Phosphate Assay Analyzer — Methods
The statistical methodology behind the phosphate assay analyzer — calibration, uncertainty, and outlier detection.
This page documents the statistical methods used by the Phosphate Assay Analyzer. The assay itself (wet-ashing followed by colorimetric quantitation of released inorganic phosphate) follows Heberle Lab SOP-PIASSAY-001 (PDF), after Bartlett (1959). The tool turns the raw absorbance readings into a phospholipid stock concentration with a defensible uncertainty, and screens the data for outliers.
The standard curve
The assay relates absorbance $A$ to the amount of inorganic phosphate $n$ (in nmol) by a straight line,
\[A = a + b\,n,\]fit by ordinary least squares to the standards. The intercept $a$ is left free — estimated from the blanks rather than forced through zero — because the blank carries a small absorbance from trace phosphate in the reagents (chiefly the hydrogen peroxide). The fit returns the slope $b$, intercept $a$, and the residual standard error
\[s = \sqrt{\frac{1}{N-2}\sum_{i=1}^{N}\bigl(A_i - \hat{A}_i\bigr)^2}, \qquad \hat{A}_i = a + b\,n_i,\]where $N$ is the number of standards. (This is the quantity Excel calls STEYX.) The amount of phosphate in each standard tube is computed from the dispensed volume of standard and its concentration, since $1~\text{mM} = 1~\text{nmol}/\mu\text{L}$.
From absorbance to stock concentration
Each sample’s absorbance $A_0$ is converted back to phosphate by inverting the curve, $n_0 = (A_0 - a)/b$, and the lipid stock concentration follows from the dispensed sample volume $V$ (µL) and the number of phosphates per lipid $p$ (one for most phospholipids; two for cardiolipin):
\[c = \frac{n_0}{V\,p}.\]Again because $1~\text{mM} = 1~\text{nmol}/\mu\text{L}$, $c$ is obtained directly in mM. The reported stock concentration is the mean $\bar{c}$ over the replicate tubes.
Uncertainty
Two independent sources of error contribute, and the tool reports both separately and combined.
Replicate scatter — the tube-to-tube spread (pipetting, ashing, reading) — is captured empirically as the standard error of the mean of the $m$ per-replicate concentrations:
\[u_\text{rep} = \frac{s_c}{\sqrt{m}},\]with $s_c$ the standard deviation of the replicate concentrations.
Standard-curve uncertainty — how well the calibration line itself is known — shifts every replicate together, so it can never show up in the replicate scatter and must be added separately. For a reading that maps to amount $n$, the line contributes
\[u_\text{cal,line}(n) = \frac{s}{|b|}\sqrt{\frac{1}{N} + \frac{(n-\bar{n})^2}{S_{nn}}}, \qquad S_{nn} = \sum_i (n_i-\bar{n})^2,\]which is converted to concentration by dividing by $V\,p$. Because this term is shared across replicates (a systematic offset, not random scatter), its contribution to the mean is the linear average of the per-replicate values rather than a $1/\sqrt{m}$ reduction. The two independent sources add in quadrature,
\[u = \sqrt{u_\text{rep}^2 + u_\text{cal}^2},\]and the reported confidence interval is $\pm\,t_{0.975,\,\nu}\,u$, with the effective degrees of freedom from the Welch–Satterthwaite formula,
\[\nu = \frac{\bigl(u_\text{rep}^2 + u_\text{cal}^2\bigr)^2} {u_\text{rep}^4/(m-1) + u_\text{cal}^4/(N-2)}.\]A useful consequence: because $u_\text{cal}$ grows with $(n-\bar{n})^2$, a sample measured far from the centre of the standards pays a larger calibration penalty — which is why the tool’s planner encourages you to bracket and centre the standards on your samples. And since running more replicate tubes shrinks $u_\text{rep}$ but leaves $u_\text{cal}$ untouched, the standard-curve term sets a floor that only better curve design can lower.
Detecting outliers
The tool screens both the standard curve and the replicate samples for outliers at a significance level of $\alpha = 0.05$. The two datasets are statistically different — the standards define a line, the replicates are repeated measurements of one quantity — so each is tested with the method appropriate to it. In both cases suspected outliers are flagged, not deleted: a statistical test only identifies a candidate, and the decision to discard a point should be made deliberately, ideally after checking the lab notebook for a physical cause (an air bubble, a mis-pipette, a cracked tube). This matters most with small samples, where discarding one of a handful of points is a large move.
Standards — studentized residuals
A bad standard is one that does not fit the line, so each standard’s residual is judged against how much scatter is expected at that point. Using the leverage $h_i = 1/N + (n_i - \bar{n})^2/S_{nn}$ and the residual standard error recomputed without point $i$,
\[s_{(i)}^2 = \frac{(N-2)\,s^2 - e_i^2/(1-h_i)}{N-3}, \qquad e_i = A_i - \hat{A}_i,\]the externally studentized (deleted) residual is
\[t_i = \frac{e_i}{s_{(i)}\sqrt{1-h_i}}.\]A standard is flagged when $|t_i|$ exceeds the Bonferroni-corrected critical value $t_{\,1-\alpha/(2N),\,N-3}$; the $1/N$ correction accounts for screening all $N$ standards at once. This is the classical single-outlier test from regression diagnostics (Cook & Weisberg, 1982). It is applied only when there are at least 5 standards.
Samples — Grubbs’ test
For the replicate concentrations the tool uses Grubbs’ test (the extreme studentized deviate test), the method recommended by ISO and IUPAC for detecting an outlier in a normally distributed sample. It takes the most extreme replicate,
\[G = \frac{\max_i \lvert c_i - \bar{c}\rvert}{s_c},\]and compares it with the critical value
\[G_\text{crit} = \frac{N-1}{\sqrt{N}} \sqrt{\frac{t^2}{\,N-2+t^2\,}}, \qquad t = t_{\,\alpha/(2N),\,N-2}.\]If $G > G_\text{crit}$ the extreme point is flagged; the test is then repeated on the remaining points (a capped, iterated Grubbs / generalized-ESD procedure), up to two flagged replicates, and is applied only when there are at least 5 replicates. Grubbs’ test is preferred here over the median/IQR rule and Dixon’s $Q$ test for its statistical rigour — though all such tests are deliberately conservative, so that genuine data is rarely discarded.
What happens in the tool
Flagged points are highlighted in amber and listed with their test statistic (for example, “Replicate #3 — 2.9σ from the mean”). You can uncheck the highlighted rows yourself, or use the “Exclude flagged & re-analyze” button, which drops them and refits; the standard curve, uncertainty, and outlier flags are all recomputed on the remaining points. Nothing is ever removed automatically.
References
- Bartlett, G. R. (1959). Phosphorus assay in column chromatography. J. Biol. Chem. 234, 466–468.
- Grubbs, F. E. (1969). Procedures for detecting outlying observations in samples. Technometrics 11, 1–21.
- Rosner, B. (1983). Percentage points for a generalized ESD many-outlier procedure. Technometrics 25, 165–172.
- Cook, R. D., & Weisberg, S. (1982). Residuals and Influence in Regression. Chapman & Hall.
- Miller, J. N., & Miller, J. C. (2018). Statistics and Chemometrics for Analytical Chemistry (7th ed.). Pearson.