a. Statistical model

Three ingredients are used in typical detection and attribution analyses.

1. 1)Observational data. These are a dataset of climate variability that potentially contains climate change signals to be detected. This can be, for example, a spatial map of long-term temperature trends. It can also be a time series of the annual mean temperature over the globe or over a region. But most often, it consists of the spatial and temporal evolutions of a climate variable that may enable detecting the effects of different external forcings separately.
2. 2)Signal(s) or expected climate response(s) to one or more external forcings. Since they are not known, they are often estimated based on the responses simulated by climate models under different external forcings, such as anthropogenic (ANT) forcing, external natural (NAT) forcing, or combined ANT and NAT (ALL) forcing.
3. 3)Information about natural internal climate variability or noise. This is typically based on climate model simulations under the “control” condition without external forcing. The residual of individual ensemble members after the removal of model-simulated responses can also be used to supplement the control simulation (in this paper, we will not make the distinction and refer to them as control simulations).

The three parts will be further explained as we introduce the notations next.

We consider a general setting with T time periods and S spatial regions (referred to sites below). For ease of notation, we assume that there are no missing data, but the method still works if there are missing values, as we will show in section 2b. Let Y_ts be the observation of the climate variable, covering time t = 1, …, T and space s = 1, …, S. Indexing the time and space separately facilitates different treatments of the temporal and spatial dependences in section 2b. Suppose that one is interested in separately detecting signals from J external forcings. Let X_tsj be the true (unobserved) fingerprint of the jth external forcing, j = 1, …, J. Let m_j be the number of simulations in the ensemble whose average ${\tilde{X}}_{\mathit{tsj}}$ is used to estimate X_tsj, j = 1, …, J. With each time period treated as a map or a cluster, define for cluster t: ${\mathbf{Y}}_t={(Y_{t1},\dots ,Y_{tS})}^{\mathrm{T}}$, ${\mathbf{X}}_{tj}={(X_{t1j},\dots ,X_{\mathit{tSj}})}^{\mathrm{T}}$, and X_t = (X_t1, …, X_tJ).

b. Estimating the scaling factor: Point estimator

Recall the EE [Eq. (6)] discards the temporal dependence, as it only uses the diagonal blocks of Σ, which means we are losing efficiency. Estimating all T blocks of the Toeplitz structure of Σ is possible, but blocks further away from the main diagonals are estimated less reliably, because the number of available pairs drops. The fully estimated Σ may still not be of full rank, a challenge similarly encountered in typical OF analyses. Discarding the off-diagonal blocks in estimation has the potential to lose efficiency, but the diagonal blocks are estimated more reliably, which leads to a more reliable weight. The gain from a more reliable weight may well exceed the loss from discarding the temporal dependence, especially when the temporal dependence is not strong, as shown in our numerical studies. Indeed, in typical OF applications where interannual variability, such as the effect of ENSO, is averaged out, the temporal dependence is weak, and the efficiency loss can be minimal.

The estimator ${\widehat{\mathit{\beta }}}_T$ in Eq. (7) has nice properties. For a large T, it is approximately unbiased (consistent) as long as the expectation in Eq. (4) holds; no distributional assumptions have been made beyond the expectation. The efficiency loss from discarding the temporal dependence in point estimation is offset by the potentially big gain from a more reliable pooled estimator of the S × S weight matrix ${\widehat{\mathbf{\Sigma }}}_{+}^{-1}$ or ${\widehat{\mathbf{\Omega }}}_{+}^{-1}$. Incorporating the temporal dependence like the prevailing TLS method does through a much bigger TS × TS weight matrix has the potential to achieve higher efficiency in point estimation, but this potential cannot be realized because of the large uncertainty in estimating a much higher-dimensional weight matrix. Further, in practice, missing data do not affect the proposed estimator. We simply use the available observations to construct the contribution of each cluster (time point) to the EEs, with the rows and columns corresponding to the missing observations removed from ${\widehat{\mathbf{\Sigma }}}_{+}$.

Although a is not needed in the point estimation of β, it is needed in constructing the confidence intervals of β, and an estimator of a provides a diagnosis of the restriction a = 1 in typical OF analyses. We give details of how to obtain an estimator ${\widehat{a}}_T$ of a with an EE in appendix C.

c. Confidence intervals

We propose a pseudo residual bootstrap approach to fully utilize the control runs in estimating $\mathsf{B}$. Define the block residual ${\mathbf{r}}_t(\mathit{\beta })=\hspace{0.17em}{\mathbf{Y}}_t-{\tilde{X}}_t\mathit{\beta }$, t = 1, … , T. Let $\mathbf{r}(\mathit{\beta })={[{\mathbf{r}}_1^{\mathrm{T}}(\mathit{\beta }),\dots ,{\mathbf{r}}_T^{\mathrm{T}}(\mathit{\beta })]}^{\mathrm{T}}$. Note that r(β) has an expectation of zero and covariance matrix δ²(β)Σ with $\delta (\mathit{\beta })={(1+a{\displaystyle {\sum }_{j=1}^J{\mathit{\beta }}_j^2}/m_j)}^{1/2}$. In a standard residual block bootstrap procedure, one would resample the residuals $\mathbf{r}({\widehat{\mathit{\beta }}}_T)$ in blocks to preserve the spatial and temporal dependences and add the bootstrap copies of the residuals to ${\tilde{X}}_t{\widehat{\mathit{\beta }}}_T$ to form bootstrap copies of Y_t, t = 1, …, T. The performance of the standard procedure here, however, is questionable, because it requires a large sample size T, whereas in a real-world application where multiyear averages are considered, T can be quite small. A valid bootstrap procedure has to preserve the spatial and temporal dependences in the data. We resort to control runs to meet this requirement, because they are assumed to have a similar covariance structure as the climate system (assumption 2).

Note that one can also incorporate the runs for the J external forcings, after centering, through the aforementioned scaling procedure to increase the number of bootstrapped copies. In our simulations and data analysis, we used L + m₁ + m₂ − 2 runs in total, where m₁ and m₂ are the numbers of runs for the ANT and NAT forcings, respectively. The subtraction of 2 is because both the ANT and NAT forcings were centered in the analysis.

d. Diagnostics of the statistical model

The first diagnosis is to test the null hypothesis H₀₁: a = 1, which is assumed in typical OF analyses. The asymptotic normal distribution of ${\widehat{a}}_T$ can be used to construct a Z statistic $Z=({\widehat{a}}_T-1)/{\widehat{\text{var}}}^{1/2}({\widehat{a}}_T)$, which follows a standard normal distribution under the null asymptotically. The variance of ${\widehat{a}}_T$ can be estimated using the same pseudo parametric bootstrap procedure for ${\widehat{\mathit{\beta }}}_T$.

Alternatively, H₀₁ can be tested without an estimation of a. Consider prewhitened residuals ${\mathbf{r}}_t^{*}=a^{-1/2}{\delta }^{-1}({\widehat{\mathit{\beta }}}_T){\widehat{\mathbf{\Psi }}}_{+}^{-1/2}{\mathbf{r}}_t$, t = 1, …, T, fixing a = 1. Define ${\mathbf{r}}^{*}={(r_1^{*},\hspace{0.5em}\dots ,\hspace{0.5em}{r}_T^{*})}^{\mathrm{T}}$. Under H₀₁, ${\mathbf{r}}^{*}$ should have a variance of 1, although there may be some temporal dependence. So we consider H₀₂ that the prewhitened residual ${\mathbf{r}}^{*}$ has a variance of 1 and use the sample variance $S^2({\mathbf{r}}^{*})$ of ${\mathbf{r}}^{*}$ as the testing statistic for H₀₂. The null distribution of $S^2({\mathbf{r}}^{*})$ depends on the unspecified temporal dependence. Denote prewhitened control runs ${\mathit{ϵ}}_t^{(l)*}={\widehat{\mathbf{\Psi }}}_{+}^{-1/2}{\mathit{ϵ}}_t^{(l)}$ and ${\mathit{ϵ}}^{(l)*}={[{ϵ}_1^{(l)},\dots ,{ϵ}_T^{(l)}]}^{\mathrm{T}}$. The variance of ${\mathit{ϵ}}^{(l)*}$, l = 1, …, L should also be 1, and its temporal dependence should be the same as that of ${\mathbf{r}}^{*}$. Therefore, the null distribution of $S^2({\mathbf{r}}^{*})$ can be approximated by the empirical distribution $\widehat{F}(\cdot )$ of the sample variances $S^2({\mathit{ϵ}}^{(l)*})$, l = 1, … , L of ${ϵ}^{(l)*}$. When L is small, to increase the accuracy, we could also generate some block bootstrapped versions of the prewhitened control runs and pool them with the original prewhitened control runs to approximate the null distribution of the testing statistic. The approximate p value is $2\min \{\widehat{F}[\text{var}({\mathbf{r}}^{*})],1-\widehat{F}[\text{var}({\mathbf{r}}^{*})]\}$. The same procedure could be applied with a evaluated at ${\widehat{a}}_T$ as a diagnostic test to check H₀₂ after allowing a ≠ 1.

a. Simulation settings

To evaluate the performance of the proposed estimator in realistic settings, we conducted simulation studies mimicking detection and attribution analyses of changes in the mean temperature. Both global and regional scales were considered. The global setting was based on S = 54 grid boxes of size 40° × 30°. The regional setting was based on eastern North America (ENA), with S = 21 grid boxes of size 5° × 5°. For ease of comparison with the results in Li et al. (2021), 5-yr mean temperatures were considered over the time period 1951–2010. Mimicking the application in the next section, the observed data and simulations under the external forcings were anomalies relative to their 30-yr average over 1961–90. Due to this centering, the period of 1961–65 was then removed from the analysis, resulting in T = 11 clusters.

For each setting, we first set the true signals X and the true covariance Σ of the TS-dimensional regression error ϵ. Two signals were considered, X₁ for the ANT forcing and X₂ for the NAT forcing. Their true values were set to be the average of 35 and 46 simulations under the ANT and NAT forcings, respectively, from phase 5 of the Coupled Model Intercomparison Project (CMIP5). The true Σ was set to be the estimate based on 223 control runs from CMIP5 with a block Toeplitz structure imposed to satisfy the temporal stationarity assumption. See appendix E for details on strategies to make it positive definite and make the temporal correlation decay as time lag increases.

With X₁, X₂, and Σ set, we can now generate the observed data and control runs. For j ∈ {1, 2}, the estimated signal ${\tilde{X}}_j={({\mathbf{X}}_{j1}^{\mathrm{T}},\dots ,{\mathbf{X}}_{jT}^{\mathrm{T}})}^{\mathrm{T}}$ was generated from a multivariate normal distribution N(X_j, aΣ/m_j), with m₁ = m₂ = m ∈ {20, 40} and a ∈ {0.5, 1}. The regression error ϵ and the control runs [ϵ⁽¹⁾, … , ϵ^(L)] were independently generated from a multivariate normal distribution N(0, Σ), with L ∈ {50, 100, 200}. Here L = 50 is relatively common in OF studies, and L = 200 is possible but not easily obtained unless runs from different climate models are pooled. The observed temperature $\mathbf{Y}={({\mathbf{Y}}_1^{\mathrm{T}},\dots ,{\mathbf{Y}}_T^{\mathrm{T}})}^{\mathrm{T}}$ was generated from the model [Eq. (1)] with $\mathit{\beta }={({\beta }_1,{\beta }_2)}^{\mathrm{T}}={(1,1)}^{\mathrm{T}}$.

For each combination of m and L, we ran 1000 replicates to evaluate the proposed method compared to two TLS-type methods. The first TLS method, denoted by TLS-TS, is the prevailing two-sample approach where the control runs were split into two samples, each giving an estimate of Σ. The first one was used to prewhiten the data in the point estimation of β; the second one was used to estimate the variance of the estimator and construct confidence intervals based on the normal approximation (DelSole et al. 2019; Li et al. 2021). Unlike the simulation-based approach of Allen and Stott (2003), no open intervals were expected. The second TLS method, denoted by TLS-PBC, uses all the control runs to estimate Σ and calibrates the confidence intervals by parametric bootstrap (Li et al. 2021) with 500 bootstrap replicates. The whole study was run on the high-performance cluster (HPC) of the University of Connecticut.

b. Results

Here we only discuss the simulation results from the setting where the true Σ is the block Toeplitz matrix, the true a is 1, and the regression error follows a multivariate normal distribution. More simulation results are provided in the supplement. Table 1 summarizes the bias and the RMSE of the three estimators in all the simulation configurations. All three point estimators appear unbiased in all settings. At the global level, the EE estimator has the smallest RMSE among the three estimators, while the TLS-TS estimator has the largest RMSE for most settings. The difference in RMSE between the EE and TLS-PBC increases as the sample size of control runs decreases or the accuracy in estimating Σ decreases. For L = 50, the EE has a much smaller RMSE for both scaling factors than TLS-PBC. The RMSE of the EE almost remains the same for different L, which means a small sample size of control runs is sufficient for an EE. In contrast, TLS-PBC needs more control runs to achieve a smaller RMSE. For the ANT forcing, the RMSE of the EE is always smaller than TLS-PBC for all m and L. For the NAT forcing, the EE is better than TLS when L = 50 or 100. In the extreme case where L = 200 and m = 20, TLS-PBC is slightly better than the EE, but this number of L is not easy to obtain. At the regional scale, where the signal-to-noise ratio is lower, the RMSE of the EE is very close to or smaller than those of TLS-TS and TLS-PBC except for the NAT forcing when m = 20. In summary, all three methods give unbiased estimators, but the EE has the smallest RMSE for most settings, especially for the ANT scaling factor.

Table 1.

Summaries of the bias and RMSE from three methods in the simulation settings.

View Table

Table 2 summarizes the average widths, the empirical coverage rates, and the average interval score (Gneiting and Raftery 2007) of the 90% confidence intervals constructed from the three methods across all the simulation settings. At both global and regional levels, EE intervals have coverage rates very close to the nominal level in almost all settings. For all three levels of L, even at L = 50, their coverage rates are close to the nominal level 90%, with a minimum of 87% for both ANT and NAT scaling factors, regardless of the noise level of the measurement error controlled by m. TLS-TS intervals have the lowest coverage, as reported in Li et al. (2021). Their coverage rate improves as L increases but remains unsatisfying and lower than 80% for both ANT and NAT even when L = 200. TLS-PBC requires L = 100 to reach the nominal level of coverage rate for the ANT scaling factor and L = 200 for the NAT scaling factor, whereas it becomes conservative for the ANT scaling factor when L = 200, especially at the regional level. In addition, EE intervals are often narrower than or comparable to TLS-PBC intervals when both methods provide desired coverage rates. The superiority of EE intervals regarding the widths and coverage rates is reflected by the lower interval scores. The undercoverage of TLS-TS intervals and the conservativeness of TLS-PBC intervals lead to larger interval scores. At the regional level, where the signal-to-noise ratio is lower, all intervals become wider than at the global level. EE intervals still give the proper coverage rate with a sample size of control runs as small as L = 50. To sum up, the proposed EE intervals provide valid confidence intervals with the desired level of coverage rate and a narrower width in most cases at a much lower computational cost than TLS-PBC intervals.

Table 2.

Summaries of the average length, empirical coverage percentages (CPs), and average interval score of the 90% confidence intervals constructed from three methods in the simulation settings.

View Table

c. Additional simulations

To investigate the effect of differences in the magnitude of climate variability between observations and model simulations and the effect of non-Gaussian residuals, we also conducted additional simulations with a ∈ {0.5, 1}, ϵ following a multivariate normal or t distribution with a degree of freedom of 15 and the true Σ set as a block Toeplitz matrix or the regularized linear shrinkage estimator from control runs (see section S2 in the online supplemental material). For the point estimators, the EE method is still almost unbiased and has the smallest RMSE across all the settings, while the two TLS methods showed a large bias when a = 0.5, especially for the NAT forcing. For the confidence interval, the EE method maintains a close-to-nominal coverage rate, while TLS-TS has a much lower coverage rate. TLS-PBC also could not reach the nominal coverage rate when the sample size of control runs is small, e.g., L = 50. These results showed the robustness of the EE method against non-Gaussian error distribution and the necessity of relaxing the well-accepted assumption a = 1. The advantage of the EE method would be even more obvious when the tails of the multivariate t distribution are heavier with smaller degrees of freedom.

We also investigated the performance of the EE method with a larger number of sites (S = 108), with ϵ following a multivariate normal or t distribution (see section S2). Results suggest that the EE method maintains its unbiasedness, efficiency, and validity in almost all cases. Even though a minor bias is observed for the NAT forcing when the measurement error is large (m = 20) and the number of control runs is small (L = 50), it becomes negligible with the increase of L. For TLS-TS and TLS-PBC, the NAT forcing is also a challenging case, especially when ϵ follows a multivariate t distribution; the bias and RMSE are significantly increased compared to their performances with S = 54. In general, the EE method retains its advantages with a large S.

In typical OF analyses, the ANT forcing is often obtained by subtracting the NAT forcing from the ALL forcing, which causes correlated measurement errors between ANT and NAT. Thus, to apply the EE method, we suggest estimating β_ALL and β_NAT first and then constructing the estimators for β_ANT and β_NAT (see details in section 2a). Although the estimating procedure is slightly different than the main simulation results presented in section 2b, which assumes that the two forcings are independent, we conducted additional simulations in section S2 mimicking the data generation and processing procedure in section 2b. The results suggest that the EE method is valid as expected and maintains its advantages over TLS methods.

As our EE method has assumed temporal stationarity (assumption 3), we further investigate if the same assumption would also improve the prevailing TLS estimator. The results in section S2 shows that the temporal stationarity assumption does have an effect in reducing the RMSE of the point estimator compared to TLS-TS and TLS-PBC at the regional scale with Gaussian errors; however, the RMSE is still larger than that of the EE estimator in most settings. Also, imposing the temporal stationarity assumption does not seem to reduce the RMSE at the global scale, possibly due to a higher spatial dimension. When the regression errors are heavy tailed (i.e., t distributed), the adapted TLS estimator is also less efficient than the EE. For confidence intervals, we assess the performance of TLS adapted with the proposed pseudo bootstrap procedure, as the prevailing TLS could not provide confidence intervals with the desired coverage rates. The results in section S2 suggest that the modified TLS method has close-to-nominal coverage rate with Gaussian errors, but it could be very conservative when the error distribution is non-Gaussian (e.g., t distribution), leading to a surprisingly large interval width. Therefore, the stationarity assumption does improve TLS-type methods overall, but our proposed EE method has clear advantages.

a. Data preparation

Our observational data Y came from the HadCRUT4 dataset (Morice et al. 2012), which contains monthly anomalies of near-surface air temperature on 5° × 5° grid boxes relative to the 30-yr average over 1961–90. The annual mean temperature anomalies were computed from the monthly values. The annual value was considered missing if there were more than 3 monthly values missing in the year. Our analyses were conducted on nonoverlapping 5-yr mean temperature anomalies. If there were at least three annual values available within a 5-yr period, we computed the 5-yr average temperatures. After removal of the 1961–65 period due to centering, we have T = 13 values of 5-yr averages at each grid box. The spatial dimension is also reduced for the global and continental scale analyses by averaging available 5-yr 5° × 5° grid boxes within a larger box. The final box sizes used in the global and continental scale analyses are as follows: GL and NH, 40° × 30°; NHM, 40° × 10°; EA, 10° × 20°; and NA, 10° × 5°. For the subcontinental scale analyses, we only aggregated the boxes of SCA to 10° × 10° boxes. Details about the boundaries of the regions and data availability are summarized in appendix F.

We obtained estimated signals and control runs from large ensemble simulations conducted with CanESM5 climate model simulations (Swart et al. 2019). Specifically, let ${\tilde{X}}_{\text{NAT}}$ be the estimated NAT signal from CanESM5 simulations with 50 runs. Since the ALL forcing in CanESM5 ended in 2014, we appended the simulations using those under the ssp245 forcing (Danabasoglu 2019). The estimated ALL signal X_ALL was obtained from 50 such appended runs. During the processing, the same missing pattern of Y was imposed, and the same average and aggregation procedures of Y were applied to obtain ${\tilde{X}}_{\text{ALL}}$ and ${\tilde{X}}_{\text{NAT}}$. Since they were centered by the 30-yr average over 1961–90, the first 5-yr period 1961–90 was excluded as well. The control runs were obtained from 50 runs under solar and volcanic forcings only, 30 runs under anthropogenic aerosols only, and 50 runs under greenhouse gas only from CanESM5 simulations; after centering under each forcing, the intraensemble variation gave 127 runs. At each grid box, a long-term linear trend was removed from the control simulations from each climate model separately.

The scaling factor of the ANT forcing is of the most interest. In typical OF analyses, one would obtain ${\tilde{X}}_{\text{ANT}}$ from subtracting ${\tilde{X}}_{\text{ALL}}$ by ${\tilde{X}}_{\text{NAT}}$ and estimate the scaling factors of ANT and NAT directly. Such transformation leads to correlated measurement errors ν₁ and ν₂, which violates assumption 1. Thus, we first fit the model and obtain the estimated scaling factors ${\widehat{\mathit{\beta }}}_{\text{ALL}}$ and ${\widehat{\mathit{\beta }}}_{\text{NAT}}$; then the estimated scaling factor for the ANT forcing is ${\widehat{\mathit{\beta }}}_{\text{ANT}}={\widehat{\mathit{\beta }}}_{\text{ALL}}$, and the estimated scaling factor for the NAT forcing is ${\widehat{\mathit{\beta }}}_{\text{ALL}}+{\widehat{\mathit{\beta }}}_{\text{NAT}}$. The variances and confidence intervals can be obtained according to the respective linear transformation of ${\widehat{\mathit{\beta }}}_{\text{ALL}}$ and ${\widehat{\mathit{\beta }}}_{\text{NAT}}$.

b. Estimation results

Figure 1 shows the estimated ANT and NAT scaling factors with 90% confidence intervals based on TLS-TS, TLS-PBC, and the proposed EE method. For both forcings, the point estimators of the three methods are different, especially on the subcontinental scale. Given the unbiasedness and robustness of the EE method demonstrated in simulation studies, its results are more trustable. It is also worth noting that the point estimators for the ANT forcing are around 0.5 across almost all regions, indicating that the signal of the simulated ANT forcing in the CanESM5 model is twice as big in magnitude as expected from observations. This discovery is consistent with existing literature indicating that CanESM4 warms foo fast (Gillett et al. 2021, Fig. 2). Regarding the confidence intervals, the EE method provides narrower intervals than TLS-PBC on most scales, which is supported by the potential conservativeness of the TLS-PBC intervals observed in simulation studies and discussed in Li et al. (2021). The often-narrower intervals of TLS-TS are questionable owing to the undercoverage issue discussed in the simulation studies.

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Estimated scaling factors with 90% confidence intervals of the ANT and NAT forcings at the global, continental, and subcontinental scales during 1951–2020. The confidence intervals for TLS-TS and TLS-PBC in WNA, ENA, and SCA regions are truncated for better visualization.

Citation: Journal of Climate 36, 20; 10.1175/JCLI-D-22-0681.1

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Across the detection and attribution analyses from the three methods, results are similar for the ANT forcing and substantially different for the NAT forcing, owing to its weaker signal. For the ANT forcing, TLS-PBC and the EE lead to both detection and attribution statements at WNA, while TLS-TS only supports the detection statement. At the ENA region, only TLS-PBC leads to the detection statement, while the other two methods do not, as their confidence intervals cover 0. In other regions, all three methods support the detection but not attribution statements for the ANT forcing. For the NAT forcing, the EE method claims its detection and attribution at NH, NHM, and NA regions; TLS-TS supports the detection and attribution statements at NA, CNA, WNA, and SEU regions; TLS-PBC does not support either detection or attribution statements in any region. Although we cannot claim which method is more accurate for this single analysis, we believe that a thorough comparison of the performances of these three methods has been reflected in the simulation studies.

c. Model diagnostics

Table 3 summarizes the p values of the three hypotheses described in section 2d. In particular, we consider two scenarios where the prewhitened residuals are calculated based on 1) estimated $\widehat{a}$ or 2) prefixed a = 1. The significance level is set as α = 0.05. The estimated $\widehat{a}$ is not close to 1 in most regions, except NHM and CNA. Accordingly, for those regions except NHM and CNA, we observe that both $H_0^1$ and $H_0^2$ (assuming a = 1) are rejected, and $H_0^2$ based on estimated $\widehat{a}$ is not rejected as expected. Thus, the assumption that a = 1 is violated for most regions in this dataset. Table 3 also supports the conclusion that there is no spatial autocorrelation in ${\mathbf{r}}^{*}(\widehat{\mathit{\beta }})$ in most regions. Hence, applying the proposed EE method to this dataset is reasonable, as all assumptions are valid, while the results of the two TLS methods are questionable owing to the violation of a = 1 in most regions.

Table 3.

Estimated a and p values of model diagnostic tests for the EE method.

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