Class notes
Lecture notes and book summary - Statistics 2 - 2023 - Grade 8.5
- Course
- Institution
- Book
Notes on the lectures from the course (2023) Statistics 2. Includes all lectures and readings
[Show more]
Connected book
Book Title:
Author(s):
- Edition:
- ISBN:
- Edition:
Seller
Follow
teaksgardens-0r
Reviews received
Content preview
Notes – Statistics 2 2023Lecture 01: 04/09/2023
Bivariate Relationships between Continuous Variables
Covariance
- Variance: how much do observations deviate from the central tendency?
- Covariance: how much do variables vary together?
- When one variable changes, how does this affect the other variable?
∑(𝑥𝑖 − 𝑥̅ ) (𝑦𝑖 − 𝑦̅)
𝑐𝑜𝑣(𝑥, 𝑦) =
𝑛−1
- Covariance does not have a set range (it depends on the variable’s scale)
- Covariance is an unstandardised measure, so we cannot compare when variables have very different
scales.
- Covariance statistic depends on the variance of x and y.
- We therefore use Correlation Coefficients: standardised covariance statistic.
- Or use Linear regression models: not standardised, but with other advantages.
Correlation coefficient, which always takes values between -1 and 1, describes the strength of the linear
relationship between two variables. We denote the correlation by 𝑅.
- The correlation coefficient is a standardised measure of the linear association between two continuous
variables. What is the direction (positive or negative) of the relationship?
- r = 1 -> a perfect positive linear relationship. All observations fall on a positively sloped line.
- r = 0 -> no linear relationship.
- r = -1 -> a perfect negative linear relationship. All observations fall on a negatively sloped line.
- Nonlinear trends, even when strong, sometimes produce correlations that do not reflect the strength.
- Always plot the data to see the distribution of the data.
- Interpreting the correlation
- r < |0.1| : very small
- |0.1| <= |0.3| : small
- |0.3| <= |0.5| : moderate
- r > |0.5|: large
- Correlation does not imply causation. Even if two variables have a strong correlation, it does not mean
that one causes the other.
Person’s r correlation
𝑐𝑜𝑣(𝑥, 𝑦)
𝑟=
𝑆𝐷(𝑥) ∗ 𝑆𝐷(𝑦)
Assumptions
- Interval-ratio (continuous) variables.
- Linear relationship between variables.
Reporting correlations:
- Higher levels of economic inequality are associated with lower levels of electoral democracy (r = -0.35).
This association is moderate in size and statistically significant (p < 0.01).
,Notes – Statistics 2 2023
Spearman’s rho correlation
- Measures the strength and direction of association between two ranked variables.
- Primarily used for discrete ordinal variables and when assumptions of Person’s r are violated.
Sample vs. Population
- Population
- Observations of relevance for our research questions.
- Sample
- Selection of observations we analyse.
We use our sample to make inferences about the population.
Linear regression is the statistical method for fitting a line to data where the relationship between two variables,
x and y, can be modelled by a straight line with some error.
- Prediction line telling us how to expect the mean/ average value of Y to change when X changes by one
unit.
A statistical model is an abstraction/ simplification that may be useful for answering our questions.
- Linear regression is a method that allows researchers to summarise how predictions or average values of
an outcome vary across observations defined by a set of predictors.
- What is our best guess about one variable if we know what the other variable equals?
𝑦𝑖 = 𝑏0 + 𝑏1 ∗ 𝑥𝑖 + 𝜖𝑖
The values 𝑏0 and 𝑏1 represent the model’s parameters, and the error is represented by 𝜖.
- i represents the individual observation.
- 𝑏0 represents the intercept/ constant term (the average value of Y we expect to observe when X = 0).
- 𝑏1 represents the slope (how we expect the mean of Y to change when X increases by one unit).
- The DV needs to be a continuous variable while the IV can have any form.
- The data fall around a straight line, even if none of the observations fall exactly on the line.
- Dependent variable
- What we want to predict
- Common labels: Y, DV, outcome variable
- Independent variable
- What we are using to predict the DV
- Common labels: X, IV, predictor variable
Main purposes of regression
- Making predictions including to new data.
- Describing relationships.
- Studying causal relationships: causal inference.
Extrapolation describes the fallacy of applying a model estimate to values outside of the realm of the original
data. It can be unreliable, as it assumes that the linear relationship continues indefinitely.
, Notes – Statistics 2 2023
Lecture 02: 11/09/2023
Bivariate Linear Regression
Ordinary Least squares (OLS) regression
Least squares regression aims to find the best-fitting linear relationship by minimising the sum of squared
residuals.
𝑦𝑖 = 𝑏0 + 𝑏1 ∗ 𝑥 + 𝜖𝑖
Error/ Residual
- e -> actual value of Y for observation i and the model’s prediction for that observation.
- Represents variation in Y not explained by our model.
- Positive error/ residual -> the actual value is higher than our predicted value (above the regression line).
- Negative error/ residual -> the actual value is lower than our predicted value (below the regression line).
Reporting OLS regression:
- A discussion about the direction of the relationship (positive or negative coefficient).
- Higher values of X are associated with higher/ lower values of Y.
- Name the value of the effect.
- Based on this model, we expect Y to increase/ decrease by … (value) on average with each one
unit increase in X.
- If it is a bivariate OLS regression: we only interpret the intercept if the predictor variable is scaled such
that the value of 0 refers to a particular category of relevance -> then the intercept is the mean of Y.
- A conclusion about the null hypothesis with reference to the p-value or the confidence interval.
- This association is (not) statistically significant (p ...).
Residuals are the leftover variation in the data after accounting for the model fit:
𝐷𝑎𝑡𝑎 = 𝐹𝑖𝑡 + 𝑅𝑒𝑠𝑖𝑑𝑢𝑎𝑙
- Each observation has a residual.
- Residuals can be used to detect outliers (large residuals show us the outliers).
- The sum of residuals in a well-fitted linear regression model should ideally be close to 0.
- The model is not systematically overestimating or underestimating the observed values.
First, we need to make predictions of certain points. Then we need to subtract the actual observed value.
Residual = Observed Value − Predicted Value
Prediction for example point (77.0, 85.3): 𝑦̂ = 41 + 0.59𝑥 = 41 + 0.59 ∗ 77.0 = 86.4
𝑒 = 𝑦𝑥 − 𝑦̂𝑥 = 85.3 − 86.4 = −1.1
Residuals are helpful in evaluating how well a linear model fits a data set. Residuals can be displayed in a residual
plot where the vertical coordinate is the value of the residual.
- A residual plot where the residuals are around zero indicates a good model fit.
- Other patterns (curves, funnels) in the residual plot can suggest violations of the regression assumption.
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller teaksgardens-0r. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $14.94. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
82871 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now