R Tutorial: Randomized distributions

Key Takeaways

the idea behind statistic inference is to understand samples from a hypothetical population where the null hypothesis is true for example from east and west coasts where Cola preference is the same as a way of summarizing each of the null samples we calculate one statistic from each sample here the statistic is the difference in the proportion of West Coast people who prefer Cola as compared with the proportion of East Coast people who prefer Cola where each of the sample proportions is denoted P hat the difference in P hats changes with each sample first it's zero then it's negative one third and it will keep changing we can build a distribution of differences in proportions assuming the null hypothesis that there is no link between location and soda preference is true that is the null samples consist of randomly shuffled soda variables so that the samples don't have any dependency between location and soda preference the original sample proportions are P hat east of 0.82 and P hat west of 0.73 a difference of negative 0.09 the first shuffle of the drink variable gives the exact same summaries as the observed data the second shuffle on the other hand gives 27 people on the East Coast who prefer Cola as compared with 20 on the West Coast to prefer Cola the difference in sample proportions for the second shuffle of the data is negative 0.02 which is less extreme than the original data note that both the original data the redline and the first to shuffle differences in proportions black dots can be plotted together the next few shuffles give differences in proportions centered around zero note that the fifth difference is negative 0.16 which is farther from zero than the original data that is the fifth shuffle gives more evidence of a difference in soda preference than the original data does and we know that the fifth shuffle was created by randomly permuting the labels so a difference of negative 0.16 is plausible under the null hypothesis generally the null differences are between negative point two and positive point two and about a third of the differences are as or more extreme than the observed difference of negative 0.09 now that we have seen a visual representation of the null distribution let's see how a null sample can be generated in our using the mutate and sample functions the vector of soda preferences is mixed up or / muted such that whether someone is on the east or west coast can't possibly be causing any difference in proportions however due to inherent natural variability there's also no expectation that soda preferences are exactly the same for any sample after grouping by the location variable summarize calculates the proportion of each coast that prefers Cola note that drink equals Cola produces a vector of trues and falses which are then courses two ones and zeros when the mean function is applied since a 1 represents an individual who prefers Cola the average of these ones and zeros represent the proportion of individuals who prefer Cola summarizes used a second time to find the difference in proportion of Cola preference across the two costal groups the DIF function is applied across the two costal groups because the data have been summarized by location notice that the output gives a per muted difference of negative 0.02 as compared to the observed difference of negative 0.09 however the per muted difference of negative 0.02 represents only one instance of the variability of soda preference under the null model to get a sense of the degree of variability under the null model it's necessary to permute the drink variable many times by repeating the permuting and difference calculations 5 times the per muted differences are seen to be sometimes positive sometimes negative sometimes close to zero sometimes far from zero however five times isn't quite enough to capture all of the variability in null differences by repeating the permutation process 100 time's the null differences are seen to range from approximately negative 0.3 to positive 0.3 although the majority of the differences are between negative 0.1 and positive 0.1 the observed data difference of negative 0.09 doesn't seem too extreme compared to this collection of null differences okay now it's your turn to practice what you've learned