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10.2 Application

Now, let’s apply path modeling to the real-world example. As explained in the introduction, we will investigate the relationships between physical health, functional health, and subjective health, as proposed by Whitelaw and Liang (1991).

We’ve formulated the following hypotheses for our investigation:

  1. Physical health influences functional health, which, in turn, predicts subjective health.

  2. Physical health directly affects subjective health.

Based on these hypotheses, we structure the relationships into the following diagram:

Variables

Loading the data

The data is provided in the book. As in the previous session, we can either load it from the local files (if you downloaded the entire book), or from the GitHub link. We will again use the second version as this would also work in Google Colab:

import pandas as pd
import semopy
import matplotlib.pyplot as plt

df = pd.read_csv("https://raw.githubusercontent.com/mibur1/psy111/main/book/statistics/5_Path_Modelling/data/data.txt", sep="\t")
print(df.head())
   1  1.1   0    13  1.2  1.3  1.4
0  1    1   7  15.5  1.0  1.0  1.0
1  1    1   3  19.0  1.0  1.0  1.0
2  1    1   6  20.0  1.0  1.0  1.0
3  1    2  10  20.5  1.0  1.0  1.0
4  1    1   7  22.5  1.0  1.0  1.0

We can see that the columns to not have names, so we assign them based on knowledge we have about the data:

df.columns = ["SubjHealth","SubjHealthChange", "PhysicHealth", "NrDoctorApp", "FunctHealth", "FunctHealth1", "FunctHealth2"]
print(df.head())
   SubjHealth  SubjHealthChange  PhysicHealth  NrDoctorApp  FunctHealth  \
0           1                 1             7         15.5          1.0   
1           1                 1             3         19.0          1.0   
2           1                 1             6         20.0          1.0   
3           1                 2            10         20.5          1.0   
4           1                 1             7         22.5          1.0   

   FunctHealth1  FunctHealth2  
0           1.0           1.0  
1           1.0           1.0  
2           1.0           1.0  
3           1.0           1.0  
4           1.0           1.0

Great, the DataFrame now looks like it is ready for use! We can continue with defining and fitting the model:

# Define the model
model = semopy.Model("""
                     SubjHealth ~ PhysicHealth + FunctHealth
                     FunctHealth ~ PhysicHealth
                     """)

# Fit the model to our data and estimate parameters
info = model.fit(df)
print(info)
Name of objective: MLW
Optimization method: SLSQP
Optimization successful.
Optimization terminated successfully
Objective value: 0.000
Number of iterations: 14
Params: -0.095 0.876 -0.096 0.172 0.432

We can then inspect the results:

estimates = model.inspect(std_est= True)
print(estimates)
          lval  op          rval  Estimate  Est. Std  Std. Err    z-value  \
0  FunctHealth   ~  PhysicHealth -0.096184 -0.451359  0.004686 -20.527585   
1   SubjHealth   ~  PhysicHealth -0.094898 -0.244325  0.008330 -11.392426   
2   SubjHealth   ~   FunctHealth  0.876182  0.480711  0.039090  22.414663   
3  FunctHealth  ~~   FunctHealth  0.171521  0.796275  0.005977  28.696690   
4   SubjHealth  ~~    SubjHealth  0.431653  0.603198  0.015042  28.696690   

   p-value  
0      0.0  
1      0.0  
2      0.0  
3      0.0  
4      0.0

This output provides a detailed breakdown of the parameter estimates from the model.

Or the fit statistics:

stats = semopy.calc_stats(model)
print(stats)
       DoF  DoF Baseline      chi2  chi2 p-value  chi2 Baseline       CFI  \
Value    1             4  0.000113      0.991515    1207.802947  1.000831   

       GFI  AGFI  NFI       TLI  RMSEA   AIC        BIC        LogLik  
Value  1.0   1.0  1.0  1.003322      0  10.0  37.033554  6.866405e-08

Finally, we can also visualise the model:

semopy.semplot(model, "figures/health.png", std_ests=True) # plot standardized estimates
Loading...

Additional information: If you want to visualise the model, you need to install Graphviz through pip (if you installed everything in the requirements.txt file you already have it). You then also need to install Graphviz as a normal program. On Windows, make sure to select the “Add Graphviz to PATH” option during the installation.

Model Output

Objective and Optimization

The first ouput we looked at after defining the model was info, which contains general information about the model fit:

Parameter Estimates

  1. Functional Health ~ Physical Health

    • Coefficient: -0.096, SE: 0.0047, z: -20.527, p: < 0.001

    • Interpretation: Physical health (the number of illnesses) has a significant negative effect on functional health. More illnesses reduce functional health.

  2. Subjective Health ~ Physical Health

    • Coefficient: -0.094, SE: 0.008, z: -11.39, p: < 0.001

    • Interpretation: Physical health has a significant negative effect on subjective health. More illnesses reduce subjective health.

  3. Subjective Health ~ Functional Health

    • Coefficient: 0.876, SE: 0.039, z: 22.41, p: < 0.001

    • Interpretation: Functional health has a significant positive effect on subjective health. Higher functional health leads to better subjective health.

Standardized Coefficients

Standardized coefficients (Est.Std) allow us to compare the relative influence of each predictor on a common scale. They can be interpreted similarly to correlation coefficients in terms of strength and direction. However, unlike correlations, they represent model-implied directional associations that depend on the assumed causal structure of the model. By themselves, they do not establish causality, but quantify effects conditional on the specified paths.

Variances

  1. Functional Health ~~ Functional Health: Residual variance = 0.172 (p < 0.001). This represents the unexplained variance in functional health.

    • Coefficient: 0.172, standardized coefficient: 0.796

    • Interpretation: The standardized Residual Variance indicates that approximately 80% of the individual variance in functional health remains unexplained by the model.

  2. Subjective Health ~~ Subjective Health: Residual variance = 0.432 (p < 0.001). This represents the unexplained variance in subjective health.

    • Coefficient: 0.432, standardized coefficient: 0.603

    • Interpretation: In this case, the standardized Residual Variance indicates that approximately 60% of the individual variance in subjective health remains unexplained by the model.

Model Evaluation

Fit Metrics

Interpretation of Model Fit

The model shows near-perfect fit indices. However, this should be interpreted with caution. The model has very few degrees of freedom (DoF = 1), meaning that global fit indices are not very informative in this context. In small or nearly just-identified models, it is common to observe extremely high fit values, including CFI and TLI values above 1 and RMSEA values close to zero.

Therefore, the excellent fit does not necessarily indicate overfitting, but rather reflects the simplicity and identification properties of the model.

Recommendation:
Model evaluation should focus primarily on parameter estimates and theoretical plausibility. External validation using an independent dataset can still be useful to assess generalizability.

Interpretation

Key Insights

  1. Functional Health ~ Physical Health: Negative relationship; more illnesses (measured as higher score in physical health) lead to reduced functional health.

  2. Subjective Health ~ Physical Health: Negative relationship; more illnesses lead to poorer subjective health.

  3. Subjective Health ~ Functional Health: Positive relationship; higher functional health improves subjective health.

Implications

Limitations

Multivariate Statistics
10.1 The semopy package
Multivariate Statistics
10.3 Exercises