## 4 Discussion
### Correlation of Error Results with Existing Non-Proportionality Indices
For unconservative errors, the curves are, in general, more closely squeezed than for conservative errors. Thus, the distinction between frame-type and shear-type system is not  
Figure 10: THMA error (type 1) max and min envelope for all cases versus index \(I_{A1}\). relevant for un-conservative errors but is relevant for conservative errors, particularly at the roof. These observations are generally consistent with those of Section 3. The max and min envelopes of all cases (outermost envelope curves) are used to obtain the maximum index value acceptable to limit the un-conservative and/or conservative THMA error; this information is listed in Table 7 (index \(I_{PS5}\)), Table 8 (\(I_{TLL1}\)), and Table 9 (\(I_{A1}\)), where no distinction has been made between frame-type and shear-type (and consequently the table is conservative for frame-type systems). For instance, to keep the un-conservative error \(\Delta_{1,\text{error}}\) below or equal \(-0.15\), the corresponding error row is located in Table 7 under header 'Error'; the value in this row on column under header 'first', sub-header '\(\Delta\)' is 0.10, indicating that \(I_{PS5}\) should be below or equal 0.10 to keep \(\Delta_{1,\text{error,first}}\) below or equal \(-0.15\); other values are found in a similar way: 0.24 (average across stories), 0.12 (central story) or 0.20 (roof). The most limitative story is the first, and the corresponding value (0.10) is found under the header 'ALL', sub-header '\(\Delta\)', at the right side of the table. To keep the conservative error \(\Delta_{1,\text{error}}\) below +0.20, the corresponding error row indicates that the index value should be limited to 0.13 (first story), 0.34 (average), 0.18 (central) or 0.10 (roof). The most limitative story is the roof and again the maximum index value (0.10) is listed under header 'ALL', sub-header '\(\Delta\)'. Thus, limiting the value of \(I_{PS5}\) to about 0.10 ensures that the max and min error fall within the [\(-0.15\); +0.20] range at all stories. A similar process with other variables renders corresponding limitative values of \(I_{PS5}\): 0.10 for \(u_{1,\text{error}}\) and \(a_{1,\text{error}}\); 0.08 for \(V_{\text{S1,error}}\) and \(V_{\text{D1, error}}\). For \(I_{TLL1}\) and \(I_{A1}\) the rationale is the same, except that Table 8 (\(I_{TLL1}\)), or Table 9 (\(I_{A1}\)) should be used; the limiting value to keep THMA error within a [\(-0.15\); +0.20] range is 0.10 (\(I_{TLL1}\)) or 0.08 (\(I_{A1}\)). A system fulfilling any one of these three conditions (\(I_{PS5}\leq 0.10\) or \(I_{TLL1}\leq 0.10\) or \(I_{A1}\leq 0.08\)) will produce THMA errors within the [\(-0.15\); +0.20] range for the five relevant variables considered. It must be kept in mind, however, that to keep the THMA error under control the maximum index value is sufficient but not necessary, as shown by all points within boundary limits at scatter plots in Figure 7. Information on the tables is limited to the maximum index value of the systems considered in the study. \begin{table}
\begin{tabular}{l  
Figure S12 (error 2, \(I_{A1}\)); Figure S13 (error 3, \(I_{PS5}\)); Figure S14 (error 3, \(I_{TL1}\)); Figure S15 (error 3, \(I_{A1}\)); Table S1 (error 2, \(I_{PSS5}\)); Table S2 (error 2, \(I_{TL1}\)); Table S3 (error 2, \(I_{A1}\)); Table S4 (error 3, \(I_{PSS5}\)); Table S5 (error 3, \(I_{TL1}\)); Table S6 (error 3, \(I_{A1}\)). For these procedures, the relationship between max and min error and the four selected NPIs is kept, but it is not always feasible to control the error for all variables; as an example, focusing on \(I_{PSS}\) and RSA-SRSS (Figure S10) it is only possible to keep the conservative error (max) for \(a_{2}\) or \(V_{\text{D2}}\) at central story or roof below a reasonable threshold if the value of \(I_{PSS5}\) is practically 0; this trend is consistent for the four selected NPIs. Even so, limitation of the indices is still valuable to keep other relevant variables under control.