---

title: "Power calculations"

output: html_document

layout: page

---

```{r options, echo=FALSE}

library(knitr)

opts_chunk$set(fig.path=paste0("figure/", sub("(.*).Rmd","\\1",basename(knitr:::knit_concord$get('infile'))), "-"))

```

## Power Calculations

#### Introduction

We have used the example of the effects of two different diets on the weight of mice. Since in this illustrative example we have access to the population, we know that in fact there is a substantial (about 10%) difference between the average weights of the two populations:

```{r,message=FALSE,echo=FALSE}

library(downloader)

url <- "https://raw.githubusercontent.com/genomicsclass/dagdata/master/inst/extdata/mice_pheno.csv"

filename <- "mice_pheno.csv"

if(!file.exists(filename)) download(url,destfile=filename)

```

```{r,message=FALSE}

library(dplyr)

dat <- read.csv("mice_pheno.csv") #Previously downloaded

controlPopulation <- filter(dat,Sex == "F" & Diet == "chow") %>%

select(Bodyweight) %>% unlist

hfPopulation <- filter(dat,Sex == "F" & Diet == "hf") %>%

select(Bodyweight) %>% unlist

mu_hf <- mean(hfPopulation)

mu_control <- mean(controlPopulation)

print(mu_hf - mu_control)

print((mu_hf - mu_control)/mu_control * 100) #percent increase

```

We have also seen that, in some cases, when we take a sample and perform a t-test, we don't always get a p-value smaller than 0.05. For example, here is a case where we take a sample of 5 mice and don't achieve statistical significance at the 0.05 level:

```{r}

set.seed(1)

N <- 5

hf <- sample(hfPopulation,N)

control <- sample(controlPopulation,N)

t.test(hf,control)$p.value

```

Did we make a mistake? By not rejecting the null hypothesis, are we

saying the diet has no effect? The answer to this question is no. All

we can say is that we did not reject the null hypothesis. But this

does not necessarily imply that the null is true. The problem is that,

in this particular instance, we don't have enough _power_, a term we

are now going to define. If you are doing scientific research, it is

very likely that you will have to do a power calculation at some

point. In many cases, it is an ethical obligation as it can help you

avoid sacrificing mice unnecessarily or limiting the number of human

subjects exposed to potential risk in a study. Here we explain what

statistical power means.

#### Types of Error

Whenever we perform a statistical test, we are aware that we may make a

mistake. This is why our p-values are not 0. Under the null, there is

always a positive, perhaps very small, but still positive chance that we

will reject the null when it is true. If the p-value is 0.05, it will

happen 1 out of 20 times. This *error* is called _type I error_ by

statisticians.

A type I error is defined as rejecting the null when we should

not. This is also referred to as a false positive. So why do we then

use 0.05? Shouldn't we use 0.000001 to be really sure? The reason we

don't use infinitesimal cut-offs to avoid type I errors at all cost is

that there is another error we can commit: to not reject the null when we

should. This is called a _type II error_ or a false negative. The R

code analysis above shows an example of a false negative: we did not

reject the null hypothesis (at the 0.05 level) and, because we happen

to know and peeked at the true population means, we know there is in fact a

difference. Had we used a p-value cutoff of 0.25, we would not have

made this mistake. However, in general, are we comfortable with a type

I error rate of 1 in 4? Usually we are not.

#### The 0.05 and 0.01 Cut-offs Are Arbitrary

Most journals and regulatory agencies frequently insist that results be significant at the 0.01 or 0.05 levels. Of course there is nothing special about these numbers other than the fact that some of the first papers on p-values used these values as examples. Part of the goal of this book is to give readers a good understanding of what p-values and confidence intervals are so that these choices can be judged in an informed way. Unfortunately, in science, these cut-offs are applied somewhat mindlessly, but that topic is part of a complicated debate.

#### Power Calculation

Power is the probability of rejecting the null when the null is

false. Of course "when the null is false" is a complicated statement

because it can be false in many ways.

$\Delta \equiv \mu_Y - \mu_X$

could be anything and the power actually depends on this parameter. It

also depends on the standard error of your estimates which in turn

depends on the sample size and the population standard deviations. In

practice, we don't know these so we usually report power for several

plausible values of $\Delta$, $\sigma_X$, $\sigma_Y$ and various

sample sizes.

Statistical theory gives us formulas to calculate

power. The `pwr` package performs these calculations for you. Here we

will illustrate the concepts behind power by coding up simulations in R.

Suppose our sample size is:

```{r}

N <- 12

```

and we will reject the null hypothesis at:

```{r}

alpha <- 0.05

```

What is our power with this particular data? We will compute this probability by re-running the exercise many times and calculating the proportion of times the null hypothesis is rejected. Specifically, we will run:

```{r}

B <- 2000

```

simulations. The simulation is as follows: we take a sample of size $N$ from both control and treatment groups, we perform a t-test comparing these two, and report if the p-value is less than `alpha` or not. We write a function that does this:

```{r}

reject <- function(N, alpha=0.05){

hf <- sample(hfPopulation,N)

control <- sample(controlPopulation,N)

pval <- t.test(hf,control)$p.value

pval < alpha

}

```

Here is an example of one simulation for a sample size of 12. The call to `reject` answers the question "Did we reject?"

```{r}

reject(12)

```

Now we can use the `replicate` function to do this `B` times.

```{r}

rejections <- replicate(B,reject(N))

```

Our power is just the proportion of times we correctly reject. So with $N=12$ our power is only:

```{r}

mean(rejections)

```

This explains why the t-test was not rejecting when we knew the null

was false. With a sample size of just 12, our power is about 23%. To

guard against false positives at the 0.05 level, we had set the

threshold at a high enough level that resulted in many type II

errors.

Let's see how power improves with N. We will use the function `sapply`, which applies a function to each of the elements of a vector. We want to repeat the above for the following sample size:

```{r}

Ns <- seq(5, 50, 5)

```

So we use `apply` like this:

```{r}

power <- sapply(Ns,function(N){

rejections <- replicate(B, reject(N))

mean(rejections)

})

```

For each of the three simulations, the above code returns the proportion of times we reject. Not surprisingly power increases with N:

```{r power_versus_sample_size, fig.cap="Power plotted against sample size."}

plot(Ns, power, type="b")

```

Similarly, if we change the level `alpha` at which we reject, power

changes. The smaller I want the chance of type I error to be, the less

power I will have. Another way of saying this is that we trade off

between the two types of error. We can see this by writing similar code, but

keeping $N$ fixed and considering several values of `alpha`:

```{r power_versus_alpha, fig.cap="Power plotted against cut-off."}

N <- 30

alphas <- c(0.1,0.05,0.01,0.001,0.0001)

power <- sapply(alphas,function(alpha){

rejections <- replicate(B,reject(N,alpha=alpha))

mean(rejections)

})

plot(alphas, power, xlab="alpha", type="b", log="x")

```

Note that the x-axis in this last plot is in the log scale.

There is no "right" power or "right" alpha level, but it is important that you understand what each means.

To see this clearly, you could create a plot with curves of power versus N. Show several curves in the same plot with color representing alpha level.

#### p-values Are Arbitrary under the Alternative Hypothesis

Another consequence of what we have learned about power is that

p-values are somewhat arbitrary when the

null hypothesis is not true and therefore

the *alternative* hypothesis is true (the

difference between the population means is not zero).

When the alternative hypothesis is true,

we can make a p-value as small as we want simply by increasing

the sample size (supposing that we have an infinite population to sample

from). We can show this property of p-values

by drawing larger and larger samples from our

population and calculating p-values. This works because, in our case,

we know that the alternative hypothesis is true, since we have

access to the populations and can calculate the difference in their means.

First write a function that returns a p-value for a given sample size $N$:

```{r}

calculatePvalue <- function(N) {

hf <- sample(hfPopulation,N)

control <- sample(controlPopulation,N)

t.test(hf,control)$p.value

}

```

We have a limit here of 200 for the high-fat diet population, but we can

see the effect well before we get to 200.

For each sample size, we will calculate a few p-values. We can do

this by repeating each value of $N$ a few times.

```{r}

Ns <- seq(10,200,by=10)

Ns_rep <- rep(Ns, each=10)

```

Again we use `sapply` to run our simulations:

```{r}

pvalues <- sapply(Ns_rep, calculatePvalue)

```

Now we can plot the 10 p-values we generated for each sample size:

```{r pvals_decrease, fig.cap="p-values from random samples at varying sample size. The actual value of the p-values decreases as we increase sample size whenever the alternative hypothesis is true."}

plot(Ns_rep, pvalues, log="y", xlab="sample size",

ylab="p-values")

abline(h=c(.01, .05), col="red", lwd=2)

```

Note that the y-axis is log scale and that the p-values show a

decreasing trend all the way to $10^{-8}$

as the sample size gets larger. The standard cutoffs

of 0.01 and 0.05 are indicated with horizontal red lines.

It is important to remember that p-values are not more interesting as

they become very very small. Once we have convinced ourselves to

reject the null hypothesis at a threshold we find reasonable, having

an even smaller p-value just means that we sampled more mice than was

necessary. Having a larger sample size does help to increase the

precision of our estimate of the difference $\Delta$, but the fact

that the p-value becomes very very small is just a natural consequence

of the mathematics of the test. The p-values get smaller and smaller

with increasing sample size because the numerator of the t-statistic

has $\sqrt{N}$ (for equal sized groups, and a similar effect occurs

when $M \neq N$). Therefore, if $\Delta$ is non-zero, the t-statistic

will increase with $N$.

Therefore, a better statistic to report is the effect size with

a confidence interval or some statistic which gives the reader a

sense of the change in a meaningful scale. We can

report the effect size as a percent by dividing the difference

and the confidence interval by the control population mean:

```{r}

N <- 12

hf <- sample(hfPopulation, N)

control <- sample(controlPopulation, N)

diff <- mean(hf) - mean(control)

diff / mean(control) * 100

t.test(hf, control)$conf.int / mean(control) * 100

```

In addition, we can report a statistic called

[Cohen's d](https://en.wikipedia.org/wiki/Effect_size#Cohen.27s_d),

which is the difference between the groups divided by the pooled standard

deviation of the two groups.

```{r}

sd_pool <- sqrt(((N-1)*var(hf) + (N-1)*var(control))/(2*N - 2))

diff / sd_pool

```

This tells us how many standard deviations of the data the mean of the

high-fat diet group is from the control group. Under the

alternative hypothesis, unlike the t-statistic which is guaranteed to

increase, the effect size and Cohen's d will become more precise.