These functions transform data between percentage, probit and Normal Equivalent Deviate (NED) (Bliss 1934; Finney 1952) .
PercentAdjust(percentage, n) Percent2NED(percentage) Percent2Probit(percentage) Probit2NED(probit) NED2Probit(NED) NED2Percent(NED) Probit2Percent(probit)
| percentage | The percentage value. |
|---|---|
| n | Sample size for estimation of percentage. |
| probit | The probit value |
| NED | The NED value. |
The transformed value.
Probit transformation can be used to transform a sigmoid curve of percentage data to a linear one. The probit transformation is defined as NED + 5. However the two terms probit and NED are used interchangeably in literature.
NED function (Φ-1) is is the inverse of the cumulative distribution function (Φ) of the standard normal distribution (z ∼ N(0,1)) or the quantile function associated with the standard normal distribution.
For percentage p,
NED(p)= Φ-1(p) = √2 erf-1 (2p-1)
and
probit(p) = NED(p) + 5
The PercentAdjust function adjusts the percentage values of 0 and 100
to 100 × [ 0.25 ⁄
n ] and
100 × [ (n − 0.25) ⁄
n ] respectively,
according to the sample size n to avoid
infinity values during probit transformation
(Miller and Tainter 1944)
.
Bliss CI (1934). “The method of probits.” Science, 79(2037), 38--39.
Miller LC, Tainter ML (1944). “Estimation of the ED50 and its error by means of logarithmic-probit graph paper.” Proceedings of the Society for Experimental Biology and Medicine, 57(2), 261--264.
Finney DJ (1952). Probit Analysis: A Statistical Treatment of the Sigmoid Response Curve. Cambridge University Press, Cambridge, England.
Percent2NED(0:100)#> [1] -Inf -2.32634787 -2.05374891 -1.88079361 -1.75068607 -1.64485363 #> [7] -1.55477359 -1.47579103 -1.40507156 -1.34075503 -1.28155157 -1.22652812 #> [13] -1.17498679 -1.12639113 -1.08031934 -1.03643339 -0.99445788 -0.95416525 #> [19] -0.91536509 -0.87789630 -0.84162123 -0.80642125 -0.77219321 -0.73884685 #> [25] -0.70630256 -0.67448975 -0.64334541 -0.61281299 -0.58284151 -0.55338472 #> [31] -0.52440051 -0.49585035 -0.46769880 -0.43991317 -0.41246313 -0.38532047 #> [37] -0.35845879 -0.33185335 -0.30548079 -0.27931903 -0.25334710 -0.22754498 #> [43] -0.20189348 -0.17637416 -0.15096922 -0.12566135 -0.10043372 -0.07526986 #> [49] -0.05015358 -0.02506891 0.00000000 0.02506891 0.05015358 0.07526986 #> [55] 0.10043372 0.12566135 0.15096922 0.17637416 0.20189348 0.22754498 #> [61] 0.25334710 0.27931903 0.30548079 0.33185335 0.35845879 0.38532047 #> [67] 0.41246313 0.43991317 0.46769880 0.49585035 0.52440051 0.55338472 #> [73] 0.58284151 0.61281299 0.64334541 0.67448975 0.70630256 0.73884685 #> [79] 0.77219321 0.80642125 0.84162123 0.87789630 0.91536509 0.95416525 #> [85] 0.99445788 1.03643339 1.08031934 1.12639113 1.17498679 1.22652812 #> [91] 1.28155157 1.34075503 1.40507156 1.47579103 1.55477359 1.64485363 #> [97] 1.75068607 1.88079361 2.05374891 2.32634787 InfPercent2Probit(0:100)#> [1] -Inf 2.673652 2.946251 3.119206 3.249314 3.355146 3.445226 3.524209 #> [9] 3.594928 3.659245 3.718448 3.773472 3.825013 3.873609 3.919681 3.963567 #> [17] 4.005542 4.045835 4.084635 4.122104 4.158379 4.193579 4.227807 4.261153 #> [25] 4.293697 4.325510 4.356655 4.387187 4.417158 4.446615 4.475599 4.504150 #> [33] 4.532301 4.560087 4.587537 4.614680 4.641541 4.668147 4.694519 4.720681 #> [41] 4.746653 4.772455 4.798107 4.823626 4.849031 4.874339 4.899566 4.924730 #> [49] 4.949846 4.974931 5.000000 5.025069 5.050154 5.075270 5.100434 5.125661 #> [57] 5.150969 5.176374 5.201893 5.227545 5.253347 5.279319 5.305481 5.331853 #> [65] 5.358459 5.385320 5.412463 5.439913 5.467699 5.495850 5.524401 5.553385 #> [73] 5.582842 5.612813 5.643345 5.674490 5.706303 5.738847 5.772193 5.806421 #> [81] 5.841621 5.877896 5.915365 5.954165 5.994458 6.036433 6.080319 6.126391 #> [89] 6.174987 6.226528 6.281552 6.340755 6.405072 6.475791 6.554774 6.644854 #> [97] 6.750686 6.880794 7.053749 7.326348 InfPercent2NED(25)#> [1] -0.6744898Percent2Probit(25)#> [1] 4.32551Percent2NED(25) +5#> [1] 4.32551NED2Probit(-0.6744898)#> [1] 4.32551# Percentage adjustment for 0 and 100 Percent2Probit(100)#> [1] InfPercent2Probit(0)#> [1] -Infn = 50 Percent2Probit(PercentAdjust(100, n))#> [1] 7.575829Percent2Probit(PercentAdjust(0, n))#> [1] 2.424171