**2. Methods**

#### *2.1. Variables of Interest*

The EP method uses a modified version of Kendall's *τ* statistics to test for independence of variables in a truncated data set. Here, *τ* is defined as:

$$\tau = \frac{\sum\_{i} \left( \mathcal{R}\_{i} - \mathcal{E}\_{i} \right)}{\sqrt{\sum\_{i} \mathcal{V}\_{i}}} \tag{2}$$

with *R<sup>i</sup>* defined as the rank E*<sup>i</sup>* = (1/2)(*i* + 1) defined as the expectation value, and V*<sup>i</sup>* = (1/12)(*i* <sup>2</sup> + 1) defined as the variance. For a more complete discussion of the method and the algebra involved, see Dainotti et al. [7,34], Efron and Petrosian [41], Singal et al. [45], Dainotti et al. [77], Petrosian et al. [78], Lloyd-Ronning and Petrosian [79]. Here, we use the EP method to determine the impact of redshift evolution and selection bias on four variables: *T* ∗ <sup>90</sup>, where the star denotes the rest-frame, *E*iso, the radio light curve break time *T* ∗ a,radio, and the radio luminosity at the time of break *L*a,radio. These variables are considered to be they are pertinent to the correlations analyzed in Levine et al. (2021 in preparation). Throughout our analysis, we consider these variables in logarithmic scale for convenience.

We look at the log *E*iso and log *T* ∗ <sup>90</sup> for a sample of 80 GRBs with observed radio afterglow published in the literature [80–98]. Values of log *E*iso are taken from the literature. If no *E*iso value could be found, the log *E*iso, in units of ergs, is calculated using the equation:

$$E\_{\rm iso} = 4\pi D\_L^2(z) \text{SK} \,\, \, \, \, \} \tag{3}$$

where *S* is the fluence in units of erg cm−<sup>2</sup> , *D*<sup>2</sup> L (*z*) is the luminosity distance assuming a flat ΛCDM model with Ω*<sup>M</sup>* = 0.3 and *H*<sup>0</sup> = 70 km s−<sup>1</sup> Mpc−<sup>1</sup> (see Equation (1)), and *K* is the correction for cosmic expansion [99]:

$$K = \frac{1}{(1+z)^{1-\beta}} \, ' \tag{4}$$

with *β* as the spectral index of the GRB. Fluence and *β* values are taken from the literature.

To analyze the impact of selection bias and redshift evolution for log *L*a,radio and log *T* ∗ a,radio, we first fit each of the 80 GRBs with a broken power law (BPL) according to the formulation:

$$F(t) = \begin{cases} F\_a(\frac{t}{T\_d^\*})^{-\alpha\_1} & t < T\_a^\* \\ F\_a(\frac{t}{T\_d^\*})^{-\alpha\_2} & t \ge T\_a^\* \end{cases} \tag{5}$$

where *F<sup>a</sup>* refers to the flux at the break, *T* ∗ *a* refers to the rest-frame time of break, *α*<sup>1</sup> refers to the slope before the break, and *α*<sup>2</sup> refers to the slope after the break. We can only obtain values of log *L*a,radio and log *T* ∗ a,radio for light curves that show a "plateau" feature, or a flattening of the light curve before a clear break. In our analysis, we consider a light curve to display a plateau if |*α*1| < 0.5. Therefore, we discard fits to light curves with scattered observations, unreliable error bars, or shapes incompatible with a BPL and plateau. In our subsequent analysis we include those light curves whose ∆*χ* <sup>2</sup> analysis of the BPL best-fit parameters are suitable following the Avni [100] methodology. After the rejection process, we are left with 18 GRBs that present a plateau and clear break in the light curve.

The luminosity log *L*a,radio in units of erg *s* −1 is computed at time log *T* ∗ a,radio using the equation:

$$L\_a = 4\pi D\_L^2(z) F\_d(T\_d) K\_\prime \tag{6}$$

where *F<sup>a</sup>* is the observed flux at *T*a,radio, *D*<sup>2</sup> L (*z*) is defined as in Equation (3), and K is the k-correction:

$$K = \frac{1}{(1+z)^{a\_1 - \beta}} \, ' \tag{7}$$

with *β* as the radio spectral index and *α*<sup>1</sup> as the fitted BPL temporal index before the break. *β* values are taken from Chandra and Frail [80] or other literature—if no *β* value could be found, the average of published spectral indices, *β* = 0.902 ± 0.17, was assigned. We show the distribution of spectral indices for the plateau sample in Figure 1.

**Figure 1.** Distribution of spectral indices (*β*) for sample of 18 GRBs that display a plateau in their light curve.
