Exercise 3
Lawrence Houston III
2025-05-06
Part I: Multilevel Modeling Based on previous research, we hypothesize that leadership climate is a group-level construct that plays a major role, along with two individual-level predictors (need for achievement and work experience), in predicting employees’ job performance. Specifically, we come up with the following hypothesized model (note: ‘H’ stands for ‘hypothesis’):
To test the model, we collected the following items and variables from a sample of 5,033 employees nested in 80 work units:
- A 5-item measure of Leadership Climate (1 = Strongly Disagree; 5 = Strongly Agree – higher scores indicate better leadership climate) – specific items include:
- The officers in my unit establish clear work objectives for me (lead1)
- The officers in my unit are interested in my personal welfare (lead2)
- The officers in my unit delegate work effectively to me (lead3)
- The officers in my unit let me know when I have done a good job (lead4)
- The officers in my unit are interested in what I think (lead5)
An overall, individual-level perceived leadership climate score (perlead), calculated as the individual means across lead1-lead5.
An overall aggregated leadership climate (leadclim), calculated as unit means in perceived leadership climate, assigned back down to individuals.
An overall individual Work Experience (workExp), calculated as number of years employed
An overall individual Need for Achievement (Nach) scale (1 = low motivation; 5 = high motivation)
An overall individual Job Performance (performance) scale (1 = very ineffective; 5 = very effective)
Use the dataset Exercise_3a.csv to answer the questions on the following page. Show your work by including all calculations, input syntax, and key output.
library(lme4)
library(lmerTest)
library(dplyr)
library(multilevel)
library(performance)
library(sjPlot)1. Using ANOVA, calculate and report ICC(1) and the respective F-value associated with leadership climate (using the ‘perlead’ scale).
# convert grouping variable to factor
Exercise_3a$group <- as.factor(Exercise_3a$group)
# Run ANOVA
anova_model <- aov(perlead ~ group, data = Exercise_3a)
summary(anova_model)## Df Sum Sq Mean Sq F value Pr(>F)
## group 79 498.5 6.309 11.77 <0.0000000000000002 ***
## Residuals 4953 2655.0 0.536
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Extract MSB, MSW, and K-bar (average group size)
anova_table <- summary(anova_model)[[1]]
MSB <- anova_table["group", "Mean Sq"] # Mean Square Between
MSW <- anova_table["Residuals", "Mean Sq"] # Mean Square Within
n_per_group <- mean(table(Exercise_3a$group)) # K-bar
# Calculate ICC(1) – Option 1
icc1 <- (MSB - MSW) / (MSB + (n_per_group - 1) * MSW)
cat("ICC(1) for Leadership Climate:", icc1, "\n")## ICC(1) for Leadership Climate: 0.1461721# Calculate ICC(1) – Option 2
cat("ICC(1) for Leadership Climate:", ICC1(anova_model), "\n")## ICC(1) for Leadership Climate: 0.14617211b. What do these statistics mean? Provide a write-up of methods and results, similar to what would appear in a published research article.
Q1b: ICC(1) values represent the strength of the non-independence and can be interpreted as the proportion of the variance in the outcome that is attributable to group membership (Bryk & Raudenbush, 1992). This value can also be interpreted as an interrater reliability measure (Bliese, 2000). Based on the former interpretation, the current results suggest a significant amount (15%) of the variance in the ratings of leadership climate is attributable to military unit (ICC[1] = .15; F(79, 4953) = 11.77, p < .001).
2. Using ANOVA, calculate and report ICC(1) and the respective F-value associated with job performance.
# convert grouping variable to factor
Exercise_3a$group <- as.factor(Exercise_3a$group)
# Run ANOVA
anova_model <- aov(performance ~ group, data = Exercise_3a)
summary(anova_model)## Df Sum Sq Mean Sq F value Pr(>F)
## group 79 260 3.291 4.113 <0.0000000000000002 ***
## Residuals 4953 3963 0.800
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Extract MSB, MSW, and K-bar (average group size)
anova_table <- summary(anova_model)[[1]]
MSB <- anova_table["group", "Mean Sq"] # Mean Square Between
MSW <- anova_table["Residuals", "Mean Sq"] # Mean Square Within
n_per_group <- mean(table(Exercise_3a$group)) # K-bar
# Calculate ICC(1) – Option 1
icc1 <- (MSB - MSW) / (MSB + (n_per_group - 1) * MSW)
cat("ICC(1) for Job Performance:", icc1, "\n")## ICC(1) for Job Performance: 0.04714684# Calculate ICC(1) – Option 2
ICC <- ICC1(anova_model)
cat("ICC(1) for Job Performance:", ICC1(anova_model), "\n")## ICC(1) for Job Performance: 0.047146842b. What are the potential implications of these results to analyzing relationships involving leadership climate and job performance using OLS regression? Provide a write-up of methods and results, similar to what would appear in a published research article.
Q2b: Five percent of the variance in job performance rating could be attributable to group membership (ICC[1] = .05; F(79, 4953) = 4.11, p < .001). Because group membership influences scores for job performance, using OLS regression to test hypotheses could produce a number of problems. These include 1) biased estimates that do not take into account between- and within-group correlations, and 2) inefficient estimates that violate the independence assumption of OLS regression. As a result, analyses will produce biased standard errors and inaccurate significance tests, leading researchers to draw erroneous conclusions from their data.
3. Report the within-group variance (σ²) and between-group variance (τ₀₀) associated with job performance, as well as the ICC(1) associated with job performance (calculate the ICC value using τ₀₀ and σ²). What are your conclusions?
# Null model (no predictors, random intercepts)
model_null <- lmer(performance ~ 1 + (1 | group), data = Exercise_3a)
tab_model(model_null)| performance | |||
|---|---|---|---|
| Predictors | Estimates | CI | p |
| (Intercept) | 2.76 | 2.70 – 2.81 | <0.001 |
| Random Effects | |||
| σ2 | 0.80 | ||
| τ00 group | 0.04 | ||
| ICC | 0.05 | ||
| N group | 80 | ||
| Observations | 5033 | ||
| Marginal R2 / Conditional R2 | 0.000 / 0.046 | ||
# Extract variance components
var_components <- as.data.frame(VarCorr(model_null))
tau_00 <- var_components$vcov[1] # Between-group variance
sigma2 <- var_components$vcov[2] # Within-group (residual) variance
# Compute ICC(1)
icc1.2 <- performance::icc(model_null)
icc1.2## # Intraclass Correlation Coefficient
##
## Adjusted ICC: 0.046
## Unadjusted ICC: 0.046Q3: For job performance, the within-group variance (σ²) was .80, and the between-group variance (τ₀₀) was .04.This value indicates that 4% of the variance in job performance is attributable to group membership. Thus, group membership exerts a small effect on our outcome of interest, and multilevel modeling is the appropriate analytical approach.
4. Test Hypotheses 1-3 (i.e., the hypotheses pertaining to the direct effects of need for achievement, work experience, and leadership climate on job performance). Report your results using an APA-style table. What are your conclusions? Please provide a write-up of methods and results, similar to what would appear in a published research article.
Note: Your APA-style table and write-up may include the methods and results for Questions 4 through 6.
# Grand-mean center the leadclim variable
Exercise_3a <- Exercise_3a %>%
mutate(leadclim_cntr = leadclim - mean(leadclim, na.rm = TRUE))
# Group-mean center the Nach and workExp variables
Exercise_3a <- Exercise_3a %>%
group_by(group) %>%
mutate(
Nach_cntr = Nach - mean(Nach, na.rm = TRUE),
workExp_cntr = workExp - mean(workExp, na.rm = TRUE)
) %>%
ungroup()
# Hypothesized model (direct effects, random intercepts & slopes)
model_ri <- lmer(performance ~ workExp_cntr + Nach_cntr + leadclim_cntr + (1 | group), data = Exercise_3a)
tab_model(model_ri)| performance | |||
|---|---|---|---|
| Predictors | Estimates | CI | p |
| (Intercept) | 2.76 | 2.72 – 2.80 | <0.001 |
| workExp cntr | -0.03 | -0.04 – -0.02 | <0.001 |
| Nach cntr | 0.24 | 0.22 – 0.27 | <0.001 |
| leadclim cntr | 0.41 | 0.29 – 0.54 | <0.001 |
| Random Effects | |||
| σ2 | 0.75 | ||
| τ00 group | 0.02 | ||
| ICC | 0.03 | ||
| N group | 80 | ||
| Observations | 5033 | ||
| Marginal R2 / Conditional R2 | 0.080 / 0.107 | ||
Q4: Results for the direct effects model can be seen in Table X. When the Level 1 variables (need for achievement and work experience) and the Level 2 variable (leadership climate) were entered simultaneously, need for achievement was positively related to job performance (b = .24, s.e. = .01, p < .01), and work experience was negatively related to job performance (b = -.03, s.e. = .01, p < .01). Results indicate that leadership climate significantly predict group mean estimates of job performance. Thus, our hypotheses (H1-H3) were supported. An examination of the residual variances indicated that there was still significant variance remaining to be explained at the individual-level (σ² = .75)
5. Test whether the need for achievement – job performance slope varies significantly across groups/units. What are your conclusions? Provide a write-up of results, similar to what would appear in a published research article.
# Hypothesized model (direct effects, random intercepts & slopes)
model.rs <- lmer(performance ~ workExp_cntr + Nach_cntr + leadclim_cntr + (1 + Nach_cntr | group), data = Exercise_3a)
tab_model(model.rs)| performance | |||
|---|---|---|---|
| Predictors | Estimates | CI | p |
| (Intercept) | 2.76 | 2.72 – 2.80 | <0.001 |
| workExp cntr | -0.03 | -0.04 – -0.02 | <0.001 |
| Nach cntr | 0.24 | 0.21 – 0.27 | <0.001 |
| leadclim cntr | 0.42 | 0.30 – 0.55 | <0.001 |
| Random Effects | |||
| σ2 | 0.75 | ||
| τ00 group | 0.02 | ||
| τ11 group.Nach_cntr | 0.01 | ||
| ρ01 group | -0.17 | ||
| ICC | 0.03 | ||
| N group | 80 | ||
| Observations | 5033 | ||
| Marginal R2 / Conditional R2 | 0.080 / 0.111 | ||
# Option 1: Compare Models
anova(model_ri,model.rs)## refitting model(s) with ML (instead of REML)## Data: Exercise_3a
## Models:
## model_ri: performance ~ workExp_cntr + Nach_cntr + leadclim_cntr + (1 | group)
## model.rs: performance ~ workExp_cntr + Nach_cntr + leadclim_cntr + (1 + Nach_cntr | group)
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## model_ri 6 12917 12956 -6452.5 12905
## model.rs 8 12917 12970 -6450.6 12901 3.7386 2 0.1542# Option 2: Test whether random (vs. fixed) slopes model is significantly better (extent of variation across groups)
ranova(model.rs) # Run a random effects ANOVA ## ANOVA-like table for random-effects: Single term deletions
##
## Model:
## performance ~ workExp_cntr + Nach_cntr + leadclim_cntr + (1 + Nach_cntr | group)
## npar logLik AIC LRT Df Pr(>Chisq)
## <none> 8 -6462.8 12942
## Nach_cntr in (1 + Nach_cntr | group) 6 -6464.8 12942 4.0144 2 0.1344Q5: Results indicated that, on average, need for achievement was positively related to job performance (.24, p < .001). To examine whether the relationship between need for achievement (Nach_cntr) and performance varied across groups, we compared a random intercepts model to a random slopes model in which the slope of need for achievement was allowed to vary by group. The change in chi-square was not significant, Δχ²(2) = 3.74, p = .154, indicating that the random slope model did not significantly improve model fit. Thus, the random intercepts only model would be retained.
6. Test Hypothesis 4 (pertaining to the interactive effect of leadership climate and need for achievement on job performance). Report your results using an APA-style table. What are your conclusions? Provide a write-up of results, similar to what would appear in a published research article.
# Hypothesized model (interactive effect, random intercepts & slopes)
model <- lmer(performance ~ workExp_cntr + Nach_cntr*leadclim_cntr + (1 + Nach_cntr | group), data = Exercise_3a)
tab_model(model.rs,model)| performance | performance | |||||
|---|---|---|---|---|---|---|
| Predictors | Estimates | CI | p | Estimates | CI | p |
| (Intercept) | 2.76 | 2.72 – 2.80 | <0.001 | 2.76 | 2.72 – 2.80 | <0.001 |
| workExp cntr | -0.03 | -0.04 – -0.02 | <0.001 | -0.03 | -0.04 – -0.02 | <0.001 |
| Nach cntr | 0.24 | 0.21 – 0.27 | <0.001 | 0.24 | 0.21 – 0.28 | <0.001 |
| leadclim cntr | 0.42 | 0.30 – 0.55 | <0.001 | 0.41 | 0.29 – 0.54 | <0.001 |
| Nach cntr × leadclim cntr | 0.10 | 0.01 – 0.20 | 0.039 | |||
| Random Effects | ||||||
| σ2 | 0.75 | 0.74 | ||||
| τ00 | 0.02 group | 0.02 group | ||||
| τ11 | 0.01 group.Nach_cntr | 0.00 group.Nach_cntr | ||||
| ρ01 | -0.17 group | -0.16 group | ||||
| ICC | 0.03 | 0.03 | ||||
| N | 80 group | 80 group | ||||
| Observations | 5033 | 5033 | ||||
| Marginal R2 / Conditional R2 | 0.080 / 0.111 | 0.080 / 0.111 | ||||
# Simple slopes analysis
library(interactions)## Warning: package 'interactions' was built under R version 4.3.3sim_slopes(model, Nach_cntr, leadclim_cntr, digits = 3) ## JOHNSON-NEYMAN INTERVAL
##
## When leadclim_cntr is OUTSIDE the interval [-47.437, -1.190], the slope of
## Nach_cntr is p < .05.
##
## Note: The range of observed values of leadclim_cntr is [-0.670, 0.883]
##
## SIMPLE SLOPES ANALYSIS
##
## Slope of Nach_cntr when leadclim_cntr = -0.3147318397362501851688 (- 1 SD):
##
## Est. S.E. t val. p
## ------- ------- -------- -------
## 0.211 0.022 9.466 0.000
##
## Slope of Nach_cntr when leadclim_cntr = 0.0000000000000001700298 (Mean):
##
## Est. S.E. t val. p
## ------- ------- -------- -------
## 0.244 0.016 14.803 0.000
##
## Slope of Nach_cntr when leadclim_cntr = 0.3147318397362505182357 (+ 1 SD):
##
## Est. S.E. t val. p
## ------- ------- -------- -------
## 0.276 0.023 11.822 0.000interact_plot(model, Nach_cntr, leadclim_cntr)
Q5: Results for the cross-level interaction model suggest that leadership climate moderated the relationship between need for achievement and job performance (b = .10, p < .05), such that the relationship was stronger for groups with high leadership climates (b = .28, p < .01) than low leadership climates (b = .21, p < .01).
Part II: Growth Modeling Employee engagement is a key factor in organizational success, influencing performance, job satisfaction, and retention. To study how engagement evolves over time, data were collected from 500 employees participating in a leadership development program. Their engagement levels (i.e., engagement1-engagement4) were measured at 1 month, 4 months, 7 months, and 10 months after entering the program. Additionally, their leadership self-efficacy (i.e., selfefficacy) was assessed during the first wave of data collection. The researcher is interested in testing the following three hypotheses:
1. Employee engagement follows a positive linear trajectory over time as employees progress in the leadership program.
Exercise_3b_long <- read.csv("Exercise_3b_long.csv")
model_h1 <- lmer(engagement ~ time + (time | id), data = Exercise_3b_long, REML = TRUE)
tab_model(model_h1)| engagement | |||
|---|---|---|---|
| Predictors | Estimates | CI | p |
| (Intercept) | 0.62 | 0.49 – 0.76 | <0.001 |
| time | 1.05 | 0.98 – 1.12 | <0.001 |
| Random Effects | |||
| σ2 | 0.63 | ||
| τ00 id | 1.93 | ||
| τ11 id.time | 0.48 | ||
| ρ01 id | 0.59 | ||
| ICC | 0.89 | ||
| N id | 500 | ||
| Observations | 2000 | ||
| Marginal R2 / Conditional R2 | 0.187 / 0.914 | ||
Q1: Results indicated that time has a significant and positive effect on psychological adjustment (1.05, p < .01); such that for every 3-month increment of time, there was a 1.05 increase in employee engagement.
2. Leadership self-efficacy is positively associated with initial levels of engagement, such that employees with greater self-efficacy start with higher engagement levels.
model_h2 <- lmer(engagement ~ selfefficacy + time + (time | id), data = Exercise_3b_long, REML = TRUE)
tab_model(model_h2)| engagement | |||
|---|---|---|---|
| Predictors | Estimates | CI | p |
| (Intercept) | 0.66 | 0.53 – 0.78 | <0.001 |
| selfefficacy | 0.50 | 0.37 – 0.62 | <0.001 |
| time | 1.05 | 0.98 – 1.12 | <0.001 |
| Random Effects | |||
| σ2 | 0.63 | ||
| τ00 id | 1.58 | ||
| τ11 id.time | 0.48 | ||
| ρ01 id | 0.49 | ||
| ICC | 0.88 | ||
| N id | 500 | ||
| Observations | 2000 | ||
| Marginal R2 / Conditional R2 | 0.238 / 0.907 | ||
Q2: Results indicated that self-efficacy was positively associated with engagement, such that individuals with higher self-efficacy reported higher initial levels of engagement (b = 0.496, p < .001). Additionally, engagement significantly increased over time (b = 1.048, p < .001), suggesting an overall positive trend in engagement over time.
The model included random intercepts and slopes for time, with significant variability observed in both the intercepts (σ² = 1.579) and the slopes (σ² = 0.483), indicating substantial between-person differences in both initial engagement levels and in how engagement changed over time.
3. Leadership self-efficacy is positively related to the rate of growth in levels of engagement, with employees high in self-efficacy showing steeper engagement growth over time.
model_h3 <- lmer(engagement ~ selfefficacy * time + (time | id), data = Exercise_3b_long, REML = TRUE)
tab_model(model_h2,model_h3)| engagement | engagement | |||||
|---|---|---|---|---|---|---|
| Predictors | Estimates | CI | p | Estimates | CI | p |
| (Intercept) | 0.66 | 0.53 – 0.78 | <0.001 | 0.67 | 0.54 – 0.79 | <0.001 |
| selfefficacy | 0.50 | 0.37 – 0.62 | <0.001 | 0.61 | 0.49 – 0.74 | <0.001 |
| time | 1.05 | 0.98 – 1.12 | <0.001 | 1.07 | 1.01 – 1.13 | <0.001 |
| selfefficacy × time | 0.30 | 0.23 – 0.36 | <0.001 | |||
| Random Effects | ||||||
| σ2 | 0.63 | 0.63 | ||||
| τ00 | 1.58 id | 1.57 id | ||||
| τ11 | 0.48 id.time | 0.40 id.time | ||||
| ρ01 | 0.49 id | 0.50 id | ||||
| ICC | 0.88 | 0.87 | ||||
| N | 500 id | 500 id | ||||
| Observations | 2000 | 2000 | ||||
| Marginal R2 / Conditional R2 | 0.238 / 0.907 | 0.352 / 0.914 | ||||
Q3: Results indicated that self-efficacy was positively associated with engagement, such that individuals with higher self-efficacy reported higher initial levels of engagement (b = 0.610, p < .001). Additionally, engagement increased significantly over time (b = 1.069, p < .001), indicating that, on average, individuals became more engaged as time progressed.
Importantly, the interaction between self-efficacy and time was also significant (b = 0.296, p < .001), suggesting that individuals with higher self-efficacy not only started out with greater engagement, but also experienced a faster rate of increase in engagement over time compared to those with lower self-efficacy.
Use the dataset Exercise_3b.csv to test these three hypotheses. Show your work by including all calculations, input syntax, and key output. Present your results in an APA-style table, ensuring clarity and proper formatting. Additionally, provide a write-up of methods and results, similar to what would appear in a published research article.
Exercise_3b_wide <- read.csv("Exercise_3b_wide.csv")library(lavaan)## Warning: package 'lavaan' was built under R version 4.3.3## This is lavaan 0.6-19
## lavaan is FREE software! Please report any bugs.# Specify model: Hypothesis 1
growth_model <- '
# Define latent intercept and slope factors
i =~ 1*engagement_T1 + 1*engagement_T2 + 1*engagement_T3 + 1*engagement_T4
s =~ 0*engagement_T1 + 1*engagement_T2 + 2*engagement_T3 + 3*engagement_T4
'
# Fit the model to data
fit <- growth(growth_model, data = Exercise_3b_wide)
summary(fit, standardized = TRUE, fit.measures = TRUE)## lavaan 0.6-19 ended normally after 30 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 9
##
## Number of observations 500
##
## Model Test User Model:
##
## Test statistic 12.136
## Degrees of freedom 5
## P-value (Chi-square) 0.033
##
## Model Test Baseline Model:
##
## Test statistic 2142.891
## Degrees of freedom 6
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.997
## Tucker-Lewis Index (TLI) 0.996
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -3423.640
## Loglikelihood unrestricted model (H1) -3417.572
##
## Akaike (AIC) 6865.280
## Bayesian (BIC) 6903.211
## Sample-size adjusted Bayesian (SABIC) 6874.645
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.053
## 90 Percent confidence interval - lower 0.014
## 90 Percent confidence interval - upper 0.093
## P-value H_0: RMSEA <= 0.050 0.382
## P-value H_0: RMSEA >= 0.080 0.145
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.026
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## i =~
## engagement_T1 1.000 1.394 0.884
## engagement_T2 1.000 1.394 0.677
## engagement_T3 1.000 1.394 0.537
## engagement_T4 1.000 1.394 0.432
## s =~
## engagement_T1 0.000 0.000 0.000
## engagement_T2 1.000 0.700 0.340
## engagement_T3 2.000 1.400 0.539
## engagement_T4 3.000 2.100 0.651
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## i ~~
## s 0.559 0.060 9.361 0.000 0.573 0.573
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## i 0.620 0.069 9.050 0.000 0.445 0.445
## s 1.049 0.035 29.977 0.000 1.499 1.499
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .engagement_T1 0.545 0.074 7.331 0.000 0.545 0.219
## .engagement_T2 0.694 0.056 12.428 0.000 0.694 0.163
## .engagement_T3 0.596 0.063 9.395 0.000 0.596 0.089
## .engagement_T4 0.681 0.115 5.942 0.000 0.681 0.066
## i 1.943 0.154 12.585 0.000 1.000 1.000
## s 0.490 0.041 12.020 0.000 1.000 1.000# Specify model: Hypotheses 2-3
growth_model <- '
# Define latent intercept and slope factors
i =~ 1*engagement_T1 + 1*engagement_T2 + 1*engagement_T3 + 1*engagement_T4
s =~ 0*engagement_T1 + 1*engagement_T2 + 2*engagement_T3 + 3*engagement_T4
# Hypothesis 2: Self-efficacy predicts initial engagement levels
i ~ selfefficacy # Test if self-efficacy influences baseline engagement
# Hypothesis 3: Self-efficacy predicts growth in engagement over time
s ~ selfefficacy # Test if self-efficacy influences rate of engagement growth
'
# Fit the model to data
fit <- growth(growth_model, data = Exercise_3b_wide)
summary(fit, standardized = TRUE, fit.measures = TRUE)## lavaan 0.6-19 ended normally after 28 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 11
##
## Number of observations 500
##
## Model Test User Model:
##
## Test statistic 13.670
## Degrees of freedom 7
## P-value (Chi-square) 0.057
##
## Model Test Baseline Model:
##
## Test statistic 2272.777
## Degrees of freedom 10
## P-value 0.000
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.997
## Tucker-Lewis Index (TLI) 0.996
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -3359.464
## Loglikelihood unrestricted model (H1) -3352.629
##
## Akaike (AIC) 6740.928
## Bayesian (BIC) 6787.289
## Sample-size adjusted Bayesian (SABIC) 6752.374
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.044
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.078
## P-value H_0: RMSEA <= 0.050 0.569
## P-value H_0: RMSEA >= 0.080 0.039
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.022
##
## Parameter Estimates:
##
## Standard errors Standard
## Information Expected
## Information saturated (h1) model Structured
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## i =~
## engagement_T1 1.000 1.395 0.884
## engagement_T2 1.000 1.395 0.677
## engagement_T3 1.000 1.395 0.537
## engagement_T4 1.000 1.395 0.433
## s =~
## engagement_T1 0.000 0.000 0.000
## engagement_T2 1.000 0.700 0.340
## engagement_T3 2.000 1.401 0.540
## engagement_T4 3.000 2.101 0.653
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## i ~
## selfefficacy 0.607 0.063 9.580 0.000 0.435 0.433
## s ~
## selfefficacy 0.298 0.033 9.159 0.000 0.425 0.423
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .i ~~
## .s 0.378 0.050 7.486 0.000 0.474 0.474
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .i 0.665 0.063 10.520 0.000 0.477 0.477
## .s 1.071 0.032 32.994 0.000 1.528 1.528
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .engagement_T1 0.543 0.074 7.329 0.000 0.543 0.218
## .engagement_T2 0.692 0.056 12.446 0.000 0.692 0.163
## .engagement_T3 0.607 0.063 9.622 0.000 0.607 0.090
## .engagement_T4 0.664 0.112 5.910 0.000 0.664 0.064
## .i 1.580 0.132 11.948 0.000 0.813 0.813
## .s 0.403 0.035 11.363 0.000 0.821 0.821