Regression Model with Log-Transformed Response in MedCalc: Step-by-Step Interpretation and Scatter Plot Analysis

 Introduction

In biomedical and biostatistical research, regression analysis is a fundamental tool to explore the relationship between dependent and independent variables. However, when the dependent variable (response) does not meet the assumptions of normality or homoscedasticity, a transformation may be required. One of the most common transformations is the logarithmic transformation, which stabilizes variance and improves model fit.

In this article, we present the results of a regression model where the response variable (Triglycerides) has been log-transformed in MedCalc. The independent variable under study is Age (years). We provide a full interpretation of the regression coefficients, statistical significance, and goodness-of-fit measures, supported by a scatter plot with regression line. This analysis follows a format similar to that used in peer-reviewed scientific journal publications.

Materials and Methods

Dataset

• Dependent variable (Y): Triglycerides (mg/dL)
• Independent variable (X): Age (years)
• Sample size: 15 subjects

Statistical Approach

• Regression model with log-transformed dependent variable
• Performed using MedCalc Statistical Software
• Model Equation:

$$\log(y) = \beta_0 + \beta_1 X$$Where:

y = Triglycerides (mg/dL)

X = Age (years)

β0​ = Intercept

β1​ = Slope

Results

Regression Model Summary

Statistic	Value
Sample size (n)	15
Coefficient of determination (R²)	0.9905
Residual standard deviation	0.01736

Regression Equation

$$\log(y) = 1.8329 + 0.01018X$$

$$ParameterCoefficientStd.\; Error95\%\; CIt-valuep-valueIntercept1.83290.014631.8013\; –\; 1.8645125.27<0.0001Slope\; (Age)0.010180.0002770.009582\; –\; 0.0107836.74<0.0001$$

$$ANOVA\; Table$$

$$SourceDFSum\; of\; SquaresMean\; SquareF-ratiop-valueRegression10.40680.40681350.02<0.0001Residual130.0039170.000301$$

Residual Analysis

• Shapiro-Wilk test for normality: W = 0.8987, p = 0.0908 → Residuals are normally distributed.
• Model assumptions satisfied.

Scatter Plot Interpretation

Figure 1. Scatter plot showing the relationship between Age and Triglycerides with log-transformed regression line and 95% CI bands.

The scatter plot (Figure 1) presents the relationship between Triglycerides (mg/dL) and Age (years). A log-transformed regression line was fitted to the data.

• The fitted regression equation:

log(y)=1.833+0.0102X

• The correlation coefficient: r = 1.00, p < 0.001, indicating an almost perfect positive linear relationship.
• The confidence band around the regression line (shaded area) demonstrates a narrow range, confirming the precision of the estimates.
• As Age increases, the log of triglyceride levels also increases significantly. This implies that triglyceride concentration rises with age in an exponential-like manner when viewed in the raw (non-log) scale.

Discussion

The analysis demonstrates a strong and statistically significant association between age and triglyceride levels. The log transformation of the dependent variable was necessary to meet regression assumptions, particularly normality of residuals.

Key points from the results include:

1. Strong Model Fit: R² = 0.9905 indicates that 99% of the variance in log-triglycerides is explained by age.
2. Significant Predictor: Age is a significant predictor of triglyceride levels (p < 0.0001).
3. Positive Relationship: The slope (0.01018) suggests that for each additional year of age, the log of triglyceride levels increases by approximately 0.010 units.
4. Clinical Implication: In raw scale, this corresponds to a consistent exponential increase in triglyceride levels with age, which has implications for cardiovascular risk monitoring.

Conclusion

This study illustrates how log-transformed regression models can be effectively applied in MedCalc to analyze skewed biomedical data. The results confirm that age is a strong determinant of triglyceride levels, with a nearly perfect correlation.

For researchers, this demonstrates the importance of transformation techniques in regression modeling to achieve valid and interpretable results. The findings support the clinical evidence that lipid metabolism changes significantly with aging, reinforcing the need for regular monitoring in older populations.