Hypothesetoets: Zijn snavels dieper in 2012?

Je ECDF-plot en het bepalen van het betrouwbaarheidsinterval maken vrij duidelijk dat de snavels van G. scandens op Daphne Major dieper zijn geworden. Maar kan het zijn dat dit effect gewoon toeval is? Met andere woorden: wat is de kans dat we het geobserveerde verschil in gemiddelde snaveldiepte krijgen als de gemiddelden gelijk zouden zijn?

Let op! De hypothese die we testen is niet dat de snaveldieptes uit dezelfde verdeling komen. Daarvoor zouden we een permutatietoets kunnen gebruiken. De hypothese is dat de gemiddelden gelijk zijn. Om deze hypothesetoets uit te voeren, moeten we de twee gegevenssets verschuiven zodat ze hetzelfde gemiddelde hebben en vervolgens bootstrap-steekproeven gebruiken om het verschil van gemiddelden te berekenen.

Deze oefening maakt deel uit van de cursus

Statistical Thinking in Python (deel 2)

Cursus bekijken

Oefeninstructies

Maak een samengevoegde array van de snaveldieptes van 1975 en 2012 en bereken en sla het gemiddelde op.
Verschuif bd_1975 en bd_2012 zodat hun gemiddelden gelijk zijn aan het zojuist berekende gemiddelde van de gecombineerde gegevensset.
Neem 10.000 bootstrap-replicaties van het gemiddelde voor zowel de snaveldieptes van 1975 als die van 2012.
Trek de replicaties van 1975 af van die van 2012 om bootstrap-replicaties van het verschil te krijgen.
Bereken en print de p-waarde. Het geobserveerde verschil in gemiddelden dat je in de vorige oefening hebt berekend staat nog in je namespace als mean_diff.

Praktische interactieve oefening

Probeer deze oefening eens door deze voorbeeldcode in te vullen.

# Compute mean of combined data set: combined_mean
combined_mean = ____(____((bd_1975, bd_2012)))

# Shift the samples
bd_1975_shifted = ____
bd_2012_shifted = ____

# Get bootstrap replicates of shifted data sets
bs_replicates_1975 = ____
bs_replicates_2012 = ____

# Compute replicates of difference of means: bs_diff_replicates
bs_diff_replicates = ____

# Compute the p-value
p = np.sum(____ >= ____) / len(____)

# Print p-value
print('p =', p)

Code bewerken en uitvoeren

Deze oefening maakt deel uit van de cursus

Statistical Thinking in Python (deel 2)

SkillTag.level.intermediateSkillTag.label

4.8+

Begin de cursus gratis

When doing statistical inference, we speak the language of probability. A probability distribution that describes your data has parameters. So, a major goal of statistical inference is to estimate the values of these parameters, which allows us to concisely and unambiguously describe our data and draw conclusions from it. In this chapter, you will learn how to find the optimal parameters, those that best describe your data.

Exercise 1: Optimal parameters Exercise 2: How often do we get no-hitters?Exercise 3: Do the data follow our story?Exercise 4: How is this parameter optimal?Exercise 5: Linear regression by least squares Exercise 6: EDA of literacy/fertility data Exercise 7: Linear regression Exercise 8: How is it optimal?Exercise 9: The importance of EDA: Anscombe's quartet Exercise 10: The importance of EDA Exercise 11: Linear regression on appropriate Anscombe data Exercise 12: Linear regression on all Anscombe data

To "pull yourself up by your bootstraps" is a classic idiom meaning that you achieve a difficult task by yourself with no help at all. In statistical inference, you want to know what would happen if you could repeat your data acquisition an infinite number of times. This task is impossible, but can we use only the data we actually have to get close to the same result as an infinitude of experiments? The answer is yes! The technique to do it is aptly called bootstrapping. This chapter will introduce you to this extraordinarily powerful tool.

Exercise 1: Generating bootstrap replicates Exercise 2: Getting the terminology down Exercise 3: Bootstrapping by hand Exercise 4: Visualizing bootstrap samples Exercise 5: Bootstrap confidence intervals Exercise 6: Generating many bootstrap replicates Exercise 7: Bootstrap replicates of the mean and the SEM Exercise 8: Confidence intervals of rainfall data Exercise 9: Bootstrap replicates of other statistics Exercise 10: Confidence interval on the rate of no-hitters Exercise 11: Pairs bootstrap Exercise 12: A function to do pairs bootstrap Exercise 13: Pairs bootstrap of literacy/fertility data Exercise 14: Plotting bootstrap regressions

You now know how to define and estimate parameters given a model. But the question remains: how reasonable is it to observe your data if a model is true? This question is addressed by hypothesis tests. They are the icing on the inference cake. After completing this chapter, you will be able to carefully construct and test hypotheses using hacker statistics.

Exercise 1: Formulating and simulating a hypothesis Exercise 2: Generating a permutation sample Exercise 3: Visualizing permutation sampling Exercise 4: Test statistics and p-values Exercise 5: Test statistics Exercise 6: What is a p-value?Exercise 7: Generating permutation replicates Exercise 8: Look before you leap: EDA before hypothesis testing Exercise 9: Permutation test on frog data Exercise 10: Bootstrap hypothesis tests Exercise 11: A one-sample bootstrap hypothesis test Exercise 12: A two-sample bootstrap hypothesis test for difference of means

As you saw from the last chapter, hypothesis testing can be a bit tricky. You need to define the null hypothesis, figure out how to simulate it, and define clearly what it means to be "more extreme" in order to compute the p-value. Like any skill, practice makes perfect, and this chapter gives you some good practice with hypothesis tests.

Exercise 1: A/B testing Exercise 2: The vote for the Civil Rights Act in 1964 Exercise 3: What is equivalent?Exercise 4: A time-on-website analog Exercise 5: What should you have done first?Exercise 6: Test of correlation Exercise 7: Simulating a null hypothesis concerning correlation Exercise 8: Hypothesis test on Pearson correlation Exercise 9: Do neonicotinoid insecticides have unintended consequences?Exercise 10: Bootstrap hypothesis test on bee sperm counts

Every year for the past 40-plus years, Peter and Rosemary Grant have gone to the Galápagos island of Daphne Major and collected data on Darwin's finches. Using your skills in statistical inference, you will spend this chapter with their data, and witness first hand, through data, evolution in action. It's an exhilarating way to end the course!

Exercise 1: Vinkensnavels en de noodzaak van statistiek Exercise 2: EDA van snaveldieptes van Darwinvinken Exercise 3: ECDF's van snavel dieptes Exercise 4: Parameterinschattingen van snaveldieptes Exercise 5: Hypothesetoets: Zijn snavels dieper in 2012?

Huidige oefening

Exercise 6: Variatie in snavelvormen Exercise 7: EDA van snavellengte en -diepte Exercise 8: Lineaire regressies Exercise 9: De resultaten van de lineaire regressie weergeven Exercise 10: Verhouding snavellengte tot -diepte Exercise 11: Hoe anders is de verhouding?Exercise 12: Berekening van erfelijkheid Exercise 13: EDA van erfelijkheid Exercise 14: Correlatie tussen nakomelingen en ouders Exercise 15: Pearson-correlatie tussen nakomelingen en ouders Exercise 16: Erfelijkheid meten Exercise 17: Is snaveldiepte überhaupt erfelijk in G. scandens?Exercise 18: Tot slot