Quantitative Methods 1
1
Introduction
1.1
Books I can (highly) recommend:
1.2
1.3
Additional Tools
1.4
Workflow suggestion
1.5
Orientation for the course and script
1.6
Warning of incompleteness
2
Probability
2.1
Frequentist vs. Bayesian statistics
2.1.1
Frequentist statistics
2.1.2
Bayesian statistics
2.2
Foundations of probability theory
2.2.1
Questions about the 1000 researcher-experiment (among many others):
2.2.2
Axioms of probability theory
2.2.3
Independence of events
2.2.4
Difference between independence and disjointness
2.2.5
Answers to questions about the 1000 researcher-experiment (among many others):
2.2.6
Addition of probabilities
2.2.7
Probabilities for health science
2.2.8
Discrete vs. continuous probability distributions
2.2.9
Examples of prominent probability distributions used in health sciences
2.3
Exercises
2.3.1
[M] Exercise 1 - Throwing a die very often
2.3.2
[D] Exercise 2 - Bayes-teaser
2.3.3
[E] Exercise 3 - Find journals
2.3.4
[M] Exercise 4 - Independent and disjoint
2.3.5
[M] Exercise 5 - Variance
2.3.6
[E] Exercise 6 - Three researchers
2.3.7
[E] Exercise 7 - Conditional probability
2.3.8
[E] Exercise 8 - Invent a discrete probability distribution
2.3.9
[E] Exercise 9 - Continuous probability distributions
2.3.10
[M] Exercise 10 - MSc-ZHAW-distribution
2.3.11
[M] Exercise 11 - Independence and disjointness for dice events
2.3.12
[D] Exercise 12 - Student’s
\(t\)
-distribution
2.4
Solutions
2.5
Sample exam questions for this chapter (in German since exam is in German)
2.5.1
Question 1 - Independence and disjointness
2.5.2
Frage 2 - Bedingte Wahrscheinlichkeit
2.5.3
Frage 3 - Erwartungswert und Varianz
2.5.4
Frage 4 - Dichtefunktion
3
Descriptive statistics
3.1
Example: Descriptive statistics in health sciences
3.2
Univarate vs. bivariate statisics
3.3
The histogram
3.3.1
Example in the wild
3.4
Q-Q Plots
3.5
Correlation
3.5.1
Example in the wild
3.5.2
Spearman correlation
3.6
Exercises
3.6.1
[D] Exercise 1 - Recreate table with fake data
3.6.2
[E] Exercise 2 - Outliers and estimates
3.6.3
[E] Exercise 3 - Recreating data in Table 2
3.6.4
[E] Exercise 4 - Z-scores
3.6.5
[M] Exercise 5 - Correlation
3.6.6
[M] Exercise 6 - Bike parking locations in Switzerland
3.6.7
[E] Exercise 7 - Median, Mean, and Mode
3.6.8
[E] Exercise 8 - Correlation by hand
3.6.9
[E] Exercise 9 - Q-Q plot
3.7
Solutions
3.8
Sample exam questions for this chapter (in German since exam is in German)
3.8.1
Frage 1 - Mittelwert, Median, Modus
3.8.2
Frage 2 - Normalverteilung
3.8.3
Frage 3 - Korrelation
3.8.4
Frage 4
4
Bayes statistics
4.1
Derivation of Bayes’ theorem
4.2
Bayes’ theorem in the context of parameter estimation
4.3
Examples
4.3.1
Example 1 - defective products
4.3.2
Example 2 - extending the defective products example
4.4
Highest Density Intervals (HDI)
4.5
Bayesian
\(t\)
-test
4.5.1
Example - Bayesian
\(t\)
-test
4.6
Bayesian updating
4.7
More complex parameter spaces
4.8
Advantages/disadvantages of Bayesian statistics
4.8.1
(Some) Advantages
4.8.2
(Some) Disadvantages
4.9
Exercises
4.10
[E] Exercise 1 - defective product rate
4.11
[H] Exercise 2 - Bayesian updating
4.12
Solutions
4.13
Sample exam questions for this chapter (in German since exam is in German)
4.13.1
Frage 1
4.13.2
Frage 2
4.13.3
Frage 3
5
Null Hypothesis Significance Testing (NHST)
5.1
Example in the literature
5.2
Binomial test
5.3
Proportions test
5.3.1
One sample case
5.3.2
More than one proportion
5.3.3
Fisher’s exact test
5.4
(Classical)
\(t\)
-test
5.4.1
One sample
\(t\)
-test
5.4.2
Two sample
\(t\)
-test
5.4.3
Paired
\(t\)
-test
5.5
Correlation test
5.5.1
Classical correlation test
5.5.2
Bootstrap confidence interval for the correlation coefficient
5.5.3
Comparison with Bayesian approach
5.6
Type 1 and Type 2 errors
5.7
The frequentist confidence interval
5.8
Simulations based approaches
5.9
Exercises
5.9.1
Exercise 1 - frequentist confidence interval
5.9.2
Exercise 2 - everything becomes “significant”
5.9.3
Exercise 3 - binomial test
5.9.4
Exercise 4 - proportions test
5.9.5
Exercise 5 - proportions test 2
5.9.6
Exercise 6 - correlation coefficent
5.9.7
Exercise 7 - coverage frequency of CI
5.9.8
Exercise 8 -
\(\chi^2\)
-distribution
5.10
Sample exam questions for this chapter (in German since exam is in German)
5.10.1
Frage 1
5.10.2
Frage 2
5.10.3
Frage 3 -
\(t\)
-Test
5.10.4
Frage 4
References
Published with bookdown
Quantitative Methods 1, ZHAW
References
Westfall, Peter. 2020.
Understanding Regression Analysis: An Conditional Distribution Approach
. Wiley.