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I'm trying to determine the effect of an interferent (hemolysis, represented "H-index") on the measurement of various substances - in this example, $K^+$ (potassium). Ultimately, I'm looking to predict $K^+_{actual}$ within a confidence interval, given $K^+_{observed}$ and a known H-index.

To do this, I have obtained samples ($S_A, S_B, S_C, ... $), each unrelated and containing varying amounts of $K^+$, and, from each, prepared a series of aliquots with increasing hemolysis, giving:

$S_{A_0}, S_{A_1}, S_{A_2}, ...S_{A_n}$
$S_{B_0}, S_{B_1}, S_{B_2}, ...S_{B_n}$
$S_{C_0}, S_{C_1}, S_{C_2}, ...S_{C_n}$

Then I measured $K^+$ for each of these aliquots and subtracted, from each, the $K^+$ obtained from the unadultered aliquot 0, giving:

$0, S_{A_{1\Delta}}, S_{A_{2\Delta}},...S_{A_{n\Delta}}$
$0, S_{B_{1\Delta}}, S_{B_{2\Delta}},...S_{B_{n\Delta}}$
$0, S_{C_{1\Delta}}, S_{C_{2\Delta}},...S_{C_{n\Delta}}$

I was envisioning doing OLS linear regression with the resulting data points, but I don't know if that test is valid for my data, given that each aliquot in a series is related to all the other aliquots from that series.

I've been searching fruitlessly for an answer to this question, probably hindered by my lack of a statistical background - most of what I've found has been well beyond my understanding, but, from what I do understand, doesn't seem to answer my question.

Below is a visualization of the data, each colored line being one series of aliquots.

Data figure with better labels

--EDIT--

Reading through this article on the OLS assumptions, it looks like my data are consistent with assumptions 1, 2, and 5.

Assumption #3 presents a problem, since the errors don't have constant variance; they increase with increasing hemolysis. The suggested fix of doing a log transformation renders the data nonlinear. Does this mean OLS regression is out the window?

Assumption #4 may also be problematic; each point in an aliquot series is probably correlated with each other point in the same series. However, the linked article says that a Durbin-Watson test of 2 indicates no autocorrelation, and that values below 1 or above 3 are problematic. Statsmodels is giving me a D-W value of 1.044 - not great, but maybe still workable?

I'd love some input as to whether I'm on the right track.

I have a plot of my residuals below:
Residuals plot with better labels

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  • $\begingroup$ Hi. Welcome to CV. Could you please explain what is x-axis and y-axis in the chart above? If I understand correctly, $S_i$ are samples with different (unknown?) amounts of $Y$? In $S_{A_j}$ and $S_{B_j}$ the amount of $X$ added is same but amount of $Y$ is different? Finally do you think measurement of $Y$ is influenced by amount of $X$ in the sample? $\endgroup$ – Dayne Oct 17 at 7:09
  • $\begingroup$ @Dayne, certainly. The X axis represents interferent $X$, and the Y axis represents the measurement error introduced by the presence of $X$ - i.e., the difference between aliquot $n$ and aliquot $0$ in a given sample (hence why all lines start at 0, 0; the first point is aliquot $0$. Let me see if I can generate a better figure with labels. $\endgroup$ – EOTech Oct 18 at 20:40
  • $\begingroup$ @Dayne, as to your other questions: yes, $S_i$ are samples with different amounts of $Y$. The amount is known from aliquot $S_{i_0}$, which has no interferent $X$. The concentration of $X$ is not precisely the same in each equivalent aliquot; I obtain it from a biological extract, which does not yield precise concentrations. And yes, measurement of $Y$ is influenced by the concentration of $X$; the goal is to determine how strong and consistent this influence is. $\endgroup$ – EOTech Oct 18 at 20:43
  • $\begingroup$ I've replaced $X$ and $Y$ with more descriptive terms in the hopes of adding clarity. $\endgroup$ – EOTech Oct 23 at 0:26
  • $\begingroup$ From what I understand does the following captures (at least in spirit) the model for your problem: $K^+_{actual} = K^+_{observed} + H \epsilon$, where $epsilon$ is white noise? When you say each aliquot series maybe correlated, which series are you considering? One possible series is $S_{A_i}$. Here $K$ is same but $H$ is increasing. If errors increase with H then the above model might work. Another series is $S_{i_1}$. If for this also you think that errors are correlated so maybe this model is better: $K^+_{actual} = K^+_{observed}*H*\epsilon$. Does this make sense? $\endgroup$ – Dayne Oct 23 at 3:39

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