# Using P-Values with Confidence

## Background

The p-value is perhaps the most ubiquitous statistical index.

It is also the most

misunderstood,

and/or misused,

and/or misaligned

depending on whom you ask.

## American Statistical Association

Wasserstein, R.L., and Lazar, N.A. 2016. The ASA’s Statement on P-Values: Context, Process, and Purpose. The American Statistician 70(2): 129–133.

## What is a p-value?

A p-value of <0.05 indicates that the 95% confidence interval excludes 0.

## What is the utility of a p-value?

P-values are useful because they indicate the confidence with which we can exclude 0.

## T-Test

```
model <- lm(Length~Flow, data = data)
round(summary(model)$coefficients,4)
```

```
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 10.1079 0.1576 64.1487 0.000
## FlowHigh -0.5200 0.2228 -2.3336 0.025
```

## Principle 1

**1. P-values can indicate how incompatible the data are with a specified statistical model.**

A p-value, or statistical significance, does not measure the size of an effect or the importance of a result.

Scientific conclusions and business or policy decisions should not be based only on whether a p-value passes a specific threshold.

Proper inference requires full reporting and transparency.

P-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone.

## Significance

In biology p-values of <0.05 are generally considered to be *significant*.

## The End

#### Conclusion

Flow is a significant predictor of fish length.

#### Appendix

```
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 10.108 0.158 64.149 0.000
## FlowHigh -0.520 0.223 -2.334 0.025
```

## Principle 2

- P-values can indicate how incompatible the data are with a specified statistical model.

**2. A p-value, or statistical significance, does not measure the size of an effect or the importance of a result.**

Scientific conclusions and business or policy decisions should not be based only on whether a p-value passes a specific threshold.

Proper inference requires full reporting and transparency.

P-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone.

## A Limitation

## p = 1

## p = 0.01

## Biological Importance

## Principle 3

P-values can indicate how incompatible the data are with a specified statistical model.

A p-value, or statistical significance, does not measure the size of an effect or the importance of a result.

**3. Scientific conclusions and business or policy decisions should not be based only on whether a p-value passes a specific threshold.**

Proper inference requires full reporting and transparency.

P-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone.

## Confidence Interval

## Effects Size

## Principle 4

P-values can indicate how incompatible the data are with a specified statistical model.

A p-value, or statistical significance, does not measure the size of an effect or the importance of a result.

Scientific conclusions and business or policy decisions should not be based only on whether a p-value passes a specific threshold.

**4. Proper inference requires full reporting and transparency.**

- P-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone.

## P-Hacking

## Principle 5

P-values can indicate how incompatible the data are with a specified statistical model.

A p-value, or statistical significance, does not measure the size of an effect or the importance of a result.

Scientific conclusions and business or policy decisions should not be based only on whether a p-value passes a specific threshold.

Proper inference requires full reporting and transparency.

**5. P-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone.**

## Conditional Probability

Much of the additional confusion around p-values stems from the fact that frequentist methods make statements about data in relation to a model.

However, intuitively what we actually want are (Bayesian) statements about models in relation to the data.

## Conclusions

Use p-values with confidence intervals.

Express confidence intervals as effects sizes.

Discuss biological importance.

Don’t p-hack.

### Further Reading

Gardner, M.J., and Altman, D.G. 1986. Confidence intervals rather than P values: estimation rather than hypothesis testing. BMJ 292(6522): 746–750. doi:10.1136/bmj.292.6522.746.

Greenland, S., and Poole, C. 2013. Living with P Values: Resurrecting a Bayesian Perspective on Frequentist Statistics. Epidemiology 24(1): 62–68. doi:10.1097/EDE.0b013e3182785741.