 LinkedIn Profile.1366 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1368 /uzparacha /uzparacha /uzparacha Patreon Page Break 22 RMSEA as 0 = Perfect fit, ≤ 0.05 = Close fit, 0.05 - 0.08 = Reasonable fit, and ≥ 0.10 = Poor fit RMSEA as 0 = Perfect fit, ≤ 0.05 = Close fit, 0.05 - 0.08 = R... RMSEA as 0 = Perfect fit, ≤ 0.05 = Close fit, 0.05 - 0.08 = Reasonable fit, and ≥ 0.10 = Poor fit RMSEA = 0.092 is high, so the fit is not great RMSEA = 0.092 is high, so the fit is not great RMSEA = 0.092 is high, so the fit is not great A goodness-of-fit measure/test in SEM that is used to determine how well a model fits the population data A goodness-of-fit measure/test in SEM that is used to determi... A goodness-of-fit measure/test in SEM that is used to determine how well a model fits the population data SRMR = 0.065 SRMR = 0.065 SRMR = 0.065 Standardized Root Mean Squared Residual (SRMR) Standardized Root Mean Squared Residual (SRMR) Standardized Root Mean Squared Residual (SRMR) Arrow 23 Patreon and LinkedIn links.1380 LinkedIn Profile.1381 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1383 /uzparacha /uzparacha /uzparacha Patreon Page Break 23 SRMR as 0 = Perfect fit, < 0.08 = Good fit SRMR as 0 = Perfect fit, < 0.08 = Good fit SRMR as 0 = Perfect fit, < 0.08 = Good fit SRMR is acceptable SRMR is acceptable SRMR is acceptable The SRMR is also used to show how well a model fits the data. It is also considered a “badness of fit” measure. The SRMR is also used to show how well a model fits the data.... The SRMR is also used to show how well a model fits the data. It is also considered a “badness of fit” measure. Loglikelihood user model (H0): -3737.745 | Akaike (AIC): 7517.49 | Bayesian (BIC): 7595.34 Loglikelihood user model (H0): -3737.745 | Akaike (AIC): 7517... Loglikelihood user model (H0): -3737.745 | Akaike (AIC): 7517.49 | Bayesian (BIC): 7595.34 Loglikelihood and Information Criteria Loglikelihood and Information Criteria Loglikelihood and Information Criteria Arrow 24 Patreon and LinkedIn links.1393 LinkedIn Profile.1394 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1396 /uzparacha /uzparacha /uzparacha Patreon Page Break 24 Lower AIC/BIC = better model (used for comparing different models). Lower AIC/BIC = better model (used for comparing different mo... Lower AIC/BIC = better model (used for comparing different models). Not interpretable alone, but useful when comparing alternative models Not interpretable alone, but useful when comparing alternativ... Not interpretable alone, but useful when comparing alternative models Akaike’s Information Criterion (AIC) and Bayesian Information Criterion (BIC) are the methods to choose the best statistical model Akaike’s Information Criterion (AIC) and Bayesian Information... Akaike’s Information Criterion (AIC) and Bayesian Information Criterion (BIC) are the methods to choose the best statistical model visual =~ x1 (1.000), x2 (0.554), x3 (0.729) visual =~ x1 (1.000), x2 (0.554), x3 (0.729) visual =~ x1 (1.000), x2 (0.554), x3 (0.729) Latent Variables (Factor Loadings) Latent Variables (Factor Loadings) Latent Variables (Factor Loadings) Arrow 25 Patreon and LinkedIn links.1409 LinkedIn Profile.1410 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1412 /uzparacha /uzparacha /uzparacha Patreon Page Break 25 These tell how much each observed variable loads on its latent factor. These tell how much each observed variable loads on its laten... These tell how much each observed variable loads on its latent factor. =~ means factor loading =~ means factor loading =~ means factor loading x1 loads on visual with standardized loading of 0.772 → strong, x1 loads on visual with standardized loading of 0.772 → stron... x1 loads on visual with standardized loading of 0.772 → strong,x2 with 0.424 → moderate.All loadings are statistically significant (p < 0.001). Std.all column = standardized loadings (like correlations) Std.all column = standardized loadings (like correlations) Std.all column = standardized loadings (like correlations) Arrow 26 visual ~~ txt = 0.408 (Std.all = 0.459) visual ~~ txt = 0.408 (Std.all = 0.459) visual ~~ txt = 0.408 (Std.all = 0.459) Covariances between Latent Variables Covariances between Latent Variables Covariances between Latent Variables Arrow 27 Patreon and LinkedIn links.1426 LinkedIn Profile.1427 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1429 /uzparacha /uzparacha /uzparacha Patreon Page Break 26 These tell how much each observed variable loads on its latent factor.1432 These tell how much each observed variable loads on its laten... These tell how much each observed variable loads on its latent factor. Positive, significant covariances → latent traits are related Positive, significant covariances → latent traits are related... Positive, significant covariances → latent traits are related visual, txt, and speed are positively correlated. Following results are from the Holzinger-Swineford example (data related to HolzingerSwineford1939) containing 301 observations... Following results are from the Holzinger-Swineford example (d... Following results are from the Holzinger-Swineford example (data related to HolzingerSwineford1939) containing 301 observations... .x1 = 0.549 (Std.all = 0.404) .x1 = 0.549 (Std.all = 0.404) .x1 = 0.549 (Std.all = 0.404) Variances of Observed Variables and Latents Variances of Observed Variables and Latents Variances of Observed Variables and Latents Arrow 28 Patreon and LinkedIn links.1441 LinkedIn Profile.1442 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1444 /uzparacha /uzparacha /uzparacha Patreon Page Break 27 . before x1 means residual variance of x1. . before x1 means residual variance of x1. Standardized resid... . before x1 means residual variance of x1.Standardized residual of 0.404 = remaining unexplained variance (after accounting for latent factor). visual = 0.809 (Std.all = 1.000) visual = 0.809 (Std.all = 1.000) visual = 0.809 (Std.all = 1.000) Arrow 29 Variance of latent variable is fixed at 1 when standardized. Variance of latent variable is fixed at 1 when standardized. Variance of latent variable is fixed at 1 when standardized. SEM Path Diagram SEM Path Diagram SEM Path Diagram Patreon and LinkedIn links.1456 LinkedIn Profile.1457 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1460 /uzparacha /uzparacha /uzparacha Patreon Page Break 28 SEM Path Diagram image → → → Single-headed arrow Single-headed arrow Single-headed arrow Arrow 30 Patreon and LinkedIn links.1472 LinkedIn Profile.1473 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1476 /uzparacha /uzparacha /uzparacha Patreon Page Break 29 A directional relationship, such as: A directional relationship, such as: Factor loading A directional relationship, such as:Factor loading (M1-M2)/SDpooled.1485 Circles (e.g., vsl, txt, spd) Circles (e.g., vsl, txt, spd) Overall Structure Overall Structure Overall Structure Arrow 31 Patreon and LinkedIn links.1488 LinkedIn Profile.1489 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1492 /uzparacha /uzparacha /uzparacha Patreon Page Break 30 Latent variables (unobserved constructs): Latent variables (unobserved constructs): vsl = Visual txt = ... Latent variables (unobserved constructs):vsl = Visualtxt = Textualspd = Speed Squares (e.g., x1, x2, ..., x9) Squares (e.g., x1, x2, ..., x9) Squares (e.g., x1, x2, ..., x9) Arrow 32 Observed variables (measured items) Observed variables (measured items) Observed variables (measured items) Straight arrows (→) from latent to observed Straight arrows (→) from latent to observed Straight arrows (→) from latent to observed Arrows Arrows Arrows Arrow 33 Patreon and LinkedIn links.1503 LinkedIn Profile.1504 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1507 /uzparacha /uzparacha /uzparacha Patreon Page Break 31 Example: From vsl → x1, x2, x3 Example: From vsl → x1, x2, x3 Numbers near these arrows = St... Example: From vsl → x1, x2, x3Numbers near these arrows = Standardized loadings (e.g., 0.77, 0.42, 0.58 for x1, x2, x3)Higher values (closer to 1) mean a stronger relationship between the factor and item. These are factor loadings, showing how strongly a latent variable explains the observed variable. These are factor loadings, showing how strongly a latent vari... These are factor loadings, showing how strongly a latent variable explains the observed variable. Curved arrows (↔) between latent variables Curved arrows (↔) between latent variables Curved arrows (↔) between latent variables Arrows.1518 Arrows Arrows Arrow 34 Patreon and LinkedIn links.1520 LinkedIn Profile.1521 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1524 /uzparacha /uzparacha /uzparacha Patreon Page Break 32 Example: vsl ↔ txt = 0.46 Example: vsl ↔ txt = 0.46 txt ↔ spd = 0.28 vsl ↔ spd = 0.47 T... Example: vsl ↔ txt = 0.46txt ↔ spd = 0.28vsl ↔ spd = 0.47 These values tell how much the factors are correlated. Values closer to 1 = strong positive relationship. These represent covariances between latent variables These represent covariances between latent variables These represent covariances between latent variables Curved arrows looping back to the same circle (e.g., vsl ↔ vsl) Curved arrows looping back to the same circle (e.g., vsl ↔ vsl) Curved arrows looping back to the same circle (e.g., vsl ↔ vsl) Arrows.1531 Arrows Arrows Arrow 35 Patreon and LinkedIn links.1533 LinkedIn Profile.1534 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1537 /uzparacha /uzparacha /uzparacha Patreon Page Break 33 Typically fixed to 1.00 for model identification in standardized solutions. Typically fixed to 1.00 for model identification in standardi... Typically fixed to 1.00 for model identification in standardized solutions. These indicate the variance of the latent variable These indicate the variance of the latent variable These indicate the variance of the latent variable Dotted arrows Dotted arrows Dotted arrows Arrows.1544 Arrows Arrows Arrow 36 Patreon and LinkedIn links.1546 LinkedIn Profile.1547 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1550 /uzparacha /uzparacha /uzparacha Patreon Page Break 34 Low values = most of the variable is explained by the factor. Low values = most of the variable is explained by the factor.... Low values = most of the variable is explained by the factor.High values = large portion unexplained. These show residual variances (errors) — the part of the observed variable not explained by the latent variable. These show residual variances (errors) — the part of the obse... These show residual variances (errors) — the part of the observed variable not explained by the latent variable. 0.77, 0.42, etc 0.77, 0.42, etc 0.77, 0.42, etc Number Type Number Type Number Type Arrow 37 Patreon and LinkedIn links.1559 LinkedIn Profile.1560 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1563 /uzparacha /uzparacha /uzparacha Patreon Page Break 35 They represent Standardized factor loadings They represent Standardized factor loadings They represent Standardized factor loadings On straight arrows from latent to observed On straight arrows from latent to observed On straight arrows from latent to observed 0.46, 0.28, etc 0.46, 0.28, etc 0.46, 0.28, etc Number Type.1570 Number Type Number Type Arrow 38 Patreon and LinkedIn links.1572 LinkedIn Profile.1573 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1576 /uzparacha /uzparacha /uzparacha Patreon Page Break 36 They represent covariances/correlations between latent factors They represent covariances/correlations between latent factors They represent covariances/correlations between latent factors On curved arrows between latent variables On curved arrows between latent variables On curved arrows between latent variables 0.40, 0.82, etc 0.40, 0.82, etc 0.40, 0.82, etc Number Type.1583 Number Type Number Type Arrow 39 Patreon and LinkedIn links.1585 LinkedIn Profile.1586 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1589 /uzparacha /uzparacha /uzparacha Patreon Page Break 37 They represent residual variances/errors of observed variables They represent residual variances/errors of observed variables They represent residual variances/errors of observed variables Near arrows looping to observed vars Near arrows looping to observed vars Near arrows looping to observed vars 1.00 1.00 1.00 Number Type.1596 Number Type Number Type Arrow 40 Patreon and LinkedIn links.1598 LinkedIn Profile.1599 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1602 /uzparacha /uzparacha /uzparacha Patreon Page Break 38 Fixed latent variances for model identification Fixed latent variances for model identification Fixed latent variances for model identification On looping arrows at latent vars On looping arrows at latent vars On looping arrows at latent vars Usman Zafar Paracha 2.1610 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1612 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1614 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1618 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1625 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1627 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1629 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1630 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1631 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1633 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1634 Usman Zafar Paracha Usman Zafar Paracha Usman Zafar Paracha 2.1636 Usman Zafar Paracha Usman Zafar Paracha Example data - Holzinger-Swineford example The model syntax used... The model syntax used... The model syntax used... visual =~ x1 + x2 + x3 visual =~ x1 + x2 + x3 txt =~ x4 + x5 + x6 speed =~ x7 + x8 +... visual =~ x1 + x2 + x3txt =~ x4 + x5 + x6speed =~ x7 + x8 + x9 Arrow Patreon and LinkedIn links.1649 LinkedIn Profile.1650 /usmanzafarparacha /usmanzafarparacha /usmanzafarparacha LinkedIn Patreon profile.1653 /uzparacha /uzparacha /uzparacha Patreon Page Break 18 Lavaan model syntax area Arrow 1
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