Steven Moran and Alena Witzlack-Makarevich (25 July, 2024)
- Introduction
- Descriptive statistics
- Inferential statistics
- Which statistical test do I use?
- Parametric versus nonparametric statistics
- Statistical assumptions
- Case studies
library(tidyverse)
library(knitr)Scientists are interested in discovering/understanding something about real world phenomena. Three types of goals:
- Describe
- Explain
- Predict
One goal of data modeling is to provide a summary of a data set, i.e., describe it. Another is to come up with hypotheses for descriptive and inference purposes.
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Descriptive statistics: organize, summarize, and communicate a group of numerical observations, e.g, with visualizations.
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Inferential statistics: use sample data to make draw conclusions about the larger population, i.e., fit statistical models to test hypothesis to infer things about the data.
Descriptive statistics use summary statistics to summarize a set of observations, including for example:
The measure of central tendency is one way to make many data points comprehensible to humans by compressing them into one value. Central tendency is a descriptive statistic that best represents the center of a data set i.e. a particular value that all the data seem to be gathering around it’s the “typical” score.
The most commonly reported measure of central tendency is the mean, the arithmetic average of a group of scores.
The median is the middle score of all the scores in a sample when the scores are arranged in ascending order.
Simply creating a visual representation of the distribution often reveals its central tendency.
Let’s look at some data about athletes.
athletes <- read_csv('data/athletes.csv')We can quickly visualize the distribution of our variables.
hist(athletes$height)
hist(athletes$weight)
hist(athletes$age)
In a histogram or a polygon of a normal distribution the central tendency is usually near the highest point.
The specific way that data cluster around a distribution’s central tendency can be measured three different ways:
Or visually:
Here is the big picture.
Here’s also a useful tutorial:
When properties of the population are inferred from a sample, we are undertaking statistical inference.
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A population consists of all the scores of some specified group of interest (in texts one uses N to refer to it)
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A sample is a subset of a population (n)
Why do we need samples?
Samples are used most often because we are rarely able to study every object in a population (e.g. every person in the world or in a country, every language, every sentence ever produced).
Historical detour: How did classical statistics come about?
In the late 19th and early 20th centuries by a small group of people, mostly in Great Britain, working for example at Guinness and in agriculture. They had:
- No computational capabilities
- No graphing capabilities
- Very small and very expensive data sets
Their situation was basically the opposite of what we have today.
Given their limitations, it took a great deal of ingenuity to solve problems that we – for the most part – no longer have, e.g.:
- We have lots of computing power
- We have lots of graphing capabilities
- There’s lots of data out there
Instead, today’s modern statistics is largely focused on areas like Bayesian reasoning, non-parametric tests, resampling, and simulations.
In contrast to variables in computer programming, variables in statistics are properties or characteristics used to measure a population of individuals. A variable is thus a quantity whose value can change across a population.
They include:
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Qualitative variables – measure non-numeric qualities and are not subject to the laws of arithmetic.
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Quantitative variables – measure numeric quantities and arithmetic can be applied to them.
Qualitative variables are also call categorical or discrete variables. Quantitative variables can be measured, so that their rank or score can tell you about the degree or amount of variable.
A hierarchy of variable types in statistics is given in the image below taken from this Stats and R blog.
As you can see, variables in statistics can be classified into four types under qualitative and quantitative variables:
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Nominal – a qualitative variable where no ordering is possible (e.g., eye color) as implied by its levels – levels can be for example binary (e.g., do you smoke?) or multilevel (e.g., what is your degree – where each degree is a level).
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Ordinal – a qualitative variable in which an order is implied in the levels, e.g., if the side effects of a drug taken are measured as “light”, “moderate”, “severe”, then this qualitative ordinal value has a clear order or ranking (but note we don’t know how different these levels are from one to the next!).
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Discrete – variables that can only take specific values (e.g., whole numbers) – no values can exist between these numbers.
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Continuous – variables can take the full range of values (e.g., floating point numbers) – there are an infinite number of potential values between values.
Consider these examples:
- What number were you wearing in the race? – “5”!
- What place did you finish in? – “5”!
- How many minutes did it take you to finish? – “5”!
The three “5”s all look the same.
However, the three variables (identification number, finish place, and time) are quite different. Because of the differences, each “5” has a different possible interpretation.
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Comparison tests: test for differences among group means (e.g., what is the difference in F1 of vowels between women and men?)
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Correlation tests: test whether variables are related without hypothesizing a cause-and-effect relationship (e.g., how are population size and phoneme inventory size related?)
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Regression tests: test for cause-and-effect relationships (e.g., does income predict happiness? do intoxicated speakers increase their articulation errors?)
As with the determining which plot to use, use a cheat sheet to determine which statistical test to use!
There are many other resources out there. Search for them!
Parametric analyses are tests for which we have prior knowledge of the population distribution, e.g., we know the distribution is normal. They also include tests in which we can approximate a normal distribution with the central limit therom.
Non-parametric analyses are tests that do not make any assumptions about the parameters of the population under study, i.e., there is no known distribution. This is why they are also called distribution-free tests.
Ask: what are the assumptions of the statistical test?
Statistical tests make assumptions about the data being tested. If the assumptions for a given statistical test are violated, then the test is not valid and the results may also not be valid.
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Independence of observations: the observations/variables you include in your test should not be related(e.g. several tests from a same test subject are not independent, while several tests from multiple different test subjects are independent)
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Homogeneity of variance: the “variance””” within each group is being compared should be similar to the rest of the group variance. If a group has a bigger variance than the other(s) this will limit the test’s effectiveness.
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Normality of data: the data follows a normal distribution, normality means that the distribution of the test is normally distributed (or bell-shaped) with mean 0, with 1 standard deviation and a symmetric bell-shaped curve.
See the case study on income vs. happiness for a detailed example of checking statistical assumption in a linear regression model:
The basic process is:
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Set up a hypothesis, and assume that it is true.
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Gather data from some real-world experiment that is relevant to the hypothesis.
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Make a determination about the hypothesis, based on the idea of “how likely is our data given the hypothesis?”
The logic of hypothesis testing is as follows. After we have identified the H0 and H1, we can do only one of two things:
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Reject the H0 in favor of the H1
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Fail to reject H0 and thus keep it
Null hypothesis testing corresponds to a reductio ad absurdum argument in logic, i.e., a claim is assumed valid if its counter claim is improbable.
The procedure for deciding is roughly as follows:
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Take a random sample from the population
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Assume that H0 holds
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If the sample data are consistent with the H0, keep H0
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If the sample data are inconsistent with the H0, reject the H0 in favor of the H1.
- Linear regression example: income vs. happiness
- Linear regression example: Does weight increase with height?
- Logistic regression example: Do intoxicated speakers increase their articulation errors?
- ANOVA example: What is the diffence in average weight of Olympic atheletes in different sporting events?
- Multiple linear regression example: What is the effect of biking and smoking on heart disease?