# Tukey correction in emmeans

I have a question about the Tukey correction in emmeans. See the example below. It says "P value adjustment: tukey method for comparing a family of 3 estimates." Does this mean that the error rate for all of these comparisons (which were run together) is 5%? What does "a family of 3 estimates" refer to?

The two factors here were "group," which had 3 levels, and "condition," which had 4 levels, so there are a total of 12 comparisons. If each comparison had been run separately, the error rate would have been 5% comparison. I'm hoping that because these were all run together with the Tukey correction, the error rate is down to 5% for all of them. What I'm confused about is the reference to "3 estimates," and I'm wondering if this means the error rate is 5% for each set of three group comparisons (one set per condition), meaning the error rate is actually 1 - 0.95^4 (about 18%)?

Thank you!

> emmeans(model_1, pairwise ~ group | condition)
$emmeans condition = EN-TVJT-R-BRF: group emmean SE df lower.CL upper.CL EN 0.0456 0.0174 219 0.01129 0.0799 HS 0.1515 0.0211 368 0.11004 0.1930 SB 0.0208 0.0246 500 -0.02745 0.0691 condition = EN-TVJT-R-BRT: group emmean SE df lower.CL upper.CL EN 0.9848 0.0174 219 0.95043 1.0191 HS 0.8712 0.0211 368 0.82974 0.9127 SB 0.9479 0.0246 500 0.89964 0.9962 condition = EN-TVJT-R-THF: group emmean SE df lower.CL upper.CL EN 0.0254 0.0174 219 -0.00891 0.0597 HS 0.1742 0.0211 368 0.13277 0.2157 SB 0.0417 0.0246 500 -0.00661 0.0899 condition = EN-TVJT-R-THT: group emmean SE df lower.CL upper.CL EN 0.9898 0.0174 219 0.95548 1.0241 HS 0.8939 0.0211 368 0.85247 0.9354 SB 0.9688 0.0246 500 0.92047 1.0170 Degrees-of-freedom method: kenward-roger Confidence level used: 0.95$contrasts
condition = EN-TVJT-R-BRF:
contrast estimate     SE  df t.ratio p.value
EN - HS   -0.1059 0.0266 783 -3.983  0.0002
EN - SB    0.0248 0.0294 783  0.842  0.6768
HS - SB    0.1307 0.0317 783  4.117  0.0001

condition = EN-TVJT-R-BRT:
contrast estimate     SE  df t.ratio p.value
EN - HS    0.1136 0.0266 783  4.271  0.0001
EN - SB    0.0368 0.0294 783  1.252  0.4230
HS - SB   -0.0767 0.0317 783 -2.417  0.0420

condition = EN-TVJT-R-THF:
contrast estimate     SE  df t.ratio p.value
EN - HS   -0.1488 0.0266 783 -5.597  <.0001
EN - SB   -0.0162 0.0294 783 -0.552  0.8454
HS - SB    0.1326 0.0317 783  4.177  0.0001

condition = EN-TVJT-R-THT:
contrast estimate     SE  df t.ratio p.value
EN - HS    0.0959 0.0266 783  3.606  0.0010
EN - SB    0.0211 0.0294 783  0.716  0.7543
HS - SB   -0.0748 0.0317 783 -2.357  0.0489

Degrees-of-freedom method: kenward-roger
P value adjustment: tukey method for comparing a family of 3 estimates


This is documented in ? summary.emmGrid. The Tukey correction is applied to each set of comparisons of three means. If you want all 12 comparisons to be adjusted as one family, you need to do something like
EMM <- emmeans(...) # as shown