Examination of test statistics and p-values
Mark Ziemann & Anusuiya Bora
2024-07-09
Source: https://github.com/markziemann/enrichment_recipe
Introduction
The issue is that the 50% difference which we used as a classifier for reproduction of pathway p-values is not great when p-values are very small.
One possible solution is to use the test statistics instead. We can use the chi-squared test for this case, with repeated tests with different hypothetical tests, to ascertain the relationship between p-values and the test statistics.
library("kableExtra")Running the chi-squared test
In the next chunk, we are running a set of chi-squred tests, starting off with balanced data and making it progressively more skewed, which will decrease p-values.
l <- lapply(0:250, function(x) {
tulip <- c(500+x, 500-x)
res <- chisq.test(tulip, p = c(1/2, 1/2))
c("stat"=res$stat,"p"=res$p.value)
})
res <- as.data.frame(do.call(rbind,l))
res$logp <- -log10(res$p)Now we can make a plot of the relationship of the chi-squared test stats againse the -log10(p-values).
plot(res$`stat.X-squared`, res$logp ,
xlab="chi-squared test stat", ylab="-log10(p-value)")
Testing the bounds
Now we’ll test plus and minus 50% the of the test statistic value.
bound=0.5
res$lower <- res$`stat.X-squared` - ( res$`stat.X-squared` * bound )
res$upper <- res$`stat.X-squared` + ( res$`stat.X-squared` * bound )
plot(res$`stat.X-squared`, res$logp,
xlab="test stat", ylab="-log10(p)",
xlim=c(0,max(res$upper)))
points(res$lower, res$logp, col="blue")
points(res$upper, res$logp, col="blue")
res$logp_lower <- approx(x = res$`stat.X-squared`, y =res$logp, xout = res$lower)[["y"]]
res$logp_upper <- approx(x = res$`stat.X-squared`, y =res$logp, xout = res$upper)[["y"]]
res$p_lower <- 10^-res$logp_lower
res$p_upper <- 10^-res$logp_upper
format(res,digits=3) %>%
kbl(caption="p-value ranges considered reproduced with a bound of 50% around the test statistic") %>%
kable_paper("hover", full_width = F)| stat.X-squared | p | logp | lower | upper | logp_lower | logp_upper | p_lower | p_upper |
|---|---|---|---|---|---|---|---|---|
| 0.000 | 1.00e+00 | 0.0000 | 0.000 | 0.000 | 0.0000 | 0.0000 | 1.00e+00 | 1.00e+00 |
| 0.004 | 9.50e-01 | 0.0225 | 0.002 | 0.006 | 0.0112 | 0.0264 | 9.74e-01 | 9.41e-01 |
| 0.016 | 8.99e-01 | 0.0461 | 0.008 | 0.024 | 0.0303 | 0.0560 | 9.33e-01 | 8.79e-01 |
| 0.036 | 8.50e-01 | 0.0708 | 0.018 | 0.054 | 0.0485 | 0.0875 | 8.94e-01 | 8.18e-01 |
| 0.064 | 8.00e-01 | 0.0968 | 0.032 | 0.096 | 0.0659 | 0.1209 | 8.59e-01 | 7.57e-01 |
| 0.100 | 7.52e-01 | 0.1239 | 0.050 | 0.150 | 0.0838 | 0.1556 | 8.25e-01 | 6.99e-01 |
| 0.144 | 7.04e-01 | 0.1522 | 0.072 | 0.216 | 0.1028 | 0.1921 | 7.89e-01 | 6.43e-01 |
| 0.196 | 6.58e-01 | 0.1818 | 0.098 | 0.294 | 0.1224 | 0.2306 | 7.54e-01 | 5.88e-01 |
| 0.256 | 6.13e-01 | 0.2126 | 0.128 | 0.384 | 0.1419 | 0.2711 | 7.21e-01 | 5.36e-01 |
| 0.324 | 5.69e-01 | 0.2447 | 0.162 | 0.486 | 0.1625 | 0.3136 | 6.88e-01 | 4.86e-01 |
| 0.400 | 5.27e-01 | 0.2781 | 0.200 | 0.600 | 0.1838 | 0.3578 | 6.55e-01 | 4.39e-01 |
| 0.484 | 4.87e-01 | 0.3128 | 0.242 | 0.726 | 0.2054 | 0.4041 | 6.23e-01 | 3.94e-01 |
| 0.576 | 4.48e-01 | 0.3488 | 0.288 | 0.864 | 0.2277 | 0.4525 | 5.92e-01 | 3.53e-01 |
| 0.676 | 4.11e-01 | 0.3862 | 0.338 | 1.014 | 0.2509 | 0.5031 | 5.61e-01 | 3.14e-01 |
| 0.784 | 3.76e-01 | 0.4249 | 0.392 | 1.176 | 0.2746 | 0.5556 | 5.31e-01 | 2.78e-01 |
| 0.900 | 3.43e-01 | 0.4650 | 0.450 | 1.350 | 0.2988 | 0.6102 | 5.03e-01 | 2.45e-01 |
| 1.024 | 3.12e-01 | 0.5064 | 0.512 | 1.536 | 0.3238 | 0.6670 | 4.74e-01 | 2.15e-01 |
| 1.156 | 2.82e-01 | 0.5493 | 0.578 | 1.734 | 0.3496 | 0.7260 | 4.47e-01 | 1.88e-01 |
| 1.296 | 2.55e-01 | 0.5936 | 0.648 | 1.944 | 0.3757 | 0.7872 | 4.21e-01 | 1.63e-01 |
| 1.444 | 2.29e-01 | 0.6392 | 0.722 | 2.166 | 0.4027 | 0.8504 | 3.96e-01 | 1.41e-01 |
| 1.600 | 2.06e-01 | 0.6863 | 0.800 | 2.400 | 0.4304 | 0.9159 | 3.71e-01 | 1.21e-01 |
| 1.764 | 1.84e-01 | 0.7349 | 0.882 | 2.646 | 0.4588 | 0.9837 | 3.48e-01 | 1.04e-01 |
| 1.936 | 1.64e-01 | 0.7849 | 0.968 | 2.904 | 0.4877 | 1.0537 | 3.25e-01 | 8.84e-02 |
| 2.116 | 1.46e-01 | 0.8363 | 1.058 | 3.174 | 0.5175 | 1.1259 | 3.04e-01 | 7.48e-02 |
| 2.304 | 1.29e-01 | 0.8893 | 1.152 | 3.456 | 0.5480 | 1.2004 | 2.83e-01 | 6.30e-02 |
| 2.500 | 1.14e-01 | 0.9437 | 1.250 | 3.750 | 0.5790 | 1.2772 | 2.64e-01 | 5.28e-02 |
| 2.704 | 1.00e-01 | 0.9996 | 1.352 | 4.056 | 0.6108 | 1.3564 | 2.45e-01 | 4.40e-02 |
| 2.916 | 8.77e-02 | 1.0570 | 1.458 | 4.374 | 0.6435 | 1.4378 | 2.27e-01 | 3.65e-02 |
| 3.136 | 7.66e-02 | 1.1159 | 1.568 | 4.704 | 0.6767 | 1.5215 | 2.11e-01 | 3.01e-02 |
| 3.364 | 6.66e-02 | 1.1763 | 1.682 | 5.046 | 0.7106 | 1.6075 | 1.95e-01 | 2.47e-02 |
| 3.600 | 5.78e-02 | 1.2382 | 1.800 | 5.400 | 0.7453 | 1.6960 | 1.80e-01 | 2.01e-02 |
| 3.844 | 4.99e-02 | 1.3017 | 1.922 | 5.766 | 0.7808 | 1.7868 | 1.66e-01 | 1.63e-02 |
| 4.096 | 4.30e-02 | 1.3667 | 2.048 | 6.144 | 0.8169 | 1.8799 | 1.52e-01 | 1.32e-02 |
| 4.356 | 3.69e-02 | 1.4332 | 2.178 | 6.534 | 0.8538 | 1.9753 | 1.40e-01 | 1.06e-02 |
| 4.624 | 3.15e-02 | 1.5013 | 2.312 | 6.936 | 0.8915 | 2.0732 | 1.28e-01 | 8.45e-03 |
| 4.900 | 2.69e-02 | 1.5709 | 2.450 | 7.350 | 0.9298 | 2.1735 | 1.18e-01 | 6.71e-03 |
| 5.184 | 2.28e-02 | 1.6421 | 2.592 | 7.776 | 0.9689 | 2.2762 | 1.07e-01 | 5.29e-03 |
| 5.476 | 1.93e-02 | 1.7149 | 2.738 | 8.214 | 1.0088 | 2.3812 | 9.80e-02 | 4.16e-03 |
| 5.776 | 1.62e-02 | 1.7892 | 2.888 | 8.664 | 1.0494 | 2.4887 | 8.92e-02 | 3.25e-03 |
| 6.084 | 1.36e-02 | 1.8651 | 3.042 | 9.126 | 1.0907 | 2.5986 | 8.12e-02 | 2.52e-03 |
| 6.400 | 1.14e-02 | 1.9426 | 3.200 | 9.600 | 1.1328 | 2.7109 | 7.36e-02 | 1.95e-03 |
| 6.724 | 9.51e-03 | 2.0217 | 3.362 | 10.086 | 1.1758 | 2.8256 | 6.67e-02 | 1.49e-03 |
| 7.056 | 7.90e-03 | 2.1024 | 3.528 | 10.584 | 1.2193 | 2.9428 | 6.03e-02 | 1.14e-03 |
| 7.396 | 6.54e-03 | 2.1846 | 3.698 | 11.094 | 1.2637 | 3.0624 | 5.45e-02 | 8.66e-04 |
| 7.744 | 5.39e-03 | 2.2685 | 3.872 | 11.616 | 1.3089 | 3.1845 | 4.91e-02 | 6.54e-04 |
| 8.100 | 4.43e-03 | 2.3539 | 4.050 | 12.150 | 1.3548 | 3.3090 | 4.42e-02 | 4.91e-04 |
| 8.464 | 3.62e-03 | 2.4410 | 4.232 | 12.696 | 1.4015 | 3.4360 | 3.97e-02 | 3.66e-04 |
| 8.836 | 2.95e-03 | 2.5297 | 4.418 | 13.254 | 1.4490 | 3.5654 | 3.56e-02 | 2.72e-04 |
| 9.216 | 2.40e-03 | 2.6200 | 4.608 | 13.824 | 1.4972 | 3.6973 | 3.18e-02 | 2.01e-04 |
| 9.604 | 1.94e-03 | 2.7119 | 4.802 | 14.406 | 1.5462 | 3.8317 | 2.84e-02 | 1.47e-04 |
| 10.000 | 1.57e-03 | 2.8054 | 5.000 | 15.000 | 1.5960 | 3.9685 | 2.54e-02 | 1.08e-04 |
| 10.404 | 1.26e-03 | 2.9005 | 5.202 | 15.606 | 1.6466 | 4.1078 | 2.26e-02 | 7.80e-05 |
| 10.816 | 1.01e-03 | 2.9973 | 5.408 | 16.224 | 1.6980 | 4.2497 | 2.00e-02 | 5.63e-05 |
| 11.236 | 8.02e-04 | 3.0957 | 5.618 | 16.854 | 1.7501 | 4.3940 | 1.78e-02 | 4.04e-05 |
| 11.664 | 6.37e-04 | 3.1957 | 5.832 | 17.496 | 1.8030 | 4.5407 | 1.57e-02 | 2.88e-05 |
| 12.100 | 5.04e-04 | 3.2974 | 6.050 | 18.150 | 1.8568 | 4.6900 | 1.39e-02 | 2.04e-05 |
| 12.544 | 3.97e-04 | 3.4007 | 6.272 | 18.816 | 1.9112 | 4.8418 | 1.23e-02 | 1.44e-05 |
| 12.996 | 3.12e-04 | 3.5056 | 6.498 | 19.494 | 1.9666 | 4.9960 | 1.08e-02 | 1.01e-05 |
| 13.456 | 2.44e-04 | 3.6122 | 6.728 | 20.184 | 2.0227 | 5.1528 | 9.49e-03 | 7.03e-06 |
| 13.924 | 1.90e-04 | 3.7204 | 6.962 | 20.886 | 2.0795 | 5.3121 | 8.33e-03 | 4.87e-06 |
| 14.400 | 1.48e-04 | 3.8303 | 7.200 | 21.600 | 2.1372 | 5.4738 | 7.29e-03 | 3.36e-06 |
| 14.884 | 1.14e-04 | 3.9418 | 7.442 | 22.326 | 2.1957 | 5.6381 | 6.37e-03 | 2.30e-06 |
| 15.376 | 8.81e-05 | 4.0550 | 7.688 | 23.064 | 2.2550 | 5.8049 | 5.56e-03 | 1.57e-06 |
| 15.876 | 6.76e-05 | 4.1699 | 7.938 | 23.814 | 2.3150 | 5.9742 | 4.84e-03 | 1.06e-06 |
| 16.384 | 5.17e-05 | 4.2863 | 8.192 | 24.576 | 2.3759 | 6.1461 | 4.21e-03 | 7.14e-07 |
| 16.900 | 3.94e-05 | 4.4045 | 8.450 | 25.350 | 2.4376 | 6.3204 | 3.65e-03 | 4.78e-07 |
| 17.424 | 2.99e-05 | 4.5243 | 8.712 | 26.136 | 2.5001 | 6.4973 | 3.16e-03 | 3.18e-07 |
| 17.956 | 2.26e-05 | 4.6458 | 8.978 | 26.934 | 2.5634 | 6.6767 | 2.73e-03 | 2.11e-07 |
| 18.496 | 1.70e-05 | 4.7689 | 9.248 | 27.744 | 2.6275 | 6.8586 | 2.36e-03 | 1.38e-07 |
| 19.044 | 1.28e-05 | 4.8937 | 9.522 | 28.566 | 2.6924 | 7.0431 | 2.03e-03 | 9.06e-08 |
| 19.600 | 9.55e-06 | 5.0201 | 9.800 | 29.400 | 2.7581 | 7.2300 | 1.75e-03 | 5.89e-08 |
| 20.164 | 7.11e-06 | 5.1483 | 10.082 | 30.246 | 2.8247 | 7.4196 | 1.50e-03 | 3.81e-08 |
| 20.736 | 5.27e-06 | 5.2781 | 10.368 | 31.104 | 2.8920 | 7.6116 | 1.28e-03 | 2.45e-08 |
| 21.316 | 3.89e-06 | 5.4095 | 10.658 | 31.974 | 2.9602 | 7.8062 | 1.10e-03 | 1.56e-08 |
| 21.904 | 2.87e-06 | 5.5427 | 10.952 | 32.856 | 3.0291 | 8.0033 | 9.35e-04 | 9.92e-09 |
| 22.500 | 2.10e-06 | 5.6775 | 11.250 | 33.750 | 3.0990 | 8.2029 | 7.96e-04 | 6.27e-09 |
| 23.104 | 1.53e-06 | 5.8140 | 11.552 | 34.656 | 3.1695 | 8.4051 | 6.77e-04 | 3.93e-09 |
| 23.716 | 1.12e-06 | 5.9521 | 11.858 | 35.574 | 3.2410 | 8.6099 | 5.74e-04 | 2.46e-09 |
| 24.336 | 8.09e-07 | 6.0920 | 12.168 | 36.504 | 3.3132 | 8.8171 | 4.86e-04 | 1.52e-09 |
| 24.964 | 5.84e-07 | 6.2335 | 12.482 | 37.446 | 3.3863 | 9.0270 | 4.11e-04 | 9.40e-10 |
| 25.600 | 4.20e-07 | 6.3767 | 12.800 | 38.400 | 3.4601 | 9.2393 | 3.47e-04 | 5.76e-10 |
| 26.244 | 3.01e-07 | 6.5216 | 13.122 | 39.366 | 3.5348 | 9.4542 | 2.92e-04 | 3.51e-10 |
| 26.896 | 2.15e-07 | 6.6682 | 13.448 | 40.344 | 3.6104 | 9.6717 | 2.45e-04 | 2.13e-10 |
| 27.556 | 1.53e-07 | 6.8164 | 13.778 | 41.334 | 3.6867 | 9.8917 | 2.06e-04 | 1.28e-10 |
| 28.224 | 1.08e-07 | 6.9663 | 14.112 | 42.336 | 3.7638 | 10.1143 | 1.72e-04 | 7.69e-11 |
| 28.900 | 7.62e-08 | 7.1180 | 14.450 | 43.350 | 3.8418 | 10.3394 | 1.44e-04 | 4.58e-11 |
| 29.584 | 5.35e-08 | 7.2713 | 14.792 | 44.376 | 3.9206 | 10.5670 | 1.20e-04 | 2.71e-11 |
| 30.276 | 3.75e-08 | 7.4263 | 15.138 | 45.414 | 4.0003 | 10.7973 | 9.99e-05 | 1.59e-11 |
| 30.976 | 2.61e-08 | 7.5830 | 15.488 | 46.464 | 4.0807 | 11.0300 | 8.30e-05 | 9.33e-12 |
| 31.684 | 1.81e-08 | 7.7413 | 15.842 | 47.526 | 4.1620 | 11.2654 | 6.89e-05 | 5.43e-12 |
| 32.400 | 1.25e-08 | 7.9014 | 16.200 | 48.600 | 4.2441 | 11.5032 | 5.70e-05 | 3.14e-12 |
| 33.124 | 8.65e-09 | 8.0632 | 16.562 | 49.686 | 4.3271 | 11.7437 | 4.71e-05 | 1.80e-12 |
| 33.856 | 5.93e-09 | 8.2266 | 16.928 | 50.784 | 4.4109 | 11.9867 | 3.88e-05 | 1.03e-12 |
| 34.596 | 4.06e-09 | 8.3918 | 17.298 | 51.894 | 4.4955 | 12.2323 | 3.20e-05 | 5.86e-13 |
| 35.344 | 2.76e-09 | 8.5586 | 17.672 | 53.016 | 4.5809 | 12.4804 | 2.62e-05 | 3.31e-13 |
| 36.100 | 1.87e-09 | 8.7271 | 18.050 | 54.150 | 4.6672 | 12.7311 | 2.15e-05 | 1.86e-13 |
| 36.864 | 1.27e-09 | 8.8973 | 18.432 | 55.296 | 4.7543 | 12.9843 | 1.76e-05 | 1.04e-13 |
| 37.636 | 8.53e-10 | 9.0693 | 18.818 | 56.454 | 4.8422 | 13.2401 | 1.44e-05 | 5.75e-14 |
| 38.416 | 5.72e-10 | 9.2429 | 19.208 | 57.624 | 4.9310 | 13.4985 | 1.17e-05 | 3.17e-14 |
| 39.204 | 3.82e-10 | 9.4182 | 19.602 | 58.806 | 5.0206 | 13.7594 | 9.54e-06 | 1.74e-14 |
| 40.000 | 2.54e-10 | 9.5952 | 20.000 | 60.000 | 5.1110 | 14.0229 | 7.74e-06 | 9.49e-15 |
| 40.804 | 1.68e-10 | 9.7739 | 20.402 | 61.206 | 5.2023 | 14.2890 | 6.28e-06 | 5.14e-15 |
| 41.616 | 1.11e-10 | 9.9544 | 20.808 | 62.424 | 5.2944 | 14.5576 | 5.08e-06 | 2.77e-15 |
| 42.436 | 7.30e-11 | 10.1365 | 21.218 | 63.654 | 5.3873 | 14.8288 | 4.10e-06 | 1.48e-15 |
| 43.264 | 4.78e-11 | 10.3203 | 21.632 | 64.896 | 5.4811 | 15.1026 | 3.30e-06 | 7.90e-16 |
| 44.100 | 3.12e-11 | 10.5058 | 22.050 | 66.150 | 5.5757 | 15.3789 | 2.66e-06 | 4.18e-16 |
| 44.944 | 2.03e-11 | 10.6930 | 22.472 | 67.416 | 5.6712 | 15.6579 | 2.13e-06 | 2.20e-16 |
| 45.796 | 1.31e-11 | 10.8820 | 22.898 | 68.694 | 5.7674 | 15.9393 | 1.71e-06 | 1.15e-16 |
| 46.656 | 8.46e-12 | 11.0726 | 23.328 | 69.984 | 5.8645 | 16.2234 | 1.37e-06 | 5.98e-17 |
| 47.524 | 5.43e-12 | 11.2649 | 23.762 | 71.286 | 5.9625 | 16.5100 | 1.09e-06 | 3.09e-17 |
| 48.400 | 3.48e-12 | 11.4590 | 24.200 | 72.600 | 6.0613 | 16.7992 | 8.68e-07 | 1.59e-17 |
| 49.284 | 2.21e-12 | 11.6547 | 24.642 | 73.926 | 6.1609 | 17.0910 | 6.90e-07 | 8.11e-18 |
| 50.176 | 1.41e-12 | 11.8521 | 25.088 | 75.264 | 6.2614 | 17.3853 | 5.48e-07 | 4.12e-18 |
| 51.076 | 8.89e-13 | 12.0513 | 25.538 | 76.614 | 6.3627 | 17.6822 | 4.34e-07 | 2.08e-18 |
| 51.984 | 5.60e-13 | 12.2522 | 25.992 | 77.976 | 6.4649 | 17.9817 | 3.43e-07 | 1.04e-18 |
| 52.900 | 3.51e-13 | 12.4547 | 26.450 | 79.350 | 6.5679 | 18.2838 | 2.70e-07 | 5.20e-19 |
| 53.824 | 2.19e-13 | 12.6590 | 26.912 | 80.736 | 6.6718 | 18.5884 | 2.13e-07 | 2.58e-19 |
| 54.756 | 1.36e-13 | 12.8650 | 27.378 | 82.134 | 6.7764 | 18.8956 | 1.67e-07 | 1.27e-19 |
| 55.696 | 8.46e-14 | 13.0727 | 27.848 | 83.544 | 6.8820 | 19.2054 | 1.31e-07 | 6.23e-20 |
| 56.644 | 5.22e-14 | 13.2821 | 28.322 | 84.966 | 6.9883 | 19.5178 | 1.03e-07 | 3.04e-20 |
| 57.600 | 3.21e-14 | 13.4932 | 28.800 | 86.400 | 7.0955 | 19.8327 | 8.03e-08 | 1.47e-20 |
| 58.564 | 1.97e-14 | 13.7060 | 29.282 | 87.846 | 7.2036 | 20.1502 | 6.26e-08 | 7.08e-21 |
| 59.536 | 1.20e-14 | 13.9205 | 29.768 | 89.304 | 7.3125 | 20.4703 | 4.87e-08 | 3.39e-21 |
| 60.516 | 7.30e-15 | 14.1368 | 30.258 | 90.774 | 7.4222 | 20.7930 | 3.78e-08 | 1.61e-21 |
| 61.504 | 4.42e-15 | 14.3547 | 30.752 | 92.256 | 7.5328 | 21.1183 | 2.93e-08 | 7.62e-22 |
| 62.500 | 2.66e-15 | 14.5744 | 31.250 | 93.750 | 7.6443 | 21.4461 | 2.27e-08 | 3.58e-22 |
| 63.504 | 1.60e-15 | 14.7958 | 31.752 | 95.256 | 7.7565 | 21.7765 | 1.75e-08 | 1.67e-22 |
| 64.516 | 9.58e-16 | 15.0189 | 32.258 | 96.774 | 7.8697 | 22.1095 | 1.35e-08 | 7.77e-23 |
| 65.536 | 5.71e-16 | 15.2437 | 32.768 | 98.304 | 7.9836 | 22.4451 | 1.04e-08 | 3.59e-23 |
| 66.564 | 3.39e-16 | 15.4702 | 33.282 | 99.846 | 8.0984 | 22.7833 | 7.97e-09 | 1.65e-23 |
| 67.600 | 2.00e-16 | 15.6984 | 33.800 | 101.400 | 8.2141 | 23.1240 | 6.11e-09 | 7.52e-24 |
| 68.644 | 1.18e-16 | 15.9283 | 34.322 | 102.966 | 8.3306 | 23.4673 | 4.67e-09 | 3.41e-24 |
| 69.696 | 6.92e-17 | 16.1600 | 34.848 | 104.544 | 8.4480 | 23.8132 | 3.56e-09 | 1.54e-24 |
| 70.756 | 4.04e-17 | 16.3933 | 35.378 | 106.134 | 8.5662 | 24.1617 | 2.72e-09 | 6.89e-25 |
| 71.824 | 2.35e-17 | 16.6284 | 35.912 | 107.736 | 8.6852 | 24.5128 | 2.06e-09 | 3.07e-25 |
| 72.900 | 1.36e-17 | 16.8652 | 36.450 | 109.350 | 8.8051 | 24.8664 | 1.57e-09 | 1.36e-25 |
| 73.984 | 7.88e-18 | 17.1037 | 36.992 | 110.976 | 8.9259 | 25.2226 | 1.19e-09 | 5.99e-26 |
| 75.076 | 4.53e-18 | 17.3440 | 37.538 | 112.614 | 9.0474 | 25.5814 | 8.97e-10 | 2.62e-26 |
| 76.176 | 2.59e-18 | 17.5859 | 38.088 | 114.264 | 9.1699 | 25.9428 | 6.76e-10 | 1.14e-26 |
| 77.284 | 1.48e-18 | 17.8296 | 38.642 | 115.926 | 9.2932 | 26.3068 | 5.09e-10 | 4.93e-27 |
| 78.400 | 8.42e-19 | 18.0749 | 39.200 | 117.600 | 9.4173 | 26.6734 | 3.83e-10 | 2.12e-27 |
| 79.524 | 4.76e-19 | 18.3220 | 39.762 | 119.286 | 9.5423 | 27.0425 | 2.87e-10 | 9.07e-28 |
| 80.656 | 2.69e-19 | 18.5708 | 40.328 | 120.984 | 9.6681 | 27.4143 | 2.15e-10 | 3.85e-28 |
| 81.796 | 1.51e-19 | 18.8214 | 40.898 | 122.694 | 9.7948 | 27.7886 | 1.60e-10 | 1.63e-28 |
| 82.944 | 8.44e-20 | 19.0736 | 41.472 | 124.416 | 9.9224 | 28.1655 | 1.20e-10 | 6.83e-29 |
| 84.100 | 4.70e-20 | 19.3276 | 42.050 | 126.150 | 10.0508 | 28.5450 | 8.90e-11 | 2.85e-29 |
| 85.264 | 2.61e-20 | 19.5832 | 42.632 | 127.896 | 10.1800 | 28.9271 | 6.61e-11 | 1.18e-29 |
| 86.436 | 1.44e-20 | 19.8406 | 43.218 | 129.654 | 10.3101 | 29.3117 | 4.90e-11 | 4.88e-30 |
| 87.616 | 7.95e-21 | 20.0997 | 43.808 | 131.424 | 10.4410 | 29.6990 | 3.62e-11 | 2.00e-30 |
| 88.804 | 4.36e-21 | 20.3606 | 44.402 | 133.206 | 10.5728 | 30.0888 | 2.67e-11 | 8.15e-31 |
| 90.000 | 2.38e-21 | 20.6231 | 45.000 | 135.000 | 10.7055 | 30.4813 | 1.97e-11 | 3.30e-31 |
| 91.204 | 1.30e-21 | 20.8874 | 45.602 | 136.806 | 10.8389 | 30.8763 | 1.45e-11 | 1.33e-31 |
| 92.416 | 7.02e-22 | 21.1534 | 46.208 | 138.624 | 10.9733 | 31.2739 | 1.06e-11 | 5.32e-32 |
| 93.636 | 3.79e-22 | 21.4211 | 46.818 | 140.454 | 11.1085 | 31.6741 | 7.79e-12 | 2.12e-32 |
| 94.864 | 2.04e-22 | 21.6905 | 47.432 | 142.296 | 11.2445 | 32.0768 | 5.69e-12 | 8.38e-33 |
| 96.100 | 1.09e-22 | 21.9617 | 48.050 | 144.150 | 11.3814 | 32.4822 | 4.15e-12 | 3.29e-33 |
| 97.344 | 5.83e-23 | 22.2345 | 48.672 | 146.016 | 11.5192 | 32.8902 | 3.03e-12 | 1.29e-33 |
| 98.596 | 3.10e-23 | 22.5091 | 49.298 | 147.894 | 11.6578 | 33.3007 | 2.20e-12 | 5.00e-34 |
| 99.856 | 1.64e-23 | 22.7854 | 49.928 | 149.784 | 11.7973 | 33.7138 | 1.59e-12 | 1.93e-34 |
| 101.124 | 8.64e-24 | 23.0635 | 50.562 | 151.686 | 11.9376 | 34.1295 | 1.15e-12 | 7.42e-35 |
| 102.400 | 4.54e-24 | 23.3432 | 51.200 | 153.600 | 12.0787 | 34.5479 | 8.34e-13 | 2.83e-35 |
| 103.684 | 2.37e-24 | 23.6247 | 51.842 | 155.526 | 12.2208 | 34.9688 | 6.02e-13 | 1.07e-35 |
| 104.976 | 1.24e-24 | 23.9079 | 52.488 | 157.464 | 12.3636 | 35.3922 | 4.33e-13 | 4.05e-36 |
| 106.276 | 6.41e-25 | 24.1928 | 53.138 | 159.414 | 12.5073 | 35.8183 | 3.11e-13 | 1.52e-36 |
| 107.584 | 3.32e-25 | 24.4794 | 53.792 | 161.376 | 12.6519 | 36.2470 | 2.23e-13 | 5.66e-37 |
| 108.900 | 1.71e-25 | 24.7678 | 54.450 | 163.350 | 12.7974 | 36.6782 | 1.59e-13 | 2.10e-37 |
| 110.224 | 8.75e-26 | 25.0579 | 55.112 | 165.336 | 12.9436 | 37.1121 | 1.14e-13 | 7.73e-38 |
| 111.556 | 4.47e-26 | 25.3497 | 55.778 | 167.334 | 13.0908 | 37.5485 | 8.11e-14 | 2.83e-38 |
| 112.896 | 2.27e-26 | 25.6432 | 56.448 | 169.344 | 13.2388 | 37.9876 | 5.77e-14 | 1.03e-38 |
| 114.244 | 1.15e-26 | 25.9385 | 57.122 | 171.366 | 13.3876 | 38.4292 | 4.10e-14 | 3.72e-39 |
| 115.600 | 5.82e-27 | 26.2354 | 57.800 | 173.400 | 13.5373 | 38.8734 | 2.90e-14 | 1.34e-39 |
| 116.964 | 2.92e-27 | 26.5341 | 58.482 | 175.446 | 13.6879 | 39.3202 | 2.05e-14 | 4.78e-40 |
| 118.336 | 1.46e-27 | 26.8345 | 59.168 | 177.504 | 13.8393 | 39.7696 | 1.45e-14 | 1.70e-40 |
| 119.716 | 7.30e-28 | 27.1367 | 59.858 | 179.574 | 13.9916 | 40.2216 | 1.02e-14 | 6.00e-41 |
| 121.104 | 3.63e-28 | 27.4406 | 60.552 | 181.656 | 14.1447 | 40.6761 | 7.17e-15 | 2.11e-41 |
| 122.500 | 1.79e-28 | 27.7461 | 61.250 | 183.750 | 14.2987 | 41.1333 | 5.03e-15 | 7.36e-42 |
| 123.904 | 8.84e-29 | 28.0535 | 61.952 | 185.856 | 14.4535 | 41.5931 | 3.52e-15 | 2.55e-42 |
| 125.316 | 4.34e-29 | 28.3625 | 62.658 | 187.974 | 14.6092 | 42.0554 | 2.46e-15 | 8.80e-43 |
| 126.736 | 2.12e-29 | 28.6732 | 63.368 | 190.104 | 14.7658 | 42.5204 | 1.71e-15 | 3.02e-43 |
| 128.164 | 1.03e-29 | 28.9857 | 64.082 | 192.246 | 14.9232 | 42.9879 | 1.19e-15 | 1.03e-43 |
| 129.600 | 5.01e-30 | 29.2999 | 64.800 | 194.400 | 15.0814 | 43.4580 | 8.29e-16 | 3.48e-44 |
| 131.044 | 2.42e-30 | 29.6159 | 65.522 | 196.566 | 15.2406 | 43.9308 | 5.75e-16 | 1.17e-44 |
| 132.496 | 1.17e-30 | 29.9335 | 66.248 | 198.744 | 15.4005 | 44.4061 | 3.98e-16 | 3.93e-45 |
| 133.956 | 5.59e-31 | 30.2529 | 66.978 | 200.934 | 15.5614 | 44.8840 | 2.75e-16 | 1.31e-45 |
| 135.424 | 2.67e-31 | 30.5740 | 67.712 | 203.136 | 15.7231 | 45.3645 | 1.89e-16 | 4.32e-46 |
| 136.900 | 1.27e-31 | 30.8968 | 68.450 | 205.350 | 15.8856 | 45.8476 | 1.30e-16 | 1.42e-46 |
| 138.384 | 6.01e-32 | 31.2214 | 69.192 | 207.576 | 16.0490 | 46.3333 | 8.93e-17 | 4.64e-47 |
| 139.876 | 2.83e-32 | 31.5477 | 69.938 | 209.814 | 16.2133 | 46.8216 | 6.12e-17 | 1.51e-47 |
| 141.376 | 1.33e-32 | 31.8757 | 70.688 | 212.064 | 16.3784 | 47.3124 | 4.18e-17 | 4.87e-48 |
| 142.884 | 6.23e-33 | 32.2054 | 71.442 | 214.326 | 16.5443 | 47.8059 | 2.86e-17 | 1.56e-48 |
| 144.400 | 2.90e-33 | 32.5369 | 72.200 | 216.600 | 16.7112 | 48.3020 | 1.94e-17 | 4.99e-49 |
| 145.924 | 1.35e-33 | 32.8700 | 72.962 | 218.886 | 16.8789 | 48.8006 | 1.32e-17 | 1.58e-49 |
| 147.456 | 6.24e-34 | 33.2050 | 73.728 | 221.184 | 17.0474 | 49.3019 | 8.97e-18 | 4.99e-50 |
| 148.996 | 2.87e-34 | 33.5416 | 74.498 | 223.494 | 17.2168 | 49.8057 | 6.07e-18 | 1.56e-50 |
| 150.544 | 1.32e-34 | 33.8799 | 75.272 | 225.816 | 17.3871 | 50.3122 | 4.10e-18 | 4.87e-51 |
| 152.100 | 6.03e-35 | 34.2200 | 76.050 | 228.150 | 17.5582 | 50.8212 | 2.77e-18 | 1.51e-51 |
| 153.664 | 2.74e-35 | 34.5618 | 76.832 | 230.496 | 17.7302 | 51.3328 | 1.86e-18 | 4.65e-52 |
| 155.236 | 1.24e-35 | 34.9054 | 77.618 | 232.854 | 17.9030 | 51.8470 | 1.25e-18 | 1.42e-52 |
| 156.816 | 5.62e-36 | 35.2506 | 78.408 | 235.224 | 18.0767 | 52.3639 | 8.38e-19 | 4.33e-53 |
| 158.404 | 2.53e-36 | 35.5976 | 79.202 | 237.606 | 18.2512 | 52.8833 | 5.61e-19 | 1.31e-53 |
| 160.000 | 1.13e-36 | 35.9464 | 80.000 | 240.000 | 18.4266 | 53.4053 | 3.74e-19 | 3.93e-54 |
| 161.604 | 5.05e-37 | 36.2968 | 80.802 | 242.406 | 18.6029 | 53.9299 | 2.50e-19 | 1.18e-54 |
| 163.216 | 2.24e-37 | 36.6490 | 81.608 | 244.824 | 18.7800 | 54.4571 | 1.66e-19 | 3.49e-55 |
| 164.836 | 9.93e-38 | 37.0029 | 82.418 | 247.254 | 18.9580 | 54.9869 | 1.10e-19 | 1.03e-55 |
| 166.464 | 4.38e-38 | 37.3585 | 83.232 | 249.696 | 19.1369 | 55.5193 | 7.30e-20 | 3.02e-56 |
| 168.100 | 1.92e-38 | 37.7158 | 84.050 | 252.150 | 19.3166 | NA | 4.82e-20 | NA |
| 169.744 | 8.42e-39 | 38.0749 | 84.872 | 254.616 | 19.4971 | NA | 3.18e-20 | NA |
| 171.396 | 3.67e-39 | 38.4357 | 85.698 | 257.094 | 19.6786 | NA | 2.10e-20 | NA |
| 173.056 | 1.59e-39 | 38.7983 | 86.528 | 259.584 | 19.8608 | NA | 1.38e-20 | NA |
| 174.724 | 6.88e-40 | 39.1625 | 87.362 | 262.086 | 20.0440 | NA | 9.04e-21 | NA |
| 176.400 | 2.96e-40 | 39.5285 | 88.200 | 264.600 | 20.2280 | NA | 5.92e-21 | NA |
| 178.084 | 1.27e-40 | 39.8962 | 89.042 | 267.126 | 20.4128 | NA | 3.87e-21 | NA |
| 179.776 | 5.42e-41 | 40.2657 | 89.888 | 269.664 | 20.5985 | NA | 2.52e-21 | NA |
| 181.476 | 2.31e-41 | 40.6368 | 90.738 | 272.214 | 20.7851 | NA | 1.64e-21 | NA |
| 183.184 | 9.78e-42 | 41.0097 | 91.592 | 274.776 | 20.9726 | NA | 1.07e-21 | NA |
| 184.900 | 4.13e-42 | 41.3844 | 92.450 | 277.350 | 21.1609 | NA | 6.90e-22 | NA |
| 186.624 | 1.73e-42 | 41.7607 | 93.312 | 279.936 | 21.3500 | NA | 4.47e-22 | NA |
| 188.356 | 7.26e-43 | 42.1388 | 94.178 | 282.534 | 21.5400 | NA | 2.88e-22 | NA |
| 190.096 | 3.03e-43 | 42.5186 | 95.048 | 285.144 | 21.7309 | NA | 1.86e-22 | NA |
| 191.844 | 1.26e-43 | 42.9002 | 95.922 | 287.766 | 21.9226 | NA | 1.20e-22 | NA |
| 193.600 | 5.21e-44 | 43.2834 | 96.800 | 290.400 | 22.1152 | NA | 7.67e-23 | NA |
| 195.364 | 2.15e-44 | 43.6684 | 97.682 | 293.046 | 22.3087 | NA | 4.91e-23 | NA |
| 197.136 | 8.81e-45 | 44.0552 | 98.568 | 295.704 | 22.5030 | NA | 3.14e-23 | NA |
| 198.916 | 3.60e-45 | 44.4436 | 99.458 | 298.374 | 22.6982 | NA | 2.00e-23 | NA |
| 200.704 | 1.47e-45 | 44.8338 | 100.352 | 301.056 | 22.8942 | NA | 1.28e-23 | NA |
| 202.500 | 5.95e-46 | 45.2257 | 101.250 | 303.750 | 23.0911 | NA | 8.11e-24 | NA |
| 204.304 | 2.40e-46 | 45.6193 | 102.152 | 306.456 | 23.2889 | NA | 5.14e-24 | NA |
| 206.116 | 9.67e-47 | 46.0147 | 103.058 | 309.174 | 23.4875 | NA | 3.25e-24 | NA |
| 207.936 | 3.87e-47 | 46.4118 | 103.968 | 311.904 | 23.6870 | NA | 2.06e-24 | NA |
| 209.764 | 1.55e-47 | 46.8106 | 104.882 | 314.646 | 23.8873 | NA | 1.30e-24 | NA |
| 211.600 | 6.15e-48 | 47.2112 | 105.800 | 317.400 | 24.0885 | NA | 8.16e-25 | NA |
| 213.444 | 2.44e-48 | 47.6135 | 106.722 | 320.166 | 24.2905 | NA | 5.12e-25 | NA |
| 215.296 | 9.60e-49 | 48.0175 | 107.648 | 322.944 | 24.4935 | NA | 3.21e-25 | NA |
| 217.156 | 3.77e-49 | 48.4232 | 108.578 | 325.734 | 24.6973 | NA | 2.01e-25 | NA |
| 219.024 | 1.48e-49 | 48.8307 | 109.512 | 328.536 | 24.9019 | NA | 1.25e-25 | NA |
| 220.900 | 5.76e-50 | 49.2399 | 110.450 | 331.350 | 25.1074 | NA | 7.81e-26 | NA |
| 222.784 | 2.23e-50 | 49.6509 | 111.392 | 334.176 | 25.3138 | NA | 4.86e-26 | NA |
| 224.676 | 8.64e-51 | 50.0635 | 112.338 | 337.014 | 25.5210 | NA | 3.01e-26 | NA |
| 226.576 | 3.33e-51 | 50.4779 | 113.288 | 339.864 | 25.7291 | NA | 1.87e-26 | NA |
| 228.484 | 1.28e-51 | 50.8940 | 114.242 | 342.726 | 25.9380 | NA | 1.15e-26 | NA |
| 230.400 | 4.88e-52 | 51.3119 | 115.200 | 345.600 | 26.1478 | NA | 7.11e-27 | NA |
| 232.324 | 1.86e-52 | 51.7315 | 116.162 | 348.486 | 26.3585 | NA | 4.38e-27 | NA |
| 234.256 | 7.03e-53 | 52.1528 | 117.128 | 351.384 | 26.5700 | NA | 2.69e-27 | NA |
| 236.196 | 2.66e-53 | 52.5758 | 118.098 | 354.294 | 26.7824 | NA | 1.65e-27 | NA |
| 238.144 | 9.99e-54 | 53.0006 | 119.072 | 357.216 | 26.9957 | NA | 1.01e-27 | NA |
| 240.100 | 3.74e-54 | 53.4271 | 120.050 | 360.150 | 27.2098 | NA | 6.17e-28 | NA |
| 242.064 | 1.40e-54 | 53.8553 | 121.032 | 363.096 | 27.4248 | NA | 3.76e-28 | NA |
| 244.036 | 5.18e-55 | 54.2853 | 122.018 | 366.054 | 27.6406 | NA | 2.29e-28 | NA |
| 246.016 | 1.92e-55 | 54.7170 | 123.008 | 369.024 | 27.8573 | NA | 1.39e-28 | NA |
| 248.004 | 7.07e-56 | 55.1504 | 124.002 | 372.006 | 28.0749 | NA | 8.42e-29 | NA |
| 250.000 | 2.60e-56 | 55.5856 | 125.000 | 375.000 | 28.2933 | NA | 5.09e-29 | NA |
That looks good but the bounds can be quite wide, approximately two orders of magnitude in either direction.
20% window
This is the range we will use for determining the similarity of p-values.
The below table will be used to judge whether the p-values are reproducible.
bound=0.2
res$lower <- res$`stat.X-squared` - ( res$`stat.X-squared` * bound )
res$upper <- res$`stat.X-squared` + ( res$`stat.X-squared` * bound )
plot(res$`stat.X-squared`, res$logp,
xlab="chisq test stat", ylab="-log10(p)",
xlim=c(0,max(res$upper)))
points(res$lower, res$logp, col="blue")
points(res$upper, res$logp, col="blue")
res$logp_lower <- approx(x = res$`stat.X-squared`, y =res$logp, xout = res$lower)[["y"]]
res$logp_upper <- approx(x = res$`stat.X-squared`, y =res$logp, xout = res$upper)[["y"]]
res$p_lower <- 10^-res$logp_lower
res$p_upper <- 10^-res$logp_upper
format(res,digits=3) %>%
kbl(caption="p-value ranges considered reproduced with a bound of 20% around the test statistic") %>%
kable_paper("hover", full_width = F)| stat.X-squared | p | logp | lower | upper | logp_lower | logp_upper | p_lower | p_upper |
|---|---|---|---|---|---|---|---|---|
| 0.000 | 1.00e+00 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.00e+00 | 1.00e+00 |
| 0.004 | 9.50e-01 | 0.0225 | 0.0032 | 0.0048 | 0.0180 | 0.0240 | 9.59e-01 | 9.46e-01 |
| 0.016 | 8.99e-01 | 0.0461 | 0.0128 | 0.0192 | 0.0398 | 0.0500 | 9.12e-01 | 8.91e-01 |
| 0.036 | 8.50e-01 | 0.0708 | 0.0288 | 0.0432 | 0.0619 | 0.0775 | 8.67e-01 | 8.37e-01 |
| 0.064 | 8.00e-01 | 0.0968 | 0.0512 | 0.0768 | 0.0849 | 0.1064 | 8.22e-01 | 7.83e-01 |
| 0.100 | 7.52e-01 | 0.1239 | 0.0800 | 0.1200 | 0.1088 | 0.1368 | 7.78e-01 | 7.30e-01 |
| 0.144 | 7.04e-01 | 0.1522 | 0.1152 | 0.1728 | 0.1337 | 0.1686 | 7.35e-01 | 6.78e-01 |
| 0.196 | 6.58e-01 | 0.1818 | 0.1568 | 0.2352 | 0.1595 | 0.2019 | 6.93e-01 | 6.28e-01 |
| 0.256 | 6.13e-01 | 0.2126 | 0.2048 | 0.3072 | 0.1863 | 0.2368 | 6.51e-01 | 5.80e-01 |
| 0.324 | 5.69e-01 | 0.2447 | 0.2592 | 0.3888 | 0.2141 | 0.2732 | 6.11e-01 | 5.33e-01 |
| 0.400 | 5.27e-01 | 0.2781 | 0.3200 | 0.4800 | 0.2428 | 0.3112 | 5.72e-01 | 4.88e-01 |
| 0.484 | 4.87e-01 | 0.3128 | 0.3872 | 0.5808 | 0.2725 | 0.3506 | 5.34e-01 | 4.46e-01 |
| 0.576 | 4.48e-01 | 0.3488 | 0.4608 | 0.6912 | 0.3032 | 0.3916 | 4.97e-01 | 4.06e-01 |
| 0.676 | 4.11e-01 | 0.3862 | 0.5408 | 0.8112 | 0.3351 | 0.4343 | 4.62e-01 | 3.68e-01 |
| 0.784 | 3.76e-01 | 0.4249 | 0.6272 | 0.9408 | 0.3680 | 0.4786 | 4.29e-01 | 3.32e-01 |
| 0.900 | 3.43e-01 | 0.4650 | 0.7200 | 1.0800 | 0.4020 | 0.5246 | 3.96e-01 | 2.99e-01 |
| 1.024 | 3.12e-01 | 0.5064 | 0.8192 | 1.2288 | 0.4371 | 0.5723 | 3.66e-01 | 2.68e-01 |
| 1.156 | 2.82e-01 | 0.5493 | 0.9248 | 1.3872 | 0.4733 | 0.6217 | 3.36e-01 | 2.39e-01 |
| 1.296 | 2.55e-01 | 0.5936 | 1.0368 | 1.5552 | 0.5106 | 0.6728 | 3.09e-01 | 2.12e-01 |
| 1.444 | 2.29e-01 | 0.6392 | 1.1552 | 1.7328 | 0.5490 | 0.7256 | 2.82e-01 | 1.88e-01 |
| 1.600 | 2.06e-01 | 0.6863 | 1.2800 | 1.9200 | 0.5885 | 0.7802 | 2.58e-01 | 1.66e-01 |
| 1.764 | 1.84e-01 | 0.7349 | 1.4112 | 2.1168 | 0.6291 | 0.8366 | 2.35e-01 | 1.46e-01 |
| 1.936 | 1.64e-01 | 0.7849 | 1.5488 | 2.3232 | 0.6709 | 0.8946 | 2.13e-01 | 1.27e-01 |
| 2.116 | 1.46e-01 | 0.8363 | 1.6928 | 2.5392 | 0.7138 | 0.9544 | 1.93e-01 | 1.11e-01 |
| 2.304 | 1.29e-01 | 0.8893 | 1.8432 | 2.7648 | 0.7579 | 1.0160 | 1.75e-01 | 9.64e-02 |
| 2.500 | 1.14e-01 | 0.9437 | 2.0000 | 3.0000 | 0.8032 | 1.0795 | 1.57e-01 | 8.33e-02 |
| 2.704 | 1.00e-01 | 0.9996 | 2.1632 | 3.2448 | 0.8496 | 1.1447 | 1.41e-01 | 7.17e-02 |
| 2.916 | 8.77e-02 | 1.0570 | 2.3328 | 3.4992 | 0.8973 | 1.2118 | 1.27e-01 | 6.14e-02 |
| 3.136 | 7.66e-02 | 1.1159 | 2.5088 | 3.7632 | 0.9461 | 1.2807 | 1.13e-01 | 5.24e-02 |
| 3.364 | 6.66e-02 | 1.1763 | 2.6912 | 4.0368 | 0.9961 | 1.3514 | 1.01e-01 | 4.45e-02 |
| 3.600 | 5.78e-02 | 1.2382 | 2.8800 | 4.3200 | 1.0472 | 1.4240 | 8.97e-02 | 3.77e-02 |
| 3.844 | 4.99e-02 | 1.3017 | 3.0752 | 4.6128 | 1.0996 | 1.4985 | 7.95e-02 | 3.17e-02 |
| 4.096 | 4.30e-02 | 1.3667 | 3.2768 | 4.9152 | 1.1532 | 1.5748 | 7.03e-02 | 2.66e-02 |
| 4.356 | 3.69e-02 | 1.4332 | 3.4848 | 5.2272 | 1.2080 | 1.6529 | 6.19e-02 | 2.22e-02 |
| 4.624 | 3.15e-02 | 1.5013 | 3.6992 | 5.5488 | 1.2640 | 1.7329 | 5.44e-02 | 1.85e-02 |
| 4.900 | 2.69e-02 | 1.5709 | 3.9200 | 5.8800 | 1.3213 | 1.8149 | 4.77e-02 | 1.53e-02 |
| 5.184 | 2.28e-02 | 1.6421 | 4.1472 | 6.2208 | 1.3798 | 1.8987 | 4.17e-02 | 1.26e-02 |
| 5.476 | 1.93e-02 | 1.7149 | 4.3808 | 6.5712 | 1.4395 | 1.9844 | 3.63e-02 | 1.04e-02 |
| 5.776 | 1.62e-02 | 1.7892 | 4.6208 | 6.9312 | 1.5005 | 2.0721 | 3.16e-02 | 8.47e-03 |
| 6.084 | 1.36e-02 | 1.8651 | 4.8672 | 7.3008 | 1.5627 | 2.1616 | 2.74e-02 | 6.89e-03 |
| 6.400 | 1.14e-02 | 1.9426 | 5.1200 | 7.6800 | 1.6261 | 2.2531 | 2.37e-02 | 5.58e-03 |
| 6.724 | 9.51e-03 | 2.0217 | 5.3792 | 8.0688 | 1.6908 | 2.3464 | 2.04e-02 | 4.50e-03 |
| 7.056 | 7.90e-03 | 2.1024 | 5.6448 | 8.4672 | 1.7567 | 2.4418 | 1.75e-02 | 3.62e-03 |
| 7.396 | 6.54e-03 | 2.1846 | 5.9168 | 8.8752 | 1.8239 | 2.5390 | 1.50e-02 | 2.89e-03 |
| 7.744 | 5.39e-03 | 2.2685 | 6.1952 | 9.2928 | 1.8924 | 2.6381 | 1.28e-02 | 2.30e-03 |
| 8.100 | 4.43e-03 | 2.3539 | 6.4800 | 9.7200 | 1.9622 | 2.7392 | 1.09e-02 | 1.82e-03 |
| 8.464 | 3.62e-03 | 2.4410 | 6.7712 | 10.1568 | 2.0332 | 2.8423 | 9.26e-03 | 1.44e-03 |
| 8.836 | 2.95e-03 | 2.5297 | 7.0688 | 10.6032 | 2.1055 | 2.9473 | 7.84e-03 | 1.13e-03 |
| 9.216 | 2.40e-03 | 2.6200 | 7.3728 | 11.0592 | 2.1790 | 3.0543 | 6.62e-03 | 8.83e-04 |
| 9.604 | 1.94e-03 | 2.7119 | 7.6832 | 11.5248 | 2.2538 | 3.1632 | 5.57e-03 | 6.87e-04 |
| 10.000 | 1.57e-03 | 2.8054 | 8.0000 | 12.0000 | 2.3299 | 3.2741 | 4.68e-03 | 5.32e-04 |
| 10.404 | 1.26e-03 | 2.9005 | 8.3232 | 12.4848 | 2.4073 | 3.3869 | 3.91e-03 | 4.10e-04 |
| 10.816 | 1.01e-03 | 2.9973 | 8.6528 | 12.9792 | 2.4860 | 3.5017 | 3.27e-03 | 3.15e-04 |
| 11.236 | 8.02e-04 | 3.0957 | 8.9888 | 13.4832 | 2.5660 | 3.6185 | 2.72e-03 | 2.41e-04 |
| 11.664 | 6.37e-04 | 3.1957 | 9.3312 | 13.9968 | 2.6472 | 3.7372 | 2.25e-03 | 1.83e-04 |
| 12.100 | 5.04e-04 | 3.2974 | 9.6800 | 14.5200 | 2.7298 | 3.8580 | 1.86e-03 | 1.39e-04 |
| 12.544 | 3.97e-04 | 3.4007 | 10.0352 | 15.0528 | 2.8137 | 3.9807 | 1.54e-03 | 1.05e-04 |
| 12.996 | 3.12e-04 | 3.5056 | 10.3968 | 15.5952 | 2.8988 | 4.1054 | 1.26e-03 | 7.85e-05 |
| 13.456 | 2.44e-04 | 3.6122 | 10.7648 | 16.1472 | 2.9853 | 4.2320 | 1.03e-03 | 5.86e-05 |
| 13.924 | 1.90e-04 | 3.7204 | 11.1392 | 16.7088 | 3.0730 | 4.3607 | 8.45e-04 | 4.36e-05 |
| 14.400 | 1.48e-04 | 3.8303 | 11.5200 | 17.2800 | 3.1621 | 4.4914 | 6.89e-04 | 3.23e-05 |
| 14.884 | 1.14e-04 | 3.9418 | 11.9072 | 17.8608 | 3.2524 | 4.6240 | 5.59e-04 | 2.38e-05 |
| 15.376 | 8.81e-05 | 4.0550 | 12.3008 | 18.4512 | 3.3441 | 4.7587 | 4.53e-04 | 1.74e-05 |
| 15.876 | 6.76e-05 | 4.1699 | 12.7008 | 19.0512 | 3.4371 | 4.8953 | 3.66e-04 | 1.27e-05 |
| 16.384 | 5.17e-05 | 4.2863 | 13.1072 | 19.6608 | 3.5314 | 5.0339 | 2.94e-04 | 9.25e-06 |
| 16.900 | 3.94e-05 | 4.4045 | 13.5200 | 20.2800 | 3.6270 | 5.1746 | 2.36e-04 | 6.69e-06 |
| 17.424 | 2.99e-05 | 4.5243 | 13.9392 | 20.9088 | 3.7240 | 5.3172 | 1.89e-04 | 4.82e-06 |
| 17.956 | 2.26e-05 | 4.6458 | 14.3648 | 21.5472 | 3.8222 | 5.4619 | 1.51e-04 | 3.45e-06 |
| 18.496 | 1.70e-05 | 4.7689 | 14.7968 | 22.1952 | 3.9218 | 5.6085 | 1.20e-04 | 2.46e-06 |
| 19.044 | 1.28e-05 | 4.8937 | 15.2352 | 22.8528 | 4.0226 | 5.7572 | 9.49e-05 | 1.75e-06 |
| 19.600 | 9.55e-06 | 5.0201 | 15.6800 | 23.5200 | 4.1248 | 5.9079 | 7.50e-05 | 1.24e-06 |
| 20.164 | 7.11e-06 | 5.1483 | 16.1312 | 24.1968 | 4.2284 | 6.0606 | 5.91e-05 | 8.70e-07 |
| 20.736 | 5.27e-06 | 5.2781 | 16.5888 | 24.8832 | 4.3332 | 6.2153 | 4.64e-05 | 6.09e-07 |
| 21.316 | 3.89e-06 | 5.4095 | 17.0528 | 25.5792 | 4.4394 | 6.3720 | 3.64e-05 | 4.25e-07 |
| 21.904 | 2.87e-06 | 5.5427 | 17.5232 | 26.2848 | 4.5469 | 6.5308 | 2.84e-05 | 2.95e-07 |
| 22.500 | 2.10e-06 | 5.6775 | 18.0000 | 27.0000 | 4.6558 | 6.6915 | 2.21e-05 | 2.03e-07 |
| 23.104 | 1.53e-06 | 5.8140 | 18.4832 | 27.7248 | 4.7660 | 6.8543 | 1.71e-05 | 1.40e-07 |
| 23.716 | 1.12e-06 | 5.9521 | 18.9728 | 28.4592 | 4.8775 | 7.0191 | 1.33e-05 | 9.57e-08 |
| 24.336 | 8.09e-07 | 6.0920 | 19.4688 | 29.2032 | 4.9903 | 7.1859 | 1.02e-05 | 6.52e-08 |
| 24.964 | 5.84e-07 | 6.2335 | 19.9712 | 29.9568 | 5.1045 | 7.3548 | 7.86e-06 | 4.42e-08 |
| 25.600 | 4.20e-07 | 6.3767 | 20.4800 | 30.7200 | 5.2200 | 7.5257 | 6.03e-06 | 2.98e-08 |
| 26.244 | 3.01e-07 | 6.5216 | 20.9952 | 31.4928 | 5.3368 | 7.6986 | 4.60e-06 | 2.00e-08 |
| 26.896 | 2.15e-07 | 6.6682 | 21.5168 | 32.2752 | 5.4550 | 7.8735 | 3.51e-06 | 1.34e-08 |
| 27.556 | 1.53e-07 | 6.8164 | 22.0448 | 33.0672 | 5.5745 | 8.0505 | 2.66e-06 | 8.90e-09 |
| 28.224 | 1.08e-07 | 6.9663 | 22.5792 | 33.8688 | 5.6954 | 8.2295 | 2.02e-06 | 5.90e-09 |
| 28.900 | 7.62e-08 | 7.1180 | 23.1200 | 34.6800 | 5.8176 | 8.4105 | 1.52e-06 | 3.89e-09 |
| 29.584 | 5.35e-08 | 7.2713 | 23.6672 | 35.5008 | 5.9411 | 8.5935 | 1.15e-06 | 2.55e-09 |
| 30.276 | 3.75e-08 | 7.4263 | 24.2208 | 36.3312 | 6.0660 | 8.7786 | 8.59e-07 | 1.66e-09 |
| 30.976 | 2.61e-08 | 7.5830 | 24.7808 | 37.1712 | 6.1922 | 8.9658 | 6.42e-07 | 1.08e-09 |
| 31.684 | 1.81e-08 | 7.7413 | 25.3472 | 38.0208 | 6.3198 | 9.1549 | 4.79e-07 | 7.00e-10 |
| 32.400 | 1.25e-08 | 7.9014 | 25.9200 | 38.8800 | 6.4487 | 9.3461 | 3.56e-07 | 4.51e-10 |
| 33.124 | 8.65e-09 | 8.0632 | 26.4992 | 39.7488 | 6.5790 | 9.5394 | 2.64e-07 | 2.89e-10 |
| 33.856 | 5.93e-09 | 8.2266 | 27.0848 | 40.6272 | 6.7106 | 9.7346 | 1.95e-07 | 1.84e-10 |
| 34.596 | 4.06e-09 | 8.3918 | 27.6768 | 41.5152 | 6.8435 | 9.9320 | 1.43e-07 | 1.17e-10 |
| 35.344 | 2.76e-09 | 8.5586 | 28.2752 | 42.4128 | 6.9778 | 10.1313 | 1.05e-07 | 7.39e-11 |
| 36.100 | 1.87e-09 | 8.7271 | 28.8800 | 43.3200 | 7.1135 | 10.3327 | 7.70e-08 | 4.65e-11 |
| 36.864 | 1.27e-09 | 8.8973 | 29.4912 | 44.2368 | 7.2505 | 10.5362 | 5.62e-08 | 2.91e-11 |
| 37.636 | 8.53e-10 | 9.0693 | 30.1088 | 45.1632 | 7.3888 | 10.7416 | 4.08e-08 | 1.81e-11 |
| 38.416 | 5.72e-10 | 9.2429 | 30.7328 | 46.0992 | 7.5285 | 10.9492 | 2.96e-08 | 1.12e-11 |
| 39.204 | 3.82e-10 | 9.4182 | 31.3632 | 47.0448 | 7.6696 | 11.1587 | 2.14e-08 | 6.94e-12 |
| 40.000 | 2.54e-10 | 9.5952 | 32.0000 | 48.0000 | 7.8120 | 11.3704 | 1.54e-08 | 4.26e-12 |
| 40.804 | 1.68e-10 | 9.7739 | 32.6432 | 48.9648 | 7.9557 | 11.5840 | 1.11e-08 | 2.61e-12 |
| 41.616 | 1.11e-10 | 9.9544 | 33.2928 | 49.9392 | 8.1009 | 11.7997 | 7.93e-09 | 1.59e-12 |
| 42.436 | 7.30e-11 | 10.1365 | 33.9488 | 50.9232 | 8.2473 | 12.0175 | 5.66e-09 | 9.61e-13 |
| 43.264 | 4.78e-11 | 10.3203 | 34.6112 | 51.9168 | 8.3951 | 12.2373 | 4.03e-09 | 5.79e-13 |
| 44.100 | 3.12e-11 | 10.5058 | 35.2800 | 52.9200 | 8.5443 | 12.4592 | 2.86e-09 | 3.47e-13 |
| 44.944 | 2.03e-11 | 10.6930 | 35.9552 | 53.9328 | 8.6948 | 12.6831 | 2.02e-09 | 2.07e-13 |
| 45.796 | 1.31e-11 | 10.8820 | 36.6368 | 54.9552 | 8.8467 | 12.9090 | 1.42e-09 | 1.23e-13 |
| 46.656 | 8.46e-12 | 11.0726 | 37.3248 | 55.9872 | 9.0000 | 13.1370 | 1.00e-09 | 7.29e-14 |
| 47.524 | 5.43e-12 | 11.2649 | 38.0192 | 57.0288 | 9.1546 | 13.3671 | 7.01e-10 | 4.29e-14 |
| 48.400 | 3.48e-12 | 11.4590 | 38.7200 | 58.0800 | 9.3105 | 13.5992 | 4.89e-10 | 2.52e-14 |
| 49.284 | 2.21e-12 | 11.6547 | 39.4272 | 59.1408 | 9.4678 | 13.8333 | 3.41e-10 | 1.47e-14 |
| 50.176 | 1.41e-12 | 11.8521 | 40.1408 | 60.2112 | 9.6265 | 14.0695 | 2.36e-10 | 8.52e-15 |
| 51.076 | 8.89e-13 | 12.0513 | 40.8608 | 61.2912 | 9.7866 | 14.3078 | 1.63e-10 | 4.92e-15 |
| 51.984 | 5.60e-13 | 12.2522 | 41.5872 | 62.3808 | 9.9480 | 14.5481 | 1.13e-10 | 2.83e-15 |
| 52.900 | 3.51e-13 | 12.4547 | 42.3200 | 63.4800 | 10.1107 | 14.7905 | 7.75e-11 | 1.62e-15 |
| 53.824 | 2.19e-13 | 12.6590 | 43.0592 | 64.5888 | 10.2748 | 15.0349 | 5.31e-11 | 9.23e-16 |
| 54.756 | 1.36e-13 | 12.8650 | 43.8048 | 65.7072 | 10.4403 | 15.2814 | 3.63e-11 | 5.23e-16 |
| 55.696 | 8.46e-14 | 13.0727 | 44.5568 | 66.8352 | 10.6071 | 15.5299 | 2.47e-11 | 2.95e-16 |
| 56.644 | 5.22e-14 | 13.2821 | 45.3152 | 67.9728 | 10.7753 | 15.7805 | 1.68e-11 | 1.66e-16 |
| 57.600 | 3.21e-14 | 13.4932 | 46.0800 | 69.1200 | 10.9449 | 16.0331 | 1.14e-11 | 9.27e-17 |
| 58.564 | 1.97e-14 | 13.7060 | 46.8512 | 70.2768 | 11.1158 | 16.2878 | 7.66e-12 | 5.15e-17 |
| 59.536 | 1.20e-14 | 13.9205 | 47.6288 | 71.4432 | 11.2881 | 16.5446 | 5.15e-12 | 2.85e-17 |
| 60.516 | 7.30e-15 | 14.1368 | 48.4128 | 72.6192 | 11.4618 | 16.8034 | 3.45e-12 | 1.57e-17 |
| 61.504 | 4.42e-15 | 14.3547 | 49.2032 | 73.8048 | 11.6368 | 17.0643 | 2.31e-12 | 8.62e-18 |
| 62.500 | 2.66e-15 | 14.5744 | 50.0000 | 75.0000 | 11.8132 | 17.3272 | 1.54e-12 | 4.71e-18 |
| 63.504 | 1.60e-15 | 14.7958 | 50.8032 | 76.2048 | 11.9909 | 17.5922 | 1.02e-12 | 2.56e-18 |
| 64.516 | 9.58e-16 | 15.0189 | 51.6128 | 77.4192 | 12.1700 | 17.8593 | 6.76e-13 | 1.38e-18 |
| 65.536 | 5.71e-16 | 15.2437 | 52.4288 | 78.6432 | 12.3505 | 18.1284 | 4.46e-13 | 7.44e-19 |
| 66.564 | 3.39e-16 | 15.4702 | 53.2512 | 79.8768 | 12.5324 | 18.3996 | 2.94e-13 | 3.99e-19 |
| 67.600 | 2.00e-16 | 15.6984 | 54.0800 | 81.1200 | 12.7156 | 18.6728 | 1.92e-13 | 2.12e-19 |
| 68.644 | 1.18e-16 | 15.9283 | 54.9152 | 82.3728 | 12.9002 | 18.9481 | 1.26e-13 | 1.13e-19 |
| 69.696 | 6.92e-17 | 16.1600 | 55.7568 | 83.6352 | 13.0861 | 19.2254 | 8.20e-14 | 5.95e-20 |
| 70.756 | 4.04e-17 | 16.3933 | 56.6048 | 84.9072 | 13.2734 | 19.5049 | 5.33e-14 | 3.13e-20 |
| 71.824 | 2.35e-17 | 16.6284 | 57.4592 | 86.1888 | 13.4621 | 19.7863 | 3.45e-14 | 1.64e-20 |
| 72.900 | 1.36e-17 | 16.8652 | 58.3200 | 87.4800 | 13.6521 | 20.0699 | 2.23e-14 | 8.51e-21 |
| 73.984 | 7.88e-18 | 17.1037 | 59.1872 | 88.7808 | 13.8436 | 20.3555 | 1.43e-14 | 4.41e-21 |
| 75.076 | 4.53e-18 | 17.3440 | 60.0608 | 90.0912 | 14.0363 | 20.6431 | 9.20e-15 | 2.27e-21 |
| 76.176 | 2.59e-18 | 17.5859 | 60.9408 | 91.4112 | 14.2305 | 20.9329 | 5.88e-15 | 1.17e-21 |
| 77.284 | 1.48e-18 | 17.8296 | 61.8272 | 92.7408 | 14.4260 | 21.2247 | 3.75e-15 | 5.96e-22 |
| 78.400 | 8.42e-19 | 18.0749 | 62.7200 | 94.0800 | 14.6229 | 21.5185 | 2.38e-15 | 3.03e-22 |
| 79.524 | 4.76e-19 | 18.3220 | 63.6192 | 95.4288 | 14.8212 | 21.8144 | 1.51e-15 | 1.53e-22 |
| 80.656 | 2.69e-19 | 18.5708 | 64.5248 | 96.7872 | 15.0208 | 22.1124 | 9.53e-16 | 7.72e-23 |
| 81.796 | 1.51e-19 | 18.8214 | 65.4368 | 98.1552 | 15.2218 | 22.4125 | 6.00e-16 | 3.87e-23 |
| 82.944 | 8.44e-20 | 19.0736 | 66.3552 | 99.5328 | 15.4242 | 22.7146 | 3.77e-16 | 1.93e-23 |
| 84.100 | 4.70e-20 | 19.3276 | 67.2800 | 100.9200 | 15.6279 | 23.0187 | 2.36e-16 | 9.58e-24 |
| 85.264 | 2.61e-20 | 19.5832 | 68.2112 | 102.3168 | 15.8330 | 23.3250 | 1.47e-16 | 4.73e-24 |
| 86.436 | 1.44e-20 | 19.8406 | 69.1488 | 103.7232 | 16.0395 | 23.6333 | 9.13e-17 | 2.33e-24 |
| 87.616 | 7.95e-21 | 20.0997 | 70.0928 | 105.1392 | 16.2473 | 23.9437 | 5.66e-17 | 1.14e-24 |
| 88.804 | 4.36e-21 | 20.3606 | 71.0432 | 106.5648 | 16.4566 | 24.2561 | 3.49e-17 | 5.54e-25 |
| 90.000 | 2.38e-21 | 20.6231 | 72.0000 | 108.0000 | 16.6672 | 24.5706 | 2.15e-17 | 2.69e-25 |
| 91.204 | 1.30e-21 | 20.8874 | 72.9632 | 109.4448 | 16.8791 | 24.8872 | 1.32e-17 | 1.30e-25 |
| 92.416 | 7.02e-22 | 21.1534 | 73.9328 | 110.8992 | 17.0925 | 25.2058 | 8.08e-18 | 6.23e-26 |
| 93.636 | 3.79e-22 | 21.4211 | 74.9088 | 112.3632 | 17.3072 | 25.5265 | 4.93e-18 | 2.98e-26 |
| 94.864 | 2.04e-22 | 21.6905 | 75.8912 | 113.8368 | 17.5233 | 25.8493 | 3.00e-18 | 1.41e-26 |
| 96.100 | 1.09e-22 | 21.9617 | 76.8800 | 115.3200 | 17.7407 | 26.1741 | 1.82e-18 | 6.70e-27 |
| 97.344 | 5.83e-23 | 22.2345 | 77.8752 | 116.8128 | 17.9595 | 26.5010 | 1.10e-18 | 3.15e-27 |
| 98.596 | 3.10e-23 | 22.5091 | 78.8768 | 118.3152 | 18.1798 | 26.8300 | 6.61e-19 | 1.48e-27 |
| 99.856 | 1.64e-23 | 22.7854 | 79.8848 | 119.8272 | 18.4013 | 27.1610 | 3.97e-19 | 6.90e-28 |
| 101.124 | 8.64e-24 | 23.0635 | 80.8992 | 121.3488 | 18.6243 | 27.4941 | 2.38e-19 | 3.21e-28 |
| 102.400 | 4.54e-24 | 23.3432 | 81.9200 | 122.8800 | 18.8486 | 27.8293 | 1.42e-19 | 1.48e-28 |
| 103.684 | 2.37e-24 | 23.6247 | 82.9472 | 124.4208 | 19.0743 | 28.1666 | 8.43e-20 | 6.81e-29 |
| 104.976 | 1.24e-24 | 23.9079 | 83.9808 | 125.9712 | 19.3014 | 28.5059 | 5.00e-20 | 3.12e-29 |
| 106.276 | 6.41e-25 | 24.1928 | 85.0208 | 127.5312 | 19.5298 | 28.8473 | 2.95e-20 | 1.42e-29 |
| 107.584 | 3.32e-25 | 24.4794 | 86.0672 | 129.1008 | 19.7596 | 29.1907 | 1.74e-20 | 6.45e-30 |
| 108.900 | 1.71e-25 | 24.7678 | 87.1200 | 130.6800 | 19.9908 | 29.5362 | 1.02e-20 | 2.91e-30 |
| 110.224 | 8.75e-26 | 25.0579 | 88.1792 | 132.2688 | 20.2234 | 29.8838 | 5.98e-21 | 1.31e-30 |
| 111.556 | 4.47e-26 | 25.3497 | 89.2448 | 133.8672 | 20.4573 | 30.2335 | 3.49e-21 | 5.84e-31 |
| 112.896 | 2.27e-26 | 25.6432 | 90.3168 | 135.4752 | 20.6927 | 30.5852 | 2.03e-21 | 2.60e-31 |
| 114.244 | 1.15e-26 | 25.9385 | 91.3952 | 137.0928 | 20.9294 | 30.9390 | 1.18e-21 | 1.15e-31 |
| 115.600 | 5.82e-27 | 26.2354 | 92.4800 | 138.7200 | 21.1674 | 31.2949 | 6.80e-22 | 5.07e-32 |
| 116.964 | 2.92e-27 | 26.5341 | 93.5712 | 140.3568 | 21.4069 | 31.6528 | 3.92e-22 | 2.22e-32 |
| 118.336 | 1.46e-27 | 26.8345 | 94.6688 | 142.0032 | 21.6477 | 32.0128 | 2.25e-22 | 9.71e-33 |
| 119.716 | 7.30e-28 | 27.1367 | 95.7728 | 143.6592 | 21.8899 | 32.3749 | 1.29e-22 | 4.22e-33 |
| 121.104 | 3.63e-28 | 27.4406 | 96.8832 | 145.3248 | 22.1335 | 32.7390 | 7.35e-23 | 1.82e-33 |
| 122.500 | 1.79e-28 | 27.7461 | 98.0000 | 147.0000 | 22.3784 | 33.1053 | 4.18e-23 | 7.85e-34 |
| 123.904 | 8.84e-29 | 28.0535 | 99.1232 | 148.6848 | 22.6247 | 33.4736 | 2.37e-23 | 3.36e-34 |
| 125.316 | 4.34e-29 | 28.3625 | 100.2528 | 150.3792 | 22.8725 | 33.8439 | 1.34e-23 | 1.43e-34 |
| 126.736 | 2.12e-29 | 28.6732 | 101.3888 | 152.0832 | 23.1215 | 34.2164 | 7.56e-24 | 6.08e-35 |
| 128.164 | 1.03e-29 | 28.9857 | 102.5312 | 153.7968 | 23.3720 | 34.5909 | 4.25e-24 | 2.57e-35 |
| 129.600 | 5.01e-30 | 29.2999 | 103.6800 | 155.5200 | 23.6238 | 34.9674 | 2.38e-24 | 1.08e-35 |
| 131.044 | 2.42e-30 | 29.6159 | 104.8352 | 157.2528 | 23.8770 | 35.3461 | 1.33e-24 | 4.51e-36 |
| 132.496 | 1.17e-30 | 29.9335 | 105.9968 | 158.9952 | 24.1316 | 35.7268 | 7.39e-25 | 1.88e-36 |
| 133.956 | 5.59e-31 | 30.2529 | 107.1648 | 160.7472 | 24.3876 | 36.1096 | 4.10e-25 | 7.77e-37 |
| 135.424 | 2.67e-31 | 30.5740 | 108.3392 | 162.5088 | 24.6449 | 36.4945 | 2.27e-25 | 3.20e-37 |
| 136.900 | 1.27e-31 | 30.8968 | 109.5200 | 164.2800 | 24.9036 | 36.8814 | 1.25e-25 | 1.31e-37 |
| 138.384 | 6.01e-32 | 31.2214 | 110.7072 | 166.0608 | 25.1637 | 37.2704 | 6.86e-26 | 5.37e-38 |
| 139.876 | 2.83e-32 | 31.5477 | 111.9008 | 167.8512 | 25.4252 | 37.6615 | 3.76e-26 | 2.18e-38 |
| 141.376 | 1.33e-32 | 31.8757 | 113.1008 | 169.6512 | 25.6881 | 38.0547 | 2.05e-26 | 8.82e-39 |
| 142.884 | 6.23e-33 | 32.2054 | 114.3072 | 171.4608 | 25.9523 | 38.4499 | 1.12e-26 | 3.55e-39 |
| 144.400 | 2.90e-33 | 32.5369 | 115.5200 | 173.2800 | 26.2179 | 38.8472 | 6.05e-27 | 1.42e-39 |
| 145.924 | 1.35e-33 | 32.8700 | 116.7392 | 175.1088 | 26.4849 | 39.2466 | 3.27e-27 | 5.67e-40 |
| 147.456 | 6.24e-34 | 33.2050 | 117.9648 | 176.9472 | 26.7533 | 39.6480 | 1.76e-27 | 2.25e-40 |
| 148.996 | 2.87e-34 | 33.5416 | 119.1968 | 178.7952 | 27.0230 | 40.0515 | 9.48e-28 | 8.88e-41 |
| 150.544 | 1.32e-34 | 33.8799 | 120.4352 | 180.6528 | 27.2941 | 40.4571 | 5.08e-28 | 3.49e-41 |
| 152.100 | 6.03e-35 | 34.2200 | 121.6800 | 182.5200 | 27.5666 | 40.8648 | 2.71e-28 | 1.37e-41 |
| 153.664 | 2.74e-35 | 34.5618 | 122.9312 | 184.3968 | 27.8405 | 41.2745 | 1.44e-28 | 5.31e-42 |
| 155.236 | 1.24e-35 | 34.9054 | 124.1888 | 186.2832 | 28.1158 | 41.6863 | 7.66e-29 | 2.06e-42 |
| 156.816 | 5.62e-36 | 35.2506 | 125.4528 | 188.1792 | 28.3924 | 42.1002 | 4.05e-29 | 7.94e-43 |
| 158.404 | 2.53e-36 | 35.5976 | 126.7232 | 190.0848 | 28.6704 | 42.5162 | 2.14e-29 | 3.05e-43 |
| 160.000 | 1.13e-36 | 35.9464 | 128.0000 | 192.0000 | 28.9498 | 42.9342 | 1.12e-29 | 1.16e-43 |
| 161.604 | 5.05e-37 | 36.2968 | 129.2832 | 193.9248 | 29.2306 | 43.3543 | 5.88e-30 | 4.42e-44 |
| 163.216 | 2.24e-37 | 36.6490 | 130.5728 | 195.8592 | 29.5128 | 43.7765 | 3.07e-30 | 1.67e-44 |
| 164.836 | 9.93e-38 | 37.0029 | 131.8688 | 197.8032 | 29.7963 | 44.2008 | 1.60e-30 | 6.30e-45 |
| 166.464 | 4.38e-38 | 37.3585 | 133.1712 | 199.7568 | 30.0812 | 44.6271 | 8.29e-31 | 2.36e-45 |
| 168.100 | 1.92e-38 | 37.7158 | 134.4800 | 201.7200 | 30.3675 | 45.0555 | 4.29e-31 | 8.80e-46 |
| 169.744 | 8.42e-39 | 38.0749 | 135.7952 | 203.6928 | 30.6552 | 45.4860 | 2.21e-31 | 3.27e-46 |
| 171.396 | 3.67e-39 | 38.4357 | 137.1168 | 205.6752 | 30.9443 | 45.9185 | 1.14e-31 | 1.21e-46 |
| 173.056 | 1.59e-39 | 38.7983 | 138.4448 | 207.6672 | 31.2347 | 46.3532 | 5.83e-32 | 4.43e-47 |
| 174.724 | 6.88e-40 | 39.1625 | 139.7792 | 209.6688 | 31.5265 | 46.7899 | 2.98e-32 | 1.62e-47 |
| 176.400 | 2.96e-40 | 39.5285 | 141.1200 | 211.6800 | 31.8197 | 47.2287 | 1.51e-32 | 5.91e-48 |
| 178.084 | 1.27e-40 | 39.8962 | 142.4672 | 213.7008 | 32.1143 | 47.6695 | 7.69e-33 | 2.14e-48 |
| 179.776 | 5.42e-41 | 40.2657 | 143.8208 | 215.7312 | 32.4102 | 48.1124 | 3.89e-33 | 7.72e-49 |
| 181.476 | 2.31e-41 | 40.6368 | 145.1808 | 217.7712 | 32.7076 | 48.5574 | 1.96e-33 | 2.77e-49 |
| 183.184 | 9.78e-42 | 41.0097 | 146.5472 | 219.8208 | 33.0063 | 49.0045 | 9.86e-34 | 9.90e-50 |
| 184.900 | 4.13e-42 | 41.3844 | 147.9200 | 221.8800 | 33.3064 | 49.4537 | 4.94e-34 | 3.52e-50 |
| 186.624 | 1.73e-42 | 41.7607 | 149.2992 | 223.9488 | 33.6079 | 49.9049 | 2.47e-34 | 1.24e-50 |
| 188.356 | 7.26e-43 | 42.1388 | 150.6848 | 226.0272 | 33.9107 | 50.3582 | 1.23e-34 | 4.38e-51 |
| 190.096 | 3.03e-43 | 42.5186 | 152.0768 | 228.1152 | 34.2150 | 50.8136 | 6.10e-35 | 1.54e-51 |
| 191.844 | 1.26e-43 | 42.9002 | 153.4752 | 230.2128 | 34.5206 | 51.2711 | 3.02e-35 | 5.36e-52 |
| 193.600 | 5.21e-44 | 43.2834 | 154.8800 | 232.3200 | 34.8276 | 51.7306 | 1.49e-35 | 1.86e-52 |
| 195.364 | 2.15e-44 | 43.6684 | 156.2912 | 234.4368 | 35.1360 | 52.1922 | 7.31e-36 | 6.42e-53 |
| 197.136 | 8.81e-45 | 44.0552 | 157.7088 | 236.5632 | 35.4457 | 52.6559 | 3.58e-36 | 2.21e-53 |
| 198.916 | 3.60e-45 | 44.4436 | 159.1328 | 238.6992 | 35.7569 | 53.1217 | 1.75e-36 | 7.56e-54 |
| 200.704 | 1.47e-45 | 44.8338 | 160.5632 | 240.8448 | 36.0694 | 53.5895 | 8.52e-37 | 2.57e-54 |
| 202.500 | 5.95e-46 | 45.2257 | 162.0000 | 243.0000 | 36.3833 | 54.0594 | 4.14e-37 | 8.72e-55 |
| 204.304 | 2.40e-46 | 45.6193 | 163.4432 | 245.1648 | 36.6986 | 54.5314 | 2.00e-37 | 2.94e-55 |
| 206.116 | 9.67e-47 | 46.0147 | 164.8928 | 247.3392 | 37.0153 | 55.0055 | 9.65e-38 | 9.87e-56 |
| 207.936 | 3.87e-47 | 46.4118 | 166.3488 | 249.5232 | 37.3333 | 55.4816 | 4.64e-38 | 3.30e-56 |
| 209.764 | 1.55e-47 | 46.8106 | 167.8112 | 251.7168 | 37.6528 | NA | 2.22e-38 | NA |
| 211.600 | 6.15e-48 | 47.2112 | 169.2800 | 253.9200 | 37.9736 | NA | 1.06e-38 | NA |
| 213.444 | 2.44e-48 | 47.6135 | 170.7552 | 256.1328 | 38.2958 | NA | 5.06e-39 | NA |
| 215.296 | 9.60e-49 | 48.0175 | 172.2368 | 258.3552 | 38.6194 | NA | 2.40e-39 | NA |
| 217.156 | 3.77e-49 | 48.4232 | 173.7248 | 260.5872 | 38.9443 | NA | 1.14e-39 | NA |
| 219.024 | 1.48e-49 | 48.8307 | 175.2192 | 262.8288 | 39.2707 | NA | 5.36e-40 | NA |
| 220.900 | 5.76e-50 | 49.2399 | 176.7200 | 265.0800 | 39.5984 | NA | 2.52e-40 | NA |
| 222.784 | 2.23e-50 | 49.6509 | 178.2272 | 267.3408 | 39.9275 | NA | 1.18e-40 | NA |
| 224.676 | 8.64e-51 | 50.0635 | 179.7408 | 269.6112 | 40.2580 | NA | 5.52e-41 | NA |
| 226.576 | 3.33e-51 | 50.4779 | 181.2608 | 271.8912 | 40.5899 | NA | 2.57e-41 | NA |
| 228.484 | 1.28e-51 | 50.8940 | 182.7872 | 274.1808 | 40.9231 | NA | 1.19e-41 | NA |
| 230.400 | 4.88e-52 | 51.3119 | 184.3200 | 276.4800 | 41.2577 | NA | 5.52e-42 | NA |
| 232.324 | 1.86e-52 | 51.7315 | 185.8592 | 278.7888 | 41.5938 | NA | 2.55e-42 | NA |
| 234.256 | 7.03e-53 | 52.1528 | 187.4048 | 281.1072 | 41.9312 | NA | 1.17e-42 | NA |
| 236.196 | 2.66e-53 | 52.5758 | 188.9568 | 283.4352 | 42.2700 | NA | 5.37e-43 | NA |
| 238.144 | 9.99e-54 | 53.0006 | 190.5152 | 285.7728 | 42.6101 | NA | 2.45e-43 | NA |
| 240.100 | 3.74e-54 | 53.4271 | 192.0800 | 288.1200 | 42.9517 | NA | 1.12e-43 | NA |
| 242.064 | 1.40e-54 | 53.8553 | 193.6512 | 290.4768 | 43.2946 | NA | 5.07e-44 | NA |
| 244.036 | 5.18e-55 | 54.2853 | 195.2288 | 292.8432 | 43.6389 | NA | 2.30e-44 | NA |
| 246.016 | 1.92e-55 | 54.7170 | 196.8128 | 295.2192 | 43.9846 | NA | 1.04e-44 | NA |
| 248.004 | 7.07e-56 | 55.1504 | 198.4032 | 297.6048 | 44.3317 | NA | 4.66e-45 | NA |
| 250.000 | 2.60e-56 | 55.5856 | 200.0000 | 300.0000 | 44.6802 | NA | 2.09e-45 | NA |
Session information
sessionInfo()## R version 4.4.1 (2024-06-14)
## Platform: x86_64-pc-linux-gnu
## Running under: Ubuntu 22.04.4 LTS
##
## Matrix products: default
## BLAS: /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.10.0
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.10.0
##
## locale:
## [1] LC_CTYPE=en_AU.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_AU.UTF-8 LC_COLLATE=en_AU.UTF-8
## [5] LC_MONETARY=en_AU.UTF-8 LC_MESSAGES=en_AU.UTF-8
## [7] LC_PAPER=en_AU.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=en_AU.UTF-8 LC_IDENTIFICATION=C
##
## time zone: Australia/Melbourne
## tzcode source: system (glibc)
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] kableExtra_1.4.0
##
## loaded via a namespace (and not attached):
## [1] vctrs_0.6.5 svglite_2.1.3 cli_3.6.3 knitr_1.47
## [5] rlang_1.1.4 xfun_0.45 highr_0.11 stringi_1.8.4
## [9] jsonlite_1.8.8 glue_1.7.0 colorspace_2.1-0 htmltools_0.5.8.1
## [13] sass_0.4.9 scales_1.3.0 rmarkdown_2.27 evaluate_0.24.0
## [17] munsell_0.5.1 jquerylib_0.1.4 fastmap_1.2.0 yaml_2.3.8
## [21] lifecycle_1.0.4 stringr_1.5.1 compiler_4.4.1 rstudioapi_0.16.0
## [25] systemfonts_1.1.0 digest_0.6.36 viridisLite_0.4.2 R6_2.5.1
## [29] magrittr_2.0.3 bslib_0.7.0 tools_4.4.1 xml2_1.3.6
## [33] cachem_1.1.0