Table 6 The application of the case-only approach for the first and second example

logit P(D = 1) = β₀ + β₂’E

OR for 1 unit difference of environmental exposure = exp.(β₂’)

OR for a 1 μg/L difference of blood chromium level: 2.20 (95% CI 1.48–3.32)

Effect estimate

When a 1 μg/L of blood chromium level (μg/L) differs, the fold-difference in the odds of albuminuria is 2.20 (95% CI 1.48–3.32) times.

Explanation

logit P(D = 1) = β₀ + β₁’S

OR for 1 unit difference of susceptibility factor = exp.(β₁’)

OR for 1% difference of glycohemoglobin level: 1.57 (95% CI 1.44–1.73)

Effect estimate

When a 1% of blood glycohemoglobin level differs, the fold-difference in the odds of albuminuria is 1.57 (95% CI 1.44–1.73) times.

Explanation

logit P(D = 1) = β₀ + β₁S + β₂E + β₃SE

ICR_c/nc = exp.(β₃)

Eq. (8)

Eq. (9)

ICR_c/nc: 0.72 (95% CI 0.35–1.60)

Effect estimate

When a 1 μg/L of both blood chromium level and 1% of blood glycohemoglobin level coincide, the multiplicative ICR is 0.72 (95% CI 0.35–1.60), with statistical insignificance.

Explanation

logit P(S = 1) = η₀ + η₁E

S-E OR_c/nc = exp.(η₁)

Eq. (6)

Eq. (7)

S-E OR_c/nc: 0.76 (95% CI 0.47–1.06)

Effect estimate

In the the population with cases and non-cases, blood chromium levels and blood glycohemoglobin levels are independent. Therefore, the case-only ICR can be a good substitute for the ICR acquired from the population with cases and non-cases.

Explanation

logit P(S = 1) = γ₀ + γ₁E

ICR_CO = exp.(γ₁)

Eq. (4)

Eq. (5)

ICR_CO: 0.59 (95% CI 0.28–0.95)

Effect estimate

When only the cases are analyzed (case-only approach), the case-only ICR is 0.59 (95% CI 0.28–0.95), with a statistical significance (a negative interaction effect).

Explanation

logit P(D = 1) = β₀ + β₂’E

OR for 1 unit difference of environmental exposure = exp.(β₂’)

OR for 1 μg/L difference of blood cobalt level: 1.09 (95% CI 0.98–1.20)

Effect estimate

When a 1 μg/L of blood cobalt level (μg/L) differs, the fold-difference in the odds of albuminuria is 1.09 (95% CI 1.31–1.57) times.

Explanation

logit P(D = 1) = β₀ + β₁’S

OR for 1 unit difference of susceptibility factor = exp.(β₁’)

OR for a 1-year difference of age: 1.05 (95% CI 1.04–1.05)

Effect estimate

When 1-year in age differs, the fold-difference in the odds of albuminuria is 1.05 (95% CI 1.04–1.05) times.

Explanation

logit P(D = 1) = β₀ + β₁S + β₂E + β₃SE

ICR_c/nc = exp.(β₃)

Eq. (8)

Eq. (9)

ICR_c/nc: 1.13 (95% CI 0.99–1.37)

Effect estimate

When a 1 μg/L difference of both blood cobalt level and 1-year difference of age coincide, the multiplicative ICR is 1.13 (95% CI 0.99–1.37), with statistical insignificance.

Explanation

logit P(S = 1) = η₀ + η₁E

S-E OR_c/nc = exp.(η₁)

Eq. (6)

Eq. (7)

S-E OR_c/nc: 1.06 (95% CI 1.03–1.10)

Effect estimate

In the a population with cases and non-cases, blood cobalt level and age in years show a slight association (not completely independent). Therefore, the case-only ICR must be multiplied by the S-E OR_c/nc to be ICR_c/nc according to Eq. (3).

Explanation

logit P(S = 1) = γ₀ + γ₁E

ICR_CO = exp.(γ₁)

Eq. (4)

Eq. (5)

ICR_CO: 1.14 (95% CI 1.03–1.37)

Effect estimate

When only the cases were analyzed (case-only approach), the case-only ICR was 1.14 (1.03–1.37), with a statistical significance (a positive interaction effect).

Explanation

\({\mathrm{ICR}}_{\mathrm{c}/\mathrm{nc}}=\frac{{\mathrm{RR}}_{\mathrm{s}\mathrm{e}}}{{\mathrm{RR}}_{\mathrm{s}}{\mathrm{RR}}_{\mathrm{e}}}=\left(\frac{\mathrm{ag}}{\mathrm{c}\mathrm{e}}\right)\left(\frac{\left(\mathrm{c}+\mathrm{D}\right)\left(\mathrm{e}+\mathrm{F}\right)}{\left(\mathrm{a}+\mathrm{B}\right)\left(\mathrm{g}+\mathrm{H}\right)}\right)=\left({\mathrm{ICR}}_{\mathrm{CO}}\right)\left(\mathrm{S}-\mathrm{E}\ {\mathrm{OR}}_{\mathrm{c}/\mathrm{nc}}\right)\)

Eq. (2)

ICR_CO: 1.14 (1.03–1.37) × S-E OR_c/nc: 1.06 (95% CI 1.03–1.10)

ICR_c/nc: 1.21 (95% CI 1.06–1.51)

Effect estimate

The ICR_CO multiplied by the S-E OR_c/nc produced the ICR_c/nc of 1.21 (95% CI 1.06–1.51).

Explanation