Statistics Predictor Variables Quiz

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Questions and Answers

What does the covariance measure?

  • The sum of squared errors between each data point and the mean.
  • The strength and direction of a linear relationship between multiple variables.
  • The ratio of standard deviations of two variables.
  • The average product of deviations from the mean for two variables. (correct)

Why is it necessary to standardize covariance?

  • To make the covariance value independent of sample size.
  • To ensure the variables are normally distributed.
  • To express the relationship in terms of standard deviations.
  • To eliminate the effect of different units of measurement. (correct)

What is the range of values for Pearson's correlation coefficient (r)?

  • -1 to +1 (correct)
  • 0 to 1
  • 0 to infinity
  • -1 to 0

What does a correlation coefficient of ±0.3 indicate?

<p>A medium effect size. (B)</p> Signup and view all the answers

In the simple bivariate model, what does the parameter 'b1' represent?

<p>The relationship between the predictor variable and the outcome. (B)</p> Signup and view all the answers

What does the coefficient of determination (r²) represent?

<p>The percentage of variance in the outcome variable explained by the predictor variable. (A)</p> Signup and view all the answers

If the correlation coefficient (r) is 0.5, what is the coefficient of determination (r²)?

<p>0.25 (A)</p> Signup and view all the answers

When calculating covariance, what are the error values that are multiplied together?

<p>The difference between each score and the mean for each variable. (A)</p> Signup and view all the answers

What does Spearman’s rank correlation method primarily rely on?

<p>Running Pearson’s r on the ranked data (D)</p> Signup and view all the answers

Kendall’s τ is preferred over Spearman’s ρ primarily when dealing with what type of data?

<p>Small datasets with many tied ranks (D)</p> Signup and view all the answers

In a decision tree framework, how many predictor variables are used for categorical measurements?

<p>One or more (C)</p> Signup and view all the answers

What is the minimum number of levels required for a categorical predictor variable in a decision tree?

<p>Two or more (C)</p> Signup and view all the answers

In a decision tree, what does 'S' stand for in relation to participant types?

<p>Same participants for all predictor levels (D)</p> Signup and view all the answers

What is an important characteristic of the predictor variables when they are categorized as 'CONT' in a decision tree framework?

<p>They can take any numerical value (D)</p> Signup and view all the answers

For which scenario is using two predictor variables necessary in a decision tree?

<p>When analyzing categorical outcomes with multiple levels (D)</p> Signup and view all the answers

What type of analysis is primarily used in decision trees for continuous predictor variables?

<p>Regression analysis (C)</p> Signup and view all the answers

What is the mathematical relationship between variance and standard deviation?

<p>Variance is the standard deviation squared. (A)</p> Signup and view all the answers

In statistical modeling, what role does variance typically play?

<p>It represents the variability in the outcome that the model attempts to capture. (B)</p> Signup and view all the answers

If the standard deviation of a dataset is $5$, what is the corresponding variance?

<p>$25$ (A)</p> Signup and view all the answers

Based on the provided image, which type of analysis method is most closely related to 'Log-linear analysis'?

<p>Chi-Squared test (A)</p> Signup and view all the answers

According to the provided diagram, which analysis method can be used for data exhibiting an independent variable?

<p>Kruskal-Wallis (D)</p> Signup and view all the answers

Which of the following is NOT a non-parametric test from the provided methods?

<p>Pearson (B)</p> Signup and view all the answers

What is the primary function of a predictor variable in the context of variance?

<p>To capture and reduce the unexplained variance of the outcome. (C)</p> Signup and view all the answers

Why is understanding variance important when conducting statistical analyses?

<p>It allows selection of the correct statistical method. (A)</p> Signup and view all the answers

Which test is suitable for analyzing non-parametric data?

<p>Kruskal-Wallis test (A)</p> Signup and view all the answers

What is the primary use of the Spearman correlation coefficient?

<p>For ordinal scale measures (C)</p> Signup and view all the answers

Which of the following statements is true regarding logistic regression?

<p>It can handle binary outcome variables. (D)</p> Signup and view all the answers

When is a t-test considered dependent?

<p>When the same subjects are tested twice. (C)</p> Signup and view all the answers

What does ANCOVA stand for?

<p>Analysis of Covariance (B)</p> Signup and view all the answers

Which correlation method is appropriate for data that follows a normal distribution?

<p>Pearson correlation (A)</p> Signup and view all the answers

What type of ANOVA is used when there are two or more independent variables?

<p>Factorial ANOVA (A)</p> Signup and view all the answers

Which statistical test would be most appropriate for ordinal data?

<p>Mann-Whitney U test (D)</p> Signup and view all the answers

What distinguishes a mixed factorial ANOVA from a standard factorial ANOVA?

<p>It includes both within-subjects and between-subjects factors. (C)</p> Signup and view all the answers

What circumstance would best warrant the use of a Friedman test?

<p>When analyzing repeated measures on an ordinal scale. (C)</p> Signup and view all the answers

What does the term 'Mean Squares' refer to in the context of regression analysis?

<p>The ratio of the sum of squares to degrees of freedom (C)</p> Signup and view all the answers

Which of the following is an assumption regarding the outcome variable in regression analysis?

<p>The outcome must be continuous (D)</p> Signup and view all the answers

What does homoscedasticity refer to in the context of regression analysis?

<p>Variance of the error term is constant across all predictions (B)</p> Signup and view all the answers

What is indicated by a negative correlation between two variables?

<p>An increase in one variable leads to a decrease in the other (A)</p> Signup and view all the answers

Why must predictor variables not have zero variance in regression analysis?

<p>It prevents any change in the outcome variable (A)</p> Signup and view all the answers

Which of the following statements regarding correlation and causation is accurate?

<p>Correlation does not imply causation (C)</p> Signup and view all the answers

What is the purpose of the Durbin-Watson test in regression analysis?

<p>To test for independence of errors (A)</p> Signup and view all the answers

In a regression model, which of the following best describes the linearity assumption?

<p>The relationship we model is truly linear in nature (C)</p> Signup and view all the answers

Flashcards

Variance

The square of the standard deviation. It represents the average of the squared differences between each data point and the mean.

ANCOVA (Analysis of Covariance)

A statistical technique used to analyze the relationships between variables, particularly when one variable is categorical and the others are continuous.

One-way ANOVA (Analysis of Variance)

A powerful statistical test for analyzing differences between groups when the dependent variable is continuous and the independent variable has two or more levels.

Pearson Correlation

A statistical technique used to assess the relationship between two continuous variables.

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Chi-Squared Test

A statistical test used to analyze differences between groups when the dependent variable is categorical and the independent variable has two or more levels. It determines if there's a significant relationship between the categorical variables.

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Logistic Regression

A statistical technique used to analyze the relationship between a categorical dependent variable and one or more independent variables.

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Mann-Whitney U Test

A non-parametric statistical test used to compare two independent groups when the data is ordinal or continuous but not normally distributed.

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Kruskal-Wallis Test

A non-parametric statistical test used to compare three or more independent groups when the data is ordinal or continuous, but not normally distributed.

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Friedman Test

A non-parametric statistical test used to analyze differences between groups when the dependent variable is continuous or ordinal, and the independent variable is within-subjects (repeated measures) and has two or more levels.

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Wilcoxon Signed-Rank Test

A non-parametric statistical test used to compare two related groups when the data is ordinal or continuous, but not normally distributed.

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Pearson Correlation Coefficient (r)

A standardized measure of covariance, ranging from -1 to +1. It quantifies the direction and strength of the linear relationship between two variables.

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Coefficient of Determination (r²)

The amount of variability in one variable that can be explained by another variable. It's calculated as the square of the correlation coefficient (r²).

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Simple Linear Regression Model

A statistical model that predicts an outcome variable based on the value of a predictor variable. The relationship is represented by a parameter, b1, which indicates the direction and strength of the association.

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Product Deviations

The product of the errors between each observation and the mean of the two variables being considered.

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Positive Covariance

A measure of how much two variables change in the same direction. A positive value indicates that as one variable increases, the other tends to increase as well. A negative value means they tend to move in opposite directions.

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Negative Covariance

A measure of how much two variables change in opposite directions. A negative value indicates that as one variable increases, the other tends to decreases.

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Independent Samples t-test

A statistical test used to compare the means of two independent groups. The data must be normally distributed and have equal variances.

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Paired Samples t-test (Dependent Samples t-test)

A statistical test used to compare the means of two dependent groups. The data must be normally distributed.

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One-Way ANOVA

A statistical test used to compare the means of two or more independent groups. The data must be normally distributed and have equal variances.

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One-Way Repeated Measures ANOVA

A statistical test used to compare the means of two or more dependent groups. The data must be normally distributed.

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Factorial ANOVA

A statistical test used to compare the means of two or more independent groups when there are two or more independent variables. The data must be normally distributed and have equal variances.

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Factorial Repeated Measures ANOVA

A statistical test used to compare the means of two or more dependent groups when there are two or more independent variables. The data must be normally distributed.

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ANCOVA

A statistical test used to compare the means of two or more groups when there is one independent variable and one continuous covariate. The data must be normally distributed.

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Spearman Correlation

A statistical test used to examine the relationship between two ordinal variables or two continuous variables that are not normally distributed. The data must have a monotonic relationship.

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Regression

A statistical test used to predict a continuous dependent variable from one or more independent variables. The data must be normally distributed and have a linear relationship.

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Spearman's ρ

A non-parametric correlation coefficient that ranks data and then calculates Pearson's r on the ranked data.

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Kendall's τ (tau)

A non-parametric correlation coefficient that works well for small datasets with many tied ranks. It provides a more accurate estimate of the population correlation compared to Spearman's rho.

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Decision Tree

A learning framework that uses a branching structure to make predictions. It starts with a root node and splits the data based on predictor variables to create branches that lead to leaf nodes, which represent predictions.

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Node in a Decision Tree

In a decision tree, a node that represents a specific value or range of values for a predictor variable.

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Leaf Node in a Decision Tree

In a decision tree, a node that represents a prediction or outcome based on the path taken through the tree.

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Continuous Predictor Variable

A predictor variable that is measured on a continuous scale, such as temperature or height.

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Categorical Predictor Variable

A predictor variable that can only take on a limited number of distinct values, such as gender or color.

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Categorical Predictor Variable with Multiple Levels

A predictor variable that has two or more levels, representing different groups or categories.

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Categorical Predictor

A predictor variable with two or more categories or levels, such as gender (male/female), treatment type (control/experimental), or political affiliation (Democrat/Republican).

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ANOVA (Analysis of Variance)

A statistical test that compares means of two or more groups to determine if there's a significant difference between them. It analyzes the variability between groups relative to the variability within groups.

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Multiple Regression

A statistical technique used to analyze the relationship between a dependent variable and two or more independent variables. It uses a linear equation to predict the dependent variable based on the values of the independent variables.

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Mean Squared Error (MSE)

A measure of the variability of the dependent variable around the regression line. It's calculated by dividing the sum of squared errors (SSE) by the degrees of freedom (n-k-1), where n is the number of observations and k is the number of predictors.

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Mean Squared Regression (MSR)

A measure of the variability of the dependent variable explained by the regression model. It's calculated by dividing the sum of squared deviations from the mean (SSM) by the degrees of freedom (k-1), where k is the number of predictors.

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Slope (bi) in Regression

The change in the outcome associated with a one-unit increase in the predictor variable, while holding all other variables constant.

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Standard Error in Regression

A statistical measure indicating how precisely the model estimates the true population value of a parameter. It represents the average error in predicting the dependent variable.

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Beta (β) in Regression

The proportion of variance in the outcome variable that can be explained by the predictor variable.

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Durbin-Watson Test

A statistical test used to check for the presence of autocorrelation in the residuals. Autocorrelation indicates that the error terms are not independent of each other.

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Linear Regression

A type of regression analysis in which the outcome variable is continuous and at least one predictor variable is also continuous.

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Homoscedasticity

A situation where the variance of the error terms is constant at different values of the predictor variable.

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Study Notes

Research Design and Statistics [RDS] Lecture 3

  • Lecture covers correlation and simple regression
  • Topics covered include variance, covariance, correlation, parametric and non-parametric methods, assumptions, and simple regression.

What We Will Do Today

  • Variance
  • Covariance
  • Correlation (as a model)
  • Parametric analysis
  • Non-parametric analysis
  • Assumptions
  • Simple regression (as a model)

Correlation [Chapter 8 Andy Field 5th Ed]

  • Outcomes can be predicted by a model and what remains is error
  • outcome = (model) + error
  • For correlation, the model is scaling another variable: outcome = (b₁X₁) + error
  • The outcome of an entity is predicted from their score on the predictor variable plus some error.
  • The model is described by a parameter (b₁), which represents the relationship between the predictor variable (X) and the outcome.

Decision Tree - Learning Framework

  • A decision tree to determine appropriate statistical test based on data type and characteristics. Includes continuous and categorical variables, number of predictors, participant groups, and assumptions.

Variance

  • Variance is a measure of dispersion in the outcome measurements.
  • It's used to predict outcomes and model effects of predictor variables.
  • Today's lecture focuses on scenarios where outcome and predictor values are measured for individuals.

Variance Expression

  • variance(s²) = Σ [(xᵢ - x̄)²] / (N-1)
  • This is the average of squared differences between outcome values and the mean of all outcomes.
  • This expression is equivalent to standard deviation squared.

Covariance

  • covariance(x,y) = Σ [(xᵢ-x̄)(yᵢ-ȳ)] / (N-1)
  • This captures the average product deviations.
  • Covariance is similar to variance in form, but it measures the relationship between two variables (x and y).

Covariance - what we do

  • Step 1: Calculate the error in the first variable (x) between the mean and each subject's score.
  • Step 2: Calculate the error in the second variable (y) between the mean and each subject's score.
  • Step 3: Multiply the error values.
  • Step 4: Add these values to get the product deviations.
  • Step 5: The covariance is the average of product deviations.
  • Covariance is influenced by the units of measurement

Standardizing Covariance

  • Standardize covariance by dividing by the product of the standard deviations (sxsy) of the two variables.
  • This standardized version is called the correlation coefficient (r) or Pearson's r.

Pearson Correlation Coefficient

  • Pearson's r values range from -1 to +1.
  • r = 0 indicates no relationship.
  • Positive values indicate a positive relationship, and negative values indicate a negative relationship
  • Absolute values of r greater than or equal to .1, .3, or .5 are considered small, medium, or large effect sizes, respectively

Correlation

  • The correlation coefficient is the ratio of covariance to a measure of variance.
  • outcome = (b₁X₁) + error

Examples of Correlations

  • Visual representations of various correlation values (e.g., r = 0, r = .55, r = -.85) showing the relationship strength and direction between variables.

Coefficient of Determination (r²)

  • r² (r-squared) calculates the amount of shared variance between variables.
  • r² values range from 0 to 1. A higher value indicates more shared variance. (e.g., r² = .81 implies 81% of variance is shared.)

Different Types of Correlation

  • Spearman's rho (ρ or rs): used with non-parametric data and ranks
  • Kendall's tau (τ): used for small datasets with tied ranks, a better estimate in a population.

Simple Regression [Chapter 9 Field 5th Ed]

  • Regression predicts phenomena not measured.
  • Predicting an outcome variable from one predictor variable.
  • A linear model of the relationship between two variables.
    • outcome = (b0 + b1X1)+ error;

Features of the Model for Simple Regression Analysis

  • Straight line models have two parameters:
    • Gradient (how the outcome changes).
    • Intercept (value of the outcome when the predictor is zero).
  • Y₁ = (b0 + b1X1) +εi

Regression: An Example

  • Predicting outcomes with a linear model, accounting for error in the variables being modeled
  • This model describes direction and strength of the relationship between variables (e.g., advertising budget and album sales

Is the Model Any Good?

  • Model fit is evaluated by comparing the sum of squared differences between the observed data and:
    • The mean value of the outcome (SST)
    • The model (SSR)
    • The mean of the outcome and the model (SSM) - comparing the sum of squares of the model to the total sum of squares.

Capturing How Good the Model Is with r²

  • r² measures the proportion of variance accounted for by the regression model.
  • r² = $\frac{SSM}{SST}$; The closer r² is to 1, the better the fit.

Testing the Model: F-ratio

  • F-ratio tests if the model is better than just using the mean as a predictor.
  • MSM & MSR are Mean Squares for the Model (SSM) and Residual/Error (SSR). A larger F implies better model fit.

Model Parameters

  • b0 - is the intercept, the predicted value of Y when X is zero.
  • b1 - is the slope, the change in Y for a one unit increase in X
  • Standard Error provides an understanding of how much the estimate would vary around the true value.

Assumptions

  • Variable Type: Outcome is continuous. Predictors can be continuous or dichotomous.
  • Non-Zero Variance: Predictors have non-zero variance.
  • Independence: Each outcome comes from a different individual (no repeated values for one individual.)

Assumptions (continued)

  • Homoscedasticity: Variance of the error is constant across predictor values.
  • Linearity: The relationship between variables is linear in reality.
  • Independent Errors: Errors between different observations are uncorrelated.
  • Normally-distributed Errors: Error terms are normally distributed.

Does Correlation Mean Causation?

  • Correlations don't imply causation. Other factors might influence the observed relationship. Example: apparent relationship between visits to a pub and exam scores could be due to an unknown third variable, like time spent studying.

Next Week

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