Correlatie tussen nakomelingen en ouders

Om de correlatie tussen snaveldieptes van nakomelingen en ouders te kwantificeren, willen we statistieken berekenen, zoals de Pearson-correlatiecoëfficiënt, tussen ouders en nakomelingen. Om hiervoor betrouwbaarheidsintervallen te krijgen, moeten we een pairs-bootstrap uitvoeren.

Je hebt eerder al een functie geschreven om een pairs-bootstrap te doen voor schattingen van parameters afgeleid uit lineaire regressie. In deze oefening maak je een nieuwe functie met de call signature draw_bs_pairs(x, y, func, size=1) die een pairs-bootstrap uitvoert en één statistiek berekent op de gedefinieerde paarsamples. De relevante statistiek wordt berekend door func(bs_x, bs_y) aan te roepen. In de volgende oefening gebruik je pearson_r voor func.

Deze oefening maakt deel uit van de cursus

Statistical Thinking in Python (deel 2)

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Oefeninstructies

Zet een array met indices op waaruit je gaat samplen. (Onthoud: bij een pairs-bootstrap kiezen we willekeurig indices en gebruiken die om de paren te vormen.)
Initialiseer de array met bootstrap-replicaten. Dit moet een eendimensionale array zijn met lengte size.
Schrijf een for-lus om de samples te trekken.
Kies willekeurig indices uit de array met indices die je zojuist hebt opgezet.
Haal x-waarden en y-waarden uit de invoerarray met de zojuist gekozen indices om een bootstrap-sample te genereren.
Gebruik func om de relevante statistiek te berekenen uit de bootstrap-samples van x en y en sla deze op in je array met bootstrap-replicaten.
Retourneer de array met bootstrap-replicaten.

Praktische interactieve oefening

Probeer deze oefening eens door deze voorbeeldcode in te vullen.

def draw_bs_pairs(x, y, func, size=1):
    """Perform pairs bootstrap for a single statistic."""

    # Set up array of indices to sample from: inds
    inds = ____

    # Initialize replicates: bs_replicates
    bs_replicates = ____

    # Generate replicates
    for i in range(size):
        bs_inds = ____
        bs_x, bs_y = ____
        bs_replicates[i] = ____

    return bs_replicates

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Deze oefening maakt deel uit van de cursus

Statistical Thinking in Python (deel 2)

SkillTag.level.intermediateSkillTag.label

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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?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

Huidige oefening

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