Table 2 Impact of main effects on type I error and analytical bias for the case-only design
From: The case-only design is a powerful approach to detect interactions but should be used with caution
Disease prevalence | |||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\({{\varvec{\beta}}}_{{\varvec{G}}}\) | \({{\varvec{\beta}}}_{{\varvec{E}}}\) | 1.0% | 1.5% | 2.0% | 2.5% | 3.0% | 3.5% | 4.0% | 4.5% | 5.0% | 10.0% | 15.0% | 20.0% |
(a) Type I error | |||||||||||||
\(\ln(3.846)\) | \(\ln(5)\) | 0.141321 | 0.252332 | 0.396712 | 0.551859 | 0.695653 | 0.810552 | 0.891836 | 0.942837 | 0.972389 | 0.999998 | 1.000000 | 1.000000 |
\(\ln(3.846)\) | \(\ln(2)\) | 0.056403 | 0.063479 | 0.072751 | 0.085891 | 0.101210 | 0.119065 | 0.138098 | 0.160377 | 0.183840 | 0.480285 | 0.735506 | 0.873881 |
\(\ln(1.2)\) | \(\ln(2)\) | 0.050132 | 0.050603 | 0.049854 | 0.050842 | 0.050614 | 0.051195 | 0.051387 | 0.052100 | 0.052221 | 0.057378 | 0.063536 | 0.070049 |
\(\ln(1.05)\) | \(\ln(1.1)\) | 0.049890 | 0.049729 | 0.049665 | 0.049657 | 0.049644 | 0.049992 | 0.049896 | 0.049923 | 0.050019 | 0.050016 | 0.049992 | 0.049666 |
\(\ln(0.952)\) | \(\ln(0.909)\) | 0.049816 | 0.049991 | 0.049856 | 0.049738 | 0.049618 | 0.050020 | 0.050054 | 0.049950 | 0.049860 | 0.049651 | 0.049931 | 0.050423 |
\(\ln(0.833)\) | \(\ln(0.5)\) | 0.050030 | 0.049658 | 0.049459 | 0.049875 | 0.049607 | 0.049515 | 0.049339 | 0.049723 | 0.049573 | 0.049940 | 0.051270 | 0.052579 |
\(\ln(0.26)\) | \(\ln(0.5)\) | 0.048962 | 0.048521 | 0.048793 | 0.048931 | 0.049388 | 0.049655 | 0.050008 | 0.050319 | 0.051327 | 0.063964 | 0.088547 | 0.125843 |
\(\ln(0.26)\) | \(\ln(0.2)\) | 0.047237 | 0.047101 | 0.046880 | 0.047278 | 0.047025 | 0.047459 | 0.047611 | 0.048081 | 0.048999 | 0.063687 | 0.096078 | 0.152817 |
(b) Analytical bias | |||||||||||||
\(\ln(3.846)\) | \(\ln(5)\) | −0.039119 | −0.057894 | −0.076150 | −0.093890 | −0.111119 | −0.127843 | −0.144066 | −0.159797 | −0.175042 | −0.302652 | −0.391307 | −0.449137 |
\(\ln(3.846)\) | \(\ln(2)\) | −0.012563 | −0.018684 | −0.024697 | −0.030605 | −0.036407 | −0.042105 | −0.047698 | −0.053188 | −0.058576 | −0.106964 | −0.145864 | −0.175993 |
\(\ln(1.2)\) | \(\ln(2)\) | −0.001667 | −0.002479 | −0.003277 | −0.004061 | −0.004831 | −0.005588 | −0.006330 | −0.007060 | −0.007775 | −0.014227 | −0.019476 | −0.023631 |
\(\ln(1.05)\) | \(\ln(1.1)\) | .−0.000048 | −0.000072 | −0.000095 | −0.000118 | −0.000141 | −0.000164 | −0.000186 | −0.000208 | −0.000230 | −0.000434 | −0.000612 | −0.000764 |
\(\ln(0.952)\) | \(\ln(0.909)\) | −0.000044 | −0.000066 | −0.000087 | −0.000109 | −0.000130 | −0.000151 | −0.000171 | −0.000192 | −0.000212 | −0.000404 | −0.000574 | −0.000724 |
\(\ln(0.833)\) | \(\ln(0.5)\) | −0.000928 | −0.001389 | −0.001847 | −0.002303 | −0.002756 | −0.003207 | −0.003655 | −0.004101 | −0.004544 | −0.008831 | −0.012836 | −0.016536 |
\(\ln(0.26)\) | \(\ln(0.5)\) | −0.005304 | −0.007952 | −0.010597 | −0.013240 | −0.015878 | −0.018514 | −0.021146 | −0.023774 | −0.026399 | −0.052388 | −0.077767 | −0.102296 |
\(\ln(0.26)\) | \(\ln(0.2)\) | −0.008789 | −0.013196 | −0.017611 | −0.022035 | −0.026467 | −0.030907 | −0.035354 | −0.039810 | −0.044273 | −0.089276 | −0.134820 | −0.180639 |