# What statistical test is most appropriate when my data consist of multiple series, each based on an individual sample?

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. --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: • 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? – Dayne Oct 17 at 7:09
• @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. – EOTech Oct 18 at 20:40
• @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. – EOTech Oct 18 at 20:43
• I've replaced $X$ and $Y$ with more descriptive terms in the hopes of adding clarity. – EOTech Oct 23 at 0:26
• 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? – Dayne Oct 23 at 3:39