# Role of data spacing along the x-axis in linear calibrations by least-squares?

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.

• 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.