TukeyHSD p-value is less than t-test p-value I have some data where when I do t.test for a specific pair of groups, the p-value is (barely) higher than when I do TukeyHSD and include all 6 comparisons.
I was under the impression that the adjusted p-value would always be higher and I'm wondering why it isn't.
This is very small data - only 8-10 measures per group and I am thinking that perhaps the difference is in how each procedure handles groups of different sizes?
 A: For a single data set, this can happen. But the probability that the t-test will reject will be higher than that the TukeyHSD-test rejects this comparision.
One of the assumptions is the homogeneity of variance. Let's assume that even though the assumption is fulfilled the sample within-group-variance in your data is (just by chance) very high for these two groups you referenced. The single t-test then overestimates the variance and therefore gives a high p-value, while the TukeyHSD-test uses the pooled sample variance and gives therefore a lower p-value.
For example if generate data in the following way, we would expect the effect you have seen (which would violate the assumptions - but this data could of course also occur by chance under the assumptions):
set.seed(1234)
dat <- data.frame(y=c(rnorm(10), 
                       rnorm(10, mean=1),
                       rnorm(50, sd=0.5)),
                   group=rep(LETTERS[1:3], times=c(10,10,50)))

boxplot(y~group, data=dat)


 TukeyHSD(aov(y~group, data=dat))
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = y ~ group, data = dat)

$group
          diff        lwr        upr     p adj
B-A  1.2649867  0.5198000  2.0101734 0.0003695
C-A  0.2405365 -0.3366827  0.8177556 0.5801783
C-B -1.0244502 -1.6016694 -0.4472311 0.0001951

t.test(dat$y[dat$group=="A"], dat$y[dat$group=="B"])

    Welch Two Sample t-test

data:  dat$y[dat$group == "A"] and dat$y[dat$group == "B"]
t = -2.7404, df = 17.914, p-value = 0.01349
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -2.2351190 -0.2948544
sample estimates:
 mean of x  mean of y 
-0.3831574  0.8818293 

As you can see the $p$-value of the t-test for comparing group A and B is 0.01349 and higher than the adjusted $p$-value of 0.0003695, since we have a higher sample variance if we only look at these two groups.
