# How Much Does the Dynamic F0 Curve Affect the Expression of Emotion in Utterances?
## 3 Results
### Modeling Emotion-Specific, Non-Linear Time Effects  
While Model 1 incorporated a non-linear effect of time on the F0, it assumed that the temporal pattern of F0 variation was identical across all emotions, differing only in their average levels. However, different emotions may exhibit distinct temporal patterns in pitch modulation. To capture these potential differences, we extended our modeling approach by allowing the non-linear effect of time to vary by emotion. In Model 2, we introduced emotion-specific smooth functions over time, enabling the model to capture the unique F0 trajectories associated with each emotion. \[Model\ 2:\ F0=\beta 0+\sum_{i=1}^{8}\beta_{i}\cdot Emotion_{i}+\sum_{j=1}^{8}S_{ j}(Time)\cdot Emotion_{j}+\epsilon\]  
In this model, the \(\sum_{j=1}^{8}S_{j}(Time)\cdot Emotion_{j}\) term allows for a separate smooth function over time for each level of the emotions variable. This means that each emotion can have its own unique F0 contour over time, providing a more flexible and detailed modeling of the data. The parametric coefficients from Model 2 are presented in Table 7.  
The intercept represents the estimated mean F0 for the neutral emotion at the reference time point. The coefficients for each emotion indicate the difference in the mean F0 compared to the neutral emotion, controlling for the emotion-specific, non-linear effects of time. Note that the uniformity in standard errors occurs because the s(Time, by = Emotion) term does not alter the fixed-effects structure of the emotions variable itself. The fixed effect for each emotion is still estimated using a common residual variance across categories. The approximate significance of the emotion-specific smooth terms s(Time) is provided in Table 8. The edf values range from approximately 7.8 to 8.9, indicating that the smooth functions are flexible enough to capture complex, non-linear patterns for each emotion. All smooth terms are highly significant (\(p<0.001\)), suggesting that the emotion-specific, non-linear effects of time significantly improve the model fit. The key summary statistics for Model 2 are presented in Table 9. The adjusted R-squared value of 0.202 indicates that Model 2 explains 20.2% of the variance in the F0, an improvement from the 19.2% explained by Model 1. This increase demonstrates the importance of allowing for emotion-specific temporal dynamics in modeling F0 contours. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Estimate** & **Std. Error** & \(t\)**Value** & **Pr (\textgreater{}|t|)** \\ \hline (Intercept) & 7.86715 & 0.03021 & 260.43 & \textless{}2 \(\times 10^{-16}\) & *** \\ Calm & \(-0.67342\) & 0.03700 & \(-18.20\) & \textless{}2 \(\times 10^{-16}\) & *** \\ Sad & 2.83017 & 0.03700 & 198.50 & \textless{}2 \(\times 10^{-16}\) & *** \\ Fear & 7.34380 & 0.03700 & 197.3 & \textless{}2 \(\times 10^{-16}\) & *** \\ Angry & 6.97021 & 0.03700 & 188.40 & \textless{}2 \(\times 10^{-16}\) & *** \\ Happy & 5.73618 & 0.03700 & 155.04 & \textless{}2 \(\times 10^{-16}\) & *** \\ Disgust & 2.22634 & 0.03700 & 60.17 & \textless{}2 \(\times 10^{-16}\) & *** \\ Surprise & 6.32826 & 0.03700 & 171.04 & \textless{}2 \(\times 10^{-16}\) & *** \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Parametric coefficients of Model 2.