Confidence Interval of a Vector or Data Frame Column

Calculation of the mean value and its confidence limits (according to Student's t-distribution), corresponding to data frame column or vector, characterizing dispersion of the individual values such as double integrals in quantitative EPR analysis, g-values or linewidths.

Usage

eval_interval_cnfd_tVec(data.vec.col, level.cnfd = 0.95, separate = TRUE)

Arguments

data.vec.col: Numeric vector (pointing to column) of interest (within a data frame) to calculate the confidence interval or uncertainty (margin of error).
level.cnfd: Numeric (floating) value, corresponding to confidence level (default: level.cnfd = 0.95). This value is related to significance level $alpha$ defined by $1 - confidence\,\,level$.
separate: Logical, whether to separate the mean value and the uncertainty (margin of error), corresponding to non-negative right/left confidence limit of the mean. If separate = TRUE (default), the result is shown as a named vector with the (mean) value and the uncertaity. Otherwise, the result is returned in the format of $value\pm\,\,uncertainty$.

Value

Named vector of (mean) value and uncertaity or $value\pm\,\,uncertainty$ format depending on the separate argument, where the uncertainty actually represents non-negative limit for the mean (one side of the confidence interval not including the mean value).

Details

The confidence interval evaluation suggests that values/observations obey two-tailed Student's t-distribution, which for number of observations $N > 30$ approaches the normal $z$-distribution. Evaluation of the confidence interval and/or its limits can be well documented on $g$-factor series example: $$g = \overline{g}\pm (t_{(1-\alpha/2),df}\,s/\sqrt{N})$$ where $\overline{g}$ is the mean value and the $t_{(1-\alpha/2),df}$ corresponds to the quantile of the t-distribution having the significance level of $\alpha$ ($1-\alpha = condfidence\,\,level$, see the level.cnfd argument) and the degrees of freedom $df = N -1$. Finally, the $s$ represents the sample standard deviation, defined by the following relation (regarding the $g$-value series example): $$s = \sqrt{(1/(N-1))\,\sum_{i=1}^N (g_i - \overline{g})^2}$$ which is computed by the sd function. The above-mentioned $t$-quantile is actually calculated by the stats::qt. Alternatively, one could also evaluate confidence interval by the one sample t.test for a certain level of confidence, giving a descriptive output with statistical characteristics.

References

Miller JN, Miller JC, Miller RD (2018). Statistics and Chemometrics for Analytical Chemistry, 7th edition, Pearson Education. ISBN 978-1-292-18674-0, https://elibrary.pearson.de/book/99.150005/9781292186726.

Hayes A, Moller-Trane R, Jordan D, Northrop P, Lang MN, Zeileis A (2022). distributions3: Probability Distributions as S3 Objects. https://github.com/alexpghayes/distributions3, https://alexpghayes.github.io/distributions3/, see also Vignette: t-confidence interval for a mean, https://alexpghayes.github.io/distributions3/articles/one-sample-t-confidence-interval.html.

National Institute of Standards and Technology (NIST) (2012). “Confidence Limits for the Mean”, https://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm.

Kaleta J, Tarábek J, Akdag A, Pohl R, Michl J (2012). “The 16 CB11(CH3)n(CD3)12–N• Radicals with 5-Fold Substitution Symmetry: Spin Density Distribution in CB11Me12•”, Inorg. Chem., 51(20), 10819–10824, https://doi.org/10.1021/ic301236s.

Curley JP, Milewski TM (2020). “PSY317L Guidebook - Confidence Intervals”, https://bookdown.org/curleyjp0/psy317l_guides5/confidence-intervals.html.

Goodson DZ (2011). Mathematical Methods for Physical and Analytical Chemistry, 1st edition, John Wiley and Sons, Inc. ISBN 978-0-470-47354-2, https://onlinelibrary.wiley.com/doi/book/10.1002/9781118135204.

Examples

## double integral/intensity values
## coming from several experiments:
di.vec <- c(0.025,0.020,0.031,0.022,0.035)
#
## evaluation of the confidence interval
## in different formats:
eval_interval_cnfd_tVec(di.vec)
#>        value  uncertainty 
#> 0.0266000000 0.0077839559 
#
eval_interval_cnfd_tVec(di.vec,
                        level.cnfd = 0.95,
                        separate = FALSE)
#> 0.027(8)