Joint parameter confidence regions improve the power of parametric regression in method-comparison studies

Abstract

Paired data from medical laboratory method-comparison experiments are commonly analysed by parametric linear regression, and differences from the line of identity are invariably assessed by determining whether confidence intervals for slope (b) and intercept (a) enclose 1 and 0, respectively. However, it has been shown that such analyses have low statistical power when the data maximum to minimum range ratio is small (maximum to minimum ratio of X values ≤ 2–1). Sample sizes > 500 are typically required to give a worthwhile probability of detecting clinically meaningful method biases. An alternative significance test is considered here, based on whether the elliptically shaped area representing the joint slope and intercept confidence region encloses the point [b = 1, a = 0]. Simulations showed a clear range related pattern. Relative to confidence intervals, the joint parameter confidence ellipse is associated with small reductions in power when the data range ratio is very large (several thousand-fold), but this is more than compensated by spectacular improvements in power with small range ratios. Sample size reductions were typically >20-fold for a 2–1 range ratio. Worthwhile sample size reductions are possible for all but the largest range ratios and in the particular case of the narrow physiological ranges associated with a number of medical laboratory tests (or their equivalents in other measurement contexts), the reductions could convert sample size requirements from impossibly large to eminently manageable. A Win32 computer program is provided, which converts data range, uncertainty characteristics and user-defined power settings into estimates of minimum sample size.

Appendix

Parameter estimates

With reference to Eq. 5, direct estimates of intercept and slope, $\hat{a}$ and $\hat{b}$, are available for OLS and WLS,

$$ \hat{b} = \sum xy/\sum x^{2} $$

(6)

$$ \hat{a} = \bar{Y} - \hat{b}\bar{X} $$

(7)

$$ \hat{S} = \sum y^{2} - (\sum xy)^{2} /\sum x^{2} $$

(8)

where $\bar{X} = \sum W_{i} X_{i} /\sum W_{i}$, $\bar{Y} = \sum W_{i} Y_{i} /\sum W_{i}$, $\sum x^{2} = \sum W_{i} (X_{i} - \bar{X})^{2}$), $\sum y^{2} = \sum W_{i} (Y_{i} - \bar{Y})^{2}$, $\sum xy = \sum W_{i} (X_{i} - \bar{X})(Y_{i} - \bar{Y})$, $\hat{S}$ denotes the minimised sum-of-squares and all summations (Σ) are over i = 1, 2, …, N paired X, Y values. WLS calculations are continued iteratively, successively updating the $\sigma_{{Y_{i} }}^{2}$ (i.e. W _i ) according to predicted Y _i values (Eq. 4), until $\hat{b}$ converges to relative change <10⁻⁵. Direct solutions are not available for Deming regression because the W _i are functions of slope. The method of Press et al. [13] was applied iteratively, successively updating the $\sigma_{{X_{i} }}^{2}$ and $\sigma_{{Y_{i} }}^{2}$ according to adjusted X _i and Y _i values [15], until $\hat{b}$ converged to relative change <10⁻⁵.

Confidence intervals

OLS confidence intervals for slope (CI _b ) and intercept (CI _a ) are given by,

$$ {\text{CI}}_{b} = t\left\{ \hat{S}/[(N - 2)\sum x^{2} ]\right\}^{1/2} \quad {\text{and}}\quad {\text{CI}}_{a} = t\left\{ {\hat{S}\sum X_{i}^{2} /[N(N - 2)\sum x^{2} ]} \right\}^{1/2} $$

(9)

where t is the critical value of Student’s t-distribution at the (1 − α) confidence level with N − 2 degrees-of-freedom (df) and the terms inside the braces are parameter standard errors. Measurement uncertainty is assumed to be known in the WLS case and confidence intervals for slope and intercept are given by,

$$ {\text{CI}}_{b} = Z\left\{ 1/\sum x^{2} \right\}^{1/2} \quad {\text{and}}\quad {\text{CI}}_{a} = Z\left\{ {\sum W_{i} X_{i}^{2} /(\sum W_{i} \sum x^{2} )} \right\}^{1/2} $$

(10)

where Z is the Normal deviate representing the (1 − α) confidence level and the terms inside the braces are parameter standard errors. Measurement uncertainty is incorporated into Eq. 10 via the $W_{i} ( = 1/\sigma_{{Y_{i} }}^{2} )$. Equation 10 applies also to Deming regression, but standard error calculations are substantially more complicated. The iterative scheme devised by Press et al. [13] was adopted here.

Joint slope, intercept confidence region

The joint slope and intercept confidence region about the point [$\hat{a}$, $\hat{b}$] is an elliptically shaped area defined [14] by the set of slope and intercept values that satisfy,

$$ \sum W_{i} (Y_{i} - a - bX_{i} )^{2} - \hat{S}[1 + 2F/(N - 2)] = 0 $$

(11)

where F is the critical value of the F-distribution with 2 and N − 2 df at the (1 − α) confidence level. Analytical solutions are available for OLS and WLS. Differentiation with respect to a yields,

$$ a = \bar{Y} - b\bar{X} $$

and substituting into Eq. 11 yields a quadratic in b,

$$ b^{2} \sum x^{2} - 2b\sum xy + \sum y^{2} - \hat{S}[1 + 2F/(N - 2)] = 0. $$

Solving by the usual formula yields,

$$ b = \hat{b} \pm \left\{ 2F\hat{S}/[(N - 2)\sum x^{2} ]\right\}^{1/2} $$

(12)

which defines the minimum and maximum slope values (b _min, b _max) of the joint (1 − α) confidence ellipse. That completes accuracy or power calculations if these limits fail to enclose the target slope. If enclosure occurs, the evaluation continues by substituting the target slope value, say B, into Eq. 11, solving the resulting quadratic to find the pair of intercept coordinates at slope B, then testing for enclosure of the target intercept. The quadratic solution reduces to,

$$ a = \bar{Y} - B\bar{X} \pm \left\{ [2F\hat{S}/(N - 2) - (B - \hat{b})^{2} \sum x^{2} ]/\sum W_{i} \right\}^{1/2} . $$

(13)

The terms inside the braces sum to a negative value (hence undefined) when B is outside the range b _min to b _max. Confidence ellipse plotting coordinates are found by defining a suitable series of values of B, between b _min and b _max, substituting each into Eq. 13 to obtain corresponding pairs of values of a.

Equation 11 also applies to Deming regression, but analytical solutions are not available because the W _i are functions of slope. However, enclosure of any particular target slope, say B, is easily determined by first updating the $W_{i} ( = 1/(\sigma_{{Y_{i} }}^{2} + B^{2} \sigma_{{X_{i} }}^{2} ))$, calculating,

$$ A = \bar{Y} - B\bar{X} $$

then evaluating Eq. 11 with the substitutions a = A and b = B. A result ≤0 signals enclosure of B. If necessary, enclosure of the target intercept is determined by substituting target slope B into Eq. 11 and solving the resulting quadratic to obtain the pair of intercept values at slope B. The quadratic solution (analogous to Eq. 13) reduces to,

$$ a = \bar{Y} - B\bar{X} \pm \left\{ [\hat{S}[1 + 2F/(N - 2)] + 2B\sum xy - B^{2} \sum x^{2} - \sum y^{2} ]/\sum W_{i} \right\}^{1/2} . $$

(14)

The approach used here to obtaining plotting coordinates is to firstly identify b _min and b _max, which are the minimum and maximum slope values that satisfy Eq. 11. They are not necessarily equidistant from $\hat{b}$ in the Deming case. The locations of b _min and b _max are first bracketed by using large outward steps from $\hat{b}$ until Eq. 11 evaluates to a positive value, then the bisection method is used in both cases to specify an efficient sequence of trial values of b until the relative difference between successive values is <10⁻⁵. Equation 14 is then evaluated for a suitable series of values of B, between b _min and b _max, noting that the W _i require updating for each new B value, thus requiring recalculation of all summation terms. Terms inside the braces in Eq. 14 sum to a negative value (hence undefined) when B is outside the range b _min to b _max.

Robustness

Table 1 results are a measure of the reliability of the methods described in preceding paragraphs. They were obtained under the ideal conditions of precisely known values of the $\sigma_{{X_{i} }}^{2}$ and $\sigma_{{Y_{i} }}^{2}$ (since the simulation data were randomly generated using predefined values of the $\sigma_{{X_{i} }}^{2}$ and $\sigma_{{Y_{i} }}^{2}$). However, the form of Eqs. 6, 7 demonstrates that WLS estimates of slope and intercept are unaffected by linear scaling of the $\sigma_{{Y_{i} }}^{2}$ because the W _i appear in both the numerator and denominator and linear scaling effects cancel out. Likewise, WLS joint parameter confidence ellipses are invariant to linear scaling of the $\sigma_{{Y_{i} }}^{2}$ because the left and right terms in Eq. 11 are effected identically. WLS slope, intercept and confidence ellipse are therefore unaffected by systematic over or underestimation of measurement uncertainty. They require only that the “shape” of the variance function is accurate. This protection also extends to Deming regression provided that linear scaling (relative inaccuracies) is the same for the X and Y variance functions. In practice, variance functions such as Eqs. 1–3 are obtained by analysis of precision data, and it is conceivable that procedural shortcomings, which cause systematic variance function inaccuracies, might exert similar effects on a pair of analytical tests under comparison. In contrast, confidence intervals estimated by Eq. 10 (WLS) and the methods in [2, 13] (Deming) rely implicitly on the absolute accuracy of the $\sigma_{{X_{i} }}^{2}$ and $\sigma_{{Y_{i} }}^{2}$. Inaccuracies translate into over- or underestimated confidence intervals and therefore erroneous significance test results.