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What is the role of data spacing along the $x$-axis in linear regression by least-squares? Is it an important criterion to consider in the experimental design of linear calibrations?

Linear regression by the method of least-squares is based on several implicit assumptions, of which the most important include:

  1. Only one variable ($y$-axis) is subject to error, the other ($x$-axis) being known exactly.
  2. The variables $x$ and $y$ must be independent.
  3. The errors in $y$ are normally distributed.
  4. Each data point has equal significance, which means that least-squares treats all deviations equally. Otherwise the correct weights $w_i$ need to be known exactly: this requires that the functional form of dependence on $y_i$ of the variance of $y_i$ be known.
  5. The errors are random rather than systematic (they are unbiased, i.e. the errors have zero mean).
  6. The errors in the observations are independent of one another.

I would like to know what is the role of data spacing in the abscissa in linear least-squares regressions and its statistical impact on the standard errors of the coefficients $m$ and $b$ of the linear relation: $y = mx + b$ and coefficients $R$ and/or $R2$? Do these 4 parameters still retain statistical meaning if the data are unevenly distributed in the abscissa? And what is the relevance of this in linear calibrations by least-squares assuming the errors in the observations are normally distributed with an approximately constant variance?

For irregularly spaced data points with heteroscedasticity in the errors of the $y_i$ variable the linear regression will yield biased values of $m$ and $b$, incl. biased standard deviations of the two linear coefficients. One typical example is the Lineweaver-Burk linearization of the Michaelis-Menten equation used in enzymatic catalysis to determine values of Km and Vmax (also called double-reciprocal plot). The linear transformation of hyperbolic data using reciprocals compresses values in the left side of the plot and extends the distance between points at the right side making the analysis essentially worthless if the original data is affected by measurement error. Transforming to linearity is not appropriate unless the error variance of the modified variable is stabilized or it is desirable to normalize the error distribution.

I was therefore wondering about the effect of equispaced data vs. random data spacing vs. data separation by a factor (or following a function), and how these changes in the $x$-scale affect the statistical reliability of the linear regression in terms of minimizing uncertainty in the coefficients, say considering the same number of data points? Is there any paper or book that focuses on this aspect particularily?

Reference material for consultation, such as scientific research papers and books, would also be highly appreciated directly related or within the context of my query. Thank you in advance.

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2 Answers 2

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After analysis of the standard errors expressions of the linear coefficients (m = slope) and (b = y-intercept), it can be concluded that not only the number of data points n used to fit the straight-line has an influence on the result but the data spacing also impacts the resulting uncertainty!

In the designing of linear calibrations, the goal is to minimize the degree of uncertainty of the two predicted coefficients. The only way one can do this is by uniformly spacing points over a wide range of analysis. For example, the standard deviations of m and b can be minimized by increasing the value of the term Σ(x_i – x_avg)^2, which is present in the denominators of both equations. The larger the value of this term (which we accomplish by increasing the range of x around its mean value), the smaller the standard deviations in m and b will be. Furthermore, to minimize the uncertainty in b, it is also required to decrease the value of the term Σx_i^2 in the equation describing the standard deviation of b. This is achieved by regularly spreading the x-axis coordinates over their range of analysis. To minimize uncertainty in b, it also helps to decrease the value of the right-hand term Σx_i in the denominator of the equation, which can be realized by including points with x-coordinates progressively closer to the y-axis: a blank point at y(x = 0) = a (with a approx. equal to b) is usually taken to achieve this.

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    $\begingroup$ This doesn't look right. When the only goals are to minimize the SEs of the estimates, then the optimal solution puts half the $x$ values at one extreme and the other half at the other extreme, which is almost the opposite of equal spacing. Equal spacing has other merits, such as providing data useful for checking linearity. $\endgroup$
    – whuber
    Jan 7, 2021 at 17:21
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    $\begingroup$ Here's a nice article by Gelman to provide background to @whuber's comment: stat.columbia.edu/~gelman/research/published/27.pdf . Specifically, he writes: "Under reasonable assumptions about nonlinearity, we find that, unless sample size is very large, the design with no interior measurements is best, because with moderate total sample sizes, any nonlinearity in the dose–response will be difficult to detect." $\endgroup$ Aug 20, 2022 at 17:53
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@whuber: an interesting analysis worth noting indeed, but doesn't appear to be realistic experimentally within the context of linear calibrations by least-squares regression. What is the point of placing half of the points in one extreme and the other half in the other extreme with nothing in between (for the same number of points)? Indeed it gives an "optimal" theoretical solution for the minimization of the SE's of the coefficient estimates but does not guarantee that a linear association between y and x exists in the middle part of the range, even if the points are clustered at the margins, which is clearly one option of random spacing. From the analysis of the SE's expressions of the linear coefficients, the most realistic way to maximize precision of the predicted variable y is to minimize the SE's of the two linear coefficients and the best way to achieve this is by spreading the n points equidistantly along the x-axis, starting with y(x = 0) = a.

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