Latent variable modeling RANCH

Latent variable modeling RANCH Ruth Koole 29-04-2020 We have imported the data of RANCH from the SAS file. Data contains 3207 rows and 949 variables. In the RANCH study mental health was measured using the parent-rated version of the Strengths and Difficulties Questionnaire (SDQ). The SDQ is a 25 item behavioural screening questionnaire consisting of five subscales (emotional symptoms, conduct problems, hyperactivity/inattention, peer relationship problems and prosocial behaviour). A total SDQ score can be computed using 20 of these variables. ## Error in pryr::where(obj_name) : length(name) == 1 is not TRUE Data preparation for analysis The 20 SDQ variables that are used for the computation of the total SDQ score are showed in the table below. variable_name variable_label p_20_7neg Obedient p_20_11neg Friendly p_20_14neg Liked p_20_21neg Thinker p_20_25neg Good attention p_20_2pos Restless p_20_3pos Headaches p_20_5pos Tempers p_20_6pos Solitary p_20_8pos Worrier p_20_10pos Fidget p_20_12pos Fights p_20_13pos Unhappy p_20_15pos Lacks concentration p_20_16pos Nervous p_20_18pos Lies_cheats p_20_19pos Picked on p_20_22pos Steals p_20_23pos Prefers adults p_20_24pos Fearful 1 Table 1: General statistics on SDQ score value Mean 9.785 Std.Dev 5.693 Min 0.000 Q1 6.000 Median 9.000 Q3 13.000 Max 34.000 MAD 5.930 IQR 7.000 CV 0.582 Skewness 0.652 SE.Skewness 0.052 Kurtosis 0.134 N.Valid 2237.000 Pct.Valid 69.754 Based on the guide of the GGD, the SDQ subscales can only be computed in case 3 or more questions regarding that subscale have been completed. The total SDQ score can only be computed if all subscales have been computed. Since the tot SDQ scores have already been computed, we will remove all rows for which the total SDQ score is a missing value. The cleaned data then contains 2237 rows and 949 variables. For applying the latent variable modeling, we make a subset of the data containing the 20 variables that are used to compute the total SDQ score. Factor analysis Before we apply factor analysis, all missing values in this dataset are replaced with the mean of the column for all items. Before applying factor analysis, we compute the correlation matrix. 2 Table 2: Correlation matrix of SDQ variables Obedient Friendly Liked Thinker Good attention Restless Headaches Tempers Solitary Worrier Fidget Fights Unhappy Lacks concentration Nervous Lies_cheats Picked on Steals Prefers adults Fearful Obedient 1.00 0.12 0.23 0.29 0.25 0.26 0.03 0.30 -0.03 0.10 0.22 0.24 0.13 0.24 0.10 0.29 0.15 0.13 0.07 0.07 Friendly 0.12 1.00 0.33 0.12 0.14 0.05 0.04 0.06 0.16 0.10 0.09 0.15 0.14 0.07 0.09 0.09 0.18 0.13 0.13 0.05 Liked 0.23 0.33 1.00 0.20 0.23 0.09 0.08 0.16 0.15 0.19 0.13 0.23 0.16 0.17 0.13 0.15 0.32 0.16 0.19 0.11 Thinker 0.29 0.12 0.20 1.00 0.44 0.26 0.02 0.21 0.05 0.06 0.25 0.14 0.10 0.38 0.12 0.25 0.16 0.11 0.03 0.07 Good attention 0.25 0.14 0.23 0.44 1.00 0.25 0.05 0.17 0.07 0.12 0.24 0.16 0.12 0.57 0.20 0.25 0.19 0.12 0.10 0.12 Restless 0.26 0.05 0.09 0.26 0.25 1.00 0.06 0.33 -0.02 0.15 0.62 0.22 0.14 0.39 0.13 0.26 0.15 0.13 0.12 0.16 Headaches 0.03 0.04 0.08 0.02 0.05 0.06 1.00 0.18 0.07 0.27 0.07 0.06 0.24 0.10 0.17 0.06 0.13 0.01 0.12 0.14 Tempers 0.30 0.06 0.16 0.21 0.17 0.33 0.18 1.00 0.04 0.21 0.28 0.29 0.26 0.24 0.17 0.28 0.21 0.12 0.13 0.18 Solitary -0.03 0.16 0.15 0.05 0.07 -0.02 0.07 0.04 1.00 0.20 0.04 0.05 0.18 0.06 0.14 0.01 0.19 0.03 0.19 0.15 Worrier 0.10 0.10 0.19 0.06 0.12 0.15 0.27 0.21 0.20 1.00 0.19 0.19 0.39 0.19 0.32 0.09 0.23 0.08 0.16 0.34 Fidget 0.22 0.09 0.13 0.25 0.24 0.62 0.07 0.28 0.04 0.19 1.00 0.22 0.16 0.39 0.17 0.23 0.15 0.12 0.07 0.13 Fights 0.24 0.15 0.23 0.14 0.16 0.22 0.06 0.29 0.05 0.19 0.22 1.00 0.17 0.23 0.13 0.33 0.20 0.21 0.12 0.11 Unhappy 0.13 0.14 0.16 0.10 0.12 0.14 0.24 0.26 0.18 0.39 0.16 0.17 1.00 0.16 0.27 0.18 0.31 0.07 0.19 0.29 Lacks concentration 0.24 0.07 0.17 0.38 0.57 0.39 0.10 0.24 0.06 0.19 0.39 0.23 0.16 1.00 0.26 0.28 0.23 0.13 0.14 0.16 Nervous 0.10 0.09 0.13 0.12 0.20 0.13 0.17 0.17 0.14 0.32 0.17 0.13 0.27 0.26 1.00 0.12 0.17 0.06 0.10 0.35 Lies_cheats 0.29 0.09 0.15 0.25 0.25 0.26 0.06 0.28 0.01 0.09 0.23 0.33 0.18 0.28 0.12 1.00 0.21 0.29 0.10 0.09 Picked on 0.15 0.18 0.32 0.16 0.19 0.15 0.13 0.21 0.19 0.23 0.15 0.20 0.31 0.23 0.17 0.21 1.00 0.12 0.23 0.23 Steals 0.13 0.13 0.16 0.11 0.12 0.13 0.01 0.12 0.03 0.08 0.12 0.21 0.07 0.13 0.06 0.29 0.12 1.00 0.09 0.02 Prefers adults 0.07 0.13 0.19 0.03 0.10 0.12 0.12 0.13 0.19 0.16 0.07 0.12 0.19 0.14 0.10 0.10 0.23 0.09 1.00 0.18 Fearful 0.07 0.05 0.11 0.07 0.12 0.16 0.14 0.18 0.15 0.34 0.13 0.11 0.29 0.16 0.35 0.09 0.23 0.02 0.18 1.00 3 We will apply two some tests to see if the dataset is suitable for factor analysis. ## # Is the data suitable for Factor Analysis? ## ## - KMO: The Kaiser, Meyer, Olkin (KMO) measure of sampling adequacy suggests that data seems appropriate for factor analysis (KMO = 0.83). ## - Sphericity: Bartlett's test of sphericity suggests that there is sufficient significant correlation in the data for factor analaysis (Chisq(210) = 8705.01, p < .001). We will use factor analysis using the factanal function in R. 1 factor ## ## Call: ## factanal(x = subset(ranch_MH_for_FA, select = -studno), factors = 1, scores = "regression", rotation = "varimax") ## ## Uniquenesses: ## Obedient Friendly Liked Thinker ## 0.806 0.939 0.846 0.781 ## Good attention Restless Headaches Tempers ## 0.711 0.697 0.954 0.753 ## Solitary Worrier Fidget Fights ## 0.971 0.841 0.703 0.809 ## Unhappy Lacks concentration Nervous Lies_cheats ## 0.830 0.613 0.852 0.777 ## Picked on Steals Prefers adults Fearful ## 0.808 0.928 0.926 0.884 ## ## Loadings: ## Factor1 ## Obedient 0.440 ## Friendly 0.247 ## Liked 0.392 ## Thinker 0.468 ## Good attention 0.538 ## Restless 0.551 ## Headaches 0.214 ## Tempers 0.497 ## Solitary 0.170 ## Worrier 0.399 ## Fidget 0.545 ## Fights 0.437 ## Unhappy 0.413 ## Lacks concentration 0.622 ## Nervous 0.385 ## Lies_cheats 0.472 ## Picked on 0.438 ## Steals 0.267 ## Prefers adults 0.273 ## Fearful 0.341 4 ## ## Factor1 ## SS loadings 3.573 ## Proportion Var 0.179 ## ## Test of the hypothesis that 1 factor is sufficient. ## The chi square statistic is 3305.25 on 170 degrees of freedom. ## The p-value is 0 2 factors ## ## Call: ## factanal(x = subset(ranch_MH_for_FA, select = -studno), factors = 2, scores = "regression", rotation = "none") ## ## Uniquenesses: ## Obedient Friendly Liked Thinker ## 0.787 0.931 0.840 0.701 ## Good attention Restless Headaches Tempers ## 0.643 0.633 0.872 0.759 ## Solitary Worrier Fidget Fights ## 0.886 0.619 0.656 0.819 ## Unhappy Lacks concentration Nervous Lies_cheats ## 0.636 0.541 0.776 0.768 ## Picked on Steals Prefers adults Fearful ## 0.753 0.931 0.881 0.750 ## ## Loadings: ## Factor1 Factor2 ## Obedient 0.428 -0.174 ## Friendly 0.243 0.102 ## Liked 0.387 0.102 ## Thinker 0.466 -0.287 ## Good attention 0.541 -0.254 ## Restless 0.549 -0.255 ## Headaches 0.229 0.275 ## Tempers 0.490 ## Solitary 0.182 0.285 ## Worrier 0.436 0.437 ## Fidget 0.544 -0.218 ## Fights 0.426 ## Unhappy 0.444 0.409 ## Lacks concentration 0.630 -0.249 ## Nervous 0.404 0.247 ## Lies_cheats 0.458 -0.147 ## Picked on 0.445 0.221 ## Steals 0.257 ## Prefers adults 0.277 0.205 ## Fearful 0.365 0.342 ## ## Factor1 Factor2 ## SS loadings 3.637 1.180 5 ## Proportion Var 0.182 0.059 ## Cumulative Var 0.182 0.241 ## ## Test of the hypothesis that 2 factors are sufficient. ## The chi square statistic is 2030.43 on 151 degrees of freedom. ## The p-value is 0 0.2 0.3 0.4 0.5 0.6 −0.2 0.0 0.2 0.4 Factor1 Factor2 Obedient Friendly Liked Thinker Good attention Restless Headaches Tempers Solitary Worrier Fidget Fights Unhappy Lacks concentration Nervous Lies_cheats Picked on Steals Prefers adults Fearful −1 0 1 2 3 −2 −1 0 1 2 3 Factor1 Factor2 3 factors ## ## Call: ## factanal(x = subset(ranch_MH_for_FA, select = -studno), factors = 3, scores = "regression", rotation = "none") ## ## Uniquenesses: ## Obedient Friendly Liked Thinker ## 0.814 0.909 0.800 0.662 ## Good attention Restless Headaches Tempers ## 0.410 0.296 0.874 0.753 ## Solitary Worrier Fidget Fights ## 0.880 0.627 0.467 0.832 ## Unhappy Lacks concentration Nervous Lies_cheats ## 0.634 0.492 0.781 0.800 ## Picked on Steals Prefers adults Fearful ## 0.741 0.939 0.880 0.759 ## ## Loadings: ## Factor1 Factor2 Factor3 ## Obedient 0.420 ## Friendly 0.207 0.211 ## Liked 0.338 0.273 -0.105 ## Thinker 0.477 -0.329 6 ## Good attention 0.571 -0.513 ## Restless 0.687 -0.402 0.265 ## Headaches 0.186 0.259 0.156 ## Tempers 0.473 0.135 ## Solitary 0.132 0.316 ## Worrier 0.369 0.414 0.255 ## Fidget 0.636 -0.291 0.208 ## Fights 0.396 ## Unhappy 0.372 0.422 0.222 ## Lacks concentration 0.655 -0.279 ## Nervous 0.356 0.296 ## Lies_cheats 0.441 ## Picked on 0.388 0.327 ## Steals 0.239 ## Prefers adults 0.242 0.233 ## Fearful 0.319 0.315 0.199 ## ## Factor1 Factor2 Factor3 ## SS loadings 3.589 1.252 0.808 ## Proportion Var 0.179 0.063 0.040 ## Cumulative Var 0.179 0.242 0.282 ## ## Test of the hypothesis that 3 factors are sufficient. ## The chi square statistic is 1237.02 on 133 degrees of freedom. ## The p-value is 2.76e-178 7 0.2 0.3 0.4 0.5 0.6 0.7 −0.4 −0.2 0.0 0.2 0.4 Factor1 Factor2 Obedient Friendly Liked Thinker Good attention Restless Headaches Tempers Solitary Worrier Fidget Fights Unhappy Lacks concentration Nervous Lies_cheats Picked on Steals Prefers adults Fearful −1 0 1 2 3 −2 −1 0 1 2 3 Factor1 Factor2 0.2 0.3 0.4 0.5 0.6 0.7 −0.4 −0.2 0.0 0.2 Factor1 Factor3 Obedient Friendly Liked Thinker Good attention Restless Headaches Tempers Solitary Worrier Fidget Fights Unhappy Lacks concentration Nervous Lies_cheats Picked on Steals Prefers adults Fearful −1 0 1 2 3 −2 −1 0 1 2 3 Factor1 Factor3 −0.4 −0.2 0.0 0.2 0.4 −0.4 −0.2 0.0 0.2 Factor2 Factor3 Obedient Friendly Liked Thinker Good attention Restless Headaches Tempers Solitary Worrier Fidget Fights Unhappy Lacks concentration Nervous Lies_cheats Picked on Steals Prefers adults Fearful −2 −1 0 1 2 3 −2 −1 0 1 2 3 Factor2 Factor3 8 4 factor ## ## Call: ## factanal(x = subset(ranch_MH_for_FA, select = -studno), factors = 4, scores = "regression", rotation = "varimax") ## ## Uniquenesses: ## Obedient Friendly Liked Thinker ## 0.748 0.854 0.718 0.677 ## Good attention Restless Headaches Tempers ## 0.314 0.320 0.866 0.725 ## Solitary Worrier Fidget Fights ## 0.880 0.595 0.458 0.730 ## Unhappy Lacks concentration Nervous Lies_cheats ## 0.644 0.456 0.725 0.704 ## Picked on Steals Prefers adults Fearful ## 0.725 0.854 0.879 0.705 ## ## Loadings: ## Factor1 Factor2 Factor3 Factor4 ## Obedient 0.409 0.210 0.199 ## Friendly 0.142 0.343 ## Liked 0.214 0.462 0.147 ## Thinker 0.275 0.166 0.469 ## Good attention 0.108 0.197 0.794 ## Restless 0.155 0.784 0.185 ## Headaches 0.361 ## Tempers 0.234 0.342 0.314 ## Solitary 0.320 ## Worrier 0.618 0.101 0.111 ## Fidget 0.128 0.148 0.682 0.195 ## Fights 0.147 0.457 0.189 ## Unhappy 0.553 0.207 ## Lacks concentration 0.191 0.157 0.312 0.621 ## Nervous 0.481 0.184 ## Lies_cheats 0.465 0.213 0.179 ## Picked on 0.371 0.352 0.106 ## Steals 0.370 ## Prefers adults 0.282 0.196 ## Fearful 0.526 0.116 ## ## Factor1 Factor2 Factor3 Factor4 ## SS loadings 1.863 1.589 1.501 1.469 ## Proportion Var 0.093 0.079 0.075 0.073 ## Cumulative Var 0.093 0.173 0.248 0.321 ## ## Test of the hypothesis that 4 factors are sufficient. ## The chi square statistic is 663.69 on 116 degrees of freedom. ## The p-value is 1.12e-77 9 −4 −2 0 2 4 0.0 0.6 Item Response Category Characteristic Curves Item: Solitary Latent variable Probability Not true Partly true Certainly true −4 −2 0 2 4 0.0 0.6 Item Response Category Characteristic Curves Item: Worrier Latent variable Probability Not true Partly true Certainly true −4 −2 0 2 4 0.0 0.6 Item Response Category Characteristic Curves Item: Fidget Latent variable Probability Not true Partly true Certainly true −4 −2 0 2 4 0.0 0.6 Item Response Category Characteristic Curves Item: Fights Latent variable Probability Not true Partly true Certainly true 16 −4 −2 0 2 4 0.0 0.6 Item Response Category Characteristic Curves Item: Unhappy Latent variable Probability Not true Partly true Certainly true −4 −2 0 2 4 0.0 0.6 Item Response Category Characteristic Curves Item: Lacks concentration Latent variable Probability Not true Partly true Certainly true −4 −2 0 2 4 0.0 0.6 Item Response Category Characteristic Curves Item: Nervous Latent variable Probability Not true Partly true Certainly true −4 −2 0 2 4 0.0 0.6 Item Response Category Characteristic Curves Item: Lies_cheats Latent variable Probability Not true Partly true Certainly true 17 −4 −2 0 2 4 0.0 0.6 Item Response Category Characteristic Curves Item: Picked on Latent variable Probability Not true Partly true Certainly true −4 −2 0 2 4 0.0 0.6 Item Response Category Characteristic Curves Item: Steals Latent variable Probability Not true Partly true Certainly true −4 −2 0 2 4 0.0 0.6 Item Response Category Characteristic Curves Item: Prefers adults Latent variable Probability Not true Partly true Certainly true −4 −2 0 2 4 0.0 0.6 Item Response Category Characteristic Curves Item: Fearful Latent variable Probability Not true Partly true Certainly true In Item Response Theory (IRT) the concept of information is used to reflect how precisely an item can measure the underlying trait. Greater information is associated with greater measurement precision. The information is inversely related to the standard error of the estimate. Over the range of the underlying trait, an information function curve can be derived for each item to reveal how measurement precision can vary across different levels of the trait. 18 19 −4 −2 0 2 4 123456 Test Information Function Ability Information Total information in ( −4 , 4 ): 34.96 ( 85.34 %) Total information in ( 0 , 4 ): 34.96 ( 61.07 %) Based on the fitted model, we can compute factor scores for each undividual. Distribution latent variable Latent trait Frequency −2 −1 0 1 2 3 0 100 200 300 400 20 −2 −1 0 1 2 3 Boxplot latent variable 21