I was not able to reproduce the results you got from WebPower using the pilot data you supplied.  I was able to reproduce your R code however.

You are correct that you can't use the $\eta^2$ for Cohen's f, but $f^2 = \frac{\eta^2}{1-\eta^2}$

1. "However, how should I compute the effect size from the pilot study" - use the $\eta^2$ from the pilot study.

2. "Why are there interaction effect sizes, i.e, the effect size for group x vs group y?" Those are the effect sizes for the pair-wise comparisons (if you were using a t-test or a TukeyHSD)

``` r
require(dplyr)
require(reshape2)

pilot <- data.frame(option1 = c(6.3, 2.8, 7.8, 7.9, 4.9),
                    option2 = c(9.9, 4.1, 3.9, 6.3, 6.9),
                    option3 = c(5.1, 2.9, 3.6, 5.7, 4.5),
                    option4 = c(1.0, 2.8, 4.8, 3.9, 1.6))
pilot2 <- pilot %>% 
  reshape2::melt(value.name = "y") %>%
  dplyr::rename("option" = "variable")

lm1 <- lm(y ~ option, data = pilot2)
aov1 <- aov(lm1)

means <- apply(pilot, 2, mean)
vs <- apply(pilot, 2, var)

# cohen's f for overall anova
# eta^2 = SSR / SST
eta.sq <- anova(lm1)$`Sum Sq`[2] / sum(anova(lm1)$`Sum Sq`)
f <- sqrt(eta.sq / (1-eta.sq))

# cohen's d for pairwise
d <- abs(means[c(1,1,1,2,2,3)] - means[c(2,3,4,3,4,4)]) / 
    sqrt(((5-1)*vs[c(1,1,1,2,2,3)] + 
    (5-1)*vs[c(2,3,4,3,4,4)])/ (5+5))
names(d) <- c("1-2", "1-3", "1-4", "2-3", "2-4", "3-4")

require(pwr)

# with 5 samples, we have the power to detect effect size f = 0.835
#  i.e. with only 5 samples, we need a large effect to detect

pwr::pwr.anova.test(k = 4, n = 5, sig.level = 0.05, power = 0.80)
#> 
#>      Balanced one-way analysis of variance power calculation 
#> 
#>               k = 4
#>               n = 5
#>               f = 0.8352722
#>       sig.level = 0.05
#>           power = 0.8
#> 
#> NOTE: n is number in each group

# since we have a really large effect in the pilot for f = 1.2,
#   we only need 3 per group to detect with 80% power

pwr::pwr.anova.test(k = 4, f = 1.2414, sig.level = 0.05, power = 0.80)
#> 
#>      Balanced one-way analysis of variance power calculation 
#> 
#>               k = 4
#>               n = 2.950833
#>               f = 1.2414
#>       sig.level = 0.05
#>           power = 0.8
#> 
#> NOTE: n is number in each group
```