# How Much Does the Dynamic F0 Curve Affect the Expression of Emotion in Utterances?
## 3 Results
### Incorporating Random Effects for Speakers  
While Models 1 and 2 accounted for non-linear time effects--both general and emotion specific--they did not consider individual variability among speakers (actors). In speech data, especially emotional speech, individual differences can significantly impact acoustic features like the fundamental frequency (F0). Speakers may have different baseline pitch levels and may express emotions with varying degrees of intensity and modulation. Ignoring this variability can lead to biased estimates and reduced model accuracy. To address this, we extended our modeling approach by incorporating random effects for speakers in Model 3. This allowed us to account for both the random intercepts (differences in baseline F0 levels among speakers) and random slopes (differences in how speakers' F0 contours change over time and across emotions). Model 3 introduces a random intercept for each speaker (actor) to capture individual baseline differences in the F0. \[Model\ 3:\ F0=\beta 0+\sum_{i=1}^{8}\beta_{i}\cdot Emotion_{i}+\sum_{j=1}^{8}S_{ j}(Time)\cdot Emotion_{j}+u_{Actor}+\epsilon\]  
In the model, the term \(u_{Actor}\sim\mathcal{N}\big{(}0,\sigma_{Actor}^{2}\big{)}\) represents the random effect for each actor, capturing individual variability in the baseline F0. This models the variability in the baseline F0 levels across different speakers. The term \(\sum_{j=1}^{8}S_{j}(Time)\cdot Emotion_{j}\)s allows each emotion to have its own non-linear F0 contour over time, as in Model 2. The parametric coefficients from Model 3 are presented in Table 11. The intercept now reflects the mean F0 for the neutral emotion, adjusted for the random effects of speakers. The standard errors of the emotion coefficients have decreased compared to previous models, indicating more precise estimates after accounting for speaker variability. Note that the standard errors of the fixed effects are still identical because of the way the model treats the random effects and smooth terms. The significance of the smooth terms in Model 3 is provided in Table 12. The edf values for the emotion-specific smooths remain high (around 8.4 to 8.9), indicating complex, non-linear F0 contours for each emotion. The term s(Actor) has an edf of approximately 23, reflecting the 24 actors in the dataset (since edf = number of levels \(-\) 1). The extremely high F-value and significant \(p\)-value indicate substantial variability among speakers. Table 13 presents the summary statistics for Model 3. The adjusted R-squared has increased dramatically to 0.579 from 0.202 in Model 2, indicating that Model 3 explains 57.9% of the variance in the F0. This substantial improvement underscores the importance of accounting for speaker variability. We compared Model 3 with Model 2 as in Table 14 to assess the impact of including random effects for speakers. The large reduction in fREML score (\(\Delta\)fREML = 241,865, \(p<\) 0.001) indicates that including random intercepts for actors significantly improves the model fit. The increase in edf by 1 corresponds to the addition of the random intercept term. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Estimate** & **Std. Error** & **\(t\) Value** & **Pr(\textgreater{1}1)** \\ \hline (Intercept) & 7.86671 & 0.97132 & 8.099 & 5.55 \(\times\) 10\({}^{-16}\) & *** \\ Calm & \(-\)0.67314 & 0.02688 & \(-\)25.043 & \(<\)2 \(\times\) 10\({}^{-16}\) & *** \\ Sad & 2.83025 & 0.02688 & 105.295 & \(<\)2 \(\times\) 10\({}^{-16}\) & *** \\ Fear & 7.34407 & 0.02688 & 273.224 & \(<\)2 \(\times\) 10\({}^{-16}\) & *** \\ Angry & 6.96962 & 0.02688 & 259.292 & \(<\)2 \(\times\) 10\({}^{-16}\) & *** \\ Happy & 5.73642 & 0.02688 & 213.413 & \(<\)2 \(\times\) 10\({}^{-16}\) & *** \\ Disgust & 2.22613 & 0.02688 & 82.819 & \(<\)2 \(\times\) 10\({}^{-16}\) & *** \\ Surprise & 6.32624 & 0.02688 & 235.357 & \(<\)2 \(\times\) 10\({}^{-16}\) & *** \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Parametric coefficients of Model 3.