Measurement error and misclassification

Correct answer: 1.775. For the same drop, specificity does more damage, because the unexposed are the majority and false positives are manufactured out of the larger group

1.775 sits below the 1.911 from the sensitivity row, so in this simulation an equal drop in specificity really does cost more. The reason is prevalence: exposure runs at three tenths, the unexposed are the majority, and a percentage point off specificity manufactures false positives out of the larger group while a percentage point off sensitivity misses true positives in the smaller one. 1.472 is the row where sensitivity and specificity both fall further, and the claim attached to it - that the ordering holds regardless of prevalence - is wrong, because the ordering reverses when exposure is common. 1.904 comes from the separate curve that varies sensitivity only, and calling the two parameters symmetric is the opposite of what this section demonstrates. What does not depend on prevalence is the direction, not which parameter hurts more, and even the direction carries three preconditions: a binary exposure, misclassification of exposure independent of misclassification of outcome, and sensitivity plus specificity above 1.

Correct answer: 1.227. One true value, one average quality of instrument, and the direction can reverse - what makes differential misclassification hard is not how wrong it is but how unevenly

1.227 pushes the truth down to less than half, while the first row's 2.392 exaggerates the same truth - the two rows use the same sensitivities and specificities, with only the question of who is more accurate swapped. Differential misclassification therefore has no predictable direction, which is what makes it harder to live with than the non-differential kind: you cannot even fall back on saying the truth must lie further from the null. 1.695 is the fourth row, which deliberately sets both groups to the same parameters and so removes the differential part, which is why it obediently biases towards the null; taking it as the general rule for differential misclassification reverses the section's conclusion. 2.294 is the third row, where the specificity among cases slips a little through over-reporting, and it does exaggerate - but it is one of four possibilities, not the rule. The practical test is when the exposure information was collected: an exposure asked about after the outcome is known makes differential misclassification a default topic.

Correct answer: 2.134, higher than perfect measurement. Non-differential misclassification has pushed the middle stratum away from the null, so the bias-towards-the-null rule fails once there are more than two categories

2.134 exceeds the 2.028 from perfect measurement and exceeds this comparison's own truth as well - a non-differential error has pushed the estimate past the null on the far side. The mechanism is that the asymmetric matrix moves a large share of the top stratum into the middle one, so the middle stratum fills up with genuinely high-exposure people and its risk rises. 3.105 and 3.200 both belong to the high-versus-low row rather than the one asked about, and the claim attached to 3.200 - that a matching correct-classification rate forces a matching answer - is exactly what this table refutes, since the two matrices were given identical accuracy on purpose and differ only in the direction of the error. 1.716 is the symmetric column, which does follow the rule. The practical consequence lands on dose-response: a tidy staircase may owe part of its shape to the top stratum being under-read, so observing a dose-response trend and therefore inferring causation deserves a discount whenever the exposure categories are coarse or self-reported.

Correct answer: 0.754, about half the true slope. Classical measurement error multiplies the coefficient by the reliability, and that amount is algebra rather than an empirical regularity

0.754 is almost exactly half of 1.5, and the reliability is 0.5 - the coefficient is multiplied by the reliability, a closed form rather than an empirical regularity, and the near-identical estimated-slope and reliability-times-true-slope columns are the evidence. 1.201 belongs to a row with a smaller error standard deviation, and treating dilution as proportional to the size of the error badly understates the damage once the error grows. 0.300 is the row with the largest error, where reliability is only two tenths; mapping a reliability of 0.5 onto a fall to a fifth confuses two rows. Two practical consequences: a repeat-measurement substudy lets the reliability be estimated and the coefficient divided back out, which is regression calibration, and without that substudy there is nothing to calibrate against. And because dilution shrinks the test statistic along with the point estimate, a poorly measured exposure systematically makes a real association harder to detect, so reading such a result as no association turns a measurement problem into a biological conclusion.

Correct answer: 1.494. The baseline risk is only 2%, so a false-positive rate of two percentage points acting on the overwhelmingly healthy majority manufactures spurious events on the same scale as the real ones

1.494 is the specificity 0.98 row, and it falls far further than the sensitivity 0.70 row at 1.985. This is peculiar to rare outcomes: real events are scarce to begin with, so a false-positive rate acting on the overwhelmingly healthy majority creates spurious events on the same scale as the real ones, and both arms are diluted down towards the same background noise. 1.999 comes from the row with perfect specificity and sensitivity alone reduced - the first three rows really do barely move, but that is a statement about sensitivity, since missed events are proportional across the arms and the ratio survives, and carrying it over to specificity reverses the conclusion. 1.654 is the row where specificity drops by one point only, and it sits above 1.494, so the table refutes the claim that a two point drop hurts less. The practical consequence is for database studies that define outcomes with diagnosis codes: the usual worry is that the codes miss patients, but for a rare outcome what destroys the estimate is false positives, so a validation study should report positive predictive value and specificity.

Correct answer: 1.676. Adjustment removes only part of the confounding, and what remains is indistinguishable from a real effect

1.676 is still a long way from the truth, and it is an adjusted number - most of the adjustment was eaten by measurement error in the confounder. 1.086 belongs to the row with very little error, and taking it as the general case yields in the model is enough, which is the opposite of what this table says row after row. 2.090 is the row with the most error; it does approach the unadjusted 2.380 without reaching it, because this column varies continuously rather than switching between adjusted and unadjusted. The practical version: adjusted for smoking is not the same as smoking is no longer a confounder, because smoking recorded as a yes-or-no box carries no information about pack-years, and the confounding that survives is indistinguishable from a real effect. So the limitations paragraph should not say that all known confounders were adjusted for; it should say how those confounders were measured and roughly how much confounding is likely to remain.

Correct answer: It reads 0.370, above the truth. Losing sensitivity pushes the apparent prevalence down and losing specificity pushes it up, and the upward push is the larger of the two

0.370 sits above the true 0.30, while the same drop in sensitivity pushes it down to 0.269: the same ten points move it up more than twice as far as they move it down. The reason is the one behind the odds ratio column - only three in ten are exposed, so the false positives created by imperfect specificity come out of the large unexposed group, while the true positives missed by imperfect sensitivity can only come out of the smaller one. 0.239 is a row further down the sensitivity-only block, and it too sits below the truth, but using it to claim that misclassification always pulls the prevalence down ignores the false-positive route, which is what the lower half of the same table is doing. 0.313 is the row where specificity slips by two points only, and reading it as the answer for a ten-point drop is how the symmetric conclusion arises, which this entire column refutes. The practical use is direct: when a study reports an exposure prevalence that will not line up with a registry or a national survey, ask how exposure was measured - too high points at specificity, too low at sensitivity.