Exercise 1 Descriptive statistics INPUT Variable: Names are - - PDF document

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Exercise 1 Descriptive statistics

INPUT Variable: Names are mahcig01 MeHGsx1 DePicSTS ZDePicSAS DeNamVTS ZDeNamVAS mdhgsx1 sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros male pregsmok missingd; USEVAR are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros male pregsmok ZDePicSAS ZDeNamVAS ; CATEGORICAL are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros male pregsmok ; Missing are all (-999) ; Analysis: Type = basic ; Exercise 1: OUTPUT (abridged) SUMMARY OF ANALYSIS Number of groups 1 Number of observations 2624 Observed dependent variables Continuous ZDEPICSAS ZDENAMVAS Binary and ordered categorical (ordinal) SW4EMO SW4CON SW4HYP SW4PEER SW4PROS SW5EMO

2 SW5CON SW5HYP SW5PEER SW5PROS MALE PREGSMOK SUMMARY OF MISSING DATA PATTERNS MISSING DATA PATTERNS (x = not missing) 1 SW4EMO x SW4CON x SW4HYP x SW4PEER x SW4PROS x SW5EMO x SW5CON x SW5HYP x SW5PEER x SW5PROS x MALE x PREGSMOK x ZDEPICSA x ZDENAMVA x SUMMARY OF CATEGORICAL DATA PROPORTIONS SW4EMO Category 1 0.938 Category 2 0.039 Category 3 0.023 SW4CON Category 1 0.711 Category 2 0.162 Category 3 0.127 SW4HYP Category 1 0.832 Category 2 0.072 Category 3 0.095 SW4PEER Category 1 0.854 Category 2 0.082 Category 3 0.064 SW4PROS Category 1 0.903 Category 2 0.068 Category 3 0.029 SW5EMO

3 Category 1 0.929 Category 2 0.039 Category 3 0.032 SW5CON Category 1 0.767 Category 2 0.133 Category 3 0.101 SW5HYP Category 1 0.816 Category 2 0.075 Category 3 0.109 SW5PEER Category 1 0.874 Category 2 0.063 Category 3 0.063 SW5PROS Category 1 0.929 Category 2 0.054 Category 3 0.016 MALE Category 1 0.500 Category 2 0.500 PREGSMOK Category 1 0.792 Category 2 0.208 MEANS/INTERCEPTS/THRESHOLDS MALE$1 PREGSMOK ZDEPICSA ZDENAMVA ________ ________ ________ ________ 1 0.000 0.814 0.024 0.022

• 1. How many observations are in the dataset?

2624

• 2. What is the proportion of males and females? 50% / 50%
• 3. What is the proportion of children whose mother reported smoking during pregnancy? 21%
• 4. What are the means of the Picture Similarity and the Naming vocabulary z scores? 0.024 ;

0.022

• 5. Are there missing data in any of these variables ? NO

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EXERCISE 2

INPUT 2-class model (abridged) Variable: Names are mahcig01 MeHGsx1 DePicSTS ZDePicSAS DeNamVTS ZDeNamVAS mdhgsx1 sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros male pregsmok missingd; USEVAR are sw4emo sw4con sw4hyp sw4peer sw4pros; CATEGORICAL are sw4emo sw4con sw4hyp sw4peer sw4pros; Missing are all (-999) ; classes are x(2); !estimates a latent categorical variable with two classes, !the indicators are sw4emo-sw4pros, specified in categorical option of variable Analysis: Type = MIXTURE ; ! invokes a mixture model algorithm OUTPUT: TECH1 TECH10 TECH11 TECH14; PLOT:Type is PLOT3; series is sw4emo (1) sw4con (2) sw4hyp (3) sw4peer (4) sw4pros (5); DEFINE: CUT sw4emo-sw4pros (0); !creates binary variables using 0 as cut-point OUTPUT (abridged) RANDOM STARTS RESULTS RANKED FROM THE BEST TO THE WORST LOGLIKELIHOOD VALUES Final stage loglikelihood values at local maxima, seeds, and initial stage start numbers:

• 5066.709 195873 6
• 5066.709 253358 2

5 The best loglikelihood is replicated and no error messages or warnings. Appears to be a trustworthy-enough solution.

• FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES

BASED ON THE ESTIMATED MODEL Latent Classes 1 566.72122 0.21598 2 2057.27878 0.78402

• 22% of individuals are in latent class 1 and 78% are in latent class 2, based on estimated model

RESULTS IN PROBABILITY SCALE Latent Class 1 SW4EMO Category 1 0.812 0.023 35.084 0.000 Category 2 0.188 0.023 8.123 0.000 SW4CON Category 1 0.190 0.060 3.136 0.002 Category 2 0.810 0.060 13.400 0.000 SW4HYP Category 1 0.489 0.047 10.367 0.000 Category 2 0.511 0.047 10.843 0.000 SW4PEER Category 1 0.660 0.031 21.150 0.000 Category 2 0.340 0.031 10.914 0.000 SW4PROS Category 1 0.758 0.026 29.003 0.000 Category 2 0.242 0.026 9.249 0.000 Latent Class 2 SW4EMO

6 Category 1 0.973 0.006 159.352 0.000 Category 2 0.027 0.006 4.497 0.000 SW4CON Category 1 0.855 0.017 49.465 0.000 Category 2 0.145 0.017 8.403 0.000 SW4HYP Category 1 0.927 0.011 88.091 0.000 Category 2 0.073 0.011 6.941 0.000 SW4PEER Category 1 0.908 0.010 90.681 0.000 Category 2 0.092 0.010 9.232 0.000 SW4PROS Category 1 0.943 0.007 130.536 0.000 Category 2 0.057 0.007 7.941 0.000 Individuals in class 1 have a higher probability of borderline /abnormal scores (category 2) in the SDQ scales compared to individual in latent class 2. The conditional probability of abnormal scores is particularly high for the conduct problems scale for individuals in class 1 (0.81). Individuals in class 2 have, in general, low probabilities of reporting abnormal scores in the SDQ scales. Class 1 can be identified as the “problem” class and class 2 as the “difficulty” class.

• PARAMETRIC BOOTSTRAPPED LIKELIHOOD RATIO TEST FOR 1 (H0) VERSUS 2 CLASSES

H0 Loglikelihood Value -5301.974 2 Times the Loglikelihood Difference 470.531 Difference in the Number of Parameters 6 Approximate P-Value 0.0000 Successful Bootstrap Draws 5 WARNING: THE BEST LOGLIKELIHOOD VALUE WAS NOT REPLICATED IN 5 OUT OF 5 BOOTSTRAP DRAWS. THE P-VALUE MAY NOT BE TRUSTWORTHY DUE TO LOCAL

• MAXIMA. INCREASE THE NUMBER OF RANDOM STARTS USING THE LRTSTARTS OPTION.
• The low p value indicates that the model with one less class (1-class model) is rejected in favour of

the estimated 2-class model. However, the warning message suggests increasing the random starts of the bootstrapped test. The default for LRTSTARTS = 0 0 20 5; More thorough is

LRTSTARTS = 2 1 50 15; This would be added in the ANALYSIS: command.

INPUT 3-class model (abridged) Variable: Names are mahcig01 MeHGsx1 DePicSTS ZDePicSAS DeNamVTS ZDeNamVAS mdhgsx1 sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros male pregsmok missingd; USEVAR are sw4emo sw4con sw4hyp sw4peer sw4pros;

7 CATEGORICAL are sw4emo sw4con sw4hyp sw4peer sw4pros; Missing are all (-999) ; classes are x(3); !estimates a latent categorical variable with three classes, !the indicators are sw4emo-sw4pros, specified in categorical option of variable Analysis: Type = MIXTURE ; ! invokes a mixture model algorithm OUTPUT: TECH1 TECH10 TECH11 TECH14; PLOT:Type is PLOT3; series is sw4emo (1) sw4con (2) sw4hyp (3) sw4peer (4) sw4pros (5); DEFINE: CUT sw4emo-sw4pros (0); !creates binary variables using 0 as cut-point OUTPUT (abridged)

RANDOM STARTS RESULTS RANKED FROM THE BEST TO THE WORST LOGLIKELIHOOD VALUES Final stage loglikelihood values at local maxima, seeds, and initial stage start numbers:

• 5049.029 608496 4
• 5049.946 253358 2

WARNING: WHEN ESTIMATING A MODEL WITH MORE THAN TWO CLASSES, IT MAY BE NECESSARY TO INCREASE THE NUMBER OF RANDOM STARTS USING THE STARTS OPTION TO AVOID LOCAL MAXIMA. WARNING: THE BEST LOGLIKELIHOOD VALUE WAS NOT REPLICATED. THE SOLUTION MAY NOT BE TRUSTWORTHY DUE TO LOCAL MAXIMA. INCREASE THE NUMBER OF RANDOM STARTS.

• The log-likelihood is not replicated. Furthermore, the software provides warnings.

Increase the start values and the number of iterations in the initial stage:

• Type = MIXTURE ;

STARTS = 300 10; STITERATIONS = 20;

• Which provides this solution:
• RANDOM STARTS RESULTS RANKED FROM THE BEST TO THE WORST LOGLIKELIHOOD

VALUES Final stage loglikelihood values at local maxima, seeds, and initial stage start numbers:

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• 5049.029 608496 4
• 5049.029 840078 203
• 5049.029 476498 179
• 5049.029 190339 102
• 5049.029 545140 278
• 5049.029 136842 58
• 5049.029 797594 234
• 5049.029 848890 95
• 5049.029 499150 216
• 5049.946 645664 39
• This solution appears more trustworthy (the best loglikelihood value has been replicated

in most of the final stage optimisations), however, it still warrants further checks (e.g. use the OPTSEED option in ANALYSIS command, using the seeds of the loglikelihood values above).

2-class solution 3-class solution TESTS OF MODEL FIT TESTS OF MODEL FIT Loglikelihood Loglikelihood H0 Value -5066.709 H0 Value -5049.029 H0 Scaling Correction Factor 1.047 H0 Scaling Correction Factor 1.039 for MLR for MLR Information Criteria Information Criteria Number of Free Parameters 11 Number of Free Parameters 17 Akaike (AIC) 10155.417 Akaike (AIC) 10132.059 Bayesian (BIC) 10220.014 Bayesian (BIC) 10231.891 Sample-Size Adjusted BIC 10185.064 Sample-Size Adjusted BIC 10177.877 (n* = (n + 2) / 24) (n* = (n + 2) / 24) Chi-Square Test of Model Fit for the Binary and Ordered Categorical Chi-Square Test of Model Fit for the Binary and Ordered Categorical (Ordinal) Outcomes (Ordinal) Outcomes Pearson Chi-Square Pearson Chi-Square Value 81.543 Value 37.836 Degrees of Freedom 20 Degrees of Freedom 14 P-Value 0.0000 P-Value 0.0006 Likelihood Ratio Chi-Square Likelihood Ratio Chi-Square Value 72.475 Value 37.116 Degrees of Freedom 20 Degrees of Freedom 14

9 P-Value 0.0000 P-Value 0.0007 CLASSIFICATION QUALITY CLASSIFICATION QUALITY Entropy 0.792 Entropy 0.640 The 3-class model has higher (=better fit) log-likelihood, lower (=better fit) according to AIC and SSA BIC , but 2-class has a lower (=better fit) BIC. The 3-class solution provides better classification: clearer class separation. It is important to also consider the number of significant bivariate residuals of the two models (see TECH10 output). Consider the results of Bootstrap Likelihood Ratio Test (BLRT) too. To inspect the graph produced by the output of the 3-class solution, open from the FILE option in the main menu the Mplus graph file (*.gph) : it has the same name as the input and output (only the extension changes). You should then see a dialog window with some options (e.g. Histograms). Select “Estimated probabilities”. You shall see another dialog window, which asks you to specify probabilities plot for one category. Select category 2 (in the example, is the “abnormal score” category for each scale). You should see a graph like this: One class (the largest one) includes individuals with low probabilities of being in the abnormal range of the scales. Another class includes individuals with high probability of being in the abnormal range of the Conduct Problems scale (indicator 2) and with higher probabilities to be in

10 the abnormal range of the Hyperactivity scale (indicator 3). Another class includes individuals with high probability (actually 100% probability) of being in the abnormal range of the Emotional Symptoms scale (Indicator 1) and with high probability of being in the abnormal range of the Peer Problems scale (Indicator 4)

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Exercise 3

INPUT: FULL MEASUREMENT INVARIANCE Variable: [..] usevar are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros ; categorical are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros; Missing are all (-999) ; classes are x(2) y(2); Analysis: Type = mixture ; starts = 200 20; stiterations = 20; MODEL: %overall% MODEL x: %x#1% [sw4emo$1-sw4pros$1] (1-5); !this is assigning numbers 1 to 5 to thresholds of indicators !sw4emo to sw4pros in class 1 of x %x#2% [sw4emo$1-sw4pros$1] (6-10); !this assigns numbers 6 to 10 to thresholds in class 2 of x MODEL y: %y#1% [sw5emo$1-sw5pros$1] (1-5); !wave 5 indicators in class 1 of y have the same thresholds ! as wave 4 indicators have in class 1 of x (1 to 5 ) %y#2% [sw5emo$1-sw5pros$1] (6-10); !wave 5 indicators in class 2 of y have the same thresholds !as wave 5 indicators in class 2 of x (6-10 respectively)

12 OUPUT – FULL MEASUREMENT INVARIANCE TESTS OF MODEL FIT Loglikelihood H0 Value -9817.307 H0 Scaling Correction Factor 1.223 for MLR Information Criteria Number of Free Parameters 12 Akaike (AIC) 19658.614 Bayesian (BIC) 19729.084 Sample-Size Adjusted BIC 19690.956 (n* = (n + 2) / 24)

• The output below shows that the same indicators in class 1 of latent variable x and in class 1 of

latent variable y have the identical thresholds. Same indicators in class 2 of x and y have identical thresholds, which are different from those of class 1 . MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value Latent Class Pattern 1 1 Thresholds SW4EMO$1 1.520 0.113 13.395 0.000 SW4CON$1 -1.169 0.247 -4.735 0.000 SW4HYP$1 -0.083 0.148 -0.559 0.576 SW4PEER$1 0.686 0.107 6.427 0.000 SW4PROS$1 1.238 0.112 11.021 0.000 SW5EMO$1 1.520 0.113 13.395 0.000 SW5CON$1 -1.169 0.247 -4.735 0.000 SW5HYP$1 -0.083 0.148 -0.559 0.576 SW5PEER$1 0.686 0.107 6.427 0.000 SW5PROS$1 1.238 0.112 11.021 0.000 Latent Class Pattern 2 2 Thresholds SW4EMO$1 3.374 0.154 21.844 0.000 SW4CON$1 2.074 0.130 15.907 0.000 SW4HYP$1 2.538 0.120 21.090 0.000

13 SW4PEER$1 2.486 0.110 22.516 0.000 SW4PROS$1 3.130 0.119 26.398 0.000 SW5EMO$1 3.374 0.154 21.844 0.000 SW5CON$1 2.074 0.130 15.907 0.000 SW5HYP$1 2.538 0.120 21.090 0.000 SW5PEER$1 2.486 0.110 22.516 0.000 SW5PROS$1 3.130 0.119 26.398 0.000

• Equality of thresholds determine equality of conditional item probabilities for same indicators.

Individuals in class 1 of x have the same probability of being in category 2 (abnormal score) of indicator emotional symptoms (for example) as individuals in class 1 of y, and so on for the other indicators. Latent Class Pattern 1 1 SW4EMO Category 1 0.820 0.017 49.104 0.000 Category 2 0.180 0.017 10.743 0.000 SW4CON Category 1 0.237 0.045 5.308 0.000 Category 2 0.763 0.045 17.088 0.000 SW4HYP Category 1 0.479 0.037 12.977 0.000 Category 2 0.521 0.037 14.096 0.000 SW4PEER Category 1 0.665 0.024 27.971 0.000 Category 2 0.335 0.024 14.086 0.000 SW4PROS Category 1 0.775 0.020 39.603 0.000 Category 2 0.225 0.020 11.486 0.000 SW5EMO Category 1 0.820 0.017 49.104 0.000 Category 2 0.180 0.017 10.743 0.000 SW5CON Category 1 0.237 0.045 5.308 0.000 Category 2 0.763 0.045 17.088 0.000 SW5HYP Category 1 0.479 0.037 12.977 0.000 Category 2 0.521 0.037 14.096 0.000 SW5PEER Category 1 0.665 0.024 27.971 0.000 Category 2 0.335 0.024 14.086 0.000 SW5PROS Category 1 0.775 0.020 39.603 0.000 Category 2 0.225 0.020 11.486 0.000

14 INPUT: FULL MEASUREMENT NON- INVARIANCE Variable: [..] usevar are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros ; categorical are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros; Missing are all (-999) ; classes are x(2) y(2); Analysis: Type = mixture ; starts = 200 20; stiterations = 20; MODEL: %overall% MODEL x: %x#1% [sw4emo$1-sw4pros$1] ; !no constraints are imposed on these and the other parameters %x#2% [sw4emo$1-sw4pros$1] ; MODEL y: %y#1% [sw5emo$1-sw5pros$1] ; %y#2% [sw5emo$1-sw5pros$1] ; OUTPUT - FULL MEASUREMENT NON-INVARIANCE TESTS OF MODEL FIT Loglikelihood

15 H0 Value -9791.638 H0 Scaling Correction Factor 1.055 for MLR Information Criteria Number of Free Parameters 22 Akaike (AIC) 19627.276 Bayesian (BIC) 19756.470 Sample-Size Adjusted BIC 19686.570

• Inspection of thresholds of indicators in class 1 of x and in class 1 of y show differences in

thresholds for indicators of these classes. While some thresholds seem remarkably similar (e.g. 1.463 for emotional symptoms in class 1 of x, 1.499 for emotional symptoms in class 1 of y), others differ to some degree (e.g. conduct problems). MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value Latent Class Pattern 1 1 Thresholds SW4EMO$1 1.463 0.152 9.650 0.000 SW4CON$1 -1.452 0.394 -3.691 0.000 SW4HYP$1 -0.045 0.189 -0.238 0.812 SW4PEER$1 0.662 0.139 4.763 0.000 SW4PROS$1 1.143 0.143 8.015 0.000 SW5EMO$1 1.499 0.138 10.891 0.000 SW5CON$1 -0.938 0.289 -3.244 0.001 SW5HYP$1 -0.260 0.200 -1.304 0.192 SW5PEER$1 0.656 0.128 5.108 0.000 SW5PROS$1 1.329 0.153 8.713 0.000 Below we can see the thresholds of indicators in class 2 of x and in class2 di1 of y. Latent Class Pattern 2 2 Thresholds SW4EMO$1 3.568 0.229 15.603 0.000 SW4CON$1 1.773 0.139 12.733 0.000 SW4HYP$1 2.541 0.155 16.349 0.000 SW4PEER$1 2.285 0.119 19.143 0.000 SW4PROS$1 2.800 0.134 20.956 0.000 SW5EMO$1 3.214 0.192 16.744 0.000 SW5CON$1 2.302 0.171 13.490 0.000

16 SW5HYP$1 2.549 0.162 15.702 0.000 SW5PEER$1 2.700 0.178 15.183 0.000 SW5PROS$1 3.482 0.187 18.610 0.000

• Below we can see the conditional item probabilities for indicators of class 1 of latent variable x

and for the indicators of class 1 of latent variable y. Class 1 is the “difficulty” class (=higher probabilities of being in the abnormal range score for SDQ scales). RESULTS IN PROBABILITY SCALE Latent Class Pattern 1 1 SW4EMO Category 1 0.812 0.023 35.084 0.000 Category 2 0.188 0.023 8.123 0.000 SW4CON Category 1 0.190 0.060 3.136 0.002 Category 2 0.810 0.060 13.400 0.000 SW4HYP Category 1 0.489 0.047 10.367 0.000 Category 2 0.511 0.047 10.843 0.000 SW4PEER Category 1 0.660 0.031 21.150 0.000 Category 2 0.340 0.031 10.914 0.000 SW4PROS Category 1 0.758 0.026 29.003 0.000 Category 2 0.242 0.026 9.249 0.000 SW5EMO Category 1 0.817 0.021 39.792 0.000 Category 2 0.183 0.021 8.891 0.000 SW5CON Category 1 0.281 0.058 4.810 0.000 Category 2 0.719 0.058 12.293 0.000 SW5HYP Category 1 0.435 0.049 8.870 0.000 Category 2 0.565 0.049 11.508 0.000 SW5PEER Category 1 0.658 0.029 22.790 0.000 Category 2 0.342 0.029 11.824 0.000 SW5PROS Category 1 0.791 0.025 31.321 0.000 Category 2 0.209 0.025 8.288 0.000

• Below we can see the conditional item probabilities for indicators of class 2 of latent variable x

and for the indicators of class 1 of latent variable y. Class 2 is the “normative” class (=higher probabilities of being in the normal range score for SDQ scales).

17 Latent Class Pattern 2 2 SW4EMO Category 1 0.973 0.006 159.352 0.000 Category 2 0.027 0.006 4.497 0.000 SW4CON Category 1 0.855 0.017 49.465 0.000 Category 2 0.145 0.017 8.403 0.000 SW4HYP Category 1 0.927 0.011 88.091 0.000 Category 2 0.073 0.011 6.941 0.000 SW4PEER Category 1 0.908 0.010 90.681 0.000 Category 2 0.092 0.010 9.232 0.000 SW4PROS Category 1 0.943 0.007 130.536 0.000 Category 2 0.057 0.007 7.941 0.000 SW5EMO Category 1 0.961 0.007 134.859 0.000 Category 2 0.039 0.007 5.418 0.000 SW5CON Category 1 0.909 0.014 64.436 0.000 Category 2 0.091 0.014 6.445 0.000 SW5HYP Category 1 0.928 0.011 84.970 0.000 Category 2 0.072 0.011 6.642 0.000 SW5PEER Category 1 0.937 0.010 89.278 0.000 Category 2 0.063 0.010 6.002 0.000 SW5PROS Category 1 0.970 0.005 179.138 0.000 Category 2 0.030 0.005 5.509 0.000

• To run a Likelihood Ratio Test (LRT) we need to retrieve the LogLikelihood values of the two models

above (full measurement invariance vs. full measurement non-invariance). We also need to consider their respective scaling correction factor and the number of parameters in the models to calculate the degrees of freedom of the test. The formula for the test is: LR = -2 *( L0-L1 / cd) cd = [(p0*c0)-(p1*c1)]/(p0-p1) L0 =log-likelihood of the null model (the model with equality constraints: full measurement invariance) L1 = log-likelihood of alternative model (full non-measurement invariance) c0 = scaling factor null model c1 = scaling factor alternative model

18 p0 = parameters in null model p1 = parameters in alternative model Considering the values of the two models (see above: TESTS OF MODEL FIT) we can calculate: cd = [ ( 12*1.223) – (22*1.055) ] / (12-22) = 0.8534 LR = -2 * [-9817.31 – (-9791.64)] / 0.8534 = 60.15702 df = p1 – p0 = (22-12) = 10 The Chi Square p value for (60.15, 10) is < .001. The null model (full measurement invariance) is rejected since it results in a significant worsening of model fit.

• INPUT – PARTIAL MEASUREMENT INVARIANCE

Variable: [..] usevar are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros ; categorical are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros; Missing are all (-999) ; classes are x(2) y(2); Analysis: Type = mixture ; starts = 200 20; stiterations = 20; MODEL: %overall% MODEL x: %x#1% [sw4emo$1-sw4pros$1*3] ; !item condit prob freely estimated in class 1 (normative class) %x#2% [sw4emo$1-sw4pros$1*-1] (6-10); !item condit prob is constrained equal for same indicators in !class 2 (difficulty class) MODEL y: %y#1% [sw5emo$1-sw5pros$1*3] ; %y#2% [sw5emo$1-sw5pros$1*-1] (6-10);

19 OUTPUT – PARTIAL MEASUREMENT INVARIANCE TESTS OF MODEL FIT Loglikelihood H0 Value -9794.566 H0 Scaling Correction Factor 1.091 for MLR Information Criteria Number of Free Parameters 17 Akaike (AIC) 19623.133 Bayesian (BIC) 19722.965 Sample-Size Adjusted BIC 19668.951 (n* = (n + 2) / 24)

• Thresholds differ for indicators of class 1 (normative class) of latent variable x and latent variable y

MODEL RESULTS Two-Tailed Estimate S.E. Est./S.E. P-Value Latent Class Pattern 1 1 Thresholds SW4EMO$1 3.609 0.215 16.749 0.000 SW4CON$1 1.728 0.110 15.753 0.000 SW4HYP$1 2.652 0.152 17.439 0.000 SW4PEER$1 2.308 0.108 21.342 0.000 SW4PROS$1 2.793 0.120 23.225 0.000 SW5EMO$1 3.178 0.157 20.256 0.000 SW5CON$1 2.336 0.155 15.105 0.000 SW5HYP$1 2.442 0.125 19.546 0.000 SW5PEER$1 2.642 0.140 18.846 0.000 SW5PROS$1 3.467 0.170 20.384 0.000

• Thresholds are constrained equal for same indicators of class 2 (difficulty class) of latent variable x

and latent variable y Latent Class Pattern 2 2 Thresholds SW4EMO$1 1.477 0.111 13.352 0.000 SW4CON$1 -1.181 0.242 -4.880 0.000 SW4HYP$1 -0.170 0.147 -1.153 0.249

20 SW4PEER$1 0.658 0.104 6.346 0.000 SW4PROS$1 1.232 0.112 10.979 0.000 SW5EMO$1 1.477 0.111 13.352 0.000 SW5CON$1 -1.181 0.242 -4.880 0.000 SW5HYP$1 -0.170 0.147 -1.153 0.249 SW5PEER$1 0.658 0.104 6.346 0.000 SW5PROS$1 1.232 0.112 10.979 0.000

• Item response probability conditional on being in class 1 (normative class) of latent variable x and
• f latent variable y

RESULTS IN PROBABILITY SCALE Latent Class Pattern 1 1 SW4EMO Category 1 0.974 0.006 176.051 0.000 Category 2 0.026 0.006 4.766 0.000 SW4CON Category 1 0.849 0.014 60.437 0.000 Category 2 0.151 0.014 10.736 0.000 SW4HYP Category 1 0.934 0.009 99.858 0.000 Category 2 0.066 0.009 7.038 0.000 SW4PEER Category 1 0.910 0.009 102.240 0.000 Category 2 0.090 0.009 10.164 0.000 SW4PROS Category 1 0.942 0.007 144.091 0.000 Category 2 0.058 0.007 8.825 0.000 SW5EMO Category 1 0.960 0.006 159.333 0.000 Category 2 0.040 0.006 6.640 0.000 SW5CON Category 1 0.912 0.012 73.318 0.000 Category 2 0.088 0.012 7.093 0.000 SW5HYP Category 1 0.920 0.009 100.031 0.000 Category 2 0.080 0.009 8.700 0.000 SW5PEER Category 1 0.934 0.009 107.282 0.000 Category 2 0.066 0.009 7.642 0.000 SW5PROS Category 1 0.970 0.005 194.257 0.000 Category 2 0.030 0.005 6.063 0.000 Item response probability conditional on being in class 2 (difficulty class) of latent variable x and

• f latent variable y

21 Latent Class Pattern 2 2 SW4EMO Category 1 0.814 0.017 48.632 0.000 Category 2 0.186 0.017 11.107 0.000 SW4CON Category 1 0.235 0.043 5.398 0.000 Category 2 0.765 0.043 17.592 0.000 SW4HYP Category 1 0.458 0.037 12.531 0.000 Category 2 0.542 0.037 14.847 0.000 SW4PEER Category 1 0.659 0.023 28.269 0.000 Category 2 0.341 0.023 14.640 0.000 SW4PROS Category 1 0.774 0.020 39.462 0.000 Category 2 0.226 0.020 11.512 0.000 SW5EMO Category 1 0.814 0.017 48.632 0.000 Category 2 0.186 0.017 11.107 0.000 SW5CON Category 1 0.235 0.043 5.398 0.000 Category 2 0.765 0.043 17.592 0.000 SW5HYP Category 1 0.458 0.037 12.531 0.000 Category 2 0.542 0.037 14.847 0.000 SW5PEER Category 1 0.659 0.023 28.269 0.000 Category 2 0.341 0.023 14.640 0.000 SW5PROS Category 1 0.774 0.020 39.462 0.000 Category 2 0.226 0.020 11.512 0.000 Considering parameters of the partial measurement invariance model, we can run a LRT in the same way as above comparing the null model (partial measurement invariance) to the alternative model (full non-invariance). This results in: cd = [ ( 17*1.091) – (22 * 1.055) ] / (17 – 22) = 0.9326 LR = -2 * [-9794.57 – ( - 9791.64)] / 0.9326 = 6.279 df = p1 – p0 = 22- 17 = 5 The p value for Chi Square (6.279 , 5) is approximately 0.28. The partial measurement invariance model does not provide a significant worsening of model fit, therefore we can accept this model.

22

EXERCISE 4

INPUT – LATENT TRANSITION MODEL (2 classes x – 2 classes y – partial meas. inv.) usevar are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros ; categorical are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros ; Missing are all (-999) ; classes are x(2) y(2); Analysis: Type = mixture ; starts = 200 20; stiterations = 20; MODEL: %overall% y ON x; MODEL x: %x#1% [sw4emo$1-sw4pros$1*3] ; !item condit prob freely estimated in class 1 (normative cl %x#2% [sw4emo$1-sw4pros$1*-1] (6-10); !item condit prob is constrained equal for same indi MODEL y: %y#1% [sw5emo$1-sw5pros$1*3] ; %y#2% [sw5emo$1-sw5pros$1*-1] (6-10);

• utput:

tech1 tech2 tech10 tech11 tech12 tech14 ;

23 Plot: type is plot1 plot2 plot3; series is sw4emo(1) sw4con(2) sw4hyp(3) sw4peer(4) sw4pros(5) sw5emo (6) sw5con (7) sw5hyp (8) sw5peer (9) sw5pros (10); DEFINE: CUT sw4emo-sw4pros (0); CUT sw5emo-sw5pros (0); OUTPUT – LATENT TRANSITION MODEL (2 classes x – 2 classes y – partial meas. inv.) TESTS OF MODEL FIT Loglikelihood H0 Value -9441.455 H0 Scaling Correction Factor 1.193 for MLR Information Criteria Number of Free Parameters 18 Akaike (AIC) 18918.909 Bayesian (BIC) 19024.613 Sample-Size Adjusted BIC 18967.422 (n* = (n + 2) / 24)

• MODEL RESULTS USE THE LATENT CLASS VARIABLE ORDER

X Y Latent Class Variable Patterns X Y Class Class 1 1 1 2 2 1 2 2

24 FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASS PATTERNS BASED ON THE ESTIMATED MODEL Latent Class Pattern 1 1 1845.54834 0.70333 1 2 5.19019 0.00198 2 1 44.64142 0.01701 2 2 728.62006 0.27768 LATENT TRANSITION PROBABILITIES BASED ON THE ESTIMATED MODEL X Classes (Rows) by Y Classes (Columns) 1 2 1 0.997 0.003 2 0.058 0.942

• The output above shows that individuals in class 1 of x (normative class) have a very low

probability of transitioning into class 2 of y (difficulty class). There is a substantial degree of stability also for class 2 (0.94 probability of remaining in the difficulty class across waves). Transition probabilities can be calculated using the parameters of the logistic regression coefficients: y = 1 y = 2 x = 1 a1+b11 x = 2 a1 a1 = [y#1] b11 = y#1 ON x#1 P (y1 | x=1) = exp(a1+b11) / [exp(a1+b11)+1] P (y1 | x=2) = exp(a1) / [exp(a1)+1 P (y2 | x=1) = exp(0) / [exp(a1+b11)+1] P (y2 | x=2) = exp(0) / [exp(a1)+1 In the output:

25

• Categorical Latent Variables

Y#1 ON X#1 8.666 4.238 2.045 0.041 Means X#1 0.873 0.104 8.426 0.000 Y#1 -2.792 0.555 -5.027 0.000

• Using the estimated parameters we can calculate:

P (y1 | x=1) = exp(-2.792+8.666) / [exp(-2.792+8.666) +1 ] = 0.997 P (y1 | x=2) = exp(-2.792) / [exp(-2.792)+1] = 0.058 P (y2 | x=1) = 1 / [exp(-2.792+8.666) +1 ] = 0.003 P (y2 | x=2) = 1 / [exp(-2.792)+1] = 0.942 Note that the MODEL statement in the input file was more simply “y ON x”, which implied estimate of the 3 parameters above : [x#1]; [y#1]; y#1 ON x#1; As said above, we could have written the statement “y ON x” in its full form (specifying the 3 parameters involved in estimating the regression of dependent y on x). The utility of writing the model in its full form is apparent when one needs to constrain parameters. INPUT – Latent Transition model with “normative” class the absorbing class MODEL: %overall% y#1 ON x#1 ; [y#1@-15]; [x#1] ; MODEL x: %x#1% [sw4emo$1-sw4pros$1*-1] (6-10); ! note that in this case we have made class 1 the difficulty class %x#2% [sw4emo$1-sw4pros$1*3] ; MODEL y: %y#1% [sw5emo$1-sw5pros$1*-1] (6-10);

26 %y#2% [sw5emo$1-sw5pros$1*3] ; OUTPUT – Latent Transition model with “normative” class the absorbing class Note that in this example we have made latent class 1 the “difficulty” class. This is for simplicity. In the specification we used before, where class 1 was the normative class, in order to make P (y1 | x=1) = exp(a1+b11) / [exp(a1+b11)+1] = 1 we would have had to make sure that the numerator (involving a1 and b11) of the expression was bigger than the denominator. If we reverse the order

• f the classes making class 1 the difficulty class and class 2 the normative class, we just need to

make sure that P (y1 | x=2) = exp(a1) / [exp(a1)+1 = 0, which just requires fixing a1 [y#1] to a value small enough for this purpose (e.g. -15).

• TESTS OF MODEL FIT

Loglikelihood H0 Value -9441.486 H0 Scaling Correction Factor 1.201 for MLR Information Criteria Number of Free Parameters 17 Akaike (AIC) 18916.971 Bayesian (BIC) 19016.803 Sample-Size Adjusted BIC 18962.789 MODEL RESULTS USE THE LATENT CLASS VARIABLE ORDER X Y Latent Class Variable Patterns X Y Class Class 1 1 1 2 2 1 2 2

27 FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASS PATTERNS BASED ON THE ESTIMATED MODEL Latent Class Pattern 1 1 732.65916 0.27921 1 2 44.21301 0.01685 2 1 0.00057 0.00000 2 2 1847.12727 0.70394 LATENT TRANSITION PROBABILITIES BASED ON THE ESTIMATED MODEL X Classes (Rows) by Y Classes (Columns) 1 2 1 0.943 0.057 2 0.000 1.000

• The transition probabilities are thus:

Wave 5: Difficulty Wave 5: Normative Wave 4: Difficulty 0.943 0.057 Wave 4: Normative 1 We can calculate a LRT test using the parameters of the constrained(constrained transition probability model above (null model) and the non-constrained transition model (alternative model): This results in: cd = [ ( 17*1.012) – (18 * - 1.193) ] / (17 –18) = 4.27 LR = -2 * [-9441.49 – ( - 9441.46)] / 4.27 = 0.015 df = p1 – p0 = 18- 17 = 1 The Chi Square (0.015 , 1) has p = 0.90. We can accept the model with a constrained transition probability since it does not involve a worsening a model fit.

28

Exercise 5 – Include covariates

INPUT –Effects of gender on sweep 4 classification Variable: Names are mahcig01 MeHGsx1 DePicSTS ZDePicSAS DeNamVTS ZDeNamVAS mdhgsx1 sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros male pregsmok missingd; usevar are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros male; categorical are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros ; Missing are all (-999) ; classes are x(2) y(2); Analysis: Type = mixture ; starts = 200 20; stiterations = 20; MODEL: %overall% x ON male; y#1 ON x#1 ; [y#1@-15]; [x#1] ;

29 MODEL x: %x#1% [sw4emo$1-sw4pros$1*-1] (6-10); %x#2% [sw4emo$1-sw4pros$1*3] ; MODEL y: %y#1% [sw5emo$1-sw5pros$1*-1] (6-10); %y#2% [sw5emo$1-sw5pros$1*3] ;

• utput:

tech1 tech2 tech10 tech11 tech12 tech14 ; Plot: type is plot1 plot2 plot3; series is sw4emo(1) sw4con(2) sw4hyp(3) sw4peer(4) sw4pros(5) sw5emo (6) sw5con (7) sw5hyp (8) sw5peer (9) sw5pros (10); DEFINE: CUT sw4emo-sw4pros (0); CUT sw5emo-sw5pros (0); OUTPUT: effects of gender on wave 4 classification MODEL RESULTS USE THE LATENT CLASS VARIABLE ORDER X Y Latent Class Variable Patterns X Y Class Class 1 1 1 2 2 1 2 2

30 FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASS PATTERNS BASED ON THE ESTIMATED MODEL Latent Class Pattern 1 1 738.30408 0.28137 1 2 47.56015 0.01813 2 1 0.00056 0.00000 2 2 1838.13522 0.70051 LATENT TRANSITION PROBABILITIES BASED ON THE ESTIMATED MODEL LATENT TRANSITION PROBABILITIES BASED ON THE ESTIMATED MODEL X Classes (Rows) by Y Classes (Columns) 1 2 1 0.939 0.061 2 0.000 1.000 Categorical Latent Variables Y#1 ON X#1 17.742 0.515 34.432 0.000 X#1 ON MALE 0.619 0.105 5.887 0.000 Intercepts X#1 -1.178 0.118 -10.012 0.000 Y#1 -15.000 0.000 999.000 999.000

• The parameter “x#1 ON male” represent the change in the log-odds for male children compared to

females of being in class 1 compared to class 2 (the reference class). The odds ratio for being in class 1 of latent variable x vs. class 2 of x when comparing males to females would be the exponentiation of this parameter : exp(0.619) = 1.86 . Since class 1 is the “difficulty” class (see input), we can say that compared to girls, boys are 1.86 more likely to be in the difficulty class (class 1 of x) than they are to be in the “normative” class (class 2 of x) Mplus can calculate the confidence interval of the log-odds as well as the CI for the odds ratio. To get these, include the statement CINTERVAL in the OUTPUT: command.

31 INPUT: effects of gender on transitions from wave 4 to wave 5 Variable: Names are mahcig01 MeHGsx1 DePicSTS ZDePicSAS DeNamVTS ZDeNamVAS mdhgsx1 sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros male pregsmok missingd; usevar are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros male; categorical are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros ; Missing are all (-999) ; classes are x(2) y(2); Analysis: Type = mixture ; starts = 200 20; stiterations = 20; MODEL: %overall% x ON male; y#1 ON x#1 male; [y#1@-15]; [x#1] ; MODEL x: %x#1% [sw4emo$1-sw4pros$1*-1] (6-10); %x#2% [sw4emo$1-sw4pros$1*3] ; MODEL y: %y#1% [sw5emo$1-sw5pros$1*-1] (6-10); %y#2% [sw5emo$1-sw5pros$1*3] ;

32

• utput:

tech1 tech2 tech10 tech11 tech12 tech14 ; Plot: type is plot1 plot2 plot3; series is sw4emo(1) sw4con(2) sw4hyp(3) sw4peer(4) sw4pros(5) sw5emo (6) sw5con (7) sw5hyp (8) sw5peer (9) sw5pros (10); DEFINE: CUT sw4emo-sw4pros (0); CUT sw5emo-sw5pros (0); OUPUT – effects of gender on transitions from wave 4 to wave 5 MODEL RESULTS USE THE LATENT CLASS VARIABLE ORDER X Y Latent Class Variable Patterns X Y Class Class 1 1 1 2 2 1 2 2 FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASS PATTERNS BASED ON THE ESTIMATED MODEL Latent Class Pattern 1 1 738.63490 0.28149 1 2 45.75037 0.01744 2 1 0.00045 0.00000 2 2 1839.61428 0.70107 LATENT TRANSITION PROBABILITIES BASED ON THE ESTIMATED MODEL X Classes (Rows) by Y Classes (Columns) 1 2 1 0.942 0.058

33 2 0.000 1.000 Categorical Latent Variables Y#1 ON X#1 18.183 1.191 15.269 0.000 X#1 ON MALE 0.638 0.111 5.759 0.000 Y#1 ON MALE -0.598 1.211 -0.494 0.621 Intercepts X#1 -1.192 0.120 -9.911 0.000 Y#1 -15.000 0.000 999.000 999.000

• The results indicate a non significant effect of gender on the transitions across wave 4 and wave 5

(p = 0.62). We can calculate the transition matrices for males and females using the logistic regression coefficients obtained from the output. y = 1 y = 2 x = 1 a1 + b11 + g1 (male) x =2 a1+g1 (male) a1 [y#1] b11 y#1 ON x#1 g1 y#1 ON male Considering that male=1 in the male group, the parameters necessary for calculating transitions probabilities are: male = 1 y = 1 y = 2 x = 1

• 15 + 18.183 + (-0.598*1)

x =2

• 15 + (-0.598*1)

We can therefore calculate the following transition probabilities: P (y=1 | x=1 , male = 1) = exp(-15 + 18.183– 0.598) / [exp(-15 + 18.183 – 0.598)+1] = 0.930 P (y=1 | x=2 , male = 1) = exp(-15– 0.598) / [exp(-15– 0.598)+1] = 0 P (y=2 | x=1 , male = 1) = exp(0) / [exp(-15 + 18.183– 0.598)+1] = 0.07 P (y=1 | x=2 , male = 1) = exp(1) / [exp(-15– 0.598)+1] = 1 If we consider that for females male=0, the parameters are:

34 male = 0 y = 1 y = 2 x = 1

• 15 + 18.183 + (-0.598*0)

x =2

• 15 + (-0.598*0)

therefore: P (y=1 | x=1 , male = 0) = exp(-15 + 18.183) / [exp(-15 + 18.183)+1] = 0.960 P (y=1 | x=2 , male = 0) = exp(-15) / [exp(-15)+1] = 0 P (y=2 | x=1 , male = 0) = exp(0) / [exp(-15 + 18.183)+1] = 0.04 P (y=1 | x=2 , male = 0) = exp(1) / [exp(-15)+1] = 1 Transition matrices for males and female are: MALES y=1 (difficulties class) y=2 (normative class) x=1 (difficulties class) 0.93 0.07 x=2 (normative class) 1 FEMALES y=1 (difficulties class) y=2 (normative class) x=1 (difficulties class) 0.96 0.040 x=2 (normative class) 1 The transition matrices show that there are no conspicuous differences in the transition probabilities between males and females. The same results can be obtained using the option KNOWNCLASS in the VARIABLE: command. In this case, a new latent variable is created (e.g. cmale) which represents a class for which membership is known. The option KNOWNCLASS allows to define these classes. For example:] KNOWNCLASS = cmale ( male = 0 male = 1 ); will consider females (male = 0 ) as the first category of latent variable cmale, and males (male = 1) as the second category of latent variable cmale. If we wanted males to be the first category of the latent variable, we would have written: KNOWNCLASS = cmale ( male = 1 male =0 ); The INPUT and OUTPUT of the analyses using KNOWNCLASS are reported below INPUT - Effect of gender on transition across waves using KNOWNCLASS option usevar are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros ; categorical are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros ; Missing are all (-999) ;

35 classes are cmale(2) x(2) y(2); KNOWNCLASS = cmale (male = 0 male = 1); Analysis: Type = mixture ; starts = 200 20; stiterations = 20; MODEL: %overall% x ON cmale; y#1 ON x#1 cmale; [y#1@-15]; [x#1] ; MODEL x: %x#1% [sw4emo$1-sw4pros$1*-1] (6-10); %x#2% [sw4emo$1-sw4pros$1*3] ; MODEL y: %y#1% [sw5emo$1-sw5pros$1*-1] (6-10); %y#2% [sw5emo$1-sw5pros$1*3] ;

• utput:

tech1 tech2 tech10 tech11 tech12 tech14 ; cinterval; Plot: type is plot1 plot2 plot3; series is sw4emo(1) sw4con(2) sw4hyp(3) sw4peer(4) sw4pros(5) sw5emo (6) sw5con (7) sw5hyp (8) sw5peer (9) sw5pros (10); DEFINE: CUT sw4emo-sw4pros (0); CUT sw5emo-sw5pros (0); OUTPUT - Effect of gender on transition across waves using KNOWNCLASS option MODEL RESULTS USE THE LATENT CLASS VARIABLE ORDER CMALE X Y

36 Latent Class Variable Patterns CMALE X Y Class Class Class 1 1 1 1 1 2 1 2 1 1 2 2 2 1 1 2 1 2 2 2 1 2 2 2 FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASS PATTERNS BASED ON THE ESTIMATED MODEL Latent Class Pattern 1 1 1 293.39236 0.11181 1 1 2 12.16775 0.00464 1 2 1 0.00056 0.00000 1 2 2 1006.43933 0.38355 2 1 1 445.24191 0.16968 2 1 2 33.58273 0.01280 2 2 1 0.00025 0.00000 2 2 2 833.17510 0.31752

• The table above represent estimated numbers and proportions of individuals in class 1 of cmale

(=females) that are in class 1 of x and class 1 of y, numbers and proportions of individuals in class 1

• f cmale that are in class 1 of x and in class 2 of y, and so on...

If we use these numbers to create transition matrices for females and males: FEMALES (cmale=1) y=1 y=2 TOT x =1 293.3924 12.16775 305.5601 x=2 0.00056 1006.439 1006.44 MALES (cmale=2) y=1 y=2 TOT x=1 445.2419 33.58273 478.8246 x=2 0.00025 833.1751 833.1754 We can use to calculate the numbers above transition probabilities: FEMALES (cmale=1) y=1 y=2 TOT x =1 0.96 0.04 1 x=2 1 1 MALES (cmale=2) y=1 y=2 TOT x=1 0.93 0.07 1 x=2 1 1

37 These transition probabilities are the same we calculated in the previous example where we used the covariate “male”. INPUT- effects of gender and exposure to tobacco during pregnancy on transitions across waves Variable: Names are mahcig01 MeHGsx1 DePicSTS ZDePicSAS DeNamVTS ZDeNamVAS mdhgsx1 sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros male pregsmok missingd; usevar are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros male pregsmok; categorical are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros ; Missing are all (-999) ; classes are x(2) y(2); Analysis: Type = mixture ; starts = 200 20; stiterations = 20; MODEL: %overall% x ON male pregsmok; y#1 ON x#1 male pregsmok; [y#1@-15]; [x#1] ; MODEL x: %x#1% [sw4emo$1-sw4pros$1*-1] (6-10); %x#2% [sw4emo$1-sw4pros$1*3] ; MODEL y: %y#1% [sw5emo$1-sw5pros$1*-1] (6-10);

38 %y#2% [sw5emo$1-sw5pros$1*3] ;

• utput:

tech1 tech2 tech10 tech11 tech12 tech14 ; cinterval; Plot: type is plot1 plot2 plot3; series is sw4emo(1) sw4con(2) sw4hyp(3) sw4peer(4) sw4pros(5) sw5emo (6) sw5con (7) sw5hyp (8) sw5peer (9) sw5pros (10); DEFINE: CUT sw4emo-sw4pros (0); CUT sw5emo-sw5pros (0); MODEL RESULTS USE THE LATENT CLASS VARIABLE ORDER X Y Latent Class Variable Patterns X Y Class Class 1 1 1 2 2 1 2 2 FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASS PATTERNS BASED ON THE ESTIMATED MODEL Latent Class Pattern 1 1 721.73612 0.27505 1 2 46.92272 0.01788 2 1 0.00129 0.00000 2 2 1855.33987 0.70707 LATENT TRANSITION PROBABILITIES BASED ON THE ESTIMATED MODEL

39 X Classes (Rows) by Y Classes (Columns) 1 2 1 0.939 0.061 2 0.000 1.000 Categorical Latent Variables Y#1 ON X#1 17.949 2.016 8.901 0.000 X#1 ON MALE 0.688 0.116 5.913 0.000 PREGSMOK 1.271 0.129 9.820 0.000 Y#1 ON MALE -0.930 2.068 -0.450 0.653 PREGSMOK 2.727 8.992 0.303 0.762 Intercepts X#1 -1.556 0.140 -11.119 0.000 Y#1 -15.000 0.000 999.000 999.000 The results indicate a significant effect of exposure to tobacco during gestation on latent status in wave 4 after controlling for gender differences, but there was no effect of exposure to tobacco on transitions across waves after controlling for gender and latent status at the previous wave. The parameter x#1 ON pregsmok represents the change in the log odds: in wave 4 and after controlling for gender effects, compared to children that had not been exposed to tobacco, children that had been exposed to tobacco during gestation were 3.56 times more likely [exp(1.271)] to be in the difficulty class (x=1 at wave 4) than they were to be in the normative class. Transition probability matrices can be calculated for males exposed to tobacco (male = 1 , pregsmoke = 1), females not exposed to tobacco (male = 0 , pregsmoke = 0) , and so on, using the estimated parameters of the model in the same way illustrated above for gender.

40

Exercise 6 Test differences in distal outcomes across latent statuses

INPUT – ESTIMATED ZDeNamVAS MEANS (and significance test of differences) across classes of y Variable: Names are mahcig01 MeHGsx1 DePicSTS ZDePicSAS DeNamVTS ZDeNamVAS mdhgsx1 sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros male pregsmok missingd; usevar are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros ZDeNamVAS; categorical are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros; Missing are all (-999) ; classes are x(2) y(2); Analysis: Type = mixture ; starts = 200 20; stiterations = 20; MODEL: %overall% y#1 ON x#1 ; [y#1@-15]; [x#1] ; MODEL x: %x#1% [sw4emo$1-sw4pros$1*-1] (6-10);

41 %x#2% [sw4emo$1-sw4pros$1*3] ; MODEL y: %y#1% [sw5emo$1-sw5pros$1*-1] (6-10); [ZDeNamVAS] (p1); %y#2% [sw5emo$1-sw5pros$1*3] ; [ZDeNamVAS] (p2); MODEL TEST: p1 = p2;

• utput:

tech1 tech2 tech10 tech11 tech12 tech14 ; cinterval; Plot: type is plot1 plot2 plot3; series is sw4emo(1) sw4con(2) sw4hyp(3) sw4peer(4) sw4pros(5) sw5emo (6) sw5con (7) sw5hyp (8) sw5peer (9) sw5pros (10); DEFINE: CUT sw4emo-sw4pros (0); CUT sw5emo-sw5pros (0); OUTPUT – ESTIMATED ZDeNamVAS MEANS (and significance test of differences) across classes of y Wald Test of Parameter Constraints Value 53.024 Degrees of Freedom 1 P-Value 0.0000 Latent Class Pattern 1 1 Means ZDENAMVAS -0.302 0.055 -5.527 0.000 Variances ZDENAMVAS 0.953 0.050 19.162 0.000 Latent Class Pattern 1 2 Means

42 ZDENAMVAS 0.142 0.023 6.178 0.000 Variances ZDENAMVAS 0.953 0.050 19.162 0.000

• The mean for class 1 of y (difficulty class) is -0.302, while the mean for the normative group (group

2 of y) is 0.142. This difference is significant (Wald test in the first part of the output). The Wald test was requested in the input by the MODEL TEST: statement. The MODEL TEST requested a test for the null hypothesis that the parameters indicating the means (estimated in y#1 and y#2) are equal. INPUT – ESTIMATED proportions of binary variable (ZDePicSAS: recoded) Variable: Names are mahcig01 MeHGsx1 DePicSTS ZDePicSAS DeNamVTS ZDeNamVAS mdhgsx1 sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros male pregsmok missingd; usevar are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros ZDePicSAS; categorical are sw4emo sw4con sw4hyp sw4peer sw4pros sw5emo sw5con sw5hyp sw5peer sw5pros ZDePicSAS; Missing are all (-999) ; classes are x(2) y(2); Analysis: Type = mixture ; starts = 200 20; stiterations = 20; MODEL: %overall%

43 y#1 ON x#1 ; [y#1@-15]; [x#1] ; MODEL x: %x#1% [sw4emo$1-sw4pros$1*-1] (6-10); %x#2% [sw4emo$1-sw4pros$1*3] ; MODEL y: %y#1% [sw5emo$1-sw5pros$1*-1] (6-10); [ZDePicSAS$1] (p1); %y#2% [sw5emo$1-sw5pros$1*3] ; [ZDePicSAS$1] (p2);

• utput:

tech1 tech2 tech10 tech11 tech12 tech14 ; cinterval; Plot: type is plot1 plot2 plot3; series is sw4emo(1) sw4con(2) sw4hyp(3) sw4peer(4) sw4pros(5) sw5emo (6) sw5con (7) sw5hyp (8) sw5peer (9) sw5pros (10); DEFINE: CUT sw4emo-sw4pros (0); CUT sw5emo-sw5pros (0); CUT ZDePicSAS (0); In the input above, ZDePicSAS is cut in two categories using cut point 0. The proportions of this variable across the two classes of latent variable y are estimated. Note that ZDePicSAS is indicated among the CATEGORICAL variables, as it is a categorical outcome. RESULTS IN PROBABILITY SCALE Latent Class Pattern 1 1 ZDEPICSA

44 Category 1 0.595 0.022 26.821 0.000 Category 2 0.405 0.022 18.253 0.000 Latent Class Pattern 1 2 ZDEPICSA Category 1 0.452 0.012 37.150 0.000 Category 2 0.548 0.012 45.044 0.000 The results indicate that about 40% of children in the difficulty class in wave 5 (y=1) have a score above 0 (category 2 of binary ZDEPICSA), while about 55% of children in the normative class in wave 5 (y=2) have a score above 0 (category 2 of binary ZDEPICSA).