Empirical Quantile Confidence Interval

NOTES:

• As of version 5.0, only Method 2 of the Empirical Quantile Confidence Interval (CI) calculation is provided.  This CI option is no longer considered a "Beta" feature.
  

In previous versions of Joinpoint, both Method 1 and Method 2 of the Empirical Quantile Confidence Interval were available.  As of version 5.0, only Method 2 is available and is simply called Empirical Quantile.  The help text below discusses both methods.

In Joinpoint version 4.2, a new method called the Empirical Quantile Method was implemented to construct a confidence interval for the true AAPC.  The motivation behind this was a conservative tendency of the asymptotic confidence interval for the AAPC.  As of Joinpoint version 4.6.0.0, the empirical quantile method was implemented to construct a confidence interval for the true APC and $\tau$ (the location of a joinpoint).  Along with that, two separate methods of the empirical quantile confidence interval are provided:  Method 1 and Method 2. In version 4.2, only Method 1 was provided.

Description of the Empirical Quantile Methods

The idea of the empirical quantile method is to generate resampled data by (i) generating resampled residuals as the inverse function values of the uniform random numbers over (0,1) where the function is the empirical distribution function of the original residuals and then (ii) adding resampled residuals to the original fit. For each resampled data set, the model is fit and the AAPC, APC and $\tau$ values are estimated. Then, the $100(\alpha /2)th$ and $100(1-\alpha /2)th$ percentiles of the resampled AAPC, APC and $\tau$ values are obtained as the lower and upper limits of the $100(1-\alpha )\mathrm{\%}$ empirical quantile confidence intervals for the true AAPC,  true APC and $\tau$.

For the empirical quantile method implemented in Joinpoint version 4.2, the inverse function values of the uniform random numbers in Step 2 described in Section 3 of Kim et al. (2017) are calculated using $u$ and $u{}^{\mathrm{\prime }}$, where $u$ and $u{}^{\mathrm{\prime }}$ are independent random numbers from the uniform distribution on (0,1).   This method is called Method 1.  Kim et al. (2017) also describe another way to calculate the inverse function values with   $u{}^{\mathrm{\prime }}=(u-\frac{i}{n+1})\ast (n+1)$ where $u$ is a random number from the uniform distribution on (0,1), and we call this Method 2.  Kim et al. (2017) provide simulation study results for the AAPC empirical quantile confidence intervals.  In order to improve the robustness of the resampled data, a modification is made in V 5.0 and its details can be found in Kim et al. (2022).

Additional simulation studies for the APC and $\tau$ empirical quantile confidence intervals show that (i) the coverage probabilities of the Method 2 confidence intervals for the true APC and $\tau$  are usually greater than those of the Method 1 confidence intervals, (ii) the coverage probabilities of the Method 1 confidence interval for the true APC were often much below the nominal level, especially when the number of observations is small, and (iii) the Method 2 confidence interval  for the true APC  is  conservative sometimes, although its coverage probabilities varied less than those of the Method 1 APC confidence interval.  Based on our simulation study, we recommend Method 2 based on its less liberal behavior for data over a relatively short period of time and robustness (Kim et al. (2022)).

The empirical quantile confidence interval for the true APC is shown to be more robust in terms of the segment length than the asymptotic parametric interval. In our simulation study, it was shown that the asymptotic parametric interval for the true APC is often very liberal when a segment is very short, but the empirical quantile confidence interval maintains the coverage probability to the nominal level in most cases.  For the confidence interval for $\tau$, the asymptotic confidence interval and the empirical quantile confidence interval were observed to perform comparably (Kim et al. (2022)).

For details regarding this method please see the Improved Confidence Interval for Average Annual Percent Change in Trend Analysis and Twenty years since Joinpoint 1.0: Two major enhancements, their justification, and impact articles.

To adjust the random number seed involved with producing the resampled residuals, please go to the Preferences help section.

Number of Resamples

As of version 5.1.0.0, the default number of resamples for the Empirical Quantile confidence intervals changed from 10,000 to 5,001.  This change considers computing time and the possibility of estimated p-values being exactly equal to the commonly considered value of 0.05.

The number of resampled data sets, described in the previous section, can be specified in the "# of Resamples" box under the "Empirical Quantile" option of the "APC/AAPC/Tau Confidence Intervals" within the "Method and Parameters" section of the Joinpoint session. 

Empirical Quantile Method P-Value

As of Joinpoint version 5.1.0.0, p-values can be calculated using an extension to the resampling method described by Kim et al. (2017, 2022). The "Description of the Empirical Quantile Methods'' section of this webpage provides details regarding the calculation of the lower and upper limits of the $100(1-\alpha )\mathrm{\%}$ empirical quantile confidence intervals for the true AAPC, true APC, and $\tau$. An extension of this resampling method allows for hypothesis tests involving the parameter of interest (i.e. the true AAPC or the true APC), denoted by $\theta$. Considering the null hypothesis $H_0:\theta =0$ versus the alternative hypothesis $H_1:\theta \ne 0$, the following test statistic is defined based on the resampling method:

$T=2\times \frac{min(\sum _{b=1}^{B}I({\hat{\theta }}^{(b)}<0),\sum _{b=1}^{B}I({\hat{\theta }}^{(b)}\ge 0))}{B},$

where $B$ denotes the number of resamples, $I(A)=1$ if $A$ is true and $I(A)=0$ otherwise, and ${\hat{\theta }}^{(b)}$ is the estimate of $\theta$ from resample $b$. The reported p-value is equal to the calculated value of $T$. Note that p-values that equal 0 are displayed as being less than 0.0001, and p-values that equal 1 are displayed as being greater than 0.9999.