# How Much Does the Dynamic F0 Curve Affect the Expression of Emotion in Utterances?
## 3 Results
### Incorporating Random Slopes for Emotions Within Speakers  
While Model 3 accounted for random intercepts for speakers, it did not consider the possibility that the effect of emotions on the F0 might vary across speakers. In other words, different speakers may not only have different baseline pitch levels but may also express emotions differently in terms of pitch modulation. To capture this additional layer of variability, we extended our model to include random slopes for emotions within speakers in Model 4. Model 4 includes both random intercepts and random slopes for emotions within speakers. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**R-sq. (adj)** & **Deviance** & **fREML** & **Scale est.** & **N** \\ \hline
0.579 & 57.9\% & 2.2834 \(\times\) 10\({}^{6}\) & 24.3 & 757,440 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Comparison of the two models, Model 2 and Model 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Title 1** & **edf** & **Ref.df** & **F** & \(p\)**-Value** \\ \hline s(Time): Neutral & 8.439 & 8.911 & 1053.9 & \textless{}2 \(\times\) 10\({}^{-16}\) *** \\ s(Time): Angry & 8.945 & 8.999 & 1448.7 & \textless{}2 \(\times\) 10\({}^{-16}\) *** \\ s(Time): Happy & 8.938 & 8.999 & 2058.1 & \textless{}2 \(\times\) 10\({}^{-16}\) *** \\ s(Time): Fear & 8.806 & 8.989 & 805.8 & \textless{}2 \(\times\) 10\({}^{-16}\) *** \\ s(Time): Sad & 8.580 & 8.949 & 1095.1 & \textless{}2 \(\times\) 10\({}^{-16}\) *** \\ s(Time): Calm & 8.610 & 8.956 & 1706.5 & \textless{}2 \(\times\) 10\({}^{-16}\) *** \\ s(Time): Disgust & 8.915 & 8.998 & 1476.2 & \textless{}2 \(\times\) 10\({}^{-16}\) *** \\ s(Time): Surprise & 8.991 & 9.000 & 1747.5 & \textless{}2 \(\times\) 10\({}^{-16}\) *** \\ s(Actor) & 22.999 & 23.000 & 29,457.0 & \textless{}2 \(\times\) 10\({}^{-16}\) *** \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Approximate significance of smooth terms in Model 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **Score** & **edf** & **Difference** & **Df** & \(p\)**Value** & **Sig.** \\ \hline Model 2 & 2,525,258 & 24 & & & \\ Model 3 & 2,283,393 & 25 & 241,865.61 & 1.000 & \textless{}2 \(\times\) 10\({}^{-16}\) & *** \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Summary statistics for Model 3. \[ \begin{split} model\ 4:\ F0&=\beta 0+\sum\limits_{i=1}^{8} \beta_{i}\cdot Emotion_{i}+\sum\limits_{j=1}^{8}S_{j}(Time)\cdot Emotion_{j}+u_{ Actor}\\ &\qquad+u_{Actor,\ i}\cdot Emotion_{i}+\epsilon\end{split} \]  
The term \(u_{Actor,\ i}\sim\mathcal{N}\left(0,\sigma_{Actor,\ i}^{2}\right)\) represents the random slope for each emotion within each actor, capturing actor-specific variation in the effect of emotions on the F0. This models the interaction between speakers and emotions. According to [37], it is not possible to model correlations between random intercepts and random slopes in GAMs as one might in linear mixed-effects models. Therefore, we specify random intercepts and slopes separately. The parametric coefficients from Model 4 are presented in Table 15. The effect of calm is no longer statistically significant (\(p=0.28324\)), suggesting that after accounting for random slopes, calm does not differ significantly from neutral in terms of the mean F0. Compared to Model 3, the standard errors of the emotion coefficients have increased, reflecting the additional variability introduced by allowing emotion effects to vary by speaker. The significance of the smooth terms in Model 4 is provided in Table 16. The edf values for the emotion-specific smooths remain high, indicating complex, non-linear patterns for each emotion. The term s(Actor) is highly significant, confirming substantial variability in the baseline F0 among speakers. The term s(Actor, Emotions) is not statistically significant (\(p=0.23\)), suggesting that allowing the effects of emotions on the F0 to vary across speakers does not significantly improve the model fit in this case. Table 17 presents the summary statistics for Model 4.