Course Outline

segmentGetting Started (Don't Skip This Part)

segmentStatistics and Data Science: A Modeling Approach

segmentPART I: EXPLORING VARIATION

segmentChapter 1  Welcome to Statistics: A Modeling Approach

segmentChapter 2  Understanding Data

segmentChapter 3  Examining Distributions

segmentChapter 4  Explaining Variation

segmentPART II: MODELING VARIATION

segmentChapter 5  A Simple Model

segmentChapter 6  Quantifying Error

segmentChapter 7  Adding an Explanatory Variable to the Model

segmentChapter 8  Digging Deeper into Group Models

segmentChapter 9  Models with a Quantitative Explanatory Variable

segmentPART III: EVALUATING MODELS

segmentChapter 10  The Logic of Inference

segmentChapter 11  Model Comparison with F

11.6 The FDistribution: A Mathematical Model of the Sampling Distribution of F

segmentChapter 12  Parameter Estimation and Confidence Intervals

segmentChapter 13  What You Have Learned

segmentFinishing Up (Don't Skip This Part!)

segmentResources
list High School / Advanced Statistics and Data Science I (ABC)
11.6 The FDistribution: A Mathematical Model of the Sampling Distribution of F
So far we’ve used randomization (shuffle()
) to create a sampling distribution of F. However, just like mathematicians developed mathematical models of the sampling distribution of \(b_1\) (e.g., tdistributions), they have developed a mathematical model of the sampling distribution of F. This mathematical model is called the Fdistribution.
In the same way that the mathematical tdistribution can be used as a smooth idealization to model sampling distributions of \(b_1\), the Fdistribution provides a smooth mathematical model that fits the sampling distribution of F. (It also fits the sampling distribution of PRE.)
In the figure below, we show two versions of the sampling distribution of F that both assume a DGP with no effect of Condition
(i.e., the empty model). On the left, we model the randomized sampling distribution using shuffle()
, and on the right using the F distribution, where the area greater than our sample F is represented as the purple tail.
Notice that the shapes are very similar. The Fdistribution seems like a smoothed out version of the randomized sampling distribution of F, and the pvalue calculated based on the randomized sampling distribution will be very similar to the pvalue based on the mathematical Fdistribution.
Just as the shape of the tdistribution varies slightly according to the sample size or degrees of freedom, the shape of the Fdistribution also varies by degrees of freedom. But because F is calculated as the ratio of MS Model divided by MS Error, we must specify two different degrees of freedom to get the shape of the Fdistribution: the df for MS Model (1 in the ANOVA table below); and the df for MS Error, which is 42.
Analysis of Variance Table (Type III SS)
Model: Tip ~ Condition
SS df MS F PRE p
        
Model (error reduced)  402.023 1 402.023 3.305 0.0729 .0762
Error (from model)  5108.955 42 121.642
        
Total (empty model)  5510.977 43 128.162
The xpf()
function provides one way to calculate a pvalue using the Fdistribution. It requires us to enter three arguments: the sample F, the df Model (called df1
) and df Error (called df2
). Try it out in the code window below by filling in the values of df1
and df2
from the ANOVA table above.
require(coursekata)
# we have saved the sample F for you
sample_f < f(Tip ~ Condition, data = TipExperiment)
# fill in the appropriate dfs
xpf(sample_f, df1 = , df2 = )
# we have saved the sample F for you
sample_f < f(Tip ~ Condition, data = TipExperiment)
# fill in the appropriate dfs
xpf(sample_f, df1 = 1, df2 = 42)
ex() %>%
check_function(., "xpf") %>% {
check_arg(., "df1") %>% check_equal()
check_arg(., "df2") %>% check_equal()
}
We like the xpf()
function because it shows a graph of the F distribution and marks off the region of the tail that represents the pvalue. It also tells you what the pvalue is in the legend. Notice in the plot below that the pvalue for the Condition
model of the tipping experiment data is .0762. That’s the same value reported in the ANOVA table, which is no coincidence: the supernova()
function uses the mathematical F distribution to calculate the pvalue.