Applications
Attitude toward abortion
The dataset gss08 taken from the 2008 General Social Survey (Smith et al., 2010) includes six binary manifest items measuring 355 respondents’ attitudes toward abortion following the strategy suggested by McCutcheon, 1987. For each item, respondents were asked whether abortion should be legalized under various circumstances: a strong chance of serious defect in the baby (DEFECT), pregnancy is seriously endangering the woman’s health (HLTH), pregnancy as a result of rape (RAPE), due to low income, the family cannot afford any more children (POOR), woman is unmarried and has no plans to marry the man (SINGLE), and woman is married but does not want more children (NOMORE). Each item has two possible levels of response (i.e., "YES" or "NO"). gss08 also include five covariates: AGE, GENDER, RACE, DEGREE and REGION of respondents. (?gss08)
data("gss08")
str(gss08)
#> 'data.frame': 355 obs. of 11 variables:
#> $ DEFECT: Factor w/ 2 levels "YES","NO": 1 2 2 1 1 1 1 1 1 1 ...
#> $ HLTH : Factor w/ 2 levels "YES","NO": 1 2 2 1 1 1 1 1 1 1 ...
#> $ RAPE : Factor w/ 2 levels "YES","NO": 1 2 2 1 1 1 1 1 1 1 ...
#> $ POOR : Factor w/ 2 levels "YES","NO": 2 2 2 1 1 1 1 1 1 1 ...
#> $ SINGLE: Factor w/ 2 levels "YES","NO": 2 2 2 1 1 1 1 1 NA 1 ...
#> $ NOMORE: Factor w/ 2 levels "YES","NO": 1 2 2 1 1 1 1 NA 1 1 ...
#> $ AGE : num 32 21 56 40 62 71 58 36 52 78 ...
#> $ SEX : Factor w/ 2 levels "MALE","FEMALE": 1 2 2 1 2 1 2 1 1 2 ...
#> $ RACE : Factor w/ 3 levels "WHITE","BLACK",..: 2 2 2 2 1 2 2 1 1 1 ...
#> $ DEGREE: Factor w/ 4 levels "<= HS","HIGH SCHOOL",..: 2 2 1 4 3 2 3 3 2 3 ...
#> $ REGION: Factor w/ 9 levels "NEW ENGLAND",..: 2 2 2 2 2 2 2 2 2 2 ...Selecting the number of latent classes: In the first step of the analysis, we conduct a series of standard latent class models to select a number of latent classes. The number of class in the glca() function is set to 2, 3, or 4 as following commands. It should be noted that we use 10 sets of randomly generated initial parameters to avoid the problem of local maxima (i.e., n.init = 10), and the argument is set as seed = 1 in the glca() function to ensure reproductibility of results unless otherwise noted.
f <- item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ 1
lca2 <- glca(f, data = gss08, nclass = 2, seed = 1, verbose = FALSE)
lca3 <- glca(f, data = gss08, nclass = 3, seed = 1, verbose = FALSE)
lca4 <- glca(f, data = gss08, nclass = 4, seed = 1, verbose = FALSE)gofglca(lca2, lca3, lca4, test = "boot", seed = 1)
#> Model 1: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ 1
#> nclass: 2
#> Model 2: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ 1
#> nclass: 3
#> Model 3: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ 1
#> nclass: 4
#>
#> Goodness of Fit Table :
#> logLik AIC CAIC BIC entropy Res.Df Gsq Boot p-value
#> 1 -740.10 1506.20 1569.43 1556.43 0.95 50 135.13 0.00
#> 2 -687.45 1414.90 1512.17 1492.17 0.88 43 29.83 0.36
#> 3 -684.19 1422.38 1553.70 1526.70 0.79 36 23.31 0.56
#>
#> Analysis of Deviance Table :
#> npar logLik Df Deviance Boot p-value
#> 1 13 -740.10
#> 2 20 -687.45 7 105.31 0.00
#> 3 27 -684.19 7 6.52 0.26The output from the function gofglca() comprises of two tables; goodness-of-fit table and analysis-of-deviance table. The former shows model fit criteria such as AIC, CAIC, BIC, and entropy, \(G^2\) statistic, and its bootstrap \(p\)-value for absolute model fit. In this example, the bootstrap \(p\)-values indicate that the two-class model (Model 1) fits data poorly (\(p\)-value = 0.00), but the three-class and the four-class models (Model 2 and Model 3) fit data adequately (\(p\)-value = 0.36 and 0.56, respectively). The latter table displays deviance statistic comparing two competing models and its bootstrap \(p\)-value for the relative model fit. The null hypothesis in the test for comparing Model 1 and Model 2 (i.e., the fit of the two-class model is not significantly poorer than the fit of the three-class model) should be rejected (\(p\)-value = 0.00). In the test for comparing Model 2 and Model 3, however, the bootstrap \(p\)-value (= 0.26) indicates that the fit of the four-class model has not been improved significantly compared to the fit of the three-class model. In addition, the three-class model has the smallest value among these three models in the model fit criteria. Therefore, we can conclude that the three-class model is an appropriate for the gss08 data.
summary(lca3)
#>
#> Call:
#> glca(formula = f, data = gss08, nclass = 3, seed = 1, verbose = FALSE)
#>
#> Manifest items : DEFECT HLTH RAPE POOR SINGLE NOMORE
#>
#> Categories for manifest items :
#> Y = 1 Y = 2
#> DEFECT YES NO
#> HLTH YES NO
#> RAPE YES NO
#> POOR YES NO
#> SINGLE YES NO
#> NOMORE YES NO
#>
#> Model : Latent class analysis
#>
#> Number of latent classes : 3
#> Number of observations : 352
#> Number of parameters : 20
#>
#> log-likelihood : -687.4486
#> G-squared : 29.82695
#> AIC : 1414.897
#> BIC : 1492.17
#>
#> Marginal prevalences for latent classes :
#> Class 1 Class 2 Class 3
#> 0.34467 0.46396 0.19138
#>
#> Class prevalences by group :
#> Class 1 Class 2 Class 3
#> ALL 0.34467 0.46396 0.19138
#>
#> Item-response probabilities (Y = 1) :
#> DEFECT HLTH RAPE POOR SINGLE NOMORE
#> Class 1 0.8275 0.9453 0.7960 0.0638 0.0390 0.1344
#> Class 2 1.0000 1.0000 1.0000 0.9813 0.9284 0.9657
#> Class 3 0.0466 0.3684 0.0949 0.0000 0.0000 0.0000
#>
#> Item-response probabilities (Y = 2) :
#> DEFECT HLTH RAPE POOR SINGLE NOMORE
#> Class 1 0.1725 0.0547 0.2040 0.9362 0.9610 0.8656
#> Class 2 0.0000 0.0000 0.0000 0.0187 0.0716 0.0343
#> Class 3 0.9534 0.6316 0.9051 1.0000 1.0000 1.0000
plot(lca3)
Considering group variable and testing the measurement invariance As the group information provided in the data, we can consider the multilevel data structure and compare the latent class structure between higher-level units (i.e., groups). The glca() function can incorporate group variable by setting group argument as the name of group variable in the data. For example, in order to investigate whether attitudes toward legalizing abortions vary by the final degree of respondents, we may set DEGREE as group variable by typing group = DEGREE in the glca() function . DEGREE is coded into four categories (i.e., "<= HS", "HIGH SCHOOL", "COLLEGE", and "GRADUATE") indicating from under high school diploma to graduate degree. Moreover, we can implement the test for measurement invariance across groups using the glca() function. The measurement invariance assumption can be adjusted through measure.inv argument in glca(). The default is measure.inv = TRUE, constraining item-response probabilities to be equal across groups. The following commands implement two different tests for group variable: (1) test whether class prevalence is significantly influenced by group variable under the measurement-invariant model; and (2) test whether the measurement models differ across groups.
mglca1 <- glca(f, group = DEGREE, data = gss08, nclass = 3, seed = 1, verbose = FALSE)
mglca2 <- glca(f, group = DEGREE, data = gss08, nclass = 3, measure.inv = FALSE, seed = 1, verbose = FALSE)
gofglca(lca3, mglca1, mglca2, test = "chisq")
#> Model 1: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ 1
#> nclass: 3
#> Model 2: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ 1
#> Group: DEGREE, nclass: 3, measure.inv: TRUE
#> Model 3: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ 1
#> Group: DEGREE, nclass: 3, measure.inv: FALSE
#>
#> Goodness of Fit Table :
#> logLik AIC CAIC BIC entropy Res.Df Gsq
#> 1 -687.45 1414.90 1512.17 1492.17 0.88 43 29.83
#> 2 -672.41 1396.83 1523.28 1497.28 0.88 229 87.85
#> 3 -650.95 1461.89 1850.98 1770.98 0.89 175 44.91
#>
#> Analysis of Deviance Table :
#> npar logLik Df Deviance Pr(>Chi)
#> 1 20 -687.45
#> 2 26 -672.41 6 30.07 0.00
#> 3 80 -650.95 54 42.94 0.86Since the model specified in the object lca3 (Model 1 in the output from the goflca() function) is constructed with six binary items, the number of parameters for the saturated models is \(26 − 1 = 63\). Therefore, degree of freedom for Model 1 is \(63 − 20 = 43\). The models specified in mglca1 and mglca2 (Model 2 and Model 3) are involved with group variable with four categories, and the number of parameters for the saturated models is \(26 \times 4 − 1 = 255\). Therefore, degrees of freedom for Model 2 and Model 3 are \(255 − 26 = 229\) and \(255 − 80 = 175\), respectively. It should be noted that model comparison has been conducted through chi-squares by setting test = "chisq" because Model 1 is nested in Model 2, and Model 2 is nested in Model 3. The analysis-of-deviance table provided by the gofglca() function shows that the chi-square \(p\)-value for comparing Model 1 and Model 2 is 0.00, while the \(p\)-value for comparing Model 2 and Model 3 is 0.86. Hence, we can deduce that the measurement invariance assumption can be assumed, but class prevalences vary across levels of DEGREE.
Testing the equality of coefficients across groups: We can further consider the subject-specific covariates which may influence the probability of the individual belonging to a specific class. Covariates such as AGE, RACE, and SEX in the gss08 dataset can be incorporated into the model specified in mglca1. AGE is respondent’s age and considered as a numeric variable in the glca() function. The respondent’s race, RACE has three levels (i.e., "WHITE", "BLACK", and "OTHER"), and the respondent’s gender, SEX is coded as two categories (i.e., "MALE" and "FEMALE"). We can easily implement the test for exploring group differences using the gofglca() function. For example, the following commands implement two different tests for SEX: (1) test whether the class prevalence is significantly influenced by SEX under the model where the coefficients are constrained to be identical across groups; and (2) test for assessing group difference in the effect of SEX on class prevalence by specifying the model where coefficients are allowed to vary across groups and comparing it to the model where coefficients are constrained to be identical across groups. Since the iteration is initiated with random parameters, the order of class labels can be switched under the identical ML solution. Therefore, the random seed is set as seed = 3 in the glca() function for the model specified in mglcr1 below to ensure identical class order as provided in Figure 1.
fx <- item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ SEX
mglcr1 <- glca(fx, group = DEGREE, data = gss08, nclass = 3, seed = 3, verbose = FALSE)
mglcr2 <- glca(fx, group = DEGREE, data = gss08, nclass = 3, coeff.inv = FALSE, seed = 1, verbose = FALSE)
gofglca(mglca1, mglcr1, mglcr2, test = "chisq")
#> Model 1: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ 1
#> Group: DEGREE, nclass: 3, measure.inv: TRUE
#> Model 2: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ SEX
#> Group: DEGREE, nclass: 3, measure.inv: TRUE, coeff.inv: TRUE
#> Model 3: item(DEFECT, HLTH, RAPE, POOR, SINGLE, NOMORE) ~ SEX
#> Group: DEGREE, nclass: 3, measure.inv: TRUE, coeff.inv: FALSE
#>
#> Goodness of Fit Table :
#> logLik AIC CAIC BIC entropy Res.Df Gsq
#> 1 -672.41 1396.83 1523.28 1497.28 0.88 229 87.85
#> 2 -666.71 1389.42 1525.60 1497.60 0.88 323 149.97
#> 3 -662.04 1392.09 1557.45 1523.45 0.88 317 140.64
#>
#> Analysis of Deviance Table :
#> npar logLik Df Deviance Pr(>Chi)
#> 1 26 -672.41
#> 2 28 -666.71 2 11.41 0.00
#> 3 34 -662.04 6 9.33 0.16The models specified in mglcr1 and mglcr2 (Model 2 and Model 3 in the output from the goflca() function) are involved with an additional covariate, SEX, and the number of possible cases is \(26 \times 4 \times 2 − 1 = 511\). However, as only 352 observations are used for the analysis, the number of parameters for the saturated model becomes \(352 − 1 = 351\). Therefore, degrees of freedom for Model 2 and Model 3 are \(351 − 28 = 323\) and \(351 − 34 = 317\), respectively. The analysis-of-deviance table provided by the gofglca() function shows that SEX has a significant impact on the class prevalence (\(p\)-value = 0.00) when we compare Model 1 with Model 2. However, the model without any constraint on coefficients (Model 3) is not significantly superior to Model 2 (\(p\)-value = 0.16), indicating that the impact of SEX is not group specific. Note that Model 2 is mathematically equivalent to the standard LCR with covariates DEGREE and SEX without the interaction terms, but Model 2 is more intuitive and useful configuration when comparison of latent structures by group is a major concern.
Summarizing the results from the selected model: Based on the previous analysis, we can conclude that measurement models are equivalent across groups (i.e., measurement invariance assumption is satisfied) in the three-class latent class model. In addition, there is a significant effect of SEX on the class prevalence, but there is no group difference in the amount of effect.
Figure 1 displays the estimated parameters from the selected model specified in mglcr1 using the command plot(mglcr1). The line graph in Figure 1 displays the estimated item-response probabilities. We can see that the identified three classes are clearly distinguished by item-response probabilities. Class 1 represents individuals who are in favor of all the six reasons for abortion, while Class 3 represents those who consistently oppose all the six reasons. Individuals in Class 2 seem to distinguish between favoring the first three reasons (i.e., DEFECT, HLTH, and RAPE) and opposing the last three reasons (i.e., POOR, SINGLE, and NOMORE) for abortion. The first bar graph in Figure 1 describes the estimated marginal class prevalences, and the second bar graph displays the estimated class prevalences for each category of group variable. The values in parentheses printed in the x-axis are the group prevalences. Figure 1 displays that the class prevalence varies across groups, and the respondents with higher degrees are more advocate for legalizing abortion than those with lower degrees. The covariate effect can be confirmed through the Wald test for each of estimated odds ratios and coefficient by the command coef(mglcr1).
The SEX coefficient results the same, while the intercept determining the class prevalence differs across groups in the selected model. To avoid redundancy and save space, only the results of the SEX coefficient are displayed as follows, but the entire output of the object can be displayed by the command summary(mglcr1) in the R console.
coef(mglcr1)
#> Coefficients :
#>
#> Class 1 / 3 :
#> Odds Ratio Coefficient Std. Error t value Pr(>|t|)
#> SEXFEMALE 0.32824 -1.11400 0.09062 -12.29 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Class 2 / 3 :
#> Odds Ratio Coefficient Std. Error t value Pr(>|t|)
#> SEXFEMALE 0.35678 -1.03064 0.09971 -10.34 <2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
summary(mglcr1)
#>
#> Call:
#> glca(formula = fx, group = DEGREE, data = gss08, nclass = 3,
#> seed = 3, verbose = FALSE)
#>
#> Manifest items : DEFECT HLTH RAPE POOR SINGLE NOMORE
#> Grouping variable : DEGREE
#> Covariates (Level 1) : SEX
#>
#> Categories for manifest items :
#> Y = 1 Y = 2
#> DEFECT YES NO
#> HLTH YES NO
#> RAPE YES NO
#> POOR YES NO
#> SINGLE YES NO
#> NOMORE YES NO
#>
#> Model : Multiple-group latent class analysis
#>
#> Number of latent classes : 3
#> Number of groups : 4
#> Number of observations : 352
#> Number of parameters : 28
#>
#> log-likelihood : -666.7097
#> G-squared : 149.9656
#> AIC : 1389.419
#> BIC : 1497.601
#>
#> Marginal prevalences for latent classes :
#> Class 1 Class 2 Class 3
#> 0.46144 0.33996 0.19860
#>
#> Class prevalences by group :
#> Class 1 Class 2 Class 3
#> <= HS 0.16848 0.51010 0.32143
#> HIGH SCHOOL 0.44386 0.34339 0.21275
#> COLLEGE 0.55347 0.29616 0.15036
#> GRADUATE 0.71183 0.20237 0.08580
#>
#> Logistic regression coefficients :
#> Group : <= HS
#> Class 1/3 Class 2/3
#> (Intercept) 0.0811 1.1445
#> SEXFEMALE -1.1140 -1.0306
#>
#> Group : HIGH SCHOOL
#> Class 1/3 Class 2/3
#> (Intercept) 1.424 1.1261
#> SEXFEMALE -1.114 -1.0306
#>
#> Group : COLLEGE
#> Class 1/3 Class 2/3
#> (Intercept) 2.0173 1.3487
#> SEXFEMALE -1.1140 -1.0306
#>
#> Group : GRADUATE
#> Class 1/3 Class 2/3
#> (Intercept) 2.6492 1.3620
#> SEXFEMALE -1.1140 -1.0306
#>
#> Item-response probabilities (Y = 1) :
#> DEFECT HLTH RAPE POOR SINGLE NOMORE
#> Class 1 1.0000 1.0000 1.0000 0.9836 0.9309 0.9682
#> Class 2 0.8342 0.9488 0.8086 0.0700 0.0440 0.1409
#> Class 3 0.0649 0.3825 0.0989 0.0000 0.0000 0.0000
#>
#> Item-response probabilities (Y = 2) :
#> DEFECT HLTH RAPE POOR SINGLE NOMORE
#> Class 1 0.0000 0.0000 0.0000 0.0164 0.0691 0.0318
#> Class 2 0.1658 0.0512 0.1914 0.9300 0.9560 0.8591
#> Class 3 0.9351 0.6175 0.9011 1.0000 1.0000 1.0000The estimated coefficients for SEX and their odds ratios show that females are less advocate for legalizing abortion compared to their male counterparts.
Tobacco smoking behavior
The dataset nyts18 comprises five dichotomized manifest items on the life-time experience of several types of tobacco including cigarettes (ECIGT), cigars (ECIGAR), chewing tobacco/snuff/or dip (ESLT), electronic cigarettes (EELCIGT), and hookah or water pipe (EHOOKAH) taken from the National Youth Tobacco Survey 2018 (NYTS 2018, https://www.cdc.gov/tobacco/data_statistics/surveys/nyts/). The sample considered in this study includes 1,743 non-Hispanic white students from 45 schools. The number of sampled students from each school is in the range of 30 to 50. The school membership can be identified by SCH_ID and each school is classified as either middle or high school (SCH_LEV). (?nyts18)
According to socioecological models, patterns of adolescent tobacco smoking are best understood as embedded within social contexts. These social contexts can be either proximal in terms of individuals and peer groups or more distal in terms of schools and community. Socioecological models suggest that students within the same school often share common socio-economic status (SES) and cultural characteristics that may cause different tobacco smoking patterns compared to students attending other schools. However, reflecting school (group) effect in an mgLCR is less likely to provide a meaningful summary because there are too many groups, that is, 45 schools in the nyts18 dataset. In this case, npLCR would be more appropriate model to investigate group difference in terms of a small number of latent clusters of schools.
data("nyts18")
str(nyts18)
#> 'data.frame': 1734 obs. of 8 variables:
#> $ ECIGT : Factor w/ 2 levels "Yes","No": 2 2 2 2 2 2 2 2 2 2 ...
#> $ ECIGAR : Factor w/ 2 levels "Yes","No": 2 1 2 2 2 1 2 2 2 2 ...
#> $ ESLT : Factor w/ 2 levels "Yes","No": 2 2 2 2 2 2 2 2 2 2 ...
#> $ EELCIGT: Factor w/ 2 levels "Yes","No": 2 1 2 2 2 1 2 2 2 2 ...
#> $ EHOOKAH: Factor w/ 2 levels "Yes","No": 2 2 2 2 2 2 2 2 2 2 ...
#> $ SEX : Factor w/ 2 levels "Male","Female": 2 1 1 1 1 2 1 2 1 2 ...
#> $ SCH_ID : Factor w/ 45 levels "d3c3dc","12d5ad",..: 11 11 11 11 11 11 11 11 11 11 ...
#> $ SCH_LEV: Factor w/ 2 levels "High School",..: 1 1 1 1 1 1 1 1 1 1 ...Selecting the number of latent classes: Prior to conducting npLCR, it is necessary to determine the number of level-1 latent classes. Similar to the previous example, the two-, three-, and four-class standard LCA models can be fitted and compared by the gofglca() function as follow:
f <- item(starts.with = "E") ~ 1
lca2 <- glca(f, data = nyts18, nclass = 2, seed = 1, verbose = FALSE)
lca3 <- glca(f, data = nyts18, nclass = 3, seed = 1, verbose = FALSE)
lca4 <- glca(f, data = nyts18, nclass = 4, seed = 1, verbose = FALSE)
gofglca(lca2, lca3, lca4, test = "boot", seed = 1)
#> Model 1: item(starts.with = "E") ~ 1
#> nclass: 2
#> Model 2: item(starts.with = "E") ~ 1
#> nclass: 3
#> Model 3: item(starts.with = "E") ~ 1
#> nclass: 4
#>
#> Goodness of Fit Table :
#> logLik AIC CAIC BIC entropy Res.Df Gsq Boot p-value
#> 1 -2119.91 4261.83 4332.87 4321.87 0.92 20 96.48 0.00
#> 2 -2086.86 4207.71 4317.50 4300.50 0.87 14 30.37 0.28
#> 3 -2082.87 4211.74 4360.28 4337.28 0.82 8 22.40 0.78
#>
#> Analysis of Deviance Table :
#> npar logLik Df Deviance Boot p-value
#> 1 11 -2119.91
#> 2 17 -2086.86 6 66.11 0.0
#> 3 23 -2082.87 6 7.97 0.1The output of the gofglca() function indicates that the three-class and the four-class models are adequate in terms of absolute model fit (\(p\)-value = 0.28 and 0.78, respectively), but the three-class LCA provides the lowest values in information criteria. For the relative model fit, the two-class LCA is rejected (\(p\)-value = 0.00) on comparison with the three-class LCA using the bootstrap. However, the three-class is not rejected (\(p\)-value = 0.10) on comparison with the four-class model. Thus, it seems to be reasonable to select the three-class LCA model for the nyts18 dataset.
Selecting the number of latent clusters: Now, an npLCR can be implemented by adding SCH_ID as group variable and specifying the number of latent clusters (i.e., level-2 latent classes) using the ncluster argument in the glca() function. The two-, three-, and four-cluster npLCR with three latent classes are fitted and compared using the following commands:
nplca2 <- glca(f, group = SCH_ID, data = nyts18, nclass = 3, ncluster = 2, seed = 1, verbose = FALSE)
nplca3 <- glca(f, group = SCH_ID, data = nyts18, nclass = 3, ncluster = 3, seed = 1, verbose = FALSE)
nplca4 <- glca(f, group = SCH_ID, data = nyts18, nclass = 3, ncluster = 4, seed = 1, verbose = FALSE)
gofglca(lca3, nplca2, nplca3, nplca4, test = "boot", seed = 1)
#> Model 1: item(starts.with = "E") ~ 1
#> nclass: 3
#> Model 2: item(starts.with = "E") ~ 1
#> Group: SCH_ID, nclass: 3, ncluster: 2
#> Model 3: item(starts.with = "E") ~ 1
#> Group: SCH_ID, nclass: 3, ncluster: 3
#> Model 4: item(starts.with = "E") ~ 1
#> Group: SCH_ID, nclass: 3, ncluster: 4
#>
#> Goodness of Fit Table :
#> logLik AIC CAIC BIC entropy Res.Df Gsq Boot p-value
#> 1 -2086.86 4207.71 4317.50 4300.50 0.87 14 30.37 0.20
#> 2 -1955.49 3950.97 4080.14 4060.14 0.84 1419 765.73 0.08
#> 3 -1938.73 3923.46 4072.00 4049.00 0.84 1416 732.22 0.22
#> 4 -1938.30 3928.60 4096.51 4070.51 0.84 1413 731.35 0.24
#>
#> Analysis of Deviance Table :
#> npar logLik Df Deviance Boot p-value
#> 1 17 -2086.86
#> 2 20 -1955.49 3 262.74 0.00
#> 3 23 -1938.73 3 33.51 0.00
#> 4 26 -1938.30 3 0.87 0.42The goodness-of-fit table shows that the three-cluster model (Model 3) has the smallest values in information criteria among others, and the bootstrap \(p\)-value (= 0.22) indicates that this model is appropriate for the data. The analysis-of-deviance table shows that group effect is significant as Model 1 has better fit than Model 2 (\(p\)-value = 0.00). In addition, the three-cluster model (Model 3) has better fit than the two-cluster model (Model 2, \(p\)-value = 0.00), but there is insignificant difference between the three- and four-cluster models (Model 3 and Model 4) in the model fit (\(p\)-value = 0.42). Therefore, we can conclude that the three-cluster model is most appropriate among others.
Testing the equality of coefficients for level-1 covariates across groups: Covariates can be incorporated into npLCR using the glca() function; not only the subject-specific (i.e., level-1) covariates (e.g., SEX) but also the group-specific (i.e., level-2) covariates (e.g., SCH_LEV). As shown in the previous example, subject-specific covariates are constrained to be equal across latent clusters by default (i.e., coeff.inv = TRUE). The chi-square LRT test for checking the equality of coefficients for level-1 covariate, SEX can be conducted using the gofglca() function as follows:
fx <- item(starts.with = "E") ~ SEX
nplcr1 <- glca(fx, group = SCH_ID, data = nyts18, nclass = 3, ncluster = 2, seed = 1, verbose = FALSE)
nplcr2 <- glca(fx, group = SCH_ID, data = nyts18, nclass = 3, ncluster = 2, seed = 1, coeff.inv = FALSE, verbose = FALSE)
nplcr.gof <- gofglca(nplca3, nplcr1, nplcr2, test = "chisq")
nplcr.gof$dtable
#> npar logLik Df Deviance Pr(>Chi)
#> 2 22 -1951.879 NA NA NA
#> 1 23 -1938.731 1 26.294 2.931509e-07
#> 3 24 -1949.498 1 -21.533 1.000000e+00Note that typing nplca.gof$dtable into the R console will return the analysis-of-deviance table. Considering the \(p\)-values given in the table above, the equality of SEX effect can be assumed. Therefore, we conclude that the model specified in nplcr1 should be selected for the nyts18 dataset.
coef(nplcr1)
#>
#> Level 1 Coefficients :
#>
#> Class 1 / 3 :
#> Odds Ratio Coefficient Std. Error t value Pr(>|t|)
#> SEXFemale 1.05946 0.05776 0.16491 0.35 0.726
#>
#> Class 2 / 3 :
#> Odds Ratio Coefficient Std. Error t value Pr(>|t|)
#> SEXFemale 1.9301 0.6576 0.2071 3.176 0.00152 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
plot(nplcr1)
Incorporating the level-2 covariates: In npLCR, the meaning of the latent cluster (i.e., level-2 class) is interpreted by the prevalence of latent class (i.e., level-1 class) for each of cluster membership. When level-2 covariates are incorporated into the model, this prevalence is designed to be affected by level-2 covariates as shown (5). In other words, as level-2 covariates significantly influence the prevalence of level-1 class, the meaning of latent cluster may change as the level of level-2 covariate changes. Therefore, the number of clusters could be reduced when level-2 covariates are included compared to the number of clusters of npLCR without any level-2 covariate. As we selected the three-cluster model without any level-2 covariate for the nyts18 dataset, the fit of the three-cluster model should be compared with the fit of the two-cluster model when SCH_LEV is incorporated as a level-2 covariate. Note that different seed values (seed = 3 and seed = 6) are used in the glca() function below in order to produce the class order displayed in Figure 2.
f.2 <- item(starts.with = "E") ~ SEX + SCH_LEV
nplcr3 <- glca(f.2, group = SCH_ID, data = nyts18, nclass = 3, ncluster = 2, seed = 3, verbose = FALSE)
nplcr4 <- glca(f.2, group = SCH_ID, data = nyts18, nclass = 3, ncluster = 3, seed = 6, verbose = FALSE)
gofglca(nplcr3, nplcr4, test = "boot", seed = 1)
#> Model 1: item(starts.with = "E") ~ SEX + SCH_LEV
#> Group: SCH_ID, nclass: 3, ncluster: 2, coeff.inv: TRUE
#> Model 2: item(starts.with = "E") ~ SEX + SCH_LEV
#> Group: SCH_ID, nclass: 3, ncluster: 3, coeff.inv: TRUE
#>
#> Goodness of Fit Table :
#> logLik AIC CAIC BIC entropy Res.Df Gsq Boot p-value
#> 1 -1919.94 3887.87 4042.87 4018.87 0.83 1709 1052.54 0.12
#> 2 -1916.30 3886.60 4060.97 4033.97 0.84 1706 1045.26 0.06
#>
#> Analysis of Deviance Table :
#> npar logLik Df Deviance Boot p-value
#> 1 24 -1919.94
#> 2 27 -1916.30 3 7.28 0The analysis-of-deviance table from the gofglca() function shows that three clusters are required even when a level-2 covariate SCH_LEV is incorporated (\(p\)-value = 0.00). However, information criteria and \(p\)-values for absolute model fit may let us reach a different conclusion: the two-cluster model provides smaller values in some information criteria and the \(p\)-value (= 0.12) indicates that the model is appropriate for the dataset. Without strong prior beliefs, the number of latent clusters should be chosen to strike a balance between parsimony, fit, and interpretability. Two bar graphs in Figure 2 display the estimated class prevalences for each cluster membership from the models specified in nplcr3 and nplcr4 using the plot() function, respectively. The values in parentheses printed in the x-axis are the estimated cluster prevalence. The bar graph in the right, which is generated from the object nplcr4, shows that there is not much difference in class prevalence among these three clusters, compared to the bar graph in the left generated from the two-cluster model specified in nplcr3. In other words, the three-cluster model is not substantively meaningful, and therefore, we may conclude that the two-cluster model specified in nplcr3 is more adequate to describe cluster (group) variation in the latent class distribution.
Summarizing the results from the selected model: The estimated parameters from the three-class and two-cluster npLCR model specified in nplcr3 are shown in Figure 3 using the plot() function. Based on the line graph in Figure 3, we deduce that Class 1 represents the poly-user group; Class 2 is the electronic cigarette user group; and Class 3 is the non-smoking group. We already argued that the two latent clusters were clearly distinguished by their class prevalence using the left bar graph in Figure 2. According to this stacked bar graph, the probability of engagement in a certain latent class is significantly different between these two school clusters. About 12% and 29% of students in Cluster 2 belong to Class 1 (poly-user group) and Class 2 (electronic cigarette user group), whereas only 2% and 9% of them belong to Class 1 and Class 2, respectively. The difference in class prevalences by cluster indicates that students who attend a school classified as Cluster 2 are more likely to be smokers relative to a similar student attending a school classified as Cluster 1. The full output of the glca() function for the selected model can be displayed by the summary() function as follows:
summary(nplcr3)
#>
#> Call:
#> glca(formula = f.2, group = SCH_ID, data = nyts18, nclass = 3,
#> ncluster = 2, seed = 3, verbose = FALSE)
#>
#> Manifest items : ECIGAR ECIGT EELCIGT EHOOKAH ESLT
#> Grouping variable : SCH_ID
#> Covariates (Level 1) : SEX
#> Covariates (Level 2) : SCH_LEV
#>
#> Categories for manifest items :
#> Y = 1 Y = 2
#> ECIGAR Yes No
#> ECIGT Yes No
#> EELCIGT Yes No
#> EHOOKAH Yes No
#> ESLT Yes No
#>
#> Model : Nonparametric multilevel latent class analysis
#>
#> Number of latent classes : 3
#> Number of latent clusters : 2
#> Number of groups : 45
#> Number of observations : 1734
#> Number of parameters : 24
#>
#> log-likelihood : -1919.937
#> G-squared : 1052.541
#> AIC : 3887.874
#> BIC : 4018.87
#>
#> Marginal prevalences for latent classes :
#> Class 1 Class 2 Class 3
#> 0.06455 0.18159 0.75385
#>
#> Marginal prevalences for latent clusters :
#> Cluster 1 Cluster 2
#> 0.54083 0.45917
#>
#> Class prevalences by cluster :
#> Class 1 Class 2 Class 3
#> Cluster 1 0.0199 0.08978 0.89032
#> Cluster 2 0.1175 0.29044 0.59206
#>
#> Logistic regression coefficients (level 1) :
#> Cluster 1
#> Class 1/3 Class 2/3
#> (Intercept) -2.6345 -1.0654
#> SEXFemale 0.6307 0.1159
#>
#> Cluster 2
#> Class 1/3 Class 2/3
#> (Intercept) -0.8997 0.1671
#> SEXFemale 0.6307 0.1159
#>
#> Logistic regression coefficients (level 2) :
#> Class 1/3 Class 2/3
#> SCH_LEVMiddle School -2.5656 -1.9522
#>
#> Item-response probabilities (Y = 1) :
#> ECIGAR ECIGT EELCIGT EHOOKAH ESLT
#> Class 1 0.9657 0.8914 0.9782 0.5049 0.5507
#> Class 2 0.1696 0.3227 0.7266 0.0394 0.1127
#> Class 3 0.0034 0.0033 0.0372 0.0056 0.0074
#>
#> Item-response probabilities (Y = 2) :
#> ECIGAR ECIGT EELCIGT EHOOKAH ESLT
#> Class 1 0.0343 0.1086 0.0218 0.4951 0.4493
#> Class 2 0.8304 0.6773 0.2734 0.9606 0.8873
#> Class 3 0.9966 0.9967 0.9628 0.9944 0.9926As shown in the previous example, the result of Wald test for each of estimated odds ratios and coefficients can be obtained by typing coef(nplcr3) into the R console (results not shown here). The Wald test shows that females are more likely to belong to Class 1 than Class 3, indicating that females are at a higher risk than their male counterparts. For level-2 covariate SCH_LEV, middle schools are less likely to belong to Class 1 or 2 than Class 3, that is, middle-school students tend to smoke less than high-school students.
Imputing the cluster membership: Researchers often want to explore the effects of level-2 covariates on the imputed latent cluster membership. Note that we selected the two-cluster model specified in nplcr3. We can easily impute the latent cluster membership for 45 schools using the posterior probabilities from the model specified in nplcr3 and fit the standard logistic regression with SCH_LEV as a covariate. The posterior probabilities for latent cluster can be accessed by nplcr3$posterior$cluster. The following codes generate the imputed latent cluster membership for each school and save the cluster membership in ndata with level-2 covariate SCH_LEV.
tmp1 <- unique(nyts18[c("SCH_ID", "SCH_LEV")])
tmp2 <- nplcr3$posterior$cluster
tmp3 <- data.frame(SCH_ID = rownames(tmp2), Cluster = factor(apply(tmp2, 1, which.max)))
ndata <- merge(tmp1, tmp3)
head(ndata)
#> SCH_ID SCH_LEV Cluster
#> 1 00b895 Middle School 1
#> 2 066e6c High School 2
#> 3 0690d1 High School 2
#> 4 0fc94b High School 2
#> 5 12d5ad Middle School 1
#> 6 16082c Middle School 1Using the logistic regression, the effect of level-2 covariate SCH_LEV on the latent cluster membership can be estimated as following:
fit <- glm(Cluster ~ SCH_LEV, family = binomial, data = ndata)
summary(fit)
#>
#> Call:
#> glm(formula = Cluster ~ SCH_LEV, family = binomial, data = ndata)
#>
#> Deviance Residuals:
#> Min 1Q Median 3Q Max
#> -1.4823 -0.9005 -0.9005 0.9005 1.4823
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 0.6931 0.5477 1.266 0.2057
#> SCH_LEVMiddle School -1.3863 0.6708 -2.067 0.0388 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for binomial family taken to be 1)
#>
#> Null deviance: 61.827 on 44 degrees of freedom
#> Residual deviance: 57.286 on 43 degrees of freedom
#> AIC: 61.286
#>
#> Number of Fisher Scoring iterations: 4