I’ve had a few colleagues ask me if Biostats I was a useful class, given my statistics background in grad school. It’s a requirement for the master’s degree program I’m pursuing, so I have to take it, but I have found it to be a nice refresher of the Biometry course I had years ago. Maybe I just know more about statistics now, so it makes more sense; or maybe it’s just explained better in this course, so I have a better grasp of the material. When I started grad school, statistics felt like Farsi. But not now.
Take Type I and Type II error, for example. In study design, you have to try to minimize both. Type I error is the probability of rejecting the null hypothesis when it is true. The acceptable Type I error rate is determined by alpha, which is generally fixed at 0.05 or lower in the analysis phase of a study. Type II error, or beta, is the probability of failing to reject the null hypothesis when the alternative hypothesis is true. While I understood these concepts empirically, the relationship between them had never been explained. What I had were random facts, with no framework to pin them on.
The relationship between alpha and beta.
This plot represents a one-tailed Student’s t-test of the difference in means between two independent samples, both with a sample size of 75 and with alpha set to 0.05. The probability of accepting the null hypothesis is represented by the red line, while the probability of accepting the alternative hypothesis is in blue. Notice that the null hypothesis distribution is centered at 0, meaning that you’re testing the hypothesis of no difference between means, and that the two distributions overlap. The area under the red curve which overlaps the blue curve is alpha, the chance of rejecting the null hypothesis when it is true. The area under the blue curve which overlaps the red curve is beta, the chance of failing to reject the null hypothesis when the alternative hypothesis is true.
Notice also, that you can’t change the value of alpha without affecting the value of beta. Here’s the same t-test with alpha set to 0.01.
Changing alpha affects beta.
Reducing alpha increases the critical value for rejecting the null hypothesis (from t=1.6552 to t=2.3518), thus increasing the likelihood of failing to reject the null when the alternative hypothesis is true. And the rest of the blue curve, which equals 1 – beta? That’s power, or the probability of rejecting the null hypothesis when the alternative hypothesis is true.
That’s the framework I was missing. The biostatistics course was worth that alone.
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Images generated using G*Power 3.
