```
library(marginaleffects)
dat <- transform(mtcars, cyl = factor(cyl))
mod <- lm(mpg ~ cyl, dat)
hyp <- hypotheses(mod, "cyl6 = cyl8")
hyp
#>
#> Estimate Std. Error z Pr(>|z|) S 2.5 % 97.5 %
#> 4.64 1.49 3.11 0.00186 9.1 1.72 7.57
#>
#> Term: cyl6 = cyl8
#> Columns: term, estimate, std.error, statistic, p.value, s.value, conf.low, conf.high
```

# S Values

The *S* value — “Shannon transform” or “binary surprisal value” — is a cognitive tool to help analysts make intuitive sense of p values [@RafGre2020]. It allows us to compare a p value to the outcome of a familiar game of chance.

Consider this: We toss a coin 4 times to see if we can reject the null hypothesis that the coin toss is fair. If the null is true, the probability of drawing *Heads* on any single toss is \(\frac{1}{2}\). The probability of observing 4 *Heads* in a row is \(\left (\frac{1}{2} \right )^4=\frac{1}{16}=0.0625\). This probability characterizes the “surprise” caused by observing 4 straight heads in a world where the null holds, that is, where the coin toss is fair.

Now consider a different exercise: We estimate a model and use `marginaleffects::hypotheses()`

to test if two of the estimated coefficients are equal:

The difference between `cyl6`

and `cyl8`

is 4.64, and the associated p value is 0.0018593. Again, the p value can be interpreted as a measure of the surprise caused by the data if the null were true (i.e., if the two coefficients were in fact equal).

How many consecutive *Heads* tosses would be as surprising as this test of equality? To answer this question, we solve for \(s\) in \(p=\left (\frac{1}{2} \right )^s\). The solution is the negative \(log_2\) of p:

```
-log2(hyp$p.value)
#> [1] 9.070986
```

Indeed, the probability of obtaining 9 straight *Heads* with fair coin tosses is \(\left (\frac{1}{2} \right )^9=0.0019531\), which is very close to the p value we observed in the test of coefficient equality (see the *S* column in the `marginaleffects`

printout above). Comparing our p value to the outcome of such a familiar game of chance gives us a nice intuitive interpretation:

If the

`cyl6`

and`cyl8`

coefficients were truly equal, finding an absolute difference greater than 4.64 purely by chance would be as surprising as tossing 9 straightHeadswith a fair coin toss.

The benefits of *S* values include [@ColEdwGre2021]:

- Calibrates the analyst’s intuitions by reference to a well-known physical process (coin flips).
- Avoids the problematic dichotomization of findings as “significant” and “not significant” [@Rot2021].
- Reduces the reliance on arbitrary thresholds of significance like \(\alpha=0.05\).
- Guards against the common
*mis*interpretation of p values as the “probability that the null hypothesis is true” or as the probability of the alternative hypothesis. This is in part because S is above 1 whenever p<0.5.^{1} - Refers to a more natural scale: “The difference between a p value of 0.99 and 0.90 in terms of how surprising the observed test statistic is, is not the same as the difference between 0.10 and 0.01.”
^{2}