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Questions and Answers
What is the formula for a linear function?
What is the formula for a linear function?
y = a + βx
In social sciences, what is the statistical linear model for a sample?
In social sciences, what is the statistical linear model for a sample?
ŷ = a + bx OR y = a + bx + e
In social sciences, what is the statistical linear model for a population?
In social sciences, what is the statistical linear model for a population?
E(y)= α + βx OR y = a + βx + ε
The slope 'b' of the prediction equation is independent of the units of the dependent and independent variables.
The slope 'b' of the prediction equation is independent of the units of the dependent and independent variables.
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What is the standardized slope 'b' equivalent to?
What is the standardized slope 'b' equivalent to?
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What is the formula for the standardized slope 'b'?
What is the formula for the standardized slope 'b'?
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In linear regression, how can you decompose the variance in the dependent variable?
In linear regression, how can you decompose the variance in the dependent variable?
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What is the formula for the total sum of squares (TSS) in linear regression?
What is the formula for the total sum of squares (TSS) in linear regression?
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What does 'SSE' represent in linear regression?
What does 'SSE' represent in linear regression?
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What is the formula for the coefficient of determination (r²)?
What is the formula for the coefficient of determination (r²)?
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The F-test is the most commonly used method for evaluating specific explanatory variables in linear regression.
The F-test is the most commonly used method for evaluating specific explanatory variables in linear regression.
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In hypothesis testing, what is a test statistic used for?
In hypothesis testing, what is a test statistic used for?
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What is the formula for the test statistic for the b-coefficient?
What is the formula for the test statistic for the b-coefficient?
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What does the standard error 'se' of the slope represent?
What does the standard error 'se' of the slope represent?
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What is the formula for the standard error 'se' of the slope?
What is the formula for the standard error 'se' of the slope?
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What is the formula for the standard deviation of the residuals 's'?
What is the formula for the standard deviation of the residuals 's'?
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What are the degrees of freedom for the t-distribution used in linear regression?
What are the degrees of freedom for the t-distribution used in linear regression?
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If the value of the test statistic is larger than the critical value, we reject the null hypothesis.
If the value of the test statistic is larger than the critical value, we reject the null hypothesis.
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The p-value represents the probability of obtaining the observed results if the null hypothesis is true.
The p-value represents the probability of obtaining the observed results if the null hypothesis is true.
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If the p-value is less than the significance level alpha, we reject the null hypothesis.
If the p-value is less than the significance level alpha, we reject the null hypothesis.
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What are the three criteria for establishing a causal relationship between two variables?
What are the three criteria for establishing a causal relationship between two variables?
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Experiments are the gold standard for establishing causality in social sciences because they allow for complete control over all variables.
Experiments are the gold standard for establishing causality in social sciences because they allow for complete control over all variables.
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Randomization in a social science experiment aims to ensure that the treatment and control groups have similar distributions of all variables, including unobserved ones.
Randomization in a social science experiment aims to ensure that the treatment and control groups have similar distributions of all variables, including unobserved ones.
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Statistical control is used in observational studies to mimic the control provided by experiments.
Statistical control is used in observational studies to mimic the control provided by experiments.
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What is a spurious association in the context of three variables?
What is a spurious association in the context of three variables?
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What is a chain/indirect relationship in the context of three variables?
What is a chain/indirect relationship in the context of three variables?
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In a multiple causes relationship with independent causes, the predictor variables have no relationship with each other.
In a multiple causes relationship with independent causes, the predictor variables have no relationship with each other.
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In multiple causes relationship with related causes, the predictor variables have a significant relationship with each other.
In multiple causes relationship with related causes, the predictor variables have a significant relationship with each other.
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What is a suppressor variable?
What is a suppressor variable?
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What is partial regression?
What is partial regression?
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Statistical interaction occurs when the effect of one independent variable on the dependent variable is influenced by another independent variable.
Statistical interaction occurs when the effect of one independent variable on the dependent variable is influenced by another independent variable.
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Study Notes
Linear Regression Recap
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A linear function describes a relationship where all data points fall precisely on a line: y = a + βx.
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In social sciences, a statistical linear model is used: ŷ = a + bx or y = a + bx + e
- This model finds the best-fitting line through a scatter of data points (sample).
- The population model is E(y)= α + βx or y = a + βx + ε
- The difference between the actual and predicted value of y is the error term.
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Population mean and standard deviation: μ and σ (unknown constants)
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Sample mean and standard deviation: ӯ and s (variables)
Bivariate Regression Recap
- The slope (b) of the prediction equation (ŷ = a + bx or y = a + bx + e) is dependent on the units of the dependent and independent variables.
- Standardized b is equivalent to Pearson's correlation coefficient (r). This coefficient measures the relationship strength between two variables unit-less.
- To interpret the slope (b) in bivariate regression, units of measurement are needed.
Variance Decomposition in Linear Regression
- In linear regression, the variance in the dependent variable (TSS) is broken down into the Regression Sum of Squares (SSR) and the Error Sum of Squares (SSE): TSS = SSR + SSE.
Hypothesis Testing Recap Page Two
- A test statistic is used in hypothesis testing to decide whether to accept or reject a null hypothesis.
- A test statistic is calculated from the data (like from an experiment or survey), and compared to its expected or critical value.
- If the test statistic is higher than the critical value, then the null hypothesis will be rejected.
T-test calculations and interpretations
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The t-statistic is calculated as follows: t = (b – 0) / se. 'b' represents the estimated slope and 'se' is standard error of the sample slope.
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Se (standard error) estimates the variability of values obtained if many samples were repeatedly drawn from a population..
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Standard error of the b (se) = S /√Σ(x –x)²
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S = SSE/(n − p − 1), where SSE is the sum of squared errors, n is sample size, and p is number of predictor variables.
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To test a hypothesis:
- First, set a significance level (e.g., α = 0.05).
- Then, calculate degrees of freedom: (n – p – 1)
- Use a t-distribution table to ascertain the critical t value.
- Lastly , compare the calculated t-value to the critical t-value. If the calculated t is greater than the critical t, then the null hypothesis is rejected.
Hypothesis Testing Procedures
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Option 1: Follow these steps
- Formulate the null and alternative hypotheses (and choose the type of test).
- Set the significance level (e.g., α = 0.05).
- Calculate the observed test statistic.
- Find the critical value using a t-distribution table with (n − p − 1) degrees of freedom.
- Reject the null hypothesis if the observed test statistic is greater than the critical value.
- Interpret the results.
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Option 2: Follow these steps
- Formulate the null and alternative hypotheses (and choose the type of test).
- Set the significance level (e.g., α = 0.05).
- Calculate the observed test statistic.
- Calculate and read the p-value associated with the observed test statistic.
- Reject the null hypothesis if the p-value is less than the significance level.
- Interpret the results
Interpretation and Context
- Look into correlation's strength between the variables.
- A positive correlation between time spent studying and grades means increased study time correlates with better grades.
- Important takeaway: note small samples won't likely meet the sampling assumptions.
Correlation and Causality
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Correlation does not equal causation: a correlation between variables does not prove one causes the other.
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Three criteria for establishing causality between two variables:
- Association: There must be an observed relationship between the variables.
- Time order: The cause (exposure) has to precede the effect (outcome).
- Elimination of alternative explanations: Other variables are controlled and eliminated (via a well designed experimental study).
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Statistical Control-in non experimental studies-can be used in place of experimental controls in a non experimental studies. Like controlling for age in a study comparing income and education.
Three-Variable Relationships
- Spurious association: The apparent relationship between two variables disappears when controlling for a third variable. For example, age can be a suppressor if age is related to both variables and changes the initial relationship between two variables.
- Chain/indirect relationship (mediation): The relationship between two variables is mediated (influenced) by a third variable, in a way an intervening variable.
- Interaction (moderation): The relationship between two variables differs depending on the level of a third variable. For example, the relationship between education and income might look different based on gender.
- Suppressor variables: A third variable (e.g., age in education and income), can mask the relationship between the two primary variables.
Statistical Interaction
- The effect of one independent variable on the dependent variable can be impacted by another independent variable.
- For instance, financial and technological industries may offer higher returns on education than other fields like transportation and retail.
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Description
This quiz covers key concepts of linear and bivariate regression, two fundamental statistical models used to explore relationships between variables. It discusses the equations involved, interpretation of slope, and the importance of coefficients in statistical analysis. Perfect for students looking to solidify their understanding of these concepts.
