As will be demonstrated momentarily, a version of Theorem 1 does not hold for the linear trend test. This test, versions of which were first proposed by Cochran (1954) and Armitage (1955), uses the linear contrast $C=(1-k,3-k,\dots ,k-3,k-1)$ to define the linear component of $E(Y|X)$. A direct interpretation of the test is that the null $C\mathrm{\beta}=0$ holds if and only if the least-squares slope when $({\mathrm{\beta}}_{1},\dots ,{\mathrm{\beta}}_{k})$ is regressed upon $(1,\dots ,k)$ is zero.

To gain a general sense of how this test is affected by misclassification, we compare $g({\mathrm{\alpha}}^{\ast},{\mathrm{\beta}}^{\ast},{\mathrm{\kappa}}^{\ast})$ to $g(\mathrm{\alpha},\mathrm{\beta},\mathrm{\kappa})$ for a large ensemble of $\mathrm{\alpha}$, $\mathrm{\beta}$, and *P* values. Particularly, we fix $k=6$ exposure categories and also fix $\mathrm{\kappa}=(1,\dots ,1)$. Then values of $(\mathrm{\alpha},\mathrm{\beta},P)$ are drawn from 12 different probability distributions. The first three distributions arise from fixing $\mathrm{\alpha}=(1/6,\dots 1/6)$, thinking of a uniform distribution for *X* as being a simple but important special case. A distribution on $\mathrm{\beta}$ is assigned by fixing ${\mathrm{\beta}}_{1}=0$ without loss of generality, and then taking the increments $({\mathrm{\beta}}_{2}-{\mathrm{\beta}}_{1},\dots ,{\mathrm{\beta}}_{k}-{\mathrm{\beta}}_{k-1})$ to be distributed as $\mathrm{D}\mathrm{i}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{l}\mathrm{e}\mathrm{t}\phantom{\rule{1pt}{0ex}}(c,\dots ,c)$, so that $\mathrm{\beta}$ is increasing and ${\mathrm{\beta}}_{k}=1$ is fixed. In our experiments we take $c=2$, which allows the possibility of far from linear patterns (as *c* increases the distribution puts more weight on relationships which are closer to linear). Then the three distributions are based on sampling *P* from a specified distribution over tridiagonal classification matrices, and conditioning on either (i) *P* being non-monotone, (ii) *P* being monotone but not tapered, or (iii) *P* being tapered. The distribution on *P* generates tridiagonal matrices by independently drawing $({p}_{11},{p}_{12})\sim \phantom{\rule{1pt}{0ex}}\mathrm{b}\mathrm{e}\mathrm{t}\mathrm{a}\phantom{\rule{1pt}{0ex}}(a,b)$, $({p}_{k,k-1},{p}_{kk})\sim \phantom{\rule{1pt}{0ex}}\mathrm{b}\mathrm{e}\mathrm{t}\mathrm{a}\phantom{\rule{1pt}{0ex}}(b,a)$, and $({p}_{i,i-1},{p}_{ii},{p}_{i,i+1})\sim \phantom{\rule{1pt}{0ex}}\mathrm{D}\mathrm{i}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{l}\mathrm{e}\mathrm{t}\phantom{\rule{1pt}{0ex}}(b/2,a,b/2)$, for $i=2,\dots ,k-1$. We use $(a,b)=(20,10)$, giving a mean of 0.667 and a standard deviation of 0.085 for each diagonal element of *P*, before conditioning. This ensures that a wide range of classification matrices is being generated, with the typical extent of misclassification being quite considerable.

The next three distributions arise exactly as per the first three, except $\mathrm{\alpha}$ is now also taken as random, with $\mathrm{\alpha}\sim \phantom{\rule{1pt}{0ex}}\mathrm{D}\mathrm{i}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{h}\mathrm{l}\mathrm{e}\mathrm{t}\phantom{\rule{1pt}{0ex}}(d,\dots ,d)$, so that each ${\mathrm{\alpha}}_{i}$ has mean $1/6$. This allows departures from *X* being uniformly distributed (which would correspond to $d\to \mathrm{\infty}$). We wish to take *d* small enough to engender substantial departures from uniformity, but not so small as to yield distributions that place almost no mass on one or more values. Thus we set $d=8$, which induces a standard deviation of 0.053 for each ${\mathrm{\alpha}}_{i}$.

Results for $(\mathrm{\alpha},\mathrm{\beta},P)$ arising from these six distributions appear in Figure 1. For both the general test and the trend test, results are given in terms of relative power, defined as the power achieved from misclassified data $({X}^{\ast},Y)$ at the sample size for which ideal data $(X,Y)$ yield 80% power. That is, if ${F}_{q}(;t)$ denotes the distribution function for the noncentral ${\mathrm{\chi}}_{q}^{2}$ distribution with noncentrality *t*, then the relative power is
$RP=1-{F}_{q}\left\{{F}_{q}^{-1}(0.95;0);{t}_{q}\frac{g({\mathrm{\alpha}}^{\ast},{\mathrm{\beta}}^{\ast},{\mathrm{\kappa}}^{\ast})}{g(\mathrm{\alpha},\mathrm{\beta},\mathrm{\kappa})}\right\},$where ${t}_{q}$ solves ${F}_{q}\{{F}_{q}^{-1}(0.95;0);t\}=0.2$. For the general test, Theorem 1 guarantees that the relative power cannot exceed 80%. Figure 1 shows contrary behavior for the trend test. At least when the distribution of *X* can depart from uniformity, there are values of *P* (of all three types) and $\mathrm{\alpha}$ and $\mathrm{\beta}$ for which misclassification induces power gain. Of course these results are based on first-order asymptotic theory; in Appendix A we present a small simulation indicating that the theory does indeed translate to the finite-sample case. Also, while our simulated scenarios never produced a power gain when *X* is uniformly distributed, in Appendix B we exhibit a $\mathrm{\beta}$ and a tridiagonal, tapered *P* such that there is power gain in this case.

Figure 1 Relative power for the trend test and for the general test, with tridiagonal misclassification matrices. The panels give 1,000 realizations from each of the six distributions over $(\mathrm{\alpha},\mathrm{\beta},P)$ described in the text, with $\mathrm{\alpha}$ fixed (random) in the upper (lower) panels. Quartiles of relative power are indicated (diamonds), and the reference lines correspond to 80% relative power

The remaining six distributions for $(\mathrm{\alpha},\mathrm{\beta},P)$ are constructed as per the first six described above, but with the underlying distribution of *P* modified so as to generate non-banded matrices. This is achieved by taking the rows of *P* to be independently distributed as Dirichlet over all *k* categories. To mimic the earlier construction, for each row the parameters of the Dirichlet distribution are taken to be *a* for the correct category and to sum to *b* over the $k-1$ incorrect categories, implying the same $\phantom{\rule{1pt}{0ex}}\mathrm{b}\mathrm{e}\mathrm{t}\mathrm{a}\phantom{\rule{1pt}{0ex}}(a,b)$ distribution for diagonal elements as previously. For all but the first and last rows, sub-sums of $b/2$ are forced, both below and above the correct category. The specification is completed by fixing a value of *r* such that the Dirichlet parameters for a given row decay geometrically by a factor of *r* as we move further away from the correct category. We take $(a,b,r)=(20,10,0.25)$ in generating the results of Figure 2. Qualitatively, we see very similar behavior as before with tridiagonal misclassification.

Figure 2 Relative power for the trend test and for the general test, with non-banded misclassification matrices. The panels give 1,000 realizations from each of the six distributions over $(\mathrm{\alpha},\mathrm{\beta},P)$ described in the text, with $\mathrm{\alpha}$ fixed (random) in the upper (lower) panels. Quartiles of relative power are indicated (diamonds), and the reference lines correspond to 80% relative power

To seek further insight into when misclassification increases power, we focus on the situation of non-uniform *X* distributions and tapered misclassification matrices, i.e., the settings underlying the lower right panels of both Figures 1 and 2. For these simulated values, in Figure 3 we first plot the relative power of the trend test against the absolute ratio of $C{\mathrm{\beta}}^{\ast}$ to $C\mathrm{\beta}$. Values of this ratio larger than one correspond to the misclassification inflating the slope which summarizes the linear component of the exposure–disease relationship. In situations involving linear regression of a continuous outcome on a continuous exposure, nondifferential exposure misclassification is quite widely guaranteed to attenuate the regression slope being estimated [see, for instance, Gustafson (2004); Carroll et al. (2006)]. Figure 3 shows that such a guarantee does not apply to the present situation, but it is perhaps not surprising that attenuation is much more common than inflation across the simulated scenarios. What is more surprising is that slope inflation is neither a necessary condition nor a sufficient condition for power gain. That is, in some scenarios the misclassification inflates the slope yet power is lost for detecting that the slope is not zero. And in some situations, the misclassification attenuates the slope, yet power is gained for detecting that the slope is not zero. So the behavior of the slopes alone is not determining whether power is lost or gained.

Next in Figure 3 we plot the relative power against ${min}_{i}\phantom{\rule{thinmathspace}{0ex}}{p}_{ii}$, to see whether the worst probability of correct classification across exposure categories is a driving force behind the power of the test based on misclassified data. The plots provide a negative answer here, indicating very little association, if any, between this characteristic of the classification matrix and the relative power.

Finally, we consider the extent to which misclassification pushes the exposure distribution toward or away from uniformity. We define the log sum-of-squares ratio (LSSR) as
$\mathrm{L}\mathrm{S}\mathrm{S}\mathrm{R}=log\frac{{\sum}_{i=1}^{k}{({\mathrm{\alpha}}_{i}^{\ast}-{k}^{-1})}^{2}}{{\sum}_{i=1}^{k}{({\mathrm{\alpha}}_{i}-{k}^{-1})}^{2}}.$So, for instance, a large positive value of LSSR describes a case where the misclassification induces a distribution of ${X}^{\ast}$ that is much further from uniform than the distribution of *X*. Figure 3 exhibits quite strong negative associations between relative power and LSSR, in both the tridiagonal and non-banded cases. Thus scenarios with power gain tend to have misclassification which makes ${X}^{\ast}$ much more uniformly distributed than *X*. But again this is only a stochastic tendency with respect to the chosen distribution of $(\mathrm{\alpha},\mathrm{\beta},P)$. A low LSSR is neither a necessary nor sufficient condition for power gain. In general, there seems to be considerable complexity in how the distribution of *X*, the distribution of $(Y|X)$, and the misclassification matrix *P* combine to determine the performance of the trend test applied to misclassified data.

Figure 3 Relative power of the trend test versus (i) absolute ratio of slopes (top panels), (ii) smallest diagonal element of *P* (middle panels), and (iii) LSSR (bottom panels). The misclassification matrices are tapered throughout. The left (right) panels correspond to tridiagonal (non-banded) misclassification matrices. For both distributions of $(\mathrm{\alpha},\mathrm{\beta},P)$, 1,000 realizations are simulated

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