How does one calculate the non-centrality parameter (t-distribution) in R?

Question

I’m trying to calculate this parameter to perform a power analysis. I want to know the power to detect a future hypothetical 30% increase in a continuous variable (density). I think my alternative=”less than”, but R returns a warning saying the ncp can’t be positive when alternative hypothesis is “less”, so I think my calculation of ncp may be incorrect. I’m using the pwrss::power.t.test and my ncp formula = mean2(the 30% increase of the mean) x sqrt(n) which I got from here: https://stats.stackexchange.com/questions/92993/noncentrality-parameter-what-is-it-what-does-it-do-what-would-be-a-suggested

mean1=0.38

mean2 (30% increase) = 0.4940

sample size per group = 47

> power.t.test(ncp = (0.4940*sqrt(47)),

+ df = 46,

+ alpha = 0.05,

+ alternative = "less",

+ plot = FALSE)

power ncp.alt ncp.null alpha df t.crit

0.9546192 3.387 0 0.05 46 1.67866

Warning message:

alternative = 'less' but non-centrality parameter is positive

Answer 1

The warning you're encountering in R is due to a misunderstanding of the role of the noncentrality parameter (ncp) in the context of your hypothesis testing and the direction of the alternative hypothesis.

When conducting a power analysis for a t-test with `power.t.test` in R, the `ncp` (non-centrality parameter) is used to specify the effect size you're interested in detecting. This parameter essentially shifts the mean of the distribution under the alternative hypothesis away from the null hypothesis mean, allowing you to determine the power to detect this difference.

However, the direction of this shift must align with your alternative hypothesis. Since you've set `alternative = "less"`, R expects a negative shift (i.e., you are testing for a decrease), but your `ncp` calculation suggests an increase (a positive shift), hence the warning.

Given your scenario where you expect a 30% increase, your alternative hypothesis should be `alternative = "greater"` (testing for an increase). Here's how you can adjust your `power.t.test` function:

1. Change the `alternative` parameter to `"greater"`.

2. Calculate the `ncp` as the difference between the means (mean2 - mean1), multiplied by the square root of the sample size.

Let's modify your code accordingly and calculate the power:

mean1 = 0.38

mean2 = 0.4940

sample_size = 47

ncp = (mean2 - mean1) * sqrt(sample_size)

power.t.test(ncp = ncp,

df = 46,

alpha = 0.05,

alternative = "greater",

plot = FALSE)

This should provide you with the correct power analysis for detecting a 30% increase in the density variable.