Interpreting T-Statistics and P-Values with Finite Samples

# T-Statistics and/or P-Value


# T-Statistics and/or P-Value












using models like fama-french 3 models, in some books like CFA curriculums, it was always mentioned to look at the T-statistics for the magic number '1.96' or 2 which tells us if it is statistically significant.

But i realized that sometimes even if T-statistics are very high say above 2 or 3, the p-value can still be very high as well ( above 0.05). Or it could be T-Statistics Low but p-value Low as well.

If that's the case would T-statistics still reliable or do investment practitioners also take note of p-value together with T-statistics?

These are some of the questions exercise i came across as below screenshot.

thanks a bunch - still a student but really interested in learning how you guys think

1st example...

2nd example...

## Answer by Dave (score 1)

https://quant.stackexchange.com/a/81479

The p-value is what matters, but the t-stat is a convenient proxy, especially when sample sizes are large.

You are correct that the t-stat could be quite high without giving a small p-value, 100% correct. For instance, for a t-distribution with just one degree of freedom, the critical value for $p\le 0.05$ is almost $13$ rather than $1.96$ or $2$. Let's look at a plot.

```
library(ggplot2)
dfs <- seq(1, 100, 1)
crit_vals <- rep(NA, length(dfs))
for (i in 1:length(dfs)){
  crit_vals[i] <- qt(0.975, dfs[i])
}
d <- data.frame(
  dfs = dfs,
  crit_vals = crit_vals
)
ggplot(d, aes(x = dfs, y = crit_vals)) +
  geom_line() +
  geom_point() +
  xlab("Degrees of Freedom") +
  ylab("Critical Value for 0.05-level") +
  geom_abline(slope = 0, intercept = 2)
```

Yes, at lower values, two is a rather poor approximation. However, let's zoom in.

```
library(ggplot2)
dfs <- seq(1, 100, 1)
crit_vals <- rep(NA, length(dfs))
for (i in 1:length(dfs)){
  crit_vals[i] <- qt(0.975, dfs[i])
}
d <- data.frame(
  dfs = dfs,
  crit_vals = crit_vals
)
ggplot(d, aes(x = dfs, y = crit_vals)) +
  geom_line() +
  geom_point() +
  xlab("Degrees of Freedom") +
  ylab("Critical Value for 0.05-level") +
  geom_abline(slope = 0, intercept = 2) +
  xlim(10, 100) +
  ylim(1.5, 2.5)
```

As the degrees of freedom value for the t-distribution gets to be about $50$, let alone $75$ or $100$, the approximation makes more sense.

Consider what kinds of sample sizes and degrees of freedom in t-distributions there are when you read this kind of approximation. If there are a thousand observations and close to a thousand degrees of freedom, this approximation is probably pretty good.

I suppose the reliance on critical values over explicit p-values might come from a time when it was harder to calculate the p-values. The critical value is just a bit of arithmetic: subtract a mean, divide by a standard deviation, take a square root of a sample size, multiply by that square root. Calculating the p-value these days is easy in a software package like R I am using above (call `pt`) but was probably harder a century ago. This use of such an approximation is a bit irritating, then, with the explicit p-values being so easy to calculate with a command like `pt`, but, in most cases, there is probably little to no damage, so tradition lives.