# How Much Does the Dynamic F0 Curve Affect the Expression of Emotion in Utterances?
## 3 Results
### Incorporating Non-Linear Time Effects  
We fit another simple model (Model 1) that includes the constant difference among emotions but only a single smooth. The function s sets up a smooth over the parameter (i.e., time). As such, Model 1 assumes that the pattern over time is the same for all the  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **Estimate** & **Std. Error** & \(t\)**Value** & **Pr (\textgreater{}|\textless{}|\textless{}|\textless{}|\textless{}|\textless{}| \textless{}|\textless{}|\textless{}|\textless{}|\textless{}|\textless{}| \textless{}|\textless{}|\textless{}|\textless{}|\textless{}|\textless{}| \textless{}\textless{}|\textless{}\textless{}|\textless{}\textless{}| \  
eight emotions. The modified Generalized Additive Model was specified to include a non-linear pattern over time. In this model, the Helmert contrast was applied with the order of neutral, calm, sad, fear, angry, happy, disgust, and surprise (the same order that is observed in Figure 3). \[Model\ 1:F0=\beta 0+\sum_{i=1}^{8}\beta_{i}\cdot Emotion_{i}+s(Time)+\epsilon\]  
The output of Model 1 is given in Table 4 for coefficient estimates and Table 5 for the approximate significance of smooth terms:  
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 mean F0 compared to the neutral emotion, controlling for the effect of time. The standard errors for each emotion remain identical because the categorical variable is still treated as a fixed effect without interaction with time. Figure 3: The modeled F0 trends over time only. Figure 2: Boxplot of mean F0 (semitones) for each emotion category. The dot inside each box denotes the mean F0 for that emotion. The approximate significance of the smooth term s(Time) is provided in Table 5. In the table, edf stands for the estimated degrees of freedom, which reflects the complexity of the smooth term. An edf close to 9 suggests a highly flexible function, allowing for intricate non-linear patterns in the F0 over time. Ref.df for the reference degree of freedom is used for hypothesis testing of the smooth term. The significant \(p\)-value (<0.001) confirms that the non-linear effect of time on the F0 is highly significant. The adjusted R-squared value of 0.192 indicates that Model 1 explains 19.2% of the variance in the F0, an improvement from the 14.5% explained by Model 0. This increase demonstrates the importance of modeling the temporal dynamics of F0. In Figure 3, the smooth curve represents the estimated F0 contour over the normalized time course of the utterances, averaged across all emotions. The non-linear shape of the curve reflects the dynamic changes in pitch that are characteristic of spoken utterances. Incorporating the non-linear effect of time significantly improved the model fit, as evidenced by the increase in the adjusted R-squared and deviance explained. The smooth term captures the inherent temporal dynamics of speech, which are essential for accurately modeling F0 contours. However, Model 1 assumes that the shape of the F0 contour over time is identical across all emotions, differing only in their average levels. This assumption may not hold true, as different emotions can exhibit distinct temporal patterns in pitch modulation. For instance, emotions like surprise may have abrupt pitch changes, while sadness may show more gradual contours. To address this limitation, we developed Model 2, which allows for emotion-specific smooth functions over time, enabling us to capture the unique F0 trajectories associated with each emotion. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**R-sq. (adj)** & **Deviance** & **fREML** & **Scale est.** & **N** \\ \hline
0.192 & 19.2\% & 2.5295 \(\times\) 10\({}^{6}\) & 46.58 & 757,440 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Summary statistics for Model 1.