Course Outline

list Introduction to Statistics: A Modeling Approach

The F Distribution

We’ve only gone through examples of randomization to create a sampling distribution of F values. However, we could also use bootstrapping and simulation. But these computational methods aren’t the only ways to construct sampling distributions of F. Just as mathematicians have developed probability models for the normal distribution (the z, and more useful t, distributions), mathematicians have also developed a probability distribution that does a good job modeling the sampling distribution of F. It’s called the F distribution.

Just as the t distribution changed its shape according to the degrees of freedom (n-1), the F distribution changes its shape according to both model and error degrees of freedom. You can think of it as the degrees of freedom for the numerator and degrees of freedom in the denominator of the variances used to calculate the F ratio (i.e., the MS for model and the MS for error).

In the Smiley Face experiment, you can see from the supernova() table above that you have 1 df in the numerator, and 42 in the denominator. Below, on the right, we’ve plotted the theoretical F distribution for an F ratio with 1 and 42 degrees of freedom (on the right). We’ve colored the part of this smooth theoretical distribution greater than our sample F in bluish-green. For comparison, on the left, we’ve put the sampling distribution of F we created with randomization (using shuffle()).

xpf(sampleF, df1=1, df2=42)

As you can see, the smooth F distribution does a pretty good job modeling the shape of the randomized sampling distribution we created. And because the F distribution is a mathematically defined probability distribution, R can easily calculate the probability of obtaining a score in any particular region of the curve.

Let’s break down the code we used to get the probability in the F distribution. The function xpf() is about the F distribution, and more specifically about getting the p-value from the F distribution. The x in front of the function is just there because there is another version of this function called pf() that does not also print out a picture. We prefer xpf() so that the plot of this mathematical distribution automatically gets printed as well.

The entire line of code xpf(sampleF, df1=1, df2=42) asks R to return the probability of getting an F ratio more extreme than our sample F (which was 3.305), if the empty model were true. The empty model represents a DGP where any differences between groups were the result of chance. All the Fs that could result from running this empty model DGP is modeled by the F distribution with a df1 = 1 and df2 = 42.

L_Ch11_FDis_1

The results of this command show us that the probability of getting an F of 3.305 or greater in a study with 1 and 42 degrees of freedom if the empty model is true at .076. This, again, is very close to the probability we got from the randomized sampling distribution we got above. In fact, whether we use the randomized PRE distribution, randomized F, or probability distribution of F, the probability of getting an estimate as high as the one observed in the Smiley Face study was, rounded off, .08 in all three cases.

It’s also worth noting that the .076 we got using the F probability calculator is exactly the same as we get in the p column of the supernova() table. This is not an accident. The function supernova() uses the F distribution to calculate the probability of getting the F value we calculated from the data, or one more extreme, if the empty model were true.

Exploring the F Distribution

The shape of the F distribution can change quite markedly as the degrees of freedom in the numerator and denominator of the F ratio change. Here are just two examples, produced using variants of this R code:

curve(df(x, df1=1, df2=42), from=0, to=6)
curve(df(x, df1=6, df2=42), from=0, to=6)

We know from using the xpf() function introduced previously, that the probability of getting an F of 3.305 if the empty model is true would be .076. Modify the code in the DataCamp window below to determine what the probability would be if our study had six degrees of freedom in the numerator.

#load packages require(ggformula) require(mosaic) require(supernova) require(Lock5Data) require(Lock5withR) require(okcupiddata) sampleF <- fVal(Tip ~ Condition, data = TipExperiment) # edit this to explore the F distribution with df model = 6 xpf(sampleF, df1 = 1, df2 = 42) # edit this to explore the F distribution with df model = 6 xpf(sampleF, df1 = 6, df2 = 42) test_function("xpf", args = c("df1", "df2")) test_error() success_msg("Great job!")
DataCamp: ch11-12

You can see that the probability of getting an F of 3.305 or higher in a study with more groups (more degrees of freedom in the numerator), but the same number of degrees of freedom for error (42), would be much lower (assuming that the empty model were true).

You can take a look at these F distributions as the df model increases. They were all generated with the same df2. We only changed the df1 (the number of additional parameters in our complex model). Try to describe the pattern you observe.

L_Ch11_Exploring_1

Try out this app (follow the link below) to further explore how degrees of freedom alter both the shape and probability calculations of the F distribution.

https://gallery.shinyapps.io/dist_calc/

The point we want to make is this: the F distribution has a bunch of subtly different shapes based on the df model (df1) and df error (df2). But all F distributions generated from the empty model have a mean of 1. You don’t need to know exactly how the F distributions vary with df1 and df2 (for now). We mostly want to equip you with a tool—you can always run xpf() to see a picture!

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