Rationale for small sample leads to larger effect sizes I was recently attending a seminar where the speaker mentioned something on the lines of:

Here's a statistical fallacy...smaller sample sizes will lead to
larger effect sizes

However, the speaker did not elaborate any further and there was no opportunity to follow up on it.
Could someone explain the idea behind this fallacy?

I do not see it as a fallacy, but I think I know what the speaker meant.
As the sample size increases, less effect size is demanded in order to reject at the $0.05$-level (or whatever level is desired). Therefore, if you are able to get a significant result with a small sample size, the observed effect size must be large. On the other hand, if the sample size is large, we can reject with just a small observed effect size.
library(ggplot2)
N1 <- 10
N2 <- 100
B <- 1000
e1 <- e2 <- rep(NA, B)
for (i in 1:B){
    
    x1 <- rnorm(N1, 0, 1)
    y1 <- rnorm(N1, 0.1, 1)
    
    x2 <- rnorm(N2, 0, 1)
    y2 <- rnorm(N2, 0.1, 1)
    
    # Screen for a "significant" t-test at the 0.05-level
    #
    if (t.test(x1, y1, var.equal = T)$p.value < 0.05){
    e1[i] <- abs(mean(x1) - mean(y1))
}
if (t.test(x2, y2, var.equal = T)$p.value < 0.05){
        e2[i] <- abs(mean(x2) - mean(y2))
    }
}

# Remove the NAs
#
e1 <- e1[is.na(e1) == F]
e2 <- e2[is.na(e2) == F]

# Put into a data frame and plot
#
d1 <- data.frame(effect = e1, sample = "Small")
d2 <- data.frame(effect = e2, sample = "Large")
d <- rbind(d1, d2)
ggplot(d, aes(x = effect, fill = sample)) +
    geom_density(alpha = 0.2) +
    theme_bw()

The observed effect size magnitude for the small sample size is much larger than for the large sample size. However, there are many more rejections ($alpha = 0.05$) with the large sample size (61 for the small sample size vs 124 for the large).

A practical application of this is in p-hacking (which you should not do, even if people do it). Since you are more likely to see a large effect size when you reject with a small sample size, one research strategy could be to do experiments with small sample sizes and then present the ones that are significant, wowing your audiences with the large effect sizes. This is a terrible approach to science, and I bring it up as something to keep in mind might be done when we look at literature.
