General Instructions

Rumination refers to the tendency to repetitively think about the causes, situational factors, and consequences of one’s negative emotional experience (Nolen-Hoeksema, 1991). There is amassed psychology literature showing rumination may cause immune problems, yet the underlying mechanism is still unknown. A psychologist hypothesizes that the effects of rumination on immunity may go through two mechanisms: poor sleep quality and sadness. So we investigated this research question with 300 adult participants, and measured each construct with the following items, and saved the data in the RuminationData.xlsx file, which can be downloaded from Blackboard.

Model 1. Measurement Model

Figure 1 presents the notation the 4-latent factor structure model theorized: rumination, sadness, poor sleep quality, and immune parameters.

Figure 1. Confirmatory Factor Analysis. Model 1 Notation Single Factor Model

The following R code presents the model specification and estimation


# Step 1. Specify Model

model1 <- '
# Measurement Model
           Rummination  =~ rumi1 + rumi2 + rumi3 + rumi4
           Sadness =~ sad1 + sad2 + sad3 +sad4
           Sleep =~ sleep1 + sleep2 + sleep3 
           Immunity=~ leukocyt + lymphocy
'

# Step 2: Fitting and Estimating Model 1

cfa.model1<- cfa(model1, data = data)

Table 1, presents the standardized loading factors \(\lambda_{i}\) that varies from 0.697 (rumi3) to 0.798 (rumi1) in the Rumination factor, from 0.699 (sad3) to 0.821 (sad2) in sadness factor, from 0.734 (sleep3) to 0.896 (sleep1) in sleep factor, and from 0.816 (leukocyt) to 0.921 (lymphocy) in imnune parameters factor.

Table 1. Measurement Model (Model 1)
95% CI
Item Loading SE LL UL
Rummination
rumi1 0.719 0.040 0.640 0.798
rumi2 0.659 0.043 0.574 0.744
rumi3 0.606 0.046 0.516 0.697
rumi4 0.630 0.045 0.541 0.718
Sadness
sad1 0.723 0.040 0.646 0.801
sad2 0.745 0.039 0.669 0.821
sad3 0.610 0.046 0.521 0.699
sad4 0.653 0.043 0.568 0.737
Sleep
sleep1 0.841 0.028 0.786 0.896
sleep2 0.741 0.034 0.675 0.807
sleep3 0.658 0.039 0.581 0.734
Sleep
leukocyt 0.737 0.040 0.658 0.816
lymphocy 0.845 0.039 0.769 0.921

Furthermore, Figure 2 presents the pattern of coefficients, error variances, and correlations among factors. Rumination factor has a weak and positive correlation with sadness (r =.146) and immune parameters (r = 0.34), whereas the correlation with poor sleep factors is negative and strong (r = .61). Furthermore, sadness has a negative and very weak correlation with sleep (r = -.10) and immune variables (-.09), while correlation between poor sleep factors and immune parameters is negative and strong (r =-.70).

Figure 2. Four-Factor Measurement Model (Model 1)

The goodness of model fit for measurement model was evaluated using overall and individual fit indices: (1) overall fit assessed using the \(\chi^{2}\) statistic that evaluates the magnitude of discrepancy between the sample and the model-estimated, largest values of \(\chi^{2}\) indicate a bad fit; (2) the Comparative Fit Index (CFI), Tucker-Lewis index (TLI), and Normed Fit Index (NFI) with values >.90 indicating an acceptable fit and values >.95 indicating a good fit; (3) the Standardized Root Mean Square Residual (SRMR) <.08 being indicative of good fit; and (4) the Root Mean Square Error of Approximation (RMSEA) with values <.08 being indicative of reasonable fit, values <.05 indicating a good fit and >.10 unacceptable.

Table 3. Comparison Model Fit Statistics
RMSEA 90% CI
Model DVs Chisq df p-value CFI TLI NFI SRMR RMSEA LL UL
Model 1 32 56.194 59 0.58 1 1.003 0.955 0.032 0 0 0.032

Model 1 included all 13 items distributed in a 4-factor. Overall fit assess \(\chi^{2}\) = 56.194 with p-value = 0.580, we fail to reject the null hypothesis (see Table 2). This means we do not have sufficient evidence to say that the true model estimate variance/covariance matrix differs from the observed sample variance/covariance matrix. Furthermore, based on comparative indices (CFI = 1.0, TLI = 1.0, and NFI = .95) the model showed an good fit. Additionally, the absolute indices (SRMR = .032 and RMSEA = .00) take values below the cut-off criteria for reasonable fit (<.08).

Model 2. Structure Equation Model

Figure 3 presents the hypothesis that the effects of rumination on immunity may go through two mechanisms: poor sleep quality and sadness.

Figure 1. Confirmatory Factor Analysis. Model 1 Notation Single Factor Model

The following R code presents the model specification and estimation


# Step 1. Specify Model

model2 <- '
          # Measurement Model
           Rummination  =~ rumi1 + rumi2 + rumi3 + rumi4
           Sadness =~ sad1 + sad2 + sad3 +sad4
           Sleep =~ sleep1 + sleep2 + sleep3 
           Immunity=~ leukocyt + lymphocy
           
           # Path Model
           Immunity ~ b*Sleep + d*Sadness
           Sleep ~ a*Rummination
           Sadness ~ c*Rummination
           
           # Indirect Effect
           ind1:=a*b
           ind2:=c*d
'

# Step 2: Fitting and Estimating Model 1

cfa.model2<- cfa(model2, data = data)

Table 2 presents the standardized parameters estimate in Model 2, while Figure 3 summarize the results of measurement and structural model. Findings showed that the direct effect of Rumination on Sadness is statistically significant and positive (c = 0.152, p-value = 0.036) and statistically significant and negative on poor sleep conditions (a = -0.598, p-value = 0.000). Also, poor sleep conditions and sadness have a negative and significant effect on immunity parameters (b = -0.707, p-value = 0.000, d = -0.161, p-value = 0.008).

Table 4. Standardized Direct and Indirect Effects Model 2
Label Effect Predictor DV Path Value SE pvalue
b Sleep Immunity -0.707 0.047 0.000
d Sadness Immunity -0.161 0.060 0.008
a Rummination Sleep -0.598 0.053 0.000
c Rummination Sadness 0.152 0.073 0.036
ind1 Rummination|Sleep Inmunity 0.423 0.049 0.000
ind2 Rummination|Sadness Inmunity -0.024 0.015 0.107

The indirect effects of rumination on immunity through sleep conditions is positive and statistically significant (a*b = 0.423, p-value = 0.000). However, indirect effect through sadness is not statistically significant (p-value = 0.107). In sum, findings suggest that there are strong evidence to prove that the effects of rumination on immunity through the poor sleep quality mechanics is significant, however, the evidence for sadness mechanism was not strong enough.

Figure 3. Structural Equation Model (Model 2)

Finally, Table 4 presents the fit indices for Model 2. Overall fit assess \(\chi^{2}\) = 57.838 with p-value = 0.591, we fail to reject the null hypothesis. Then we do not have sufficient evidence to say that the true model estimate variance/covariance matrix differs from the observed sample variance/covariance matrix. Furthermore, based on comparative indices (CFI = 1.0, TLI = 1.0, and NFI = .953) the model showed an good fit. Additionally, the absolute indices (SRMR = .033 and RMSEA = .00) take values below the cut-off criteria for reasonable fit (<.08).

Table 4. Comparison Model Fit Statistics Model 2
RMSEA 90% CI
Model DVs Chisq df p-value CFI TLI NFI SRMR RMSEA LL UL
Model 2 30 57.838 61 0.591 1 1.003 0.953 0.033 0 0 0.032