The Pearson correlation coefficient squared,
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r ^{2}, is an important tool used in the analysis of neural data to quantify the similarity between neural tuning curves. Yet this metric is biased by trialtotrial variability; as trialtotrial variability increases, measured correlation decreases. Major lines of research are confounded by this bias, including those involving the study of invariance of neural tuning across conditions and the analysis of the similarity of tuning across neurons. To address this, we extend an estimator,${\widehat{r}}_{\text{ER}}^{2}$ , that was recently developed for estimating modeltoneuron correlation, in which a noisy signal is compared with a noisefree prediction, to the case of neurontoneuron correlation, in which two noisy signals are compared with each other. We compare the performance of our novel estimator to a prior method developed by Spearman, commonly used in other fields but widely overlooked in neuroscience, and find that our method has less bias. We then apply our estimator to demonstrate how it avoids drastic confounds introduced by trialtotrial variability using data collected in two prior studies (macaque, both sexes) that examined two different forms of invariance in the neural encoding of visual inputs—translation invariance and filloutline invariance. Our results quantify for the first time the gradual falloff with spatial offset of translationinvariant shape selectivity within visual cortical neuronal receptive fields and offer a principled method to compare invariance in noisy biological systems to that in noisefree models.SIGNIFICANCE STATEMENT Quantifying the similarity between two sets of averaged neural responses is fundamental to the analysis of neural data. A ubiquitous metric of similarity, the correlation coefficient, is attenuated by trialtotrial variability that arises from many irrelevant factors. Spearman recognized this problem and proposed corrected methods that have been extended over a century. We show this method has large asymptotic biases that can be overcome using a novel estimator. Despite the frequent use of the correlation coefficient in neuroscience, consensus on how to address this fundamental statistical issue has not been reached. We provide an accurate estimator of the correlation coefficient and apply it to gain insight into visual invariance. 

We report on spectroscopic measurements on the
$4{f}^{7}6{s}^{2}{\phantom{\rule{thickmathspace}{0ex}}}^{8}{\phantom{\rule{negativethinmathspace}{0ex}}S}_{7/2}^{\circ <\#comment/>}\to <\#comment/>4{f}^{7}{(}^{8}\phantom{\rule{negativethinmathspace}{0ex}}{S}^{\circ <\#comment/>})6s6p{(}^{1}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{P}^{\circ <\#comment/>}){\phantom{\rule{thinmathspace}{0ex}}}^{8}\phantom{\rule{negativethinmathspace}{0ex}}{P}_{9/2}$ transition in neutral europium151 and europium153 at 459.4 nm. The center of gravity frequencies for the 151 and 153 isotopes, reported for the first time in this paper, to our knowledge, were found to be 652,389,757.16(34) MHz and 652,386,593.2(5) MHz, respectively. The hyperfine coefficients for the$6s6p{(}^{1}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{P}^{\circ <\#comment/>}){\phantom{\rule{thinmathspace}{0ex}}}^{8}\phantom{\rule{negativethinmathspace}{0ex}}{P}_{9/2}$ state were found to be$\mathrm{A}(151)=<\#comment/>228.84(2)\phantom{\rule{thickmathspace}{0ex}}\mathrm{M}\mathrm{H}\mathrm{z}$ ,$\mathrm{B}(151)=226.9(5)\phantom{\rule{thickmathspace}{0ex}}\mathrm{M}\mathrm{H}\mathrm{z}$ and$\mathrm{A}(153)=<\#comment/>101.87(6)\phantom{\rule{thickmathspace}{0ex}}\mathrm{M}\mathrm{H}\mathrm{z}$ ,$\mathrm{B}(153)=575.4(1.5)\phantom{\rule{thickmathspace}{0ex}}\mathrm{M}\mathrm{H}\mathrm{z}$ , which all agree with previously published results except for A(153), which shows a small discrepancy. The isotope shift is found to be 3163.8(6) MHz, which also has a discrepancy with previously published results.